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NLOGIT Student User`s Manual - NYU Stern School of Business

1. PREDICTED PROBABILITIES marks actual marks prediction Indiv AIR TRAIN BUS CAR 1 0543 0445 7540 1472 2 2402 2189 2014 3395 3 0137 0885 8571 0406 4 0203 0890 8287 0620 5 4058 1092 3745 1105 6 2766 3248 2189 b201 7 6129 1446 1240 1185 8 0824 5444 0648 3084 9 1815 3629 mils Bc Fo et OLX 10 1958 1863 0514 5665 Chapter 6 NLOGIT Commands and Results 80 This arrangement of the model may also include Describe Show Model to display the model configuration Effects desired elasticities or marginal effects Prob name to save probabilities Ivb name to save inclusive values All of these computations are done for the current sample This process is the same as the full model computations listed earlier But with Prlist in place the model estimated previously is used it is not reestimated 6 6 Testing Hypotheses We consider two types of hypothesis tests The first is a specification test of the IID extreme value specification The model assumptions induce the most prominent shortcoming of the multinomial logit model the independence from irrelevant alternatives IIA property The fact that the ratio of any two probabilities in the model involves only the utilities for those two models produces a number of undesirable implications including the striking pattern in the elasticities in the model shown earlier We consider a
2. ccccssssseeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeneseeeeees 30 A L e ey emer nenenter wewestcr ene rtereeurre Cemear ry veey Veneer trremr re Veererce rent eer r cerae Teeter crear erent 30 A2 The Mulinomil Loe Medel isammincsr ent ceaneuanee ceie i anems a1 4 3 Model Command for the Multinomial Logit Model ssi cccvscss eccoscie scesscas seca seussceionuesecovens scosius 32 Aa RODUS Oy aa hies MINCE erar E N 32 4 5 Output for the Multinomial Losi Model as uicsveasinstsessspe raran eas Eea anaa 34 46 Marginal ENEC as saecscsesssiesincsvaraiiesrsctsvavaicantopnrseeuasvsiaciciyssieawiaiea ness E 37 A Computime Predicted Probable asica A 40 Ghapter 5 Data Setup for NLOGIH acca ack cepts eens eecreneree nets ncedrcbereecteceteececrerncercees 41 Dl IPO MC HIOM senera Ei 41 2a Base Daro mor ALUC aaa a merettenrr tr caetn reeeer see etr Teer 41 5 3 Fixed and Variable Numbers of CHOI C68 occ sss cers ces ceedseus i eeecacsetavisvscussdecsbctouayeeed nian eain ia 42 3 4 Types of Datron the Choice Variable cnt pia veers aa ryan Renee eean en mapas 46 Sa Dan tor TS Appie CIS erraren p Eaa E AA EEEE EE 47 NLOGIT Student Reference Guide Table of Contents vii Chapter 6 NLOGIT Commands and Results ccccccesseeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeenees 49 O1 TUG EOE koipeen EE RR N 49 o NEUC T Cnm aN n A 49 6 2 1 Other Optional Specifications on NLOGIT Commands sssssersisuiceesisoeiaicoe 52 6 2 2 Specifying the Choice Variable a
3. Cannot locate size variable specified odel is too large Number of betas up to 90 odel is too large Number of alphas up to 30 odel is too large Number of gammas up to 15 odel is too large Number of thetas up to 10 Number of RHS variables is not a multiple of of choices This occurs when you are using a one line setup for your data Expected FIX name Did not find or In FIX name name does not exist lt name is given gt Error in fixed parameter given for lt name is given gt Wrong number of start values given This occurs with nested logit and other models not the random parameters logit model Command has both RHS and Model U alts Inconsistent Syntax problem in USET names list list of values USET list of parms contains an unrecognized name Warning IUSET values not equal to names Warning IUSET values not equal to names Spec for RPL model is label type or type Type N C or L Expected in COR SDV HFN REM AR1 list not found Invalid value given for correl or std dev in list Chapter 12 Diagnostics and Error Messages 172 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 COR SDV list did not give enough values for matrix Error Expected in EQC list value not found Error Value
4. HHKIDS 0092 0033 0023 0125 0045 0162 Averages of Individual Elasticities of Probabilities Variable Y 00 Y 01 Y 02 Y 03 Y 04 Y 05 ONE 4050 2 4807 1 2472 9593 6112 8796 EDUC 8732 0764 2041 3227 2638 0545 HHNINC 4847 3904 2011 0252 0907 0566 AGE 3315 7974 4894 0827 1406 1645 HHKIDS 0571 0330 0110 0300 0077 0092 iil Matrix B_LLOGIT 1 77566 0 0732571 0 285721 0 00565832 0 271876 0 542169 0 0615164 0 859294 0 000897661 0 13921 0 254329 0 109956 1 54517 0 00955207 0 081778 0 0937819 0 104535 1 74362 0 0143038 0 195486 1 56459 0 0752677 1 6403 0 0148114 0 199883 0 712012 0 873102 0 480807 0 539688 0 0561038 2 48767 0 0763641 0 386522 0 805587 0 0340761 1 25418 0 204054 0 197249 0 497504 0 00992853 0 966341 0 322768 0 0290829 0 0908123 0 0289784 0 61823 0 263811 0 0945706 0 132481 0 00873817 0 872575 0 054497 0 0604751 0 156338 0 0101966 iil Matrix PARTIALS Cel 0 0347176 v x 1 o0376 000354578 0 065477 0 000527146 0 00750032 00512738 0 000156155 0 024176 0 000353446 0 00200267 E 3 00843077 0 00123466 0 0402582 0 000704008 0 00203835 4 0146084 000402383 0 00667303 0 00034874 0 0125056 5 01233917 000415089 0 0436 0 00039811 0 00424322 6 0 4403 0 00323814 0 0796381 0
5. where P is the population mean vx is the individual specific heterogeneity with mean zero and standard deviation one and o is the standard deviation of the distribution of Bs around Bz The term mixed logit is often used in the literature for this model The choice specific constants aj and the elements of B are distributed randomly across individuals with fixed means A refinement of the Chapter 10 The Random Parameters Logit Model 121 model is to allow the means of the parameter distributions to be heterogeneous with observed data Zi which does not include one This would be a set of choice invariant characteristics that produce individual heterogeneity in the means of the randomly distributed coefficients so that Bu Be 8 Z Ones and likewise for the constants The model is not limited to the normal distribution We consider several alternatives below One important variation is the lognormal model Brui exp Pe 82 Okr The vs are individual and choice specific unobserved random disturbances the source of the heterogeneity Thus as stated above in the population if the random terms are normally distributed 2 oj Or By Normal or Lognormal P ork 8 ork Zi Ojork Other distributions may be specified For the full vector of K random coefficients in the model we may write the full set of random parameters as P p Az aa Iv where T is a diagonal matrix which contains on its d
6. where y is the index of the choice made Regardless of the number of choices there is a single vector of K parameters to be estimated This model does not suffer from the proliferation of parameters that appears in the logit model described in Chapter 4 It does however make the very strong Independence from Irrelevant Alternatives assumption which will be discussed below Prob y j NOTE The distinction made here between discrete choice and multinomial logit is not hard and fast It is made purely for convenience in the discussion By interacting the characteristics with the alternative specific constants the discrete choice model of this chapter becomes the multinomial logit model From this point in the remainder of this reference guide for NLOGIT we will refer to the model described in this chapter with mathematical formulation as given above as the multinomial logit model or MNL model as is common in the literature Chapter 8 The Multinomial Logit Model 95 The basic setup for this model consists of observations on n individuals each of whom makes a single choice among J choices or alternatives There is a subscript on J because we do not restrict the choice sets to have the same number of choices for every individual The data will typically consist of the choices and observations on K attributes for each choice The attributes that describe each choice i e the arguments that
7. Fen ge n ttme n 01 101 Now the variance for gc includes all three variables but the variance for ttme excludes family NOTE The model with both correlated parameters Correlated and heteroscedastic random parameters is not estimable If your model command contains both Correlated and Hfr list the heteroscedasticity takes precedence and the Correlated is ignored 10 5 Controlling the Simulations There are two parameters of the simulations that you can change the number of draws used in the replications and the type of sequence used to effect the integration 10 5 1 Number and Initiation of the Random Draws R is the number of points replications in the simulation Authors differ in the appropriate value Generally the more complex the model is and the greater the number of random parameters in it the larger will be the number of draws required to stabilize the estimates Train recommends several hundred Bhat suggests 1 000 is an appropriate value The program default is 100 You can choose the value with Pts number of draws R The RPL model is fairly time consuming to estimate For exploratory work while you develop a final model specification you will find that setting R to a small value such as 10 or 20 as we do in the examples in this chapter will be a useful time saver Once a specification is finalized a larger value will be appropriate In order to replicate an estimation you must us
8. MATRIX List 1 partials will display a row matrix of zeros The elasticities of the probabilities OP Ox x x Pj are placed in a J 1 xK matrix named elast_ml The format of the results is illustrated in the example below Chapter 4 The Basic Multinomial Logit Model Partial derivatives of probabilities with respect to the vector of characteristics They are computed at the means of the Xs Observations used for means are All Obs A full set is given for the entire set of outcomes NEWHSAT 0 to NEWHSAT 5 Probabilities at the mean vector are O 052 1 030 2 078 3 145 4 171 5 523 Variable Coefficient Standard Error b St Er P Z gt z Elasticity SSeS Sso S44 Marginal effects on Prob Y 0 Constant 03681271 02185753 1 684 0921 EDUC 00415059 00144841 2 866 0042 87310224 HHNINC 307533229 01759541 4 281 0000 48080659 AGE 00059378 00025180 2 398 0184 53968780 HHKIDS 00874507 00608176 1 438 1505 05610378 SqeSnsts Marginal effects on Prob Y 1 Constant 07581474 01624087 4 668 0000 EDUC 00021399 00101558 Sa 2d 8331 07636415 HHNINC 03569724 01353007 2 638 0083 38652184 AGE 00052245 00019922 23622 0087 80558651 HHKIDS 00313091 00463577 2675 4994 03407609 SSeS Se Marginal effects on Prob Y 2 Constant 09814200 02502533 232 922 0001 EDUC
9. Nested logit model must include MODEL or RHS spec Found neither Model nor RhS Rh2 Your model specification is incomplete There is an unidentified variable name in the equation In the Model U part of the command one of your specified utility functions contains a variable name that is not in your data set Model specification exceeds an internal limit See documentation RANK data can only be used for 1 level nonnested models You have specified a nested logit model and requested rank data for the observed outcomes The nested logit model cannot be estimated with ranks data Not used specifically May show up with a self explanatory Chapter 12 Diagnostics and Error Messages 169 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 message Using Box Cox function on a variable that equals 0 Insufficient valid observations to fit a model Mismatch between current and last models This occurs when you are using the Simulation part of NLOGIT Failure estimating DISCRETE CHOICE model Since this occurs during an attempt to compute the starting values for other models if it fails here it won t succeed in the more complicated model Failed to fit model S arlier diagnostic This is a general diagnostic that precedes exit from the estimator An error condition has occurred generally during estimation not setup
10. 00146816 00158947 924 73597 20405436 HHNINC 04677448 02027747 2 307 VOZIL 19724874 AGE 00082844 00031003 2 672 0075 49750446 HHKIDS 00234229 00728521 4322 7478 00992853 4 Marginal effects on Prob Y 3 Constant 13990259 03064835 4 565 0000 EDUC 00429655 00187257 2 294 0218 322 16832 HHNINC 01275949 02392200 533 5938 02908292 AGE 00027978 00039814 703 4822 09081229 HHKIDS 01264824 00934649 15 353 1760 02897839 aaa ee Y Marginal effects on Prob Y 4 Constant 10599103 03277396 3 234 0012 EDUC 00415859 00200931 2 070 0385 26381106 HHNINC 04913321 02486677 1 976 0482 09457056 AGE 00048333 00042477 1 138 225592 13248126 HHKIDS 00451648 00978660 461 6444 00873817 SsS sesS54 Marginal effects on Prob Y 5 Constant 45666308 04483400 10 186 0000 EDUC 00262240 00279117 940 3475 05449699 HHNINC 09591130 03450901 211719 0054 06047510 AGE 00174112 00056626 3 0 75 0021 15633760 HHKIDS 01608821 01313247 1 225 2205 01019657 arginal Effects Averaged Over Individuals Variable Y 00 Y 01 Y 02 Y 03 Y 04 Y 05 ONE 50377 50772 0975 1380 1051 4556 EDUC 0044 0002 0014 0043 0042 0025 HHNINC 0786 0361 0459 0136 0494 0977 AGE 0006 0005 0008 0003 0005 0018 38 Chapter 4 The Basic Multinomial Logit Model 39
11. 10 6 Model Estimates Because of the numerous components of the model the results for a random parameters model are somewhat more involved than for other specifications For an example we use the command below which specifies a fairly involved heterogeneous RPL model with two error components RPLOGIT s Lhs mode Choices air train bus car Rhs gc ttme one Effects gc air RPL hinc Pts 25 Maxit 100 Halton Ecn gc n ttme n Correlated ECM air car train bus The initial display options for the model requested with Show are the same as in other cases The Describe and Crosstab are as well These were not requested below As usual the estimates for the MNL model are given first These are used as starting values for the estimates Other parameters of the distributions of the random components are started at zeros Start values obtained using MNL model Dependent variable Choice Number of observations 210 Log likelihood function 199 9766 Number of parameters 5 Info Criterion AIC 1 95216 Finite Sample AIC 1 95356 Info Criterion BIC 2 03185 Info Criterion HQIC 1 98438 Chapter 10 The Random Parameters Logit Model R2 1 LogL LogL Log L Constants only 283 fnen R sqrd RsqAdj 7588 29526 28389 l l Chi squared 2 167 56429 Prob chi squared gt value 00000 Response data are given as ind choice Number
12. 4 e PERSON 9 o 4 E Figure 10 9 Conditional and Unconditional Distributions of Parameters In the figure each vertical leg of the centipede plot shows the conditional confidence interval for Bec for that person The dot is the midpoint of the interval which is the point estimate The center horizontal bar in the figure shows the mean of the conditional means which estimates the population mean This was reported earlier as 0 031688 The upper and lower horizontal bars show the overall mean plus and minus twice the estimated population standard deviation this was reported earlier as 0 009629 Thus the unconditional population range of variation is estimated to be about 01 to 05 Note that this is the range of variation in the kernel density estimates given in Figure 10 8 Figure 10 9 demonstrates clearly how the additional information for each individual is used to reduce the uncertainty about the individual specific estimates Chapter 10 The Random Parameters Logit Model 155 10 7 4 Willingness to Pay Estimates The previous section showed how to estimate a function of the random or nonrandom parameters using the simulation method We estimated the conditional variance using a simulation based estimator of E B lall information on individual i Another useful function of the parameters in the model is the willingness to pay function This is typically measured using WTP attribute c
13. Lhs as usual Choices Utility function specification using Rhs Rh2 or Model U to specify utilities Fen specification of random parameters The model command NLOGIT RPL is equivalent The last specification is used to define the random parameters There are many variants We begin with the simplest and add features as we proceed The Fen specification takes the basic form Fen parameter label type Chapter 10 The Random Parameters Logit Model 125 where parameter label is defined either by a variable name that you use in your Rhs specification or by the name you give in your Model definitions and the type is one of the distributions defined in the next section Alternative specific constants are a special case You will generally not want to specify the parameters that multiply Rh2 variables as random These two cases are considered specifically below For example the following specifies two normally distributed random parameters RPLOGIT Lhs mode Choices air train bus car Rhs gc ttme inve Rh2 hinc Fen ge n ttme n The type in the example is n indicating normally distributed parameters Several other specifications would probably be added Alternatively you might use the following to specify a model with two random parameters RPLOGIT Lhs mode Choices air train bus car Model U air a_air bgc ge btt ttme
14. Singular VC may mean model is unidentified Check tr What looks like convergence of a nested logit model may actually be an unidentified model In this case the covariance matrix will show up with at least one column of zeros Sometimes it is more subtle than this In a complicated model the configuration of the tree may lead to nonidentification A common source is too many constant terms in the model Models estimated variance matrix of estimates is singular Non P D 2nd derivatives Trying BHHH estimator instead This is just a notice In almost all cases the Hessian for a model that is not the simple MNL model will fail to be positive definite at the starting values This does not indicate any kind of problem In SIMULATION list of alts a name is unknown Did not find closing in labels list Error in specification of list in Choices labels list List in Choices labels list must be 1 or NALT values Merging SP and RP data Not possible with 1 line data setup Merging SP and RP data requires LHS choice NALTi ALTij form Check MERGERPSP id variable type variable for an error Indiv lt nnnnnn gt with ID lt nnnnn gt has same ID as another individual This makes it impossible to merge the data sets Specification error Scenario must begin with a colon Expected to find Scenario specification value Unbalanced parentheses in scenario specified Chapter 1
15. Start list of values provides starting values for all model parameters PRO list of values provides starting values for free parameters only Generally not used Constrained Estimation 3 CML specification constrained maximum likelihood estimator Rst list of values and symbols imposes fixed value and equality constraints Calibrate fixes parameters at previously estimated values ASC initially fit model with just ASCs Criterion Function for CLOGIT GME number of support points generalized maximum entropy Used by MLOGIT and CLOGIT Sequential sequential two step estimator for nested logit Generally not used Conditional conditional estimator for two step nested logit Generally not used Simulation Based Estimation Pts number of replications number of replications for simulation estimator Used by ECM and MNP Also used by LCM to specify number of latent classes Shuffled uses shuffled uniform draws to compute draws for simulations Halton uses Halton sequences for simulation based estimators Simulation Processor BINARY CHOICE Command for PROBIT and BLOGIT Simulation list of choices simulates effect of changes in attributes on aggregate outcomes Scenarios specifies changes in attributes for simulations Arc computes arc elasticities during simulations Merge merges revealed and stated preference data during simulations Chapter 3 Model and Command Summary for Dis
16. WALD procedure Estimates and standard errors for nonlinear functions and joint test of nonlinear restrictions Wald Statistic 57 91928 Prob from Chi squared 1 00000 H Variable Coefficient Standard Error b St Er P Z gt z Fncn 1 13 32858178 1 7513477 7 610 0000 8 4 Application The MNL model based on the CLOGIT data is estimated with the command CLOGIT Lhs mode Choices air train bus car Rhs gc ttme Rh2 one hinc Show Model Describe Crosstab Effects gc Ivb incvlu Prob pmnl List Chapter 8 The Multinomial Logit Model 98 This requests all the optional output from the model The Describe specification detailed in Chapter 6 requests a set of descriptive statistics for the variables in the model by choice The leftmost set of results gives the coefficient estimates Note that in this model they are the same for the two generic coefficients on gc and ttme but they vary by choice for the alternative specific constant and its interaction with income Also since there is no ASC for car it was dropped to avoid the dummy variable trap there are no coefficients for the car grouping The second set of values in the center section gives the mean and standard deviation for that attribute in that outcome for all observations in the sample The third set of results gives the mean and variance for the particular attribute for the indiv
17. Attribute Alternatives affected Change type Value TME AIR Scale base by value 14 250 The simulator located 209 observations for this scenario Simulated Probabilities shares for this scenario Note rounding error Choice Base Scenario Scenario Base SShare Number Share Number ChgShare ChgNumber AIR 27 619 58 LS s d1 8 32 12 501 26 TRAIN 30 000 63 33 694 71 3 694 8 BUS 14 286 30 16 126 34 1 841 4 CAR 28 095 59 354061 74 6 966 15 Total 100 000 210 100 000 211 000 1 Chapter 7 Simulating Probabilities in Discrete Choice Models 92 Specification of scenario 2 is Attribute Alternatives affected Change type Value TME TRAIN Scale base by value 1 250 The simulator located 209 observations for this scenario Simulated Probabilities shares for this scenario Choice Base Scenario Scenario Base SShare Number Share Number ChgShare ChgNumber AIR 23619 58 30 168 63 2 548 5 TRAIN 30 000 63 20 787 44 9 213 19 BUS 14 286 30 16 383 34 2 097 4 CAR 28 095 59 32 662 69 4 567 10 Total 100 000 210 100 000 210 000 0 The simulator located 209 observations for this scenario Pairwise Comparisons of Specified Scenarios Base for this comparison is scenario 1 Scenario for this comparison is scenario 2 Choice Base Scenario Scenario Base SShare Number Share Number ChgShare ChgNumber AIR 15 118 32 30 168 63 15 049 3 TRAIN 33 694
18. NsMGC 01723103 00801915 2 149 0317 According to these results the population mean of parameters on mgc computed at the mean income or an estimate of E BE z E E B lz is roughly 01317029 35 00023602 02143099 and the population standard deviation is about 01723103 Suppose in the same model we change the distribution to lognormal with Fen mge I The results change to N Random parameters in utility functions MGC 4 69325083 77314298 6 070 0000 ES Nonrandom parameters in utility functions TTME 09826080 01030574 9 539 0000 A_AIR 5 91314197 70223867 8 420 0000 A_TRAIN 4 04515395 49255837 8 213 0000 A_BUS 3 32819477 51874172 6 416 0000 Heterogeneity in mean Parameter Variable MGC HIN 01415413 01472297 961 3364 Derived standard deviations of parameter distributions LsMGC 68944036 54389813 1 268 2049 But the reported parameters are those of the underlying normal distribution In this model B exp B 62 ov v N 0 1 Chapter 10 The Random Parameters Logit Model 129 The conditional population mean of the distribution will be E Bilz exp B 8z 67 2 Inserting the estimated parameters and the mean of 35 for income we obtain an estimate of the overall population mean of 0 0190543 which is quite similar to the 02143099 for the normal distribution The variance for the lognormal is obtained as Var Bilz E Bilz exp o 1 Inserti
19. The nested logit model is requested by adding Tree definition of the tree structure to the command In order to specify the tree use these conventions specifies a trunk specifies a limb within a trunk specifies a branch within a limb in a trunk Entries in a list are separated by commas Names for trunks limbs and branches are optional before the opening or P or C If you elect not to provide names the defaults chosen will be Trunk Lmb il and Br li respectively where the numbering is developed reading from left to right in your tree definition Alternative names appear inside the parentheses Some examples are as follows One limb Tree travel fly air ground train bus car One limb With one limb the is optional Tree fly air ground train bus car One limb Branch names are optional These would be Limb 1 Br 1I1 and Br 211 Tree air train bus car One limb one branch no nesting This would be unnecessary and could be omitted Tree air train bus car Chapter 9 The Nested Logit Model 107 Nested logit model two limbs one with one branch Tree private fly air ground car_pas car_drv public train bus The fully nested 2x2x2x2 model shown in Section 9 1 could be specified with Choices al a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 Tree Trunki limb branch1 al a2 branch2 a3 a4 limb2 branch3 a5 a
20. You may compare up to five scenarios in one run with this tool Use Scenario attribute name 1 list of alternatives action magnitude of action amp attribute name 2 list of alternatives action magnitude of action Use ampersands amp to separate the scenarios Within each scenario you may have up to 20 attribute specifications separated by slashes Chapter 7 Simulating Probabilities in Discrete Choice Models 87 7 4 Simulation Commands The simulation instruction does not produce new model estimates However all other NLOGIT options can be invoked with the command such as descriptive statistics and computing and retaining predicted probabilities 7 4 1 Observations Used for the Simulations The data set used in the simulation can be the original data set used to estimate the model or anew data set The base model is fit with an estimation data set After this operation Steps 1 and 2 in the introduction if desired you may respecify the sample to direct the simulator to do the calculations with a completely different set of observations This would precede Step 4 above If you do not change the sample setting the same data are used for the simulation The simulation must follow the estimation In any case it will require a second command which will generally be identical to the first save for the specification of the simulation 7 4 2 Variables Used for the Simulations If a new data set is used
21. an expanded set of choice specific constants is not likely to be meaningful Also in the absence of a universal choice set the variable altij will not be meaningful The IIA test described later is carried out by fitting the model to a restricted choice set then comparing the two sets of parameter estimates You can restrict the choice set used in estimation irrespective of the IIA test by a slight change in the command In the Choices list of alternatives specification enclose any choices to be excluded in parentheses For example in our CLOGIT application the specification Choices air train bus car produces in part most of the results are omitted the following display in the model output WARNING Bad observations were found in the sample Found 93 bad observations among 210 individuals You can use CheckData to get a list of these points Sample proportions are marginal not conditional Choices marked with are excluded for the IIA test Choice prop Weight IIA AIR 49573 1 000 TRAIN 00000 1 000 gum BUS 00000 1 000 CAR 50427 1 000 Model Specification Table entry is the attribute that multiplies the indicated parameter Choice Parameter Row 1 INVC INVT GC E A_AIR Row 2 A_TRAIN A_BUS AIR 1 INVC INVT GC E Constant 2 none none TRAIN INVC INVT GC E none 2 Constant none BUS 1 INV
22. because the population variance is not the average of the conditional variances Rather the variance we seek equals the average of the conditional variances squares of the elements in sdbeta_i plus the variance of the conditional means The computation can be done a bit inelegantly with MATRIX vi Dirp sdbeta_i sdbeta_i MATRIX evi 1 70 vi l vei 1 70 beta_i beta_i ebi ebi MATRIX v evi vei Peek sd Sqrt v The result of this computation is 0 015192655 Recall the counterpart for the normal distribution that we examined at the outset was 01723103 Chapter 10 The Random Parameters Logit Model 132 10 3 3 Alternative Specific Constants If you have used the Rhs list specification with choices specific constants then the constants will be labeled a_name For example if you have used Choices bus train car Rhs one cost then to specify the model for random ASCs you might use Fen a_bus n a_train n If you are using the Model form then you will have supplied your own names for the ASCs Random choice specific constants in the random utility model with cross section data produce a random term that is a convolution of the original extreme value random variable and the one specified in your model command Suppose for example that you specify a normally distributed random constant for car Then the utility function for car will be U car Qear the rest of the utility function
23. estimating discrete choice models that are built around the logit and multinomial logit form This is a superset of LIMDEP s models NLOGIT 4 0 is all of LIMDEP 9 0 plus the set of tools and estimators described in this manual LIMDEP 9 0 contains the CLOGIT command and the estimator for the conditional logit or multinomial logit model CLOGIT is the same as the most basic form of the NUOGIT command described in Chapter 6 This manual will describe the tools and estimators that extend the multinomial logit model These include for example extensions of the multinomial logit model such as the nested logit mixed logit and multinomial probit models We emphasize NLOGIT Version 4 0 is a superset of LIMDEP 9 0 It is created by adding certain features to LIMDEP Version 9 0 As such the full set of features of LIMDEP 9 0 is part of this package as well We assume that you will use the other parts of LIMDEP as part of your analysis More to the point this manual is primarily oriented to the commands added to LIMDEP that request the set of discrete choice estimators To use NLOGIT you will need to be familiar with the LIMDEP platform At various points in your operation of the program you will encounter LIMDEP rather than NLOGIT as the program name for example in certain menus dialog boxes window headers diagnostics and so on Once again these result from the fact that in obtaining NLOGIT you have installed LIMDEP plus some additional
24. hr gt where Az TQv B i diaglexp hr exp Bx Bd 0e exp 7 he En A T p y B x he E Vp en cara DoS FAD nn Sarees ai i ye tee Bi X gi a Dina am Om exp he E The simulation over B E is actually a simulation over the structural random components v and E The preceding shows how to do the simulation once the maximum likelihood estimates of the structural parameters B A Q 6 7 are in hand A final representation of the results is useful Q ly X Z he hr ben w B where x L y IB x he E 0 7 Z L y B x he E 6 7 and L y IB x he E 7 is the likelihood function for individual i computed at the maximum simulated likelihood estimates of all the parameters the individual s own data and the rth simulated draw on v E Chapter 10 The Random Parameters Logit Model 147 The preceding shows how NLOGIT simulates estimates of B These form the inputs for the computation of elasticities and partial effects There is a parameter vector computed for each individual in the sample If you include Parameters in the RPLOGIT command NLOGIT creates the matrix named beta_i that contains these estimates In the preceding any nonrandom parameter is simply identically reproduced As such beta_i contains only the conditional means for the random parameters in the model Whether this estimator E B ly x z he hr pois W B i
25. rows for choice situation it is used This specification will produce the same form of heteroscedasticity in each parameter distribution note that each parameter has its own parameter vector Yz There is a method of modifying the specification of the heterogeneous means of the parameters so that some RPL variables in z may appear in the means of some parameters and not others A similar construction may be used for the variances The general form of the specification is as follows For any parameter specification Fcn name type it may contain more information beyond just the distribution type the specification may end with an exclamation point to indicate that the particular parameter is to be homoscedastic even if others are heteroscedastic For example the following produces a model with heterogeneous means and one heteroscedastic variance RPL age sex Hfr income Fen gce n ttme n 01 Chapter 10 The Random Parameters Logit Model 138 The parameter on gc has both heterogeneous mean and heteroscedastic variance The parameter on ttme has heterogeneous mean but age is excluded and homogeneous variance Note that there are no commas before or after the As in the case of the means when there is more than one Hfr variable you may add a pattern to the specification to include and exclude them from the model To continue the previous example consider RPL age sex Hfr income family urban
26. the attributes must have the exact same names and measurement units and the alternatives must also have the same names as the full or a restricted set of those used in model estimation A natural application that would obey this convention would be to use one half of a sample to estimate the model then repeat the simulation using the other half of the same sample 7 4 3 Choices Simulated One can undertake simulation either on the full choice set used in estimation or a restricted set This latter option is very useful for modelers using mixtures of data e g combined stated and revealed preference data where some alternatives are only included in estimation but not in application An extensive example is shown below in the case study 7 4 4 Other NLOGIT Options The routine that does simulation also allows you to compute the various elasticities and or derivatives Effects and descriptive statistics Describe and Crosstab as described in Chapter 6 and will produce the standard results for these You might already have done this at the estimation step but if you change the sample you can use this simulation program to recompute those values 7 4 5 Observations Used for the Simulations This program also allows you to compute display and save fitted probabilities utilities and inclusive values for specific observations using the standard setup for these as described in the LIMDEP documentation Once again this is likely to
27. vdb Nvsm vr vu CALC List q Qfr db vdb 1 Chi q 4 Chapter 6 NLOGIT Commands and Results 83 The results are 33784450384775710D 02 82501941289780950D 06 Q Result NOTE We ve been asked this one several times The difference matrix in this calculation vdb might be nonsingular have an inverse but not be positive definite In such a case the chi squared can be negative If this happens the right conclusion is probably that it should be zero 6 6 2 Lagrange Multiplier Wald and Likelihood Ratio Tests NLOGIT keeps the usual statistics for the classical Neyman Pearson hypothesis tests After estimation the matrices b and varb will be kept as usual and can be further manipulated for any purposes for example in the WALD command You can use Test restrictions as well within the NLOGIT command to set up Wald tests of linear restrictions on the parameters Likelihood ratio tests can be carried out by using the scalar logl which will be available after estimation The value of the log likelihood function for a model which contains only J 1 alternative specific constants will be reported in the output as well see the sample outputs above If your model actually contains the ASCs NLOGIT will also report the chi squared test statistic and its significance level for the hypothesis that the other coefficients in the model are all 0 0 HINT NLOGIT can detect that a model contains a set of A
28. where B is the total number of branches in the model L is the number of limbs and R is the number of trunks in the model The x y z and h vectors in the formulation above include all basic variables as well as all variables that interact with choice branch or limb specific dummy variables etc Once again in this form there may be different utility functions for each choice and as described below different utility functions defined for branches and limbs There is a vector of shallow parameters B a 5 0 at each level which multiplies the attributes at the lowest level or e g demographics at a higher level There are also three vectors of deep parameters which multiply the inclusive values at the middle and high levels In principle there is one free inclusive value parameter for each branch in the model Jj one for each limb Cn and one for each trunk But some may have to be restricted to equal 1 0 for identification purposes There are some degenerate cases e Ifthe model has one trunk then the one equals 1 0 e Ifthe model has one limb in a trunk the one o also equals 1 0 e Ifa limb contains a single branch the t for that branch equals 1 0 9 3 Commands for FIML Estimation This section will describe how to set up a nested logit model The default estimation technique is full information maximum likelihood FIML That is the entire model is estimated in a single pass 9 3 1 Data Setup
29. x he E p B E z hr dE dB Chapter 10 The Random Parameters Logit Model 148 The second term is the square of the mean that was estimated earlier The first is the expected square which can like the mean be estimated by simulation Combining the results already obtained then we have an estimator of the conditional variance x A 2 Var B ly X Z he hr gt w Bix y be Bs The square root of this quantity provides an estimate for individual i for each random parameter an estimate of the conditional standard deviation These diagonal elements appear in the matrix sdbeta_i We illustrate this with a model that includes most of the features described above RPLOGIT Lhs mode Choices air train bus car Rhs gc ttme Rh2 one ECM air car train bus RPL hinc Fen ge n ttme n Correlated Parameters Halton Pds 3 Pts 200 Model results are omitted The elements in the matrices are shown in Figure 10 7 As shown there there is a considerable amount of variation in the estimated conditional means iii Matrix BETA_ Lf Matrix spBeTA olx 70 2 Cell 0 00918842 0 256011 0 00918842 0 0495477 0 0357957 0 201861 0 00883883 0 0404709 0 0342806 0 200457 0 00933546 0 0322379 0 0396988 0 250451 0 00931861 0 0489795 0 0395694 0 260195 0 00969315 0 0430495 0 0334948 0 161442 0 0100033 0 0680474 0 0308025 0 122828 0 00958659 0 0561653 0 017 72
30. 0000 PA Attributes of Branch Choice Equations alpha AA 3 54086522 1 20812715 2 931 0034 AH 01533132 00938134 1 634 1022 SoSa 5s5 54 IV parameters tau jli l sigma i 1l phi 1 FLY 58600939 14062118 4 167 0000 GROUND 38896192 12366583 3 145 0017 Descriptive Statistics for Alternative AIR Utility Function 58 0 observs Coefficient All 210 0 obs that chose AIR Name Value Variable Mean Std Dev Mean Std Dev BT 5 0646 TASC 000 000 000 000 BB 4 0963 BASC 000 000 000 000 BG 0316 GC 102 648 B05 15 1134552 33 198 AT 1126 TME 61 010 15 719 46 534 24 389 Descriptive Statistics for Alternative TRAIN Utility Function 63 0 observs Coefficient All 210 0 obs that chose TRAIN Name Value Variable Mean Std Dev Mean Std Dev BT 5 0646 TASC 1 000 000 1 000 000 BB 4 0963 BASC 000 000 000 s000 BG 0316 GC 130 200 58 235 106 619 49 601 AT 0126 TME 35 690 12 279 28 524 19 354 Descriptive Statistics for Alternative BUS Utility Function 30 0 observs Coefficient All 210 0 obs that chose BUS Name Value Variable Mean Std Dev Mean Std Dev BT 5 0646 TASC 000 000 000 000 BB 4 0963 BASC 1 000 000 1 000 000 BG 0316 GC 115 257 44 934 108 133 43 244 AT u 1126 TME 41 657 12 077 25 200 14 919 117 Chapter 9 The Nested Logit Model Descriptive Statistics for Alternative CA
31. 0363 GC 102 648 3075735 1137552 33 198 A_AIR 1 3160 ONE 1 000 000 1 000 000 AIR_HIN1 0065 HINC 34 548 POTI 41 724 LS pl Bal eo Descriptive Statistics for Alternative TRAIN Utility Function 63 0 observs Coefficient All 210 0 obs that chose TRAIN Name Value Variable Mean Std Dev Mean Std Dev INVC 0461 INVC 51 338 27 032 37 460 20 676 INVT 0084 INVT 608 286 251 797 532 667 249 360 GC 0363 GC 130 200 58 235 106 619 49 601 A_TRAIN 2 1071 ONE 1 000 000 1 000 000 TRA_HIN2 0506 HINC 34 548 19 711 23 063 17 287 Descriptive Statistics for Alternative BUS Utility Function 30 0 observs Coefficient All 210 0 obs that chose BUS Name Value Variable Mean Std Dev Mean Std Dev INVC 0461 INVC 33 457 12 591 33 733 11 023 INVT 0084 INVT 629 462 235 408 618 833 273 610 GC 0363 GC 115 257 44 934 108 133 43 244 A_BUS 8650 ONE 1 000 000 1 000 000 BUS_HIN3 0332 HINC 34 548 19 711 29 700 16 851 Chapter 6 NLOGIT Commands and Results 72 Descriptive Statistics for Alternative CAR Utility Function 59 0 observs Coefficient All 210 0 obs that chose CAR Name Value Variable Mean Std Dev Mean Std Dev INVC 0461 INVC 20 995 14 678 15 644 9 629 INVT 0084 INVT 573 205 274 855 527 373 301 131 GC 0363 GC 95 414 46 827 89 085 49 833 You may also request a cross tabulation of the model predictions against the actual choic
32. 25 and the probabilities are recomputed In this case a fairly strong effect is predicted 26 of 58 people who chose air are now expected to take other modes eight changing to train four to bus and 15 to car The one stray person at the end is the result of rounding error in the allocation of the probabilities You may combine up to five scenarios in each simulation This allows you to have simultaneous changes in attributes Use Scenario attribute choices in which it appears the change attribute choices in which it appears the change For example suppose terminal time for both air and train both increased by 25 We would extend our previous setup as follows SAMPLE 31 840 NLOGIT Lhs mode Rhs one ge ttme Choices air train bus car NLOGIT Lhs mode Rhs one gc ttme Choices air train bus car Simulation Scenario ttme air 1 25 ttme train 1 25 Discrete Choic One Level Model Model Simulation Using Previous Estimates Number of observations 210 Simulations of Probability Model Model Discrete Choic One Level Model Simulated choice set may be a subset of the choices Number of individuals is the probability times the number of observations in the simulated sample Column totals may be affected by rounding error The model used was simulated with 210 observations Specification of scenario 1 is Attribute Alternat
33. 6 2 4 Specifying the Utility Functions with Rhs and Rh2 There are several ways to specify the utility functions in your NUOGIT command in the text editor and in the command builder In order to provide a simple explanation that covers the cases we will develop the application that will be used in the chapters to follow to illustrate the models The application is based on the data summarized in Section 5 5 We will model travel mode choice for trips between Sydney and Melbourne with utility functions for the four choices as follows gc ttme one hinc one hinc one hinc one hinc U air GC ME AAIR AIR_HIN1 0 0 0 0 1 0 oo U train GC ME 0 0 A_TRAIN TRA_HIN2 0 0 1 0 0 U bus GC ME 0 0 0 0 A_BUS BUS_HIN3 0 0 1 U car GC ME 0 0 0 0 0 0 im 0 A ase ie te The columns are headed by the names of variables generalized cost gc terminal time ttme and household income hinc The entries in the body of the table are the names given to coefficients that will multiply the variables Note that the generic coefficients in the first two columns are given the names of the variables they multiply while the interactions with the constants are given compound names It is important to note the last two columns The last one in a set of choice specific constants or variables that are interacted with them must be dropped to avoid a problem of collinearity in the model In what follows for brevity we will omit these two columns Befor
34. B trunk R D k J B L R A k xF where Ak coefficient on x k in U JIB L R and F 1 r R x 1 7 L x 1 6 B x 1G J P JIBLR trunk effect 1 r R x 1 1 L x 1 b B P BILR x P JIBLR x TBIR limb effect 1 r R x 1 l L P LIR x P BILR x P JIBLR x TBILR X OLR branch effect 1 r R P R x P LIR x P BILR x P JIBLR x TBILR X Our x Or twig effect Note in this expression J B L and R are being used generically to indicate a particular choice branch limb and trunk not the total numbers of twigs branches limbs and trunks The marginal effect is P j b l r y x k lJ B L R PG b Lr A k F A marginal effect has four components an effect on the probability of the particular trunk one on the probability for the limb one for the branch and one for the probability for the twig Note that with one trunk P P 1 1 and likewise for limbs and branches For continuous variables such as cost you might be interested instead in the Elasticity x K IJ B L R x A KU B L R x F NLOGIT will provide either As in the case of nonnested models marginal effects are requested with Effects attribute list of outcomes or Effects attribute list for elasticities This generates a table of results for each of the outcomes listed For example NLOGIT Lhs mode Choices air train bus car Tree travel public bus train private air car Model U air ba bc
35. CGT 03348741 01506250 22A 0262 AB 117857565 73948909 1 594 s LELO CHB 03339204 01299642 2 569 0102 CGB 03455919 01516387 24219 0227 CGC 03808057 01523791 2 499 0125 Shorthand Notations for Sets of Utility Functions There are several shorthands which will allow you to make the model specification much more compact If the utility functions for several alternatives are the same you can group them in one definition Thus Model U air b0 bcost gc could be specified with U car b0 bcost gc Model U air car b0 bcost ge Chapter 6 NLOGIT Commands and Results 62 For the model we have been considering i e Choices air train bus car all of the following are the same Model U air b1 ttme bcost gc U train b1 ttme bcost gc U bus b1 ttme bcost gc U car b1 ttme beost gc and Model U air train bus car b1 ttme beost gc and Model U b1 ttme beost gc and Rhs ttme gc The last will use the variable names instead of the supplied parameter names for the two parameters but the models will be the same Alternative Specific Constants and Interactions You can also specify alternative specific constants in this format by using a special notation When you have a U al a2 aJ for J alternatives then you may specify instead of a single parameter a list of parameters enclosed in pointed bracke
36. Chapter 2 Discrete Choice Models 15 The data will appear as follows e Individual data y coded 0 1 J e Grouped data yj yi iy_ give proportions or shares The structural equations of the multinomial logit model are Uiit B Xi Eijn t E S 0 1 J i 1 N where U gives the utility of choice j by person i in period t we assume a panel data application with 1 T The model about to be described can be applied to cross sections where T 1 Note also that as usual we assume that panels may be unbalanced We also assume that g has a type 1 extreme value distribution and that the J random terms are independent Finally we assume that the individual makes the choice with maximum utility Under these IIA inducing assumptions the probability that individual i makes choice j in period t is exp B x gt exPBx ijt We now suppose that individual i has latent unobserved time invariant heterogeneity that enters the utility functions in the form of a random effect so that Vij Bi Xi Qi Ein t 1 7 J 0 1 J 7 1 N The resulting choice probabilities conditioned on the random effects are DY expe x a Pij l Qit Qj7 To complete the model we assume that the heterogeneity is normally distributed with zero means and J x J 1 covariance matrix amp For identification purposes one of the coefficient vectors B4 must be normalized to zero and one of the as
37. Chi squared 122 1013 Degrees of freedom 20 Prob ChiSqd gt value 0000000 Chapter 4 The Basic Multinomial Logit Model 35 This is based on the health satisfaction variable analyzed in the preceding chapter We reduced the sample to those with newhsat reported zero to five We would note though these make for a fine numerical example the multinomial logit model would be inappropriate for these ordered data The restricted log likelihood is computed for a model in which one is the only Rhs variable In this case log Lo 2 njlogP where nj is the number of individuals who choose outcome j and P nj n the jth sample proportion The chi squared statistic is 2 log L log Lo If your model does not contain a constant term this statistic need not be positive in which case it is not reported But even if it is the statistic is meaningless if your model does not contain a constant The diagnostic statistics are followed by the coefficient estimates These are B B Recall Bo is normalized to zero and not reported Variable Coefficient Standard Error b St Er P Z gt z Mean of X a oe ca a Characteristics in numerator of Prob Y 1 Constant 1 77566023 69486152 22955 s0106 EDUC 07325707 04476186 1 637 1017 10 8759203 HHNINC 28572052 58129003 492 s6231 32998942 AGE 00565832 00838172 675 4996 46 9925061 HHKIDS 27187563 19642471 1 384 1663 331695
38. Ckit where c is now the random element driving the random parameter Chapter 2 Discrete Choice Models 20 This produces then the full random parameters logit model exp a j Bx J d D xPO BX B B Az TQ V v with mean vector 0 and covariance matrix I P jlv The specific distributions may vary from one parameter to the next We also allow the parameters to be lognormally distributed so that the preceding specification applies to the logarithm of the specific parameter 2 8 Multinomial Probit Model In this model the individual s choice among J alternatives is the one with maximum utility where the utility functions are Uji Bx j where U utility of alternative j to individual i Xj union of all attributes that appear in all utility functions For some alternatives Xj may be zero by construction for some attribute k which does not enter their utility function for alternative j The multinomial logit model specifies that are draws from independent extreme value distributions which induces the IIA condition In the multinomial probit model we assume that are normally distributed with standard deviations Sdv s o and correlations Cor i qi Pjq the same for all individuals Observations are independent so Cor amp j O if i is not equal to s for all j and q A variation of the model allows the standard deviations and covariances to be scaled by a functi
39. J 1 K gt for the Rh2 variables The total number of utility function parameters is thus K K J 1 K gt The internal limit on K the number of utility function parameters is 100 The random utility model specified by this setup is precisely of the form Ui Biri t Borin t Brixi Vij YKK Eij where the x variables are given by the Rhs list and the z variables are in the Rh2 list By this specification the same attributes and the same characteristics appear in all equations at the same position The parameters B appear in all equations and so on There are various ways to change this specification of the utility functions i e the Rhs of the equations that underlie the model and several different ways to specify the choice set These will be discussed at several points below 6 2 1 Other Optional Specifications on NLOGIT Commands The NLOGIT command operates like other LIMDEP model commands The following lists command features and options that may or may not be specified with the NLOGIT command Features marked with are unavailable with this command Controlling Output from Model Commands Par use with the random parameters logit model to save person specific parameter vectors Margin displays marginal effects Use Effects specification OLS displays least squares starting values Not used here Table name saves model results to be combined later in output tables Robust Asymptotic Covaria
40. OcaVear Ecar Ocar the rest of the utility function Uear The random term in this equation is the sum of a normally distributed variable and one with an extreme value distribution This produces a different stochastic model but probably not a useful extension of the model in general For this reason unless you are using panel data it is generally not useful to specify random constant terms in the random parameters logit model That said however there is an exception which might prove useful Random constant terms that are correlated will produce correlation across the alternatives which is one of the oft cited virtues of the multinomial probit model In addition the error components logit specification produces a useful extension that serves much the same function as a random constant term 10 3 4 Heterogeneity in the Means of the Random Parameters The RPLOGIT command requests the random parameters model generally with the parameters specified in the Fen list varying around a mean that is the same for all individuals The variables in z provide the variation of the mean across individuals To specify the variables in z use RPL list of variables in z If you desire to specify that z enter the means of some of the coefficients but not all you can change the specification of the random coefficients in the Fen specification as follows name type implies z enters the mean name type implies that z does not enter t
41. The arrangement of the data set for estimation of the nested logit model is exactly the same as shown in Chapter 5 There is no requirement that the choice sets be the same across individuals but the nested logit model will require a definition of a universal choice set so the command must contain the Choices list of labels specification The nested model structure does mandate one special consideration if you are going to define utility functions for branches ys or limbs zs Since you have one line of data for each alternative you will have more than one line of data for the variables in any branch or limb In these cases the values of y and z must be repeated for each alternative in the branch or limb Chapter 9 The Nested Logit Model 106 The following model and setup illustrate this for a three level model all in trunk 1 xl x2 yl y2 zl 2 limb 1 branch 111 twig 111 1 6 1 3 02 104 9 twig 211 1 1 2 3 02 104 9 branch 211 twig 112 1 8 2 7 15 104 9 twig 212 1 2 3 7 AS 104 9 limb 2 branch 112 twig 111 2 9 6 11 08 96 4 twig 211 2 3 1 11 08 96 4 twig 311 2 4 0 11 08 96 4 9 3 2 Tree Definition The model command for estimating nested logit models is exactly as described in Chapter 8 for single level models where the model name is now the generic NLOGIT NLOGIT Lhs Choices definition of choice set definition of utility functions for alternatives All of the options described earlier are available
42. The autoregressive model is requested by adding ARI to the NLOGIT command You may also constrain the autoregressive model with ARI list of values where the list may contain symbols for free parameters or specific numerical values including zero if you do not wish for specific coefficients to evolve in this fashion To illustrate the panel data models we will artificially treat our clogit data as if it were a panel It is not For the first model we collect the observations in groups of three and treat it as a random effects model For the second we collect the observations in groups of six and fit an AR1 model to them Since these data are in fact a cross section we should not expect much of the estimates Chapter 11 The Multinomial Probit Model 161 Chapter 11 The Multinomial Probit Model 11 1 Introduction In the multinomial probit MNP model the individual s choice among J alternatives is the one with maximum utility where the utility functions are Uji PIX Ej where U utility of alternative j to individual i X union of all attributes that appear in all utility functions For some alternatives Xi May be zero by construction for some attribute k which does not enter their utility function for alternative j unobserved heterogeneity for individual i and alternative j The multinomial logit model specifies that g are draws from independent extreme value distributions which induces the ITA con
43. adding the specification of the random parameters The model command is RPLOGIT Lhs dependent variable Choices the names of the J alternatives Rhs list of choice specific attributes Rh2 list of choice invariant individual characteristics Fen the specifications of the random parameters other specifications for the random parameters model Once again variable choice set sizes and utility function specifications are specified as in the CLOGIT command This command is the same as NLOGIT RPL the rest of the command There is one modification that might be necessary If you are providing variables that affect the means of the random parameters you would generally use NLOGIT RPL the list of variables the rest of the command The RPL specification may still be used this way The command can be NLOGIT as above or RPLOGIT RPL the list of variables the rest of the command These are identical Chapter 3 Model and Command Summary for Discrete Choice Models 25 The random parameters model may also include an error components specification defined in the next section The command will be RPLOGIT Lhs dependent variable Choices the names of the J alternatives Rhs list of choice specific attributes Rh2 list of choice invariant individual characteristics Fen the specifications of the random parameters Other specifications for the random parameters model EC
44. and the multinomial probit model Background theory and applications for the programs described here can be found in many sources For a primer that develops the theory in detail and presents many examples and applications all using NLOGIT we suggest Applied Choice Analysis A Primer Hensher D Rose J and Greene W Cambridge University Press Cambridge 2005 It is not possible nor even desirable to present all of the necessary econometric methodology in a manual of this sort The econometric background needed for Applied Choice Analysis as well as for use of the tools to be described here can be found in many graduate econometrics books One popular choice is Econometric Analysis 7 Edition Greene W Prentice Hall Englewood Cliffs 2011 Finally this guide is primarily focused on the specialized tools in NLOGIT for extensions of the multinomial logit model Users will find the LIMDEP documentation the LIMDEP Reference Guide and Volumes 1 and 2 of the LIMDEP Econometric Modeling Guide essential for effective use of this program Chapter 1 Introduction to NLOGIT 10 It is assumed throughout that you are already a user of LIMDEP The NLOGIT Reference Guide by itself will not be sufficient documentation for you to use NLOGIT unless you are already familiar with the program platform LIMDEP on which NLOGIT is placed 1 2 NLOGIT and LIMDEP This Reference Guide describes NLOGIT Version 4 0 NLOGIT is a suite of programs for
45. and w will represent the unobserved individual heterogeneity built into models such as the error components and random parameters mixed logit models The assumption that the choice made is alternative j such that U choice j gt U choice q V q j The observed outcome variable is then y the index of the observed choice The econometric model that describes the determination of y is then built around the assumptions about the random elements in the utility functions that endow the model with its stochastic characteristics Thus where Y is the random variable that will be the observed discrete outcome Prob Y j Prob U choice j gt U choice q V q The objects of estimation will be the parameters that are built into the utility functions including possibly those of the distributions of the random components and with estimates of the parameters in hand useful characteristics of consumer behavior that can be derived from the model such as partial effects and measures of aggregate behavior Chapter 2 Discrete Choice Models 12 To consider the simplest example that will provide the starting point for our development consider a consumer s random utility derived over a single choice situation say whether to make a purchase The two outcomes are make the purchase and do not make the purchase The random utility model is simply U not purchase Bo Xo o U purchase B x amp Assuming that o and g are ra
46. are observed in sequence and there is a long enough lag between situations that the effect of the passage of time might be to allow preferences to evolve consider for example cases in which habit persistence influences the choice mode of travel to work but new information enters the system In such a case an autoregressive arrangement might be appropriate Bir Vir RV Ui B Az Tv where R is a diagonal matrix of autocorrelation coefficients and u constitutes the primitive randomness in the system The two situations are requested by first specifying the panel as usual with Pds Ti where 7 is either a fixed number of observations or a variable which gives the number of observations In this setting the panel consists of groups of 7 sets of J observations In all cases T tells the number of groups of data You may have a variable number of observations and a variable number of choices within a group or any of the other three possible combinations In our examples below J 4 a fixed number of choices In one case 7 3 so in this case there are 12 rows of data for each person In the other case there are six observations in a group so 24 rows of data per person If the number of observations in a group varies so 7 is the name of a count variable this count is repeated on every row of data within an observation and for every observation in the group Chapter 10 The Random Parameters Logit Model 160
47. be useful when your estimation and simulation steps are based on different sets of observations Chapter 7 Simulating Probabilities in Discrete Choice Models 88 7 5 Applications We compute the shares for a particular sample using the following S alternative j N x Dr Pj Thus save for the rounding error which is distributed the model predicts the number of individuals in the sample who will choose each alternative The crosstabulation described in Section 6 3 summarizes this calculation For example using the clogit dat data the following results from estimation of a simple multinomial logit model Cross tabulation of actual vs predicted choices Row indicator is actual column is predicted Predicted total is F k j i Sum i 1 N P k j i Column totals may be subject to rounding error AIR TRAIN BUS CAR Total AIR 34 0000 8 0000 4 0000 13 0000 58 0000 TRAIN 8 0000 39 0000 4 0000 12 0000 63 0000 BUS 5 0000 4 0000 17 0000 4 0000 30 0000 CAR 11 0000 13 0000 5 0000 30 0000 59 0000 Total 58 0000 63 0000 30 0000 59 0000 210 0000 The feature described here is used to examine how these predictions change when the value of an attribute changes For example how do the predictions change when the generalized cost of air travel changes The simulator is used as follows Step 1 Fit the model Step 2 Use the identical model specification but add to the command Simulation a subset of the choi
48. binve inve ghinc hinc U train bus car a_ground bgc ge Fen a_ground n btt n Note that the specifications of the random parameters are separated by commas not semicolons The next several subsections will describe the various parts of the specifications of the random parameters The last part of this section describes the command builder for this model Because so much of this model is custom made for the particular application the command builder is somewhat limited compared to the command form indicated above 10 3 1 Distributions of Random Parameters in the Model There are many distributions that can be used for the random parameters The most common will be the normal which is used in the example above Many alternatives are supported however A few of these are listed below The basic distributions are specified with the following Fen parameter name type The types are n normal B B ov v N O 1 l lognormal B exp B ov v N 0 1 u uniform B B ov v U 1 1 t triangular B B ov v triangle 1 1 d dome B B ov v 2xbeta 2 2 1 e Erlang B B ov v gamma 1 4 4 w Weibull B B tov v 2 logu 5 u U 0 1 p exponential B B ov vi exponential 1 c nonstochastic Bi B Chapter 10 The Random Parameters Logit Model 126 In the list above we have denoted the constant in the distribution as 8 However the parameter definition may involve hete
49. cost a y 0 Income j l exp B cost a y Income OIncome J Dy expBicost a y Income OIncome exp OIncome exp B cost a y Income exp Zncome _exp Bicost a y Income exp B cost a y Income Sa a a a 2 i expBicost a y Income Chapter 6 NLOGIT Commands and Results 58 So the identical model arises for any 8 This means that the model still cannot be estimated in this form The solution to this remaining issue is to normalize the coefficients so that one of the choice varying parameters is equal to zero NLOGIT sets the last one to zero The same result applies to the choice specific constant terms that you create with one This produces the data matrix shown earlier with the last two columns in the dashed box normalized to zeros Finally while it is necessary for choice invariant variables to appear in the Rh2 list it is not necessary that all variables in the Rh2 list actually be choice invariant Indeed one could specify the preceding model with choice specific coefficients on the cost variable it would appear U air Ycost air COST air Brime time air Qhair Yair income Ej airs U train Ycost train costi train Brine time train Qlirain Ytrain income E trains U bus Ycost bus COST bus Brime time pus pus Yous income Ei buss U car 7 Ycost car cost car B time time car Ei bus Note also that there is no n
50. dialog shown public and private are siblings while bus is a child node of public Tree Specification Add Sibling Node Public Bus Add Child Node Train Rename Node Private ir Remove Node Figure 9 2 Tree Specification Dialog Box for Defining the Tree Structure The remaining options for output and results to be saved are defined in the Output page as shown in Figure 9 3 NLOGIT Main Options Output Display Keep as variables I Covariance matrix Branch IVs M Crosstabs Tl Limb I s M Descriptive stats M Trunk IVs Predictions Probabilities l Branch probabilities F Utility functions ja Keep model results for table as Figure 9 3 Output Page of Command Builder for Nested Logit Models Chapter 9 The Nested Logit Model 112 9 4 Marginal Effects and Elasticities In the nested logit model with P G b l r P jlb l r x P bll r x P llr x P r the marginal effect of a change in attribute k in the utility function for alternative J in branch B of limb L of trunk R on the probability of choice j in branch b of limb of trunk r is computed using the following result Lower case letters indicate the twig branch limb and trunk of the outcome upon which the effect is being exerted Upper case letters indicate the twig branch limb and trunk which contain the outcome whose attribute is being changed Clog P alt j limb I branch b trunk r Ox k alt J limb L branch
51. em iadazetintsd 168 Chapter 1 Introduction to NLOGIT 9 Chapter 1 Introduction to NLOGIT 1 1 Discrete Choice Modeling with NLOGIT NLOGIT is a set of tools for building models of discrete choice among multiple alternatives The essential building block that underlies the set of programs is the random utility model of consumer choice U choice 1 f attributes of choice 1 characteristics of the consumer 1 V w U choice J f attributes of choice J characteristics of the consumer V wW where the functions on the right hand side describe the utility to a consumer decision maker of J possible choices as functions of the attributes of the choices the characteristics of the consumer random choice specific elements of preferences that may be known to the chooser but are unobserved by the analyst and random elements v and w that will capture the unobservable heterogeneity across individuals Finally a crucial element of the underlying theory is the assumption of utility maximization The choice made is alternative j such that U choice j gt U choice q V q j The tools provided by NLOGIT are a complete suite of estimators beginning with the simplest binary logit model for choice between two alternatives and progressing through the most recently developed models for multiple choices including random parameters mixed logit models with individual specific random effects for repeated observation choice settings
52. fixed choice set sample frequency for j sample frequency MNL models R sqrd are simpl 1 LogL model log RsqAdj 1 nJ nJ nparm 1 R sqrd L other nJ sum over i choice set sizes Variable Coefficient Standard Error b St Er P Z gt z GC 01092735 00458775 2 382 0172 TIME 09546055 01047320 941 15 0000 A_AIR 5 87481336 80209034 7 324 0000 AIR_HIN1 00537349 01152940 466 6412 A_TRAIN 5 54985728 64042443 8 666 0000 TRA_HIN2 05656186 01397335 4 048 0001 A_BUS 4 13028388 67636278 6 107 0000 BUS_HIN3 02858418 01544418 1 851 0642 PREDICTED PROBABILITIES marks actual marks prediction Indiv AIR TRAIN BUS CAR 1 0984 SSEL 1959 TILAG 2 2566 2262 0530 4641 3 1401 s1799 1997 4808 4 2 7132 40297 0211 6759 5 3421 1478 30527 4575 6 0831 3962 2673 2534 7 6066 0701 0898 12335 8 0626 6059 3192 5 13 90 9 21125 32932 SLIS 3947 10 1482 0804 1267 6447 Rows 11 210 are omitted Cross tabulation of actual vs Row indicator is actual Predicted total is F k j i Sum i 1 N Column totals may be subject to rounding error predicted choices column is predicted P k j i Matrix C AIR TRAIN BUS CAR Total rosstab has 5 rows and 5 columns AIR TRAIN BUS CAR Total 33 00000 7 00000 4 00000 14 00000 58 00000 7 00000 39 000
53. for some reason you desired to force the parameters apub and bcost to be equal you could just change apub to bcost in the utility equation for public That is you can if you wish force equality of parameters at different levels of a model once again just by using the same parameter name in the model specification Given the impact of the scale parameters this is probably inadvisable but the program will allow you to do it nonetheless Chapter 9 The Nested Logit Model 108 The interaction of alternative specific constants and branch and limb specific constants is complex and it is difficult to draw generalities As a general rule models will usually become overdetermined resulting in a singular Hessian when there are more than NALT 1 constants of all three types in the entire model Likewise interactions of attributes and choice specific dummy variables can produce this effect as well Users who encounter problems in which NLOGIT claims either that it is impossible to maximize the log likelihood function or there is a singular Hessian should examine the model for this pitfall 9 3 4 Setting and Constraining Inclusive Value Parameters There is an inclusive value parameter for each limb branch and trunk in the model For example in the tree Choices air train bus car Tree travel public bus train private air car with the other parameters we estimate Tpublicttravels Tprivateltravels Stravel Since there is only o
54. for the non independence between observations associated with the same respondent a theme of importance in stated choice studies iii decomposing the mean and standard deviation of one or more random parameters to reveal sources of systematic taste heterogeneity iv accounting for correlation of random parameters v imposing priors based on known choices in model estimation vi imposing constraints on distributions e g constraining the triangular or normal to ensure that it does not change sign over its range vii selecting subsets of pre specified variables to interact with the mean and standard deviation of random parameterized attributes and viii deriving willingness to pay estimates when both the numerator and denominator are random parameter estimates 10 2 Random Parameters Mixed Logit Models This model is somewhat similar to the random coefficients model for linear regressions The model formulation is a one level multinomial logit model for individuals i 1 V in choice setting t Neglecting for the moment the error components aspect of the model we begin with the basic form of the multinomial logit model with optional alternative specific constants a and attributes x exp a T Bix Dex a B x The RPL model emerges as the form of the individual specific parameter vector B is developed The most familiar simplest version of the model specifies Prob yi Bu Be Oik and Qj A Oj
55. is the matrix defined earlier the same for all individuals and h is an individual not alternative specific set of variables not including a constant Chapter 3 Model and Command Summary for Discrete Choice Models 22 Chapter 3 Model and Command Summary for Discrete Choice Models 3 1 Introduction The chapters to follow will provide details on the various discrete choice models you can estimate with NLOGIT and on the model commands you will use to request the estimates This chapter will provide a brief summary listing of the models and model commands The variety of logit models now use a set of specific names rather than qualifiers to more general model classes as in earlier versions For example the model name OLOGIT can be used instead of ORDERD Logit The earlier formats remain available but the newer ones may prove more convenient The full listing of these commands is also given below The commands below specify the essential parts needed to fit the model The numerous options and different forms are discussed in the chapters to follow 3 2 Model Summary The descriptions below present the different discrete choice models that are the main feature of NLOGIT Note once again NLOGIT contains all of LIMDEP so all of the models documented in the Econometric Modeling Guide including the regression models limited dependent variable models generalized linear models sample selection models and so on are supported in NLOGIT as
56. names and types of random parameters Correlation specifies that random parameters are correlated Hfr list of variables defines variables in heteroscedasticity Also used by HEV and covariance heterogeneity Multinomial Probit MNP specifies multinomial probit model Command is also MNPROBIT EQC list of choices specifies a set of choices whose pairwise correlations are all equal RCR list of specifications specifies configurations for correlations for multinomial probit model Also used by RPL SDV list of specifications specifies diagonal elements of covariance matrix Also used by RPL and HEV REM specifies random effects form of the model Chapter 4 The Basic Multinomial Logit Model 30 Chapter 4 The Basic Multinomial Logit Model 4 1 Introduction This chapter will describe a basic form of the multinomial logit model These models are also known variously as conditional logit discrete choice and universal logit models among other names All of them can be viewed as special cases of a general model of utility maximization An individual is assumed to have preferences defined over a set of alternatives travel modes occupations food groups etc U alternative 0 Bo Xo w Ujalternative 1 B Xi1 1 U alternative J By xiz E Observed Y j if U alternative j gt U alternative q V q J The disturbances in this framework individual hetero
57. of parameters from which their specific vector is drawn This is the estimate of E B4i that is in row i of beta_i We also have an estimate of the standard deviation of this conditional distribution As a general result an interval in a distribution for a continuous random variable defined by the mean plus and minus two standard deviations will encompass 95 or more of the mass of the distribution This enables us to form a sort of confidence interval for B itself conditioned on all the information known about the individual To roughly this level of confidence the interval E B all information on individual i 2xSD B lall information on individual i will contain the actual draw for individual i The probability is somewhat reduced because we are using estimates of the structural parameters not the true values The centipede plot feature of PLOT allows us to produce this figure as follows We plot the figure for Bg for the Weibull model Chapter 10 The Random Parameters Logit Model 154 The commands are CREATE lowerbge bgw 2 sgw upperbge bgw 2 sgw CREATE person Trn 1 1 CALC meanbgw Xbr bgw CALC highbgw meanbgw 2 sdbgw CALC lowbgw meanbew 2 sdbgw PLOT Lhs person Rhs lowerbgc upperbge Centipede Title Confidence Limits for b_gc for Weibull Model Bars meanbew highbgw lowbgw Endpoints 0 75 Untitled Plot 13 Confidence Limits for b_gc for Weibull Model
58. page shown in Figure 10 2 requests specification of the choice variable and the utility functions This page provides both ways to do this specification The random parameters model is set up on the Options page shown in Figure 10 3 Note that there are a few options not specified in the command builder notably the Sdv specification and the technical controls of the simulation However in the random parameters window you can add these additional specifications as text Thus where we have typed ge n ttme n we could have typed ge n ttme n Stv s1 1 0 which would have added the additional specification as a text string NLOGIT Multnomial Probit HEV RPL Main Options Output Choice variable MODE gt Data type Individual ba Choice names ir Train Bus Ca Data coded on one line Code _ aaa Utility functions a Rhs Rh2 Attributes Ahs Characteristics Rh2 Specify utility functions I Figure 10 2 Main Page of Command Builder for RPL Model Chapter 10 The Random Parameters Logit Model 136 NLOGIT Multnomial Probit HEV RPL Main Options Output Model Random Parameters Logit w RPL mol Multinomial Probit Hetero Extreme Value iia UBB Fano Parameters Logit pilities fi I Correlated parameters IV Specify random parameters geln ttmefn IV Heterogeneity in mean Variables HINC Bal gt Optimization Hypothesis Tests Figure 10 3 Options Page o
59. presence of lambda I e by this construction the Box Cox transformation is treated like the log function just a transformation In this case the model results will contain an indication that the transformation has appeared in the utility functions For example the preceding with 0 5 produces Discrete choice multinomial logit model Dependent variable Choice umber of observations 210 Log likelihood function 267 4253 Log L for Choic model 267 42533 R2 1 LogL LogL Log L fncen R sqrd RsqAdj Constants only 283 7588 05756 05154 Chi squared 1 32 66687 Prob chi squared gt value 00000 Response data are given as ind choice Box Cox model LAMBDA used is 50000 Number of obs 210 skipped 0 bad obs Variable Coefficient Standard Error b St Er P Z gt z BA 6425631472 21842858 2 942 0033 BCOST 2433427492 44558999E 01 5 461 0000 BC 8456971936 23246443 3 638 0003 BB 9996728009 22980046 4 350 0000 Do note however that the results can only indicate that a Box Cox transformation using 0 5 has appeared in the model It is not possible to report where it appears Chapter 6 NLOGIT Commands and Results 65 Command Builders The command builders provide space for you to build the utility functions in this fashion See Figure 6 5 Since this is done by typing out the functions in the windows there is no menu constru
60. the case as we now examine In order to compare these sets of estimates we propose to examine the estimated conditional means We will use two devices A direct approach is to examine the distribution of estimates of E B across the observations in the sample The averages of the conditional means will estimate the population mean averaged across z as well The variances require a bit of manipulation since as noted the variance of the conditional means underestimates the overall variance by the mean of the conditional variances We will also examine the distribution of conditional means in the sample with a kernel density estimator Chapter 10 The Random Parameters Logit Model 151 First estimate the models The parameter estimates are shown above SAMPLE All CREATE mge gc mttme ttme CLOGIT Lhs mode Choices air train bus car Rhs mgc mttme Rh2 one CALC bgmnl b 1 btmnl b 2 RPLOGIT Lhs mode Choices air train bus car Rhs mgc mttme Rh2 one 3 ECM air car train bus RPL hinc Parameters Halton Pds 3 Pts 200 Fen mgc n mttme n Correlated MATRIX bn beta_i sn sdbeta_i RPLOGIT Lhs mode Choices air train bus car Rhs mgc mttme Rh2 one 3 ECM air car train bus RPL hinc Parameters Halton Pds 3 Pts 200 Fen mgce w mttme w Correlated MATRIX bw beta_i sw sdbeta_i Now move the matrices to
61. the data area so we can examine them SAMPLE 31 70 CREATE bgn 0 btn 0 bew 0 btw 0 CREATE sgn 0 stn 0 sgw 0 stw 0 NAMELIST _ betan bgn btn betaw bew btw NAMELIST _ sbetan sgn stn sbetaw sgw stw CREATE betan bn CREATE betaw bw CREATE sbetan sn CREATE sbetaw sw Now compare the different estimates The results below show that the normal and Weibull coefficients are much more similar than the raw parameter estimates would suggest We first estimate the population means by averaging the conditional means CALC List bgmnl Xbr bgn Xbr bgw CALC List btmnl Xbr btn Xbr btw These are the three estimates of E B gc BGMNL 015784 Result 031987 lt lt Normally distributed Result 031688 Weibull distributed Chapter 10 The Random Parameters Logit Model 152 These are the three estimates of EIB rme BTMNL 097091 Result 166242 lt 4 Normally distributed Result 169459 Weibull distributed Are the correlations the same Note these are the correlations of the conditional means not the correlations of the coefficients CALC List Cor bgn btn Cor bgw btw Result 962738 Two normally distributed parameters Result 723847 Two Weibull distributed parameters What about the population standard deviations The following estimate the standard deviations of the population marginal distribution of the two parameters Once again th
62. the labels will be Last Model b1_1 b1_2 b1_3 b2_1 b2_2 b2_3 Chapter 4 The Basic Multinomial Logit Model 37 4 6 Marginal Effects The marginal effects in this model are OP Ox J 9 1 J For the present ignore the normalization Bo 0 The notation P is used for Probly j After some tedious algebra we find P B B where B ee PB It follows that neither the sign nor the magnitude of 6 need bear any relationship to those of Bj This is worth bearing in mind when reporting results The asymptotic covariance matrix for the estimator of 6 would be computed using Asy Var G Asy Var G where f is the full parameter vector It can be shown that Asy Var E 2m Vi Asy Cov l B 1B Vin i 0 where Vi AG D PiL PA 8x POx and 1g 1ifj l and 0 otherwise This full set of results is produced automatically when your LOGIT command includes Marginal Effects There is no conditional mean function in this model so marginal effects are interpreted a bit differently from the usual case What is reported are the derivatives of the probabilities Note this is the same as the ordered probability models These derivatives are saved in a matrix named partials which has J 1 rows and K columns Each row is the vector of partial effects of the corresponding probability Since the probabilities will always sum to one the column sums in this matrix will always be zero That is
63. the parameter Model results for these distributions will display the structural parameters not necessarily the means and variances of the parameter distributions Note for example that the means of the lognormal and the Weibull distributions are not equal to B for the lognormal it is exp B o7 2 while for the Weibull it is B 20T 1 1 V2 Consider an example The following estimates a model with two random parameters We will use the normal Weibull and exponentiated Weibull our Rayleigh distributions Since the exponentiated Weibull estimator forces the coefficient to be positive and the coefficients on the two variables would naturally be negative we reverse the signs on the data before estimation Chapter 10 The Random Parameters Logit Model 127 The commands are CREATE mge gc mttme ttme RPLOGIT Lhs mode Choices air train bus car Rhs mgc mttme Rh2 one Fen mgc n mttme n Normally distributed parameters Maxit 50 Pts 25 Halton Pds 3 RPLOGIT Lhs mode Choices air train bus car Rhs mgc mttme Rh2 one Fen mgce w mttme w Weibull distributed parameters Maxit 50 Pts 25 Halton Pds 3 RPLOGIT Lhs mode Choices air train bus car Rhs mgc mttme Rh2 one Fen mgc r mttme r Modified Weibull distributed parameters Maxit 50 Pts 25 Halton Pds 3 These are the reported random parameter estimates The nonrandom alternati
64. we would fix the problem If you do get one of these and you cannot get around it please contact us at support nlogit com Chapter 12 Diagnostics and Error Messages 168 12 2 Discrete Choice CLOGIT and NLOGIT 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 FIML NLogit is not enabled in this program Syntax problem in tr spec or expected or not found Model defines too many alternatives more than 100 A choice label appears more than once in the tree specification Number of observations not a multiple of of alternatives This is expected when you have a fixed choice set Problem reading labels or weights for choice based sample Number of weights given does not match number of alternatives A choice based sampling weight given is not between zero and one The choice based sampling weights given do not sum to one Expected in limb specification was not found Expected in branch specification was not found A branch label appears more than once in the tree A choice label in a branch spec is not in CHOICES list Branch specifications are not separated by commas One or more CHOICE labels does not appear in the tree Ea One or more CHOICE labels appears more than once in tree The model must have either 1 or 3 LHS variables Check spec
65. which groups of people in different locations or at different times were surveyed Finally y might be a set of ranks in which case instead of zeros and ones y would take values 1 2 not necessarily in that order within and reading down each block More specifically data on the dependent Lhs variable may come in these four forms e Individual Data The Lhs variable consists of zeros and a single one which indicates the choice that the individual made When data are individual the observations on the Lhs variable will sum exactly to 1 0 for every person in the sample A sum of 0 0 or some other value will only arise if a data error has occurred Individual choice data may also be simulated e Proportions Data The Lhs variable consists of a set of sample proportions Values range from zero to one and again they sum to 1 0 over the set of choices in the choice set Observed proportions may equal 1 0 or 0 0 for some individuals e Frequency Data The Lhs variable consists of a set of frequency counts for the outcomes Frequencies are nonnegative integers for the outcomes in the choice set and may be zero e Ranks Data The Lhs variable consists of a complete set of ranks of the alternatives in the individual s choice set Thus if there are J alternatives available the observation will consist of a full set of the integers 1 J not necessarily in that order which indicate the individual s ranking of the alternatives The num
66. x Z he hr Chapter 10 The Random Parameters Logit Model 146 The reordering of terms to obtain the second expression is permissible because E and B are independent Moreover since they are independent their joint distribution equals the product of the marginal distributions so we may rewrite the preceding in a more useful form as I f B PCy IB x he E p B E z hr dE dB E B ly x z he hr I I PLY B x he E p B E z hr dE dB This would provide the basis of the conditional estimator Note that it is precisely the form of the posterior mean if this were a Bayesian application The integrals in the conditional mean for B will not exist in closed form so some other method must be used to do the integration Note first that in the expression above the term p y B x he E is the contribution to the conditional likelihood function not its log of individual i L parameters y x z he hr and the integral is the unconditional likelihood Second integration over the range of B E with weighting function equal to the joint marginal density of B and E can be done by simulation The implication is that the preceding integrals can be approximated using the simulation method used to maximize the simulated likelihood Combining our results we have the simulation based conditional estimator lor a A noe BPO IB x he E lor DNA P Y B x he E G ly X Z he
67. y dX exp B x y z where y is the index of the choice made We note at the outset that the IID assumptions made about gj are quite stringent and induce the Independence from Irrelevant Alternatives or IIA features that characterize the model This is functionally identical to the multinomial logit model of Section 2 4 Indeed the earlier model emerges by the simple restriction y 0 We have distinguished it in this fashion because the nature of the data suggests a different arrangement than for the multinomial logit model and second the models in the section to follow are formulated as extensions of this one Chapter 2 Discrete Choice Models 17 2 6 Nested Logit Model The nested logit model is an extension of the conditional logit model The models supported by NLOGIT are based on variations of a four level tree structure such as the following ROOT root TRUNKS trunk1 trunk2 LIMBS limb1 limb2 limb3 limb4 hee whe che wh the EZ BRANCHES branch1 branch2 branch3 branch4 branch5 branch branch7 branch8 Ol fd fd da dell ofa ald ALTS al a2 a3 a4 a5 a6 a7 a8 a9 al10 a11 a12 al3 a14 al5 atl6 The choice probability under the assumption of the nested logit model is defined to be the conditional probability of alternative j in branch b limb and trunk r jlb l r exp B x exp B x P jlb L r pp ite B jols au XPP Xap exp J where Jj is the inclusive value for branch b in limb J trun
68. 0 1 0 0 0 0 Bus 0 53 25 399 85 0 30 2 0 0 I 0 0 0 Car 1 0 11 255 50 0 30 2 0 0 0 1 0 0 Air 0 69 115 125 129 0 40 12 21 O 0 0 40 1 i 3 Train 0 34 98 892 LOS 0 40 1 0 J 0 0 0 0 Bus 0 35 53 882 149 0 40 ap 0 0 1 0 0 0 Car 1 0 23 720 101 0 40 1 0 0 0 1 0 0 Air 0 64 49 68 59 0 70 3 cr 0 0 0o 70 3 i 4 Train 0 44 26 354 49 0 70 3 0 1 0 0 0 0 Bus 0 53 24 399 81 0 70 3 0 0 I 0 0 0 Car 1 0 5 180 32 0 0 3 0 0 0 1 0 0 Air 0 64 60 144 82 0 45 2 1 0 0 0 45 2 i 5 Train 0 44 32 404 93 0 45 2 0 1 0 0 0 0 Bus 0 53 26 449 94 0 45 2 0 0 T 0 0 0 Car 1 0 8 600 99 0 45 2 0 0 0 1 0 0 Air 0 69 59 100 70 0 20 J i 0 0 0o 20 1 1 6 Train 1 40 20 345 57 0 20 1 0 1 0 0 0 0 Bus 0 35 13 417 58 0 20 0 0 T 0 0 0 Car 0 0 12 284 43 0 20 1 0 0 0 1 0 0 Air 1 45 148 115 160 1 45 1 1 0 0 0o 45 1 i 7 Train 0 34 111 945 213 1 45 o 1 0 0 0 0 Bus 0 35 66 935 167 45 0 0 T 0 0 0 Car 0 0 36 821 25 1 45 0 0 0 T 0 0 Air 0 69 121 152 137 0 12 Te 0 0 o 12 gt at i 8 Train 0 34 52 889 149 0 12 0 1 0 0 0 0 Bus 0 35 50 879 146 0 12 1 0 0 I 0 0 0 Car 1 0 50 780 135 0 12 1 0 0 0 1 0 0 Air 0 69 59 100 70 0 40 ck 0 0 0o 40 1 1 9 Train 0 34 ook 3IT2 71 0 40 0 1 0 0 0 0 Bus 0 35 25 417 70 0 40 1 0 0 T 0 0 0 Car 1 0 17 210 40 0 40 1 0 0 0 1 0 0 Air 0 69 58 68 65 0 70 2 i 0 0 0 70 2 i 10 Train 0 34 3L Soi 69 0 70 2 0 I 0 0 0 0 Bus 0 35 25 402 68 0 70 2 0 0 I 0 0 0 Car 1 0 7 210 30 0 70 2 0 0 0 1 0 0 Chapter 6 NLOGIT Commands and Results 49 Chapter 6 NLOG IT Comma
69. 00 5 00000 12 00000 63 00000 3 00000 6 00000 15 00000 6 00000 30 00000 15 00000 11 00000 6 00000 27 00000 59 00000 58 00000 63 00000 30 00000 59 00000 210 00000 99 Chapter 8 The Multinomial Logit Model Elasticity Attribute is GC Direct E Choice AIR Choice TRAIN Choice BUS Choice CAR Effects on probabilities of all choices in model Elasticity effect of the attribute averaged over observations in choice AIR Mean St Dev 8019 3834 3198 3370 3198 3370 3198 3370 Elasticity Attribute is GC Direct Choice AIR x Choice TRAIN Choice BUS Choice CAR Effects on probabilities of all choices in model Elasticity effect of the attribute averaged over observations in choice TRAIN Mean St Dev 3534 3511 1 0693 7134 3534 2511 3534 3511 Elasticity Attribute is GC Direct Choice AIR Choice TRAIN x Choice BUS Choice CAR Effects on probabilities of all Elasticity effect of over observations BUS choices in model the attribute averaged in choice Mean St Dev 1679 2308 L679 2308 1 0916 5183 1679 2308 Elasticity Attribute is GC Direct Choice AIR Choice TRAIN Choice BUS X Choice CAR Effects on probabilities of all Elasticity effect of over observations CAR choices in model the attribute averaged in choice Mean St Dev 2934 2674 2934 2674 293
70. 0000 R2 1 LogL LogL Log L fncn R sqrd RsqAdj No coefficients 291 1218 37164 36253 Constants only 283 7588 35534 34599 At start values 1 99 9766 gt 08525 lt O07 199 Response data are given as ind choice Replications for simulated probs 500 Number of obs 210 skipped 0 bad obs H Variable Coefficient Standard Error b St Er P 2Z gt z peas Hesad Random parameters in utility functions GC 01871422 01712611 1 093 32745 TTME 17600015 04467395 3 940 0001 T Nonrandom parameters in utility functions A_AIR 11 0829925 2 37916582 4 658 0000 A_TRAIN 9 22867193 2 20639245 4 183 0000 A_BUS 8 19884828 2 10499796 3 895 0001 Heterogeneity in mean Parameter Variable GC HI 00029029 00036724 790 4293 TTME HIN 00060674 00061444 987 3234 Derived standard deviations of parameter distributions NsGC 01364904 02015496 ELT 4983 NSTTME 11712864 03885999 3 014 0026 Parameter Matrix for Heterogeneity in Means Matrix Delta has 2 rows and HINC GC TIME 00029 00061 1 columns 157 Chapter 10 The Random Parameters Logit Model 158 Elasticity averaged Attribute is GC in choice Effects on probabilities of all AIR over observations choices in model Direct Elasticity effect of the attribute Mean St Dev x Choice AIR 8246 4614 Choice TRAIN 7617 7464 Choice BUS 1 0439 19282 Choice CAR
71. 00153523 0 0157984 Figure 4 1 Matrices Computed by MLOGIT Marginal effects are computed by averaging the effects over individuals rather than computing them at the means The difference between the two is likely to be quite small Current practice favors the averaged individual effects rather than the effects computed at the means MLOGIT also reports elasticities with the marginal effects An example appears below Chapter 4 The Basic Multinomial Logit Model 40 4 7 Computing Predicted Probabilities Predicted probabilities can be computed automatically for the multinomial logit model Since there are multiple outcomes this must be handled a bit differently from other models The procedure is as follows Request the computation with Prob name as you would normally for a discrete choice model However for this model NLOGIT does the following 1 A namelist is created with name consisting of up to the first four letters of name and prob is appended to it Thus if you use Prob pfit the namelist will be named pfitprob 2 The set of variables one for each outcome are named with the same convention with prjj instead of prob For example in a five outcome model the specification Prob job produces a namelist Jpbprob jobpr00 jobpro1 jobpr02 jobpr03 jobpr04 The variables will then contain the respective probabilities You may also use Fill with this procedure to compute probabilities for observa
72. 094 2 553 0107 The preceding command Chapter 6 NLOGIT Commands and Results 68 NOTE This is one of our frequently asked questions The R squareds shown in the output are R s in name only They do not measure the fit of the model to the data It has become common for researchers to report these with results as a measure of the improvement that the model gives over one that contains only a constant But users are cautioned not to interpret these measures as suggesting how well the model predicts the outcome variable It is essentially unrelated to this To underscore the point we will examine in detail the computations in the diagnostic measures shown in the box that precedes the coefficient estimates Consider the example below which was produced by fitting a model with five coefficients subject to two restrictions or three free coefficients npfree 3 The effect is achieved by specifying Choices air train bus car WARNING Bad observations were found in the sample Found 93 bad observations among 210 individuals You can use CheckData to get a list of these points Sample proportions are marginal not conditional Choices marked with are excluded for the IIA test Choice prop Weight IIA AIR 49573 1 000 TRAIN 00000 1 000 BUS 00000 1 000 CAR 50427 1 000 Model Specification Table entry is the attr
73. 148 0317 s BUS 1 00000000 Fixed Parameter s CAR 1 00000000 3 Fixed Parameter Correlations in the Normal Distribution rAIR TRA 12923578 74351679 174 8620 rAIR BUS 11759913 92452141 127 8988 rTRA BUS 61859572 38300577 1 265 1063 rAIR CAR POO0000 eieaa Fixed Parameter rTRA CAR OO0O0000 3 Fixed Parameter rBUS CAR 1090 0000 sr gas ety Fixed Parameter The table below compares the elasticities from the MNP model to the MNL model The MNL results appear first They are clearly similar but the specification does make a difference Elasticity averaged over observations Attribute is GC in choice AIR Effects on probabilities of all choices in model Direct Elasticity effect of the attribute Mean St Dev x Choice AIR 8019 3834 Choice TRAIN 3198 3370 Choice BUS 3198 3370 Choice CAR 3198 3370 Effects on probabilities of all choices in model Choice AIR 1 0453 4797 Choice TRAIN 3796 3184 Choice BUS s9597 23926 Choice CAR 4221 LAIO Chapter 11 The Multinomial Probit Model 166 11 4 Testing IIA with a Multinomial Probit Model A multinomial probit model with all standard deviations equal to one and uncorrelated random terms specifies a model with the IIA property This means that you could test this property by using an LR or LM test of the assumption that all of the standard deviations in a mode
74. 2 Diagnostics and Error Messages 170 1037 Choice given in scenario attr choice is not in the model 1038 Cannot identify attribute specified in scenario 1039 Value after in scenario spec is gt 20 characters 1040 Cannot identify RHS value in scenario spec 1041 Transformation asks for divide by zero 1042 Can only analyze 5 scenarios at a time 1043 Did not find any valid observations for simulation 1044 Expected to find LIST name_x choices Not found 1045 Did not find matching or in lt scenario specification is given gt 1046 Cannot recognize the name lt AAAAAAAA gt in lt scenario specification is given gt 1047 Same as 1046 1048 None of the attributes requested appear in the model 1049 Model has no free parameters among slopes This occurs during an attempt to fit the MNL model to obtain starting values for a nested logit or some other model 1050 DISC with RANKS Obs lt nnnnnn gt Alt lt nn gt Bad rank given lt nnnn gt DISC w RANKS Incomplete set of ranks given for obs lt nnnnnn gt These are data problems with the coding of the Lhs variable 1051 Singular VC matrix trying to fit MNL model When the MNL breaks down it will be impossible to fit a more elaborate model such as a nested logit model 1052 You did not provide FCN label distn for RPL model EV RPL or MNP model PL or MNP model RPL or MNP model EV RPL or MNP model EV RPL
75. 282 Chapter 10 The Random Parameters Logit Model 143 Results saved automatically by this estimator are the same as the other estimators in NLOGIT i e Matrices b and varb Scalars logl kreg nreg Note that nreg is the number of individuals not the number of rows of data in the sample Last Model See Chapter 6 for discussion of how to recover previous results You can also save the probabilities and utilities as follows Prob saves the unconditional probabilities based on individual parameters Utility saves the values of utility functions based on individual parameters This estimator will also save various matrices These are discussed in the next section 10 7 Individual Specific Estimates If you include Parameters in your RPLOGIT command NLOGIT will create an nxK matrix named beta_i that contains in a row for each individual an estimate of the random parameters in E Bjlall data for individual i The model command RPLOGIT s Lhs mode Choices air train bus car Rhs mgc ttme one RPL hinc Pts 15 Maxit 10 Pds 3 Parameters Fen mgc n specifies one random parameter The sample in use has 210 3 70 individuals The matrix shown below contains the conditional estimates of the mean of the parameter on mgc The additional matrix sdbeta_i is explained below Chapter 10 The Random Parameters Logit Model 144 8 Matrix BETA OX Matrix spseTa_s ER
76. 2897 6635 This is a two level hierarchical model There are no random parameters but the coefficients on gc and ttme are modeled as linear functions of a constant and household income RPLOGIT Lhs mode Choices air train bus car Rhs gc ttme Rh2 one RPL hinc Fen ge c ttme c Normal exit from iterations Exit status 0 Random Parameters Logit Model Dependent variable MODE Number of observations 210 Log likelihood function 198 3960 Info Criterion AIC 1 95615 Finite Sampl AIC 1 95879 Info Criterion BIC 2 06772 Info Criterion HQIC 2 00126 Restricted log likelihood 291 1218 McFadden Pseudo R squared 3185122 Chi squared 185 4517 Degrees of freedom 7 Prob ChiSqd gt value 0000000 Response data are given as ind choice R2 1 LogL LogL Log L fncen R sqrd RsqAdj No coefficients 291 1218 31851 31086 Constants only 283 7588 30083 29297 At start values 199 9766 00790 00324 Replications for simulated probs 500 Number of obs 210 skipped 0 bad obs Variable Coefficient Standard Error b St Er P Z gt z SS asd Random parameters in utility functions GC 01139658 00920685 1 238 2158 TIME 08786478 01174507 7 481 0000 TAT Nonrandom parameters in utility functions A_AIR 5 84415090 65860452 8 874 0000 A_TRAIN 3 96545510 44224936 8 967 0000 A_BUS 3 25638033 45029696 T232 0000 Ch
77. 3 A_TRAIN 000000 3 amp Fixed Parameter A_BUS SOOO OP ary hay Fixed Parameter Note that as in the IIA test this procedure results in exclusion of some bad observations that is the ones that selected the excluded choices Because of the model specification the ASCs for bus and train have been fixed at zero You may combine the choice based sampling estimator with the restricted choice set All the necessary adjustments of the weights are made internally Thus the specification Choices air train bus car 14 13 09 64 produces the following listing Choice prop Weight IIA AIR 49573 387 TRAIN 00000 000 BUS 00000 000 CAR 50427 1 739 Chapter 5 Data Setup for NLOGIT 46 5 4 Types of Data on the Choice Variable We allow several types of data on the choice variable y If you have grouped data the values of y will be proportions or frequencies instead of individual choices In the first case within each observation J data points the values of y will sum to one when summed down the J rows This will be the only difference in the grouped data treatment In the second case y will simply be a set of nonnegative integers An example of a setting in which such data might arise would be in marketing where the proportions might be market shares of several brands of a commodity Or the data might be counts of responses to particular questions in a survey in
78. 33 sSSeSos Se Characteristics in numerator of Prob Y 2 Constant 54216913 54865993 988 3231 EDUC 06151644 03616780 167 01 0890 10 8759203 HHNINC 85929376 44943471 1 912 0559 32998942 AGE 00089766 00650574 138 8903 46 9925061 HHKIDS 13920984 15529658 896 3700 7331699533 S Characteristics in numerator of Prob Y 3 Constant 25432932 49206457 geo LU 6053 EDUC 10995580 03246796 3 387 0007 10 8759203 HHNINC 1 54516927 40166793 3 847 0001 32998942 AGE 00955207 00583708 1 636 1017 46 9925061 HHKIDS 08177804 14014086 584 5595 33169533 an Sees Characteristics in numerator of Prob Y 4 Constant 09378185 48301274 194 8461 EDUC 10453491 03201865 3 265 0011 10 8759203 HHNINC 1 74362305 39382043 4 427 0000 32998942 AGE 01430375 00571476 2 2 503 0123 46 9925061 HHKIDS 19548647 213659829 1 431 1524 33169533 H et Characteristics in numerator of Prob Y 5 Constant 1 58458651 45170179 3 508 0005 EDUC 07526768 03034831 2 480 0131 10 8759203 HHNINC 1 64030015 37209397 4 408 0000 32998942 AGE 01481141 00525964 2 816 0049 46 9925061 HHKIDS 19988328 12654882 Leb 49 1142 33169533 Chapter 4 The Basic Multinomial Logit Model 36 The prediction for any observation is the cell with the largest predicted probability for that observation NOTE If you have more than three outcomes it is very common as occurred above for the model to pred
79. 4 2674 7492 4430 Descriptive Statistics for Alternative AIR Utility Function 58 0 observs Coefficient All 210 0 obs that chose AIR Name Value Variable Mean Std Dev Mean Std Dev GC 0109 GC 102 648 30 575 LAB S552 33 198 TTME 0955 TIME 61 010 ISLS 46 534 24 389 A_AIR 5 8748 ONE 1 000 000 1 000 000 AIRXHIN1 0054 HINC 34 548 19 711 41 724 19 115 100 Chapter 8 The Multinomial Logit Model 101 Descriptive Statistics for Alternative TRAIN Utility Function 63 0 observs Coefficient All 210 0 obs that chose TRAIN Name Value Variable Mean Std Dev Mean Std Dev GC 0109 GC 130 200 58 235 106 619 49 601 TME 0955 TTME 35 690 12 279 28 524 19 354 A_TRAIN 5 5499 ONE 1 000 000 1 000 000 TRAXHIN2 0566 HINC 34 548 19 711 23063 17 287 Descriptive Statistics for Alternative BUS Utility Function 30 0 observs Coefficient All 210 0 obs that chose BUS Name Value Variable Mean Std Dev Mean Std Dev GC 0109 GC 1155257 44 934 108 133 43 244 TIME 0955 TIME 41 657 12 077 25 200 14 919 A_BUS 4 1303 ONE 1 000 000 1 000 000 BUSXHIN3 0286 HINC 34 548 19 711 29 700 16 851 Descriptive Statistics for Alternative CAR Utility Function 59 0 observs Coefficient All 210 0 obs that chose CAR Name Value Variable Mean Std Dev Mean Std Dev GC 0109 GC 95 414 46 827 89 085 49 833 TTME 0955 T
80. 45 0 0115196 0 00746315 0 0424947 0 0232686 0 0615077 0 00992424 0 0499549 0 0285992 0 102599 0 00876961 0 0442738 0 0295077 0 159558 0 00933539 0 0484009 ANNATA MAMA AT 19 nninia nneqaest Mi Figure 10 7 Conditional Means and Standard Deviations Chapter 10 The Random Parameters Logit Model 149 10 7 2 Examining the Distribution of the Parameters The structural parameters often give a misleading picture of the parameters in a model Consider the following modification of the model estimated in the previous section We are going to fit the model as above but change the distribution of the random parameters from normal to Weibull The Weibull model forces parameters to be positive so we also reverse the signs on the two attributes in the model CREATE mgc gc mttme ttme RPLOGIT s Lhs mode Choices air train bus car Rhs mgc mttme Rh2 one ECM air car train bus RPL hinc Parameters Halton Pds 3 Pts 200 Fen mgc n mttme n Correlated MATRIX bn beta_i sn sdbeta_i The estimation and analysis is repeated with the Weibull distribution Fen mgc w ttme w Correlated MATRIX bw beta_i sw sdbeta_i The unconditional values in the first column of the matrix in Figure 10 7 and the nonstochastic estimates for the MNL model should suggest the likely values of the two random parameters However it would be difficult to deduce this from the estimated
81. 52295E 01 10435090E 01 9 304 0000 A_AIR 5 776358875 65591872 8 807 0000 A_TRAIN 3 923001236 44199360 8 876 0000 A_BUS 3 210734711 44965283 7 140 0000 Discrete Choic One Level Model Model Simulation Using Previous Estimates Number of observations 210 Simulations of Probability Model Model Discrete Choic One Level Model Simulated choice set may be a subset of the choices Number of individuals is the probability times the number of observations in the simulated sample Column totals may be affected by rounding error The model used was simulated with 210 observations Specification of scenario 1 is Attribut Alternatives affected Change type Value TME AIR Scale base by value 1 250 The simulator located 209 observations for this scenario Simulated Probabilities shares for this scenario Choice Base Scenario Scenario Base sShare Number Share Number ChgShare ChgNumber AIR 21 e619 58 15 118 32 12 501 TRAIN 30 000 63 33 694 UE 3 694 BUS 14 286 30 16 126 34 1 841 CAR 28 095 59 35 061 74 6 966 Total 100 000 210 100 000 211 000 gt 26 8 4 15 I 89 Chapter 7 Simulating Probabilities in Discrete Choice Models 90 The model predicts the base case using the actual data shown in the left side and what would become of this case if the scenario is assumed In this case each person s ttme for air travel is increased by
82. 6 branch4 a7 a8 Trunk2 limb3 branch5 a9 a10 branch a11 a12 limb4 branch7 a13 a14 branch8 a15 a16 9 3 3 Utility Functions You may define the utility functions exactly as described in Chapter 3 for one level models You may also define utility functions for branches and limbs and trunks but note that in order to do so you must use the explicit form These are specified exactly the same as those for elemental alternatives For example in a two level model you might put demographic characteristics such as income or family size at the top level A complete model might appear as follows NLOGIT Lhs mode Choices air train bus car Tree travel public bus train private air car s Model U air ba bcost gc btime ttme U train bt bcost gc btime ttme U car be bcost gc btime ttme U bus bcost gc btime ttme U public ap apub hinc U private aprv hinc This model can be considerably collapsed Model U air train bus car lt ba bc 0 bt gt bcost gc btime ttme U public private lt ap 0 gt lt apub aprv gt income Note that the same function specification U is used for all three kinds of equations for alternatives branches and limbs Finally as noted earlier you may impose equality constraints at any points in the model just by using the same parameter name where you want the equality imposed For example if
83. 6 0000 A_TRAIN 6 95973395 1 03548341 6 721 0000 A_BUS 6 12199207 1 13357506 5 401 0000 TTT AT Diagonal values in Cholesky matrix L NsGC 01100134 01124017 979 3277 NsTIME 03678160 03024421 1 216 2239 SoS SSSses Below diagonal values in L matrix V L Lt TIME GC 07457516 02353048 3 169 0015 a r Standard deviations of parameter distributions sdGC 01100134 01124017 979 3217 sdTIME 08315251 01967123 4 227 0000 Correlation Matrix for Random Parameters Matrix COR MAT has 2 rows and 2 columns GC TIME GC 1 00000 89685 TTME 89685 1 00000 Chapter 10 The Random Parameters Logit Model 135 Covariance Matrix for Random Parameters Matrix COV MAT has 2 rows and 2 columns GC TTME GC 00012 00082 TTME 00082 00691 Cholesky Matrix for Random Parameters Matrix Cholesky has 2 rows and 2 columns GC TTME GC 01100 0000000D 00 TTME 07458 03678 We emphasize these results apply to the linear functions of the underlying random variables not necessarily to the implied distributions of the random parameters themselves In most of the specifications the parameters involve nonlinear transformations of these variables 10 3 6 Command Builders for the RPL Models With a few important exceptions the random parameters logit RPL model can be specified with the command builder by selecting Model Discrete Choice Multinomial Probit HEV RPL The Main
84. 70 1 Cel 0 0274657 70 1 Cel 0 0101905 0 0274657 3 0 0101905 0 0158211 J 0 00785383 0 024132 0 0120969 0 0271134 0 0108668 0 0168981 0 00880909 0 015507 0 00729046 0 015860 0 00725378 0 0180881 0 0120018 0 031572 0 0136248 0 0139563 0 00803417 Figure 10 6 Estimated Conditional Means and Standard Deviations The next section will describe how these matrices are computed 10 7 1 Computing Individual Specific Parameter Estimates The random parameters model and the simulation based estimator used to estimate it allow the analyst to derive more information from the data than is usually available from models with fixed parameters In particular the model specifies that B B Az T Q v where for simplicity if there are any we include the alternative specific constants in B and where if there are nonrandom parameters in the model these are accommodated simply by having rows and columns of zeros in the appropriate places in T and Q There may also be rows of zeros in A for parameters that have homogeneous means We are interested in learning as much as possible about P and functions of B from the data The unconditional mean of B is Ep Zi p AZ Absent any other information this provides the template that one would use to form their best estimate of B However there is other information about individual i in the sample namely the choices they made y and other information about their heterogeneity h
85. 71 20 787 44 12 907 2l BUS 16 126 34 16 383 34 291 0 CAR 35 061 74 32 662 69 2 399 5 Total 100 000 211 100 000 210 0003 i Finally you can use the simulator to restrict the choice set The computed probabilities are computed assuming only the specified alternatives are available To do this use Scenario the subset of alternatives To continue the example we simulate the model assuming that people could not drive and examine what the effect of increasing terminal time in airports would do to the market shares for the remaining three alternatives SAMPLE 31 840 NLOGIT Lhs mode Rhs one gc ttme Choices air train bus car NLOGIT Lhs mode Rhs one gc ttme Choices air train bus car Simulation air train bus Scenario ttme air 1 25 Chapter 7 Simulating Probabilities in Discrete Choice Models Discrete Choic One Level Model Model Simulation Using Previous Estimates Number of observations 210 Simulations of Probability Model Model Discrete Choic One Level Model Simulated choice set may be a subset of the choices Number of individuals is the probability times the number of observations in the simulated sample Column totals may be affected by rounding error The model used was simulated with 210 observations Specification of scenario 1 is Attribute Alternatives affected Change type Value TME AIR Scale
86. 8840 Chi squared 202 5532 Degrees of freedom 13 Prob ChiSqd gt value 0000000 R2 1 LogL LogL Log L fnen R sqrd RsqAdj No coefficients 291 1218 34788 33414 Constants only 283 7588 33096 31687 At start values 216 5343 12326 10478 These are the estimates for the multinomial logit model Chapter 11 The Multinomial Probit Model 165 Variable Coefficient Standard Error b St Er P Z gt z GC 01092735 00458775 2 382 0172 TTME 09546055 01047320 O 115 0000 A_AIR 5 87481336 80209034 7 324 0000 AIR_HIN1 00537349 01152940 466 6412 A_TRAIN 5 54985728 64042443 8 666 0000 TRA_HIN2 05656186 01397335 4 048 0001 A_BUS 4 13028388 67636278 6 107 0000 BUS_HIN3 02858418 01544418 1 851 0642 These are the estimates for the multinomial probit model Variable Coefficient Standard Error b St Er P Z gt z SoSsSSsS 4 Attributes in the Utility Functions beta GC 02333086 00896463 2 603 0093 TIME 09131236 03629673 2 516 SOTTO A_AIR 4 68057508 191530359 2 444 0145 AIR_HIN1 00832932 02520384 7330 7410 A_TRAIN 5 90782858 1 92699048 3 066 0022 TRA_HIN2 06016958 02223662 2 706 0068 A_BUS 4 40097868 1 27339698 3 456 0005 BUS_HIN3 01884772 01615587 1 167 2434 Soe Std Devs of the Normal Distribution s AIR 2 85536857 1 29978748 24 LOJ 0280 s TRAIN 1 96198515 91344112 2
87. 9 164 0000 A_AIR 6 00304781 69769310 8 604 0000 A_TRAIN 4 10077954 47428938 8 646 0000 A_BUS 3 39796835 48316868 7 033 0000 Heterogeneity in mean Parameter Variable MGC HIN 00021077 00023228 907 3642 Derived standard deviations of parameter distributions TsMGC 05487307 02445605 2 244 0248 Now the mean is 02134585 and the standard deviation is 05487307 V6 022401837 The preceding serves to emphasize the need to interpret the estimated model parameters on a case by case basis Each distribution has different characteristics Worse yet in some of those cases we do not even have the convenient formulas given above to use to convert the parameters to population moments Consider the Weibull distribution which we obtain with Fen mge w For this model exp B 5z ov v 2log u u U O 1 The estimated parameters of the model are as follows Se STS En Random parameters in utility functions MGC 3 44822322 1 06929334 3 2295 0013 BSR Saori tNonrandom parameters in utility functions TIME 09807615 01018490 9 630 0000 A_AIR 5 90493475 69570219 8 488 0000 A_TRAIN 4 04347670 49138509 8 229 0000 A_BUS 3 32608885 51475257 6 462 0000 Heterogeneity in mean Parameter Variable MGC HIN 01555286 01425775 t09 2753 Derived standard deviations of parameter distributions WsMGC 79003797 83918130 941 3465 There is no obvious way to translate these back to a mean and
88. AIR 000 000 1 015 L971 X Choice CAR 000 000 1 015 4338 Total Mean 857 897 444 af 2353 532 532 746 1 059 Effect St Dev Note that across a row the effects sum to the total effect given The default method of computing the elasticities is to average the observation specific results The results show the mean and the sample standard deviations If you use the Means specification then the elasticities are computed once and the results reflect the change as shown below The differences are noticeably large Elasticity computed at sample means Attribute is GC in choice CAR Effects on probabilities of all choices in the model indicates direct Elasticity effect of the attribute Decomposition of Effect if Nest Trunk Limb Branch Choice Trunk Trunk 1 Limb TRAVEL Branch PUBLIC Choice BUS 000 000 584 000 Choice TRAIN 000 000 584 000 Branch PRIVATE Choice AIR 000 000 ArT 303 Choice CAR 000 000 411 605 Total Mean 584 584 107 016 Effect St Dev 000 000 000 000 Chapter 9 The Nested Logit Model 114 9 5 Inclusive Values Utilities and Probabilities You can request a listing of the actual outcomes and predicted probabilities with List For large nested logit models the listing would be extremely cumbersome so a list can only be produced for models with seven or fewer elemental alternative
89. At the next level up the tree we define the conditional probability of choosing a particular branch in limb J trunk r P blhr EXPCO Y prr TJ our _ EXPO yy Ty rI or Dy yy PXPCOY s TF anr exp J where Jj is the inclusive value for limb in trunk r In log Xs exp Q Ysi Tss The probability of choosing limb in trunk r is P llr exp 8Z Ondu z exp 6 Z OnLy ae exp 6 z slr Oglar exp H where H is the inclusive value for trunk r H log Xs exp Zs Osr Is Finally the probability of choosing a particular limb r is exp h H ey exp h 0 H By the laws of probability the unconditional probability of the observed choice made by an individual is PQ b Lr P jlb l r x P bll r x Pdr x P r This is the contribution of an individual observation to the likelihood function for the sample Chapter 9 The Nested Logit Model 105 The nested logit aspect of the model arises when any of the t or oj or Q differ from 1 0 If all of these deep parameters are set equal to 1 0 the unconditional probability specializes to exp B X j O 82 Oh 2 ya yy Diy exp B X np m r OV 5 z 6 h which is the probability for a one level model The model is written in a very general form The parameters of the model are in exactly this order PG DjLr B1 B2 Bnx O11 0L25 Onys01 025 0n2 91 09 0 nhs T1 TB 01 O7 015 50R
90. B 6z o and B 6z o Themeanis B 5z but the variance is 407 12 with a standard deviation of o V3 The estimated parameters are as follows Sasa SS Ss4 Random parameters in utility functions MGC 00893792 00978908 913 3612 SSeS a4 tNonrandom parameters in utility functions TIME 09779935 01063867 9 193 0000 A_AIR 5 86320087 68262859 8 589 0000 A_TRAIN 3 99147415 46159989 8 647 0000 A_BUS 3 28433873 47262187 6 949 0000 Heterogeneity in mean Parameter Variable MGC HIN 00021461 00022919 936 3491 Derived standard deviations of parameter distributions UsMGC 102222135 01975890 1 125 2607 Based on these results the overall mean is about 00893792 35 00021461 0164492 again comparable and the standard deviation is 01289502 What is reported is a scale factor or spread parameter not the standard deviation of the distribution Chapter 10 The Random Parameters Logit Model 130 The triangular distribution presents the same ambiguity In this model B B 6z ov v Triangular 1 1 The mean is B z but the variance is 0 6 which is one half the variance of the uniform distribution with the same spread and mean Repeating the previous estimation now with Fen mgc t we obtain the results below Sasa Seae4 Random parameters in utility functions MGC 01396869 01082759 1 290 1970 SSeS tNonrandom parameters in utility functions TIME 09931295 01083732
91. C INVT GC E none 2 none Constant CAR INVC INVT GC E none 2 none none Normal exit from iterations Exit status 0 Discrete choice multinomial logit model Maximum Likelihood Estimates Dependent variable Choice Weighting variable None Number of observations 117 Iterations completed 6 Log likelihood function 52 79148 Number of parameters 5 Chapter 5 Data Setup for NLOGIT 45 Info Criterion AIC 98789 Finite Sample AIC 99251 Info Criterion BIC 1 10593 Info Criterion HQIC 1 03581 R2 1 LogL LogL Log L fncen R sqrd RsqAdj Constants only 81 0939 34901 31995 Chi squared 4 56 60494 Prob chi squared gt value 00000 Response data are given as ind choice Number of obs 210 skipped 93 bad obs Restricted choice set Excluded choices are TRAIN BUS T Notes No coefficients gt P i j 1 J i Constants only gt P i j uses ASCs only N j N if fixed choice set N j total sample frequency for j N total sample frequency These 2 models are simple MNL models R sqrd 1 LogL model logL other RsqAdj 1 nJ nJ nparm 1 R sqrd nJ sum over i choice set sizes Variable Coefficient Standard Error b St Er P Z gt z INVC 04871233 02756765 1 767 0772 INVT 01195151 00394602 3 029 0025 GC 08575924 02654046 3u23 L 0012 TIME 08221552 01854075 4 434 0000 A_AIR 2 12899069 1 20530610 1 766 077
92. CML specification for imposing linear constraints Hold for using the multinomial logit model as a sample selection equation In addition if your model size exceeds 150 parameters the matrices b and varb cannot be retained Chapter 4 The Basic Multinomial Logit Model 32 4 3 Model Command for the Multinomial Logit Model The command for fitting this form of multinomial logit model is MLOGIT Lhs y or y0 yl yJ Rhs regressors The command may also be LOGIT which is what has always been used in previous versions of LIMDEP and NLOGIT All general options for controlling output and iterations are available except Keep name A program which can be used to obtain the fitted probabilities is listed below There are internally computed predictions for the multinomial logit model The Rst list form of restrictions is supported for imposing constraints on model parameters either fixed value or equality One possible application of the constrained model involves making the entire vector of coefficients in one probability equal that in another You can do this as follows NAMELIST _ x the entire set of Rhs variables CALC k Col x LOGIT Lhs y Rhs x Rst k_b k_b k_b This would force the corresponding coefficients in all probabilities to be equal You could also apply this to some but not all of the outcomes as in Rst k_b k_b k_b2 k_b3 HINT The coefficients in this model are not th
93. Command Builder The command builders can be used to specify the nested logit models Select Model Discrete Choice Nested Logit to access the command builder The choice variable is defined on the Main page and the rest of the model may be specified on the Options page See Figure 9 1 NLOGIT Main Options Output Choice variable Choice variable MODE v Data type Individual choice z J Use ordinary weights v m Choice set Fixed number of choices Choice names air T rain Bus Car F Use choice based sampling weights I Data coded on one line Code Variable number of choices Count variable J Use universal choice set indicato Choice names I Perform IIA test on choices I Use data scaling NLOGIT Main Options Output I Use one line setup Labels m Utility functions m Attributes m Interact with ASC GC ONE TTME HINC t gt gt gt gt 7 Specify utility functions T Box Cox jo Inclusive value setup I SET J Perform Lagrange Multiplier test at start values Tree Specification Optimization Run Cancel Figure 9 1 Command Builder for Nested Logit Models Chapter 9 The Nested Logit Model 111 The tree is specified in a subsidiary dialog box by selecting Tree Specification at the bottom of the Options page The dialog box shown in Figure 9 2 allows you to define the tree graphically Note in the
94. EE 107 9 3 4 Setting and Constraining Inclusive Value Parameters cisssicscissisersssiricissesrecirres 108 NLOGIT Student Reference Guide Table of Contents viii B34 Comunid Ber cracar in annn ine 110 9A Marginal Effects and Blasueiies scssscccscatacessacsvaceinusessesaustssesvoesiagesunessaesuvetos Eara ea ieie 112 9 5 Inclusive Values Utilities and Probabdlitess ccnwiciaiienueiintanneuniceawevied 114 96 Apphceaiomofa Nested Logit Model orcsoun nan ie 115 Chapter 10 The Random Parameters Logit MOdel ssscesseeeeeeeeeeeeeeeeeeeeeeeeeeeeeneesenneees 120 OE Ne Oa rtaren i cmrets sewer rtmrnt perrr a eree eeenr or eenre ae 120 10 2 Random Parameters Mined Logit Models ac pisciasete rnin rtneta heed AE St 120 10 3 Command Tor the Random Parameters Logit MOGEIS vias casesesssserenzssnsbvass vasvebessnssseasenmvnase 124 10 3 1 Distributions of Random Parameters in the Model ceesceeseceesteceeeeeeees 125 10 3 2 Spreads Scaling Parameters and Standard DevidtionS ssissisisieisssssscsssivssersssssses 128 13 2 Akernmaiye Speci Aon stants ani 132 10 3 4 Heterogeneity in the Means of the Random ParameterS eccceesseeeenteeees 132 DS Cre ate FIn EiS enirir ka RE EE ETI EAEAN 133 10 30 Command Builders Tor the RPL Mode lS ursiosineke e 135 10 4 Heterosc dasticity and Heterogeneity in the Var antes is 2 iscsssciesvsssssevussasd sewsssesvanssncesuvesses 137 10 5 Controlling the Siawlan its ess ics secisssa
95. IME 000 000 000 000 8 5 Marginal Effects We define the marginal effects in the multinomial logit model as the derivatives of the probability of choice j with respect to attribute k in alternative m This is P 1 j m P PB Xm where the function 1 7 m equals one if j equals m and zero otherwise These are naturally scaled since the probability is bounded They are usually very small so NLOGIT reports 100 times the value obtained as in the example below which is produced by Effects gc air Chapter 8 The Multinomial Logit Model 102 Derivative times 100 averaged over observations Attribute is GC in choice AIR Effects on probabilities of all choices in model Direct Derivative effect of the attribute Mean St Dev Choice AIR S339 0880 Choice TRAIN 0362 0309 Choice BUS 0204 0204 Choice CAR 0773 0763 Derivatives and elasticities are obtained by averaging the observation specific values rather than by computing them at the sample means The listing reports the sample mean average partial effect and the sample standard deviation It is common to report elasticities rather than the derivatives These are ee ieee tae eS m P x B a log Xin J m km k The example below shows the counterpart to the preceding results produced by Effects gc air which requests a table of elasticities for the effect of changing gc in the air alternative Ela
96. L LogL Log L fnen R sqrd RsqAdj Constants only 283 7588 08303 07124 Log L for Branch model 118 3945 Response data are given as ind choice Number of obs 210 skipped 0 bad obs Chapter 9 The Nested Logit Model Constants only only N j N Notes No coefficients gt P i j 1 J i gt P i j uses ASCs if fixed choice set sample frequency for j sample frequency are simple MNL models N j total N total These 2 models R sqrd nJ 1 LogL model logL other RsqAdj 1 nJ nJ nparm 1 R sqrd sum over i choice set sizes Variable Coefficient Standard Error b St Er P Z gt z Model for Choice Among Alternatives 741 572 SD 216 428 BT 77778700 20792992 BB 13076048 22872416 BG 01773795 00405470 4 AT 01340138 00317904 4 Model for Choice Among Branches AA 1 92254215 35420335 AH 02612091 00817431 195 0002 5675 0000 0000 0000 0014 The MNL estimates are followed by the nested logit estimates Normal exit from iterations Exit status 0 FIML Nested Multinomial Logit Model Dependent variable MODE Number of observations 210 Log likelihood function 193 6561 Number of parameters 8 Info Criterion AIC 1 92053 Finite Sample AIC 1 92395 Info Criterion BIC 2 04804 Info Criterion HQIC 1 97208 Restricted log likelihoo
97. LOGIT Commands and Results This procedure is automated as shown in the following example CLOGIT Lhs mode Choices air train bus car Rhs inve invt gc ttme CLOGIT Lhs mode Choices air train bus car Ias car Rhs invc invt gc ttme Discrete choice multinomial logit model Dependent variable Choice Number of observations 210 Log likelihood function 244 1342 Number of parameters 4 R2 1 LogL LogL Log L fnen R sqrd RsqAdj Constants only 283 7588 13964 13414 Response data are given as ind choice Number of obs 210 skipped 0 bad obs Variable Coefficient Standard Error b St Er P Z gt z INVC 02242963 01435409 1 563 1181 INVT 00634473 00184168 3 445 0006 GC 03182946 01372856 2 318 0204 TTME 03480667 00469397 SIVALS 0000 WARNING Bad observations were found in the sample Found 59 bad observations among 210 individuals You can use CheckData to get a list of these points Normal exit from iterations Exit status 0 Discrete choice multinomial logit model Dependent variable Choice Number of observations 15 1 Log likelihood function 103 2012 Number of parameters 4 R2 1 LogL LogL Log L fncen R sqrd RsqAdj Constants only 159 0502 35114 34243 Response data are given as ind choice Number of obs 210 skipped 59 bad obs Hausman test for IIA Excluded choices are CAR Ch
98. M specification 3 5 3 Multinomial Probit The multinomial probit model is described in Chapter 11 The essential command is MNPROBIT _ Lhs dependent variable Choices the names of the J alternatives Rhs list of choice specific attributes Rh2 list of choice invariant individual characteristics Variable choice set sizes and utility function specifications are specified as in the CLOGIT command This command is the same as NLOGIT MNP the rest of the command 3 6 Command Summary The following lists the current and where applicable alternative forms of the discrete choice model commands The two sets of commands are identical and for each model in NLOGIT 4 0 either command may be used for that model Models Command Alternative Command Form Binary Choice Models Binary Probit PROBIT PROBIT Binary Logit BLOGIT LOGIT Multinomial Logit Models Multinomial Logit MLOGIT LOGIT Conditional Logit CLOGIT DISCRETE CHOICE Conditional Logit Extensions Conditional Logit CLOGIT CLOGIT Multinomial Logit NLOGIT NLOGIT Same as CLOGIT Nested Logit NLOGIT Tree NLOGIT Tree Random Parameters Logit RPLOGIT NLOGIT RPL Multinomial Probit MNPROBIT NLOGIT MNP Chapter 3 Model and Command Summary for Discrete Choice Models 26 3 7 Subcommand Summary The following subcommands are used in NLOGIT model commands The BLOGIT BPROBIT BVPROBIT MVPROBIT OLOGIT and OPROBIT commands have additiona
99. NLOGIT Version 4 0 Student Reference Guide by William H Greene Econometric Software Inc 1986 2010 Econometric Software Inc All rights reserved This software product including both the program code and the accompanying documentation is copyrighted by and all rights are reserved by Econometric Software Inc No part of this product either the software or the documentation may be reproduced stored in a retrieval system or transmitted in any form or by any means without prior written permission of Econometric Software Inc LIMDEP and NLOGIT are registered trademarks of Econometric Software Inc All other brand and product names are trademarks or registered trademarks of their respective companies Econometric Software Inc 15 Gloria Place Plainview NY 11803 USA Tel 1 516 938 5254 Fax 1 516 938 2441 Email sales limdep com Websites www limdep com and www nlogit com Econometric Software Australia 215 Excelsior Avenue Castle Hill NSW 2154 Australia Tel 61 0 4 1843 3057 Fax 61 0 2 9899 6674 Email hgroup optusnet com au End User License Agreement This is a contract between you and Econometric Software Inc The software product refers to the computer software and documentation as well as any upgrades modified versions copies or supplements supplied by Econometric Software By installing downloading accessing or otherwise using the software product you agree to be bound b
100. R Utility Function 59 0 observs Coefficient All 210 0 obs that chose CAR Name Value Variable Mean Std Dev Mean Std Dev BT 5 0646 TASC 000 000 000 000 BB 4 0963 BASC 000 000 000 000 BG 0316 GC 95 414 46 827 89 085 49 833 AT 1126 TME 000 000 000 000 PREDICTED PROBABILITIES marks actual marks prediction Indiv AIR TRAIN BUS CAR al 1515 3518 1232 3734 2 2676 1949 0260 5114 3 1563 1040 1509 5888 4 3998 1180 0153 4669 5 3418 s350 A 0469 2603 6 1323 3423 4 2212 3043 7 4186 0815 aTT82 3817 8 0955 4956 4 1848 2241 9 1685 3915 4 1371 3030 10 2484 3203 4 ELZ 3191 Observations 11 210 are omitted Part ial ffects averag dinP alt j br b lmb 1 tr r over observations D k J B L R delta k F dx k alt J br B lmb L tr R delta k coefficient on x k in U J B L R F r R 1 L b B j J P J BLR r R 1 L b B P B LR P J BLR t B LR r R 1 L P LIR P B LR P J BLR t B LR s LR r R P R P LIR P BIIR P J BIR t B LR s L R f R P J BLR Prob choice J branch B limb L trunk R P B LR P L R P R defined likewis n N 1 if n N 0 else for n j b 1 r and N J B L R Elasticity x k D 3j B L R arginal effect P JBLR D P J BLR P B LR P L R P R D F is decomposed into the 4 parts in the tables 118 Note t
101. Random Parameters Logit Model 128 10 3 2 Spreads Scaling Parameters and Standard Deviations The RPL model is complicated It is also necessary to note that the interpretation of the parameters is partly a function of the specification chosen What are described earlier as the means and variances are actually only those parameters in the simplest cases The reported parameters may need to be interpreted and manipulated further to obtain the expected results We consider several examples In a model with a normally distributed parameter Bi B 82 OV vi N O 1 B 6z is indeed the conditional mean and o is the standard deviation The model results might appear as follows in which the parameter on variable mgc is specified to have a normal distribution with a mean that is a function of hinc which has a mean of about 35 The specification is RPLOGIT Lhs mode Choices air train bus car Rhs mgc ttme one RPL hinc Pts 15 Maxit 10 Pds 3 Fen mgc n SSeSSe 4 Random parameters in utility functions MGC 01317029 01052638 1 251 2109 Sass a4 tNonrandom parameters in utility functions TIME 09909916 01076768 9 203 0000 A_AIR 6 00438917 69301957 8 664 0000 A_TRAIN 4 09897595 47136837 8 696 0000 A_BUS 3 39330467 48215182 7 038 0000 Heterogeneity in mean Parameter Variable MGC HIN 00023602 00023165 1 019 3083 Derived standard deviations of parameter distributions
102. SCs if you have used one in an Rhs specification But it cannot determine from a set of dummy variables that you yourself provide if they are a set of ASCs because it inspects the model not the data to make the determination As such there is an advantage when possible to letting NLOGIT set up the set of alternative specific constants for you Finally an LM statistic for testing the hypothesis that the starting values are not significantly different from the MLEs the standard LM test is requested by adding Maxit 0 to the NLOGIT command Chapter 7 Simulating Probabilities in Discrete Choice Models 84 Chapter 7 Simulating Probabilities in Discrete Choice Models 7 1 Introduction The simulation program described here allows you to fit a model use it to predict the set of choices for your sample then examine how those choices would change if the attributes of the choices changed You can also examine scenarios that involve restricting the choice set from the original one Finally you can use your estimated model and this simulator to do these analyses with data sets that were not actually used to fit the model The calculation proceeds as follows Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Set the desired sample for the model estimation Estimate the model using NLOGIT This processor is supported for the following discrete choice models that are specific to NLOGIT Model Command Alternati
103. The model specification thus far builds the utility functions from the common Rhs and Rh2 specification For example in a four outcome model which contains cost time one and income the data for the choice variable and the utility functions are contained in choice cost time constants income Voir a t 1 0 O income 0 0 Z Yoran G t 0 1 0 0 income 0 Yus Gt 0 0 1 0 0 income Ya C t O 0 0 0 0 0 Chapter 6 NLOGIT Commands and Results 60 The utility functions are all the same Ui Bicost Botime a yjincome amp One might want to have different attributes appear in the different utility functions or impose other kinds of constraints on the parameters This section will describe how to structure the utility functions individually rather than generically with the Rhs and Rh2 lists The utility functions need not be the same for all choices Different attributes may enter and the coefficients may be constrained in different ways The following more flexible format can be used instead of the Rhs list and Rh2 list parts of the command described above This format also provides a way to provide starting values for parameters so this can also replace the Start list specification Finally you will also be able to use this format to fix coefficients so it will be an easy way to replace the Rst list specification We begin with the case of a fixed and named set of choices then turn to the cases of variable numbers of
104. ameterize this is to write the full model as pi p Az TOw where Q is the diagonal matrix of individual specific variance terms 0 exp hrj Chapter 10 The Random Parameters Logit Model 122 The list of variations above produces an extremely flexible general model Typically you would use only some of them though in principle all could appear in the model at once We will develop them in parts in the sections to follow A convenient form of the full random parameters logit model to begin with is exp a Bix it Prob yi j Sea es Dig exp a B x gie Finally an additional layer of individual heterogeneity may be added to the model in the form of the error components detailed below The full model with all components is exp a B X a Znad nO CXPCY he E Prob j i DA exp a Bix ane dyn 0 exp y he E i where the components of the model are as follows Random Alternative Specific Constants and Taste Parameters ip aB Az TOv Q diag n Wp or Q diag o 0 B a constant terms in the distributions of the random taste parameters Uncorrelated Parameters with Homogeneous Means and Variances Bix Bx OkWik when A 0 T I O diag o 0 Xj all observed choice attributes and individual characteristics Vi random unobserved taste variation with mean vector 0 and covariance matrix I Uncorrelated Parameters with Heterogeneous Means and Variance
105. and in LIMDEP Version 6 0 which provided some of the features that are described with the estimator presented in Chapter N13 of this reference guide NLOGIT itself began in 1996 with the development of the nested logit command originally an extension of the multinomial logit model With the additions of the multinomial probit model and the mixed logit model among several others NLOGIT has now grown to a self standing superset of LIMDEP The focus of most of the recent development is the random parameters logit model or mixed logit model as it is frequently called in the literature NLOGIT is now the only generally available package that contains panel data repeated measures versions of this model in random effects and autoregressive forms We note the technology used in the random parameters model originally proposed by Dan McFadden and Kenneth Train has proved so versatile and robust that we have been able to extend it into most of the other modeling platforms that are contained in LIMDEP They like NLOGIT now contain random parameters versions Finally a major feature of NLOGIT is the simulation package With this program you can use any model that you have estimated to do what if sorts of simulations to examine the effects on predicted behavior of changes in the attributes of choices in your model NLOGIT Version 4 0 is the result of an ongoing since 1985 collaboration of William Greene Econometric Software Inc and David Hens
106. apter 10 The Random Parameters Logit Model 159 Heterogeneity in mean Parameter Variable GC HI 00010094 00020965 481 6302 TTME HIN 00028132 00017922 1 5 70 1165 Derived standard deviations of parameter distributions CsGC 000000 gt sea ates Fixed Parameter CsTTME 000000 W saawx Fixed Parameter Parameter Matrix for Heterogeneity in Means Matrix Delta has 2 rows and 1 columns HINC GC 00010 TTME 00028 10 9 Panel Data The random parameters model includes a treatment for panel data Two forms are accommodated For a simple clustering of T choice situations by the same individual for example a stated preference survey in which several different scenarios are offered then a random effects type of treatment might be appropriate For example the sequencing of choices might be unknown In this case the usual random effects setup would apply Bi B Az Tv where f indexes the multiple observations for individual i The connection to time might not hold here but we use the same index regardless Note that the heterogeneity in the mean may change from one observation to the next or not depending on your situation but the random term v is the same for all observations As in all panel data situations in NLOGIT the number of observations 7 on individual i may vary by individual An alternative situation might arise when choice situations
107. ariable and type of choice set in the three sections of this dialog box Chapter 6 NLOGIT Commands and Results 54 DISCRETE CHOICE Main Options Output Choice variable Choice variable MODE v Data type Individual choice M Use ordinary weights Choice set Fixed number of choices Choice names I Use choice based sampling weights Data coded on one line Code Variable number of choices Count variable r Choice names Perform IIA test on choices Use data scaling Cancel Figure 6 3 Main Page of Command Builder for Conditional Logit Model NOTE The command builder for the multinomial probit HEV and RPL models requires you to provide a fixed sized choice set This is a limitation of the command builder window not the estimator With the exception of the multinomial probit model this is not a requirement of the models themselves Only the multinomial probit model requires the number of choices to be fixed For the HEV and RPL models if you build your command in the text editor rather than with the command builder you may specify a variable choice set 6 2 3 Restricting the Choice Set The IA test described in the Chapter 8 is carried out by fitting the model to a restricted choice set then comparing the two sets of parameter estimates You can restrict the choice set used in estimation irrespective of IIA by a slight change in the command In the Choices list of a
108. ata estimates are consistent and in the binomial case efficient The least squares estimates are included in the displayed results by including OLS in the model command The iterations are followed by the maximum likelihood estimates with the usual diagnostic statistics An example is shown below NOTE Minimum chi squared MCS is an estimator not a model Moreover the MCS estimator has the same properties as but is different from the maximum likelihood estimator Since the MCS estimator in NLOGIT is not iterated it should not be used as the final result of estimation Without iteration the MCS estimator is not a fixed point the weights are functions only of the sample proportions not the parameters For current purposes these are only useful as starting values Standard output for the logit model will begin with a table such as the following which results from estimation of a model in which the dependent variable takes values 0 1 2 3 4 5 LOGIT Lhs newhsat Rhs one educ hhninc age hhkids ultinomial Logit Model aximum Likelihood Estimates Dependent variable NEWHSAT Weighting variable None umber of observations 8140 Iterations completed 5 Log likelihood function 11246 97 umber of parameters 25 Info Criterion AIC 2 76953 Finite Sample AIC 2160955 Info Criterion BIC 2 79104 Info Criterion HQIC 2 77688 Restricted log likelihood 11308 02 McFadden Pseudo R squared 0053989
109. ata will appear as follows e Individual data y coded 0 1 J e Grouped data Yo y1 4i give proportions or shares In the grouped data case a weighting variable n may also be provided if the observations happen to be frequencies The proportions variables must range from zero to one and sum to one at each observation The full set must be provided even though one is redundant The data are inspected to determine which specification is appropriate The number of Lhs variables given and the coding of the data provide the full set of information necessary to estimate the model so no additional information about the dependent variable is needed This model proliferates parameters There are JxK nonzero parameters in all since there is a vector B for each probability except the first Consequently even moderately sized models quickly become very large ones if your outcome variable y takes many values The maximum number of parameters which can be estimated in a model is 150 as usual with the standard configuration However if you are able to forego certain other optional features the number of parameters can increase to 300 This is the only model in NLOGIT that extends the 150 parameter limit The model size is detected internally If your configuration contains more than 150 parameters the following options and features become unavailable marginal effects choice based sampling Rst list for imposing restrictions
110. base by value 1 250 The simulator located 209 observations for this scenario Simulated Probabilities shares for this scenario Choice Base Scenario Scenario Base sShare Number Share Number ChgShare ChgNumber AIR 39 7393 83 22 933 48 16 420 TRAIN 40 985 86 52 281 110 11 297 BUS 19 662 41 24 786 52 2 123 Total 100 000 210 100 000 210 000 35 24 TI 0 93 Chapter 8 The Multinomial Logit Model 94 Chapter 8 The Multinomial Logit Model 8 1 Introduction In the multinomial logit model there is a single vector of characteristics which describes the individual and a set of J parameter vectors In the discrete choice setting of this section these are essentially reversed The J alternatives are each characterized by a set of K attributes x Respondent i chooses among the J alternatives There is a single parameter vector B The model underlying the observed data is assumed to be the following random utility specification U choice j for individual i Uj B x 6 j 1 Ji The random individual specific terms amp 1 6 7 are assumed to be independently distributed each with an extreme value distribution Under these assumptions the probability that individual i chooses alternative j is Prob Uj gt Uu for all q j It has been shown that for independent extreme value distributions as above this probability is exp B x Ji 1 gt exp B mi
111. ber of choices may still differ by observation Thus we might have unranked 0 1 0 0 0 in the usual case and ranked 4 1 3 2 5 with ranks data Note that the positions of the ones are the same for both sets by definition You may also have partial rankings For example suppose respondents are given 10 choices and asked to rank their top three Then the remaining six choices should be coded 4 0 A set of ranks might appear thusly 1 4 2 4 3 4 4 4 4 4 The ties must only appear at the lowest level Ties in the data are detected automatically No indication is needed For later reference we note the following for the model based on ranks data You may have observation weights but no choice based sampling The IIA test described in Chapter 8 is not available The number of choices may be fixed or variable as described above You may keep probabilities or inclusive values as described in Chapter 7 Ranks data may only be used with the conditional logit model CLOGIT and the mixed logit random parameters model RPLOGIT Chapter 5 Data Setup for NLOGIT 47 The first three data types are detected automatically by NLOGIT You do not have to give any additional information about the data set since the type of data being provided can usually be deduced from the values See below for one exception The ranks data are an exception for which you would use NLOGIT as usual Ranks If you are using frequency or p
112. bservations The original choice set had J 4 choices but two were excluded leaving J 2 in the sample The log likelihood is 62 58418 The constants only log likelihood is obtained by setting each choice probability to the sample share for each outcome in the choice set For this application those are 0 49573 for air and 0 50427 for car This computation cannot be done if the choice set varies by person or if weights or frequencies are used Thus the log likelihood for the restricted model is Log Lo 117 0 49573 x log 0 49573 0 50427 x log 0 50427 81 09395 The R is 1 62 54818 81 0939 0 22869 including some rounding error The adjustment factor is K Ji n Ji n npfree 234 117 234 117 3 1 02632 and the Adjusted R is 1 K log L LogLo Adjusted R 1 1 02632 62 54818 81 0939 0 20794 6 3 1 Retained Results Results kept by this estimator are Matrices b and varb Scalars logl nreg kreg Last Model b_variable coefficient vector and asymptotic covariance matrix log likelihood function N the number of observational units the number of Rhs variables the labels kept for the WALD command Chapter 6 NLOGIT Commands and Results 70 In the Last Model groups of coefficients for variables that are interacted with constants get labels choice_variable as in trai_gco Note that the names are truncated up to four characters for the
113. by slashes Also there is no problem using this device to force IV parameters at one level to equal those at another Thus private public water forces Op pjj to equal Twateripublic ANA Oprivate In addition to the preceding you may fix inclusive value parameters The setup is the same as above with the additional specification of the value in square brackets Le Ivset the value The list in parentheses may contain a single name so as to fix a particular coefficient at a given value You might have Ivset private public fly ground 75 land 95 You will see a diagnostic message if you attempt to modify an inclusive value parameter that is fixed at 1 0 for identification purposes For example this specification of a two level model Tree travel public bus train private air car Ivset travel 75 generates an error message SINCE Otave 1 0 one limb Note also that fixed IV parameters are off limits to equality constraints as well Thus for this example the specification Ivset travel public also generates an error Error 1093 You have given a spec for an IV parm that is fixed at 1 You may not change the specification of bavet In the output of the estimation procedure inclusive value parameters are denoted by the name of the branch or limb to which they are attached or the default names given earlier Chapter 9 The Nested Logit Model 110 9 3 5
114. capabilities If you are uncertain which program is actually installed on your computer go to the About box in the main menu It will clearly indicate which program you are operating Chapter 2 Discrete Choice Models 11 Chapter 2 Discrete Choice Models 2 1 Introduction This chapter will provide a short thumbnail sketch of the discrete choice models discussed in this manual NLOGIT supports a large array of models for both discrete and continuous variables including regression models survival models models for counts and of relevance to this setting models for discrete outcomes The group of models described in this manual are those that arise naturally from a random utility framework that is those that arise from a consumer choice setting in which the model is of an individual s selection among two or more alternatives This includes several of the models described in the LIMDEP manual such as the binary logit and probit models but also excludes some others including the models for count data and some of the loglinear models such as the geometric regression model 2 2 Random Utility Models The random utility framework starts with a structural model U choice 1 f attributes of choice 1 characteristics of the consumer V W U choice J f attributes of choice J characteristics of the consumer amp V W where amp 6 denote the random elements of the random utility functions and in our later treatments v
115. ccursstseoasescavasyachgccuspeieduastatsdareiedecuvecosduasveveleadedadeanvises 138 10 3 Number and Inianon af the Random Draws sccisccdnnccnmsarnoscadsascmies 138 10 5 2 Halton Draws and Random Draws for Simulations 1 0 00 ecececeeseeesteeeeteeeeenees 139 10 e Model ESMANE ina RE E E E E vas ele et 139 10 7 individual Specihie ESti ES os sess cesssegzscasnea vbassyaisetssnsesenessndsetssmen E EESO KAEN RI EEES 143 10 7 1 Computing Individual Specific Parameter Estimates sissercisrssitcissrssiscivssseciisss 144 10 7 2 Examining the Distribtition of the Parameters sccuisccccanstncasadaancnesusnieins 149 10 7 3 Conditional Conhidence Intervals tor Paraiineters i5 sicsacscastsescansicteanatieebenisden 153 10 74 Willingness t6 Pay Estimates resourser natrie e iee 155 TOS Se DCA ONNS arresi AEE E res EEEE EESE AE 156 R a E E T EE E E E A E E E E A E E 159 Chapter 11 The Multinomial Probit Model ccccccsseeeeeeeeeeeeeeeeeeeeeeeeeeeeneeeeeeeeeneeeeneeee 161 DUA DO PET AN E AAA AEA AT entertainer AN 161 L NORTE n aaa a EN 162 DLS An Papp CAM Oi 5 ooasseseszsnsese scursenadeanasaeoasscigeaussanndusr secgsuarastedvarrecesauensheieup TENETE EES E SOS 164 114 Testing UA witha Mulinomial Probit Model icc ccicecissctas crenata nandiianinendatiieemaione 166 Chapter 12 Diagnostics and Error Messages ccccccesseeseeeeeeeeeeeeeeeeeeeeeseeneeeeeeeeeeeeeenneees 167 121 MRO OWE IOI oorr E A OTE 167 122 Dis tete Choice CLOG Y 30d EAGT naa
116. ces if desired see below Scenario what changes and how We take the base case first in which all alternatives are considered in the simulation A scenario is defined using Scenario attribute choices in which it appears the change as shown in the preceding section The results of the computation will show the market shares before and after the change For example we will refit the transport mode model examined at various points in Chapters 7 and 8 then examine the effect of increasing by 25 the terminal time spent waiting for air transport SAMPLE 1 840 NLOGIT Lhs mode Rhs one gc ttme Choices air train bus car NLOGIT Lhs mode Rhs one gc ttme Choices air train bus car Simulation Scenario ttme air 1 25 The estimated model appears first followed by the simulation Chapter 7 Simulating Probabilities in Discrete Choice Models Discrete choice multinomial logit model aximum Likelihood Estimates Dependent variable Choice umber of observations 210 Log likelihood function 199 9766 Log L for Choic model 199 97662 R2 1 LogL LogL Log L fnen R sqrd RsqAdj Constants only 283 7588 29526 28962 Chi squared 2 167 56429 Response data are given as ind choice Number of obs 210 skipped 0 bad obs Variable Coefficient Standard Error b St Er P Z gt z GC 1578374521E 01 43827919E 02 3 601 0003 TTME 97090
117. choice and three for the attribute The alternative specific constants are a_choice with names truncated to no more than six characters For example the sum of the three estimated choice specific constants could be analyzed as follows WALD Fnl a_air a_train a_bus WALD procedure Estimates and standard errors for nonlinear functions and joint test of nonlinear restrictions Wald Statistic 57 91928 Prob from Chi squared 1 00000 Variable Coefficient Standard Error b St Er P Z gt z H H Fnen 1 13 32858178 1 7513477 7 610 0000 6 3 2 Robust Standard Errors The cluster estimator described elsewhere in this document is available in NLOGIT However this routine does not support hierarchical samples There may be only one level of clustering Also the cluster specification is defined with respect to the NLOGIT groups of data not the data set NLOGIT sorts out how many clusters there are and how they are delineated But since the row count of the data set is used in constructing the estimator you must treat a group of NALT observations as one For example our sample data used in this section contain 210 groups of four rows of data Each group of four is an observation Suppose that these data were grouped in clusters of three choice situations The estimation command with the cluster estimator would appear NLOGIT the model Cluster 12 The
118. choices We replace the Rhs Rh2 setup with explicit definitions of the utility functions for the alternatives Utility functions are built up from the format Model U choice 1 linear equation U choice 2 linear equation U choice J linear equation Though we have shown all J utility functions for a given model specification you could in principle not specify a utility function in the list The implied specification would be Uj The U list is mandatory NLOGIT scans for the U and the parentheses For example Model U air ba bcost gc Note that the specification begins with Model the colon is also mandatory Parameters always come first then variables Constant terms need not multiply variables Thus ba in this model could be an air specific constant It depends on whether ba appears elsewhere in the model Notice that the utility function defines both the variables and the parameters Usually you would give an equation for each choice in the model For example NLOGIT Lhs mode Choices air train bus car Model U air ba bcost gc btime ttme U car be bcost ge U bus bb bcost gc U train bcost gc btime ttme Utility functions are separated by slashes Note also that the alternative specific constants stand alone without multiplying a variable Your utility definitions now provide the names for the parameters The estimates pr
119. cify the choice set The Rhs specification may be replaced with an explicit definition of the utility functions using Model A set of exactly J choice labels must be provided in the command These are used to label the choices in the output The number you provide is used to determine the number of choices there are in the model Therefore the set of the right number of labels is essential Use any descriptor of eight or fewer characters desired these do not have to be valid names just a set of labels separated in the list by commas The command builder for this model is found in Model Discrete Choice Discrete Choice The Main and Options pages are both used to set up the model The model and the choice set are defined in the Main page the attributes are defined in the Options page See Figure 8 1 Chapter 8 The Multinomial Logit Model DISCRETE CHOICE Main Options Output Choice variable Choice variable MODE X Data type Individual choice I Use ordinary weights z ss dividual choice m Choice set Proportion Fixed mabey Air Train Bus Car an Use choice based sampling weights l I Data coded on one line Code l Variable number of choices Count variable z F Use universal choice set indicator z Choice names I Perform IIA test on choices F Use data scaling DISCRETE CHOICE Main Options Output Model type Discrete Choice oS F Sequential est
120. crete Choice Models 29 Specific NLOGIT Model Commands LCM list of variables specifies latent class model Optionally specifies variables that enter the class probabilities Command is also LCLOGIT Also used by PROBIT and BLOGIT ECM list of specifications specifies error components logit model Command is also ECLOGIT s HEV specifies heteroscedastic extreme value model Command is also HCLOGIT Nested Logit Model Tree specification specifies tree structure in nested logit model GNL specifies generalized nested logit model Command is also GNLOGIT RU1 specifies parameterization of second and third levels of the tree s RU2 specifies parameterization of second and third levels of the tree s RU3 specifies parameterization of second and third levels of the tree IVSET specifications imposes constraints on inclusive value parameters IVB variable name keeps branch level inclusive values as a variable IVL name for limb IV keeps limb level inclusive values as a variable IVT name for trunk IV keeps trunk level inclusive values as a variable Prb name__ keeps branch level probabilities as a variable Cprob name keeps conditional probabilities for alternatives Random Parameters Logit Model RPL list of variables requests mixed logit model Optionally specifies variables to enter means of random parameters AR1 ARQ 1 structure for random terms in random parameters Fen defines
121. ction that would allow this these will not save much effort DISCRETE CHOICE Main Options Output r Choice variable Choice variable MODE v Data type Individual choice 7 J Use ordinary weights Choice set Fixed number of choices Choice names Jair train bus car I Use choice based sampling weights I Data coded on one line Code Variable number of choices Count variable Use universal choice set indicator Choice names F Perform IIA test on choices M Use data scaling DISCRETE CHOICE Main Options Output Model type Discrete Choice v I Sequential estimation I Conditional model I Use one line setup Attribute labels Utility functions p Attributes p Interact with ASC Ufair train bus car lt aa at ab 0 gt lt be be be be gt gc lt bta btg btg btg gt ttme Cancel Figure 6 5 Utility Functions Assembled in Command Builder Window Chapter 6 NLOGIT Commands and Results 66 Note that in the window you must provide the entire specification for the utility functions including the listing of which alternatives the definitions are to apply to The model shown in the window in Figure 6 5 produces these results Discrete choice multinomial logit model Maximum Likelihood Estimates Dependent variable Choice Weighting variable None Number of observations 210 Iterations completed 6 Log likeli
122. current sample For example if you happen to be estimating a model after having rejected some observations the predictions will be placed with the outcomes for the observations actually used Unused rows of the data matrix are left undefined If your model has 14 or fewer choices you can also include List in your command to request a listing of the predicted probabilities These will be listed a full observation at a time rowwise with an indicator of the choice that was made by that individual For example the first 10 observations individuals in the sample for the model above are PREDICTED PROBABILITIES marks actual marks prediction Indiv AIR TRAIN BUS CAR 1 1481 2376 1101 5042 2 1182 3694 1687 3437 3 9783 0702 0663 2853 4 2367 0725 0659 6250 5 2203 3176 1884 21 36 6 1048 4958 1589 2405 T 6500 0548 0565 2387 8 3241 3868 1472 1419 9 1824 2199 elilt2 4866 10 2863 0575 0491 GOLFE The and indicate the actual and predicted choices Where these mark the same probability the model has predicted the outcome correctly Chapter 6 NLOGIT Commands and Results 77 The inclusive value or log sum for the discrete choice model is IV log 2 exp B x Inclusive values are used for a number of purposes including computing consumer surplus measures You can keep the inclusive values for your model and data with the sp
123. d 312 5500 McFadden Pseudo R squared 3803994 Chi squared 23 TB TT Degrees of freedom 8 Prob ChiSqd gt value 0000000 R2 1 LogL LogL Log L fncen R sqrd RsqAdj No coefficients 312 5500 38040 37243 Constants only 283 7588 31753 30875 At start values 287 6816 32684 31818 Response data are given as ind choice Constants only only N j N Notes No coefficients gt P i j 1 J i gt P i j uses ASCs if fixed choice set N j total sample frequency for j N total sample frequency These 2 models are simple MNL models R sqrd 1 LogL model logL other RsqAdj 1 nJ nJ nparm 1 R sqrd nJ sum over i choice set sizes 116 Chapter 9 The Nested Logit Model and fr FIML Nested Multinomial Logit Model The model has 2 levels Nested Logit form IV parms taub 1 r sl r No normalizations imposed a priori p alt j b B 1 L r R xp bX_j BLR Sum Sum p b B 1 L r R exp aY_B LRt tauB LRIVB LR p 1 L r R exp cZ_L R sL RIVL R Sum p r R exp qH_R fRIVR Sum Number of obs 210 skipped 0 bad obs Variable Coefficient Standard Error b St Er P Z gt z 4 Attributes in the Utility Functions beta BT 5 06460277 66202159 7 650 0000 BB 4 09631480 61515554 6 659 0000 BG 03158748 00815636 314 873 0001 AT 11261749 01412912 STe 99 L
124. d the logit model based on the logistic distribution exp B x F A P x 1 exp P x Gx Chapter 2 Discrete Choice Models 13 Numerous variations on the model can be obtained A model with multiplicative heteroscedasticity is obtained with the additional assumption normal or logistic with variance exp y z where z is a set of observed characteristics of the individual A model of sample selection can be extended to the probit and logit binary choice models In both cases we depart from Prob y 1 Ix F B xi where F t W t for the probit model and A t for the logit model d a zi Uj Ui N O 1 dj 1 d gt 0 Yi Xi observed only when d 1 where z is a set of observed characteristics of the individual In both cases as stated there is no obvious way that the selection mechanism impacts the binary choice model of interest We modify the models as follows For the probit model yi B x e e N 0 1 y 1 gt 0 which is the structure underlying the probit model in any event and Ui amp i N 0 0 1 p 1 We use Np to denote the P variate normal distribution with the mean vector followed by the definition of the covariance matrix in the succeeding brackets For the logit model a similar approach does not produce a convenient bivariate model The probability is changed to exp B x 0 Prob y 11x 1 exp B x os W
125. d them useful for example for computing inclusive values in another model An example of the use of these features is shown in the next section Chapter 9 The Nested Logit Model 115 9 6 Application of a Nested Logit Model The following estimates a two level model The tree has a degenerate branch the air branch has only a single alternative fly It also uses most of the optional features mentioned above NLOGIT Lhs mode Start logit Choices air train bus car Tree travel fly air ground train bus car Model U air train bus car bt tasc bb basc bg gc at ttme U fly ground aa aasc ah hinca Describe Effects gc car Pwt List Ivb branchiv Ivl limbiv Utility u_choice Prob pkji Cprob pk_ji Starting values for the iterations are obtained by a one level multinomial logit model The MNL also reports results of estimation of the branch choice model These are the inconsistent estimates of o in the branch choice model Discrete choice and multinomial logit models Start values obtained using MNL model Maximum Likelihood Estimates Dependent variable Choice Weighting variable None Number of observations 210 Iterations completed 5 Log likelihood function 378 5920 Number of parameters 6 Info Criterion AIC 3 66278 Finite Sample AIC 3 66475 Info Criterion BIC 3 75841 Info Criterion HQIC 3 70144 Log L for Choic model 260 1975 R2 1 Log
126. dard deviations The following demonstrates the computations The command below specifies two correlated random parameters RPLOGIT Lhs mode Choices air train bus car Rhs gc ttme Rh2 one Fen ge n ttme n Correlated Mayit 50 Pts 25 Halton Output 3 Pds 3 The relevant results from estimation are as follows The coefficients reported are first B from the random parameter distributions then the nonstochastic B from the distributions of the nonrandom alternative specific constants The next results display the elements of the 2x2 lower triangular matrix I The diagonal elements appear first then the below diagonal element s The matrix I is shown again in natural form at the end of the results labeled Cholesky matrix The Standard deviations of parameter distributions are derived from T The first is 011001342 001100134 The second is 07458 03678 08315251 The standard errors for these estimators are computed using the delta method Hensher Rose and Greene 2005 discuss the Cholesky decomposition in detail with numerous examples Variable Coefficient Standard Error b St Er P 2Z gt z de L Mie mt oa SlSsrte Ss Random parameters in utility functions GC 02260684 00724332 3 121 0018 TIME 14522848 02205029 6 586 0000 SSS SSeS Ss4 tNonrandom parameters in utility functions A_AIR 8 70238058 1 22465947 7 10
127. del with the smaller number of choice sets and a smaller number of regressors There is no question of consistency or omission of a relevant attribute since if the attribute is always constant among the choices variation in it is obviously not affecting the choice After estimation the subvector of the larger parameter vector in the first model can be measured against the parameter vector from the second model using the Hausman statistic given earlier This possibility arises in the model with alternative specific constants so it is going to be a common case The examples below suggest one way you might proceed in such as case The first step is to fit the original model using the entire sample and retrieve the results NLOGIT Lhs mode Choices air train bus car Rhs inve invt ge ttme one MATRIX bu b 1 4 vu Varb 1 4 1 4 The variable choice takes values 1 2 3 4 1 2 3 4 indicating the indexing scheme for the choices CREATE choice Trn 4 0 Chair is a dummy variable that equals one for all four rows when choice made is air Now restrict the sample to the observations for choices train bus car REJECT chair 1 choice 1 Fit the model with the restricted sample choice set and one less constant term NLOGIT Lhs mode Choices train bus car Rhs inve invt ge ttme one Retrieve the restricted results and compute the Hausman statistic MATRIX br b 1 4 vr Varb 1 4 1 4 db br bu
128. dition In the multinomial probit model we assume that g are normally distributed with standard deviations Sdv s o and correlations Cor amp j Emi Pim the same for all individuals Observations are independent so Cor j 0 if i is not equal to s for all j and m A variation of the model allows the standard deviations and covariances to be scaled by a function of the data which allows some heteroscedasticity across individuals The correlations Pj are restricted to 1 lt Pim lt 1 but they are otherwise unrestricted save for a necessarily normalization The correlations in the last row of the correlation matrix must be fixed at zero The standard deviations are unrestricted with the exception of a normalization two standard deviations are fixed at 1 0 NLOGIT fixes the last two In principle up to 20 alternatives may be in the model but our experience thus far is that this model is extremely difficult to estimate and will usually not be estimable with a completely free correlation matrix even with only five alternatives The difficulty increases greatly with the number of alternatives Imposition of constraints which may improve this situation is discussed below This model may also be fit with panel data In this case the utility function is modified as follows Uji BX ine Ejiet Vine where f indexes the periods or replications There are two formulations for v Random effects Vix Vjis the sam
129. e estimated covariance matrix of the distribution of the random parameters e If I is diagonal then the diagonal elements are used to scale the random elements in the parameters However these scale parameters are only the standard deviations of the random terms when these variables are normally distributed Otherwise there is some specific scale parameter that must be added to the calculation e If T is not diagonal then I is not the covariance matrix of the random terms and the diagonal elements of I are not the standard deviations even in the normal case In this instance I is the Cholesky decomposition of the covariance matrix which must be recovered from the estimates The results given will include this decomposition as shown below for this application Partial effects for the RPL model are computed in the same fashion as for other models with one important exception As in other cases the elasticities are computed by individual and averaged to obtain the estimate However in the RPL model the individual specific estimates of the parameters described in the next section not the population averages are used to compute the estimates Elasticity averaged over observations Attribute is GC in choice AIR Effects on probabilities of all choices in model Direct Elasticity effect of the attribute Mean St Dev as Choice AIR 7700 4918 Choice TRAIN 8787 1 0465 Choice BUS 9346 1 0685 Choice CAR 6412 1 7
130. e in all periods First order autoregressive Vit Oj Viti Ajit Chapter 11 The Multinomial Probit Model 162 11 2 Model Command This is a one level nonnested model The setup is identical to the multinomial logit model with one level To request it use MNPROBIT Lhs Choices Rhs or Model U U all as usual any other options The alternative model command used in earlier versions of NLOGIT NLOGIT MNP is equivalent and may be used instead Options include Prob name to use for estimated probabilities Utility name to use for estimated utilities and the usual other options for output technical output elasticities descriptive statistics etc See Chapters 6 and 7 for details There are some special cases for this estimator The number of alternatives must be fixed it may not vary across observations The choice set must be fixed Choice based sampling is not supported though you can use ordinary weights Data may be individual proportions or frequencies The second derivatives matrix is not computed for this model so it is not possible to compute a robust covariance matrix estimator An additional option is Pts number of replications to compute multivariate normal probabilities The following features of NLOGIT are not available for this model Tree This is not a nested logit model Ivb name Ivl name Ivt name No inclusive value
131. e marginal effects But forcing the coefficient on a characteristic in probability j to equal its counterpart in probability m also forces the two marginal effects to be equal 4 4 Robust Covariance Matrices It has become common in the literature to compute a robust covariance matrix for the MLE The misspecification to which the matrix is robust is left unspecified in most cases The desired robust covariance matrix would result in the preceding computation if w equals one for all observations This suggests a simple way to obtain it just by specifying Choice Based Wts one Alternatively just use Robust which is equivalent Chapter 4 The Basic Multinomial Logit Model 33 A related calculation is used when observations occur in groups which may be correlated This is rather like a panel one might use this approach in a random effects kind of setting in which observations have a common latent heterogeneity The parameter estimator is unchanged in this case but an adjustment is made to the estimated asymptotic covariance matrix The calculation is done as follows Suppose the n observations are assembled in C clusters of observations in which the number of observations in the cth cluster is ne Thus Cc ae ne T Denote by B the full set of model parameters B B Let the observation specific gradients and Hessians for individual i in cluster c be Olog L ap O log L apop ic The uncorr
132. e of the independence from irrelevant alternatives IIA assumption of the multinomial logit model can be seen in the identical elasticities in the tables above The table also shows two aspects of the model First the meaning of the raw coefficients in a multinomial logit model all of sign magnitude and significance are ambiguous It is always necessary to do some kind of post estimation such as this to determine the implications of the estimates Second in light of this we can see that the particular model we estimated seems to be misspecified The estimates imply that as the generalized cost of each mode rises it becomes more attractive The gc coefficient has the wrong sign Chapter 6 NLOGIT Commands and Results 76 6 5 Predicted Probabilities and Inclusive Values There are some models that make use of the predicted probabilities from the discrete choice model 6 5 1 In Sample Predicted Probabilities and Inclusive Values You can compute a column of predicted probabilities for any estimated choice model Each observation consists of J rows of data where the number of choices may be fixed or variable Use the command NLOGIT Lhs 5 Prob name The variable name will contain the predicted probabilities The probabilities will sum to 1 0 for each observation that is down each set of J choices The Prob option will put the probabilities in the right places in your data set regardless of the setting of the
133. e proceeding we note the format of a set of parameter estimates for a model set up in exactly this fashion Variable Coefficient Standard Error b St Er P Z gt z GC 01092735 00458775 2 382 0172 TTME 09546055 01047320 9 115 0000 A_AIR 5 87481336 80209034 7 324 0000 AIR_HIN1 00537349 01152940 466 6412 A_TRAIN 5 54985728 64042443 8 666 0000 TRA_HIN2 05656186 01397335 4 048 0001 A_BUS 4 13028388 67636278 6 107 0000 BUS_HIN3 02858418 01544418 1 851 0642 Chapter 6 NLOGIT Commands and Results 56 Note the construction of the compound names includes what might seem to be a redundant number at the end This is necessary to avoid constructing identical names for different variables Utility Functions A basic four choice model which contains cost time one and income will have utility functions Uidir Beost COST air Brine time air Qhair Yair income Ej airs Ui train Beost costi train B time time train Qiirain Ytrain income E trains Uin Beost costi bus Brine tIME bus Opus Y bus income Ei buss U car 7 Beost cost car B time time car Ei bus The simple device you will use to construct utility functions in this fashion is Rhs list of attributes that vary across choices and Rh2 list of variables that do not vary across choices The Rh2 variables are automatically expanded into a set of J 1 interactions with the choic
134. e similarity is striking given the quite large differences in the estimates of the structural parameters CREATE vbgn sgn 2 vbtn stn 2 vbgw sgw 2 vbtw stw 2 CALC List sdbgn Sqr xbr vbgn Var bgn sdbgw Sqr xbr vbgw Var bgw sdbtn Sqr xbr vbtn Var btn sdbtw Sqr xbr vbtw Var btw SDBGN 011456 SDBGW 009629 SDBTN 089567 SDBTW 088368 A final comparison is based on the kernel density estimators for the distributions of the conditional means Only the two for B are shown KERNEL Rhs bgn Title Kernel Density for E b_gcl normal Endpoints 01 05 KERNEL Rhs bew Title Kernel Density for E b_gcl Weibull Endpoints 01 05 KERNEL Rhs btn Title Kernel Density for E b_ttmel normal KERNEL Rhs btw Title Kernel Density for E b_ttmel Weibull Based on the results obtained thus far it seems that the impact of the Weibull specification is to increase the variance of the empirical distribution Chapter 10 The Random Parameters Logit Model 153 E Untitled Plot 7 f Ex Untitled Plot 8 DER Kernel dendtectmat br BOW Figure 10 8 Kernel Densities for Parameter Distributions 10 7 3 Conditional Confidence Intervals for Parameters Finally we consider an alternative approach to examining the distribution of parameters across individuals We have for each individual an estimate of the mean of the conditional distribution
135. e specific constants as they are in the matrix shown above The implication is that generally you do not need to have these variables in your data set They are automatically created by your command Note that our clogit dat data set in Chapter 5 actually does contain the superfluous set of four choice specific constants aasc tasc basc and casc NOTE If you include one in your Rhs list it is automatically expanded to become a set of alternative specific constants That is one is automatically moved to the Rh2 list if it is placed in the Rhs list The model specification for the four utility functions shown above would be Rhs cost time Rh2 one income Note that the distinction between Rh2 and Rhs variables is that all variables in the first category are expanded by interacting with the choice specific binary variables The last term is dropped Generic Coefficients The simpler but less flexible way to specify generic coefficients in a model is to use NLOGIT s standard construction by specifying a set of Rhs variables The specification Rhs gc ttme produces the utility functions in the first two columns in the table Rhs variables are assumed to vary across the choices and will receive generic coefficients Chapter 6 NLOGIT Commands and Results 57 Alternative Specific Constants and Interactions with Constants The logit model is homogeneous of degree zero in the attributes Any attribute which does
136. e the same random draws One implication of this is that if you give the identical model command twice in sequence you will not get the identical set of results because the random draws in the sequences will be different To obtain the same results you must reset the seed of the random number generator with a command such as CALC Ran seed value We generally use CALC Ran 12345 before each of our examples precisely for this reason The specific value you use for the seed is not of consequence any odd number will do Chapter 10 The Random Parameters Logit Model 139 10 5 2 Halton Draws and Random Draws for Simulations The standard approach to simulation estimation is to use random draws from the specified distribution As suggested immediately above good performance in this connection usually requires fairly large numbers of draws The drawback to this approach is that with large samples and large models this entails a huge amount of computation and can be very time consuming A currently emerging literature has documented dramatic speed gains with no degradation in simulation performance through the use of a smaller number of Halton draws instead of a large number of random draws Some authors have found that a Halton sequence with a far small number of replications as low as a tenth for a single parameter is often as effective as a far larger number of random draws To use this approach add Halton to your model command
137. ecification vb name The specification Ivb stands for inclusive value for branch Inclusive values are stored the same way that predicted probabilities are stored Since each observation has only one inclusive value the same value will be stored for all rows choices for the observation person Figure 6 6 illustrates i Data Editor 17 900 Vars 11111 Rows 840 Obs Cell fo LOGL ops INCYLU PMNL ied ara 0 195808 0 148125 EES 0 195808 0 237597 Sra 0 0 195808 0 110103 A 0 684832 0 195808 0 504175 ie 0 239082 0 118216 ats 0 239082 0 369399 7 gt 0 239082 0 168723 EEES 1 0 239082 0 343662 9 2 17441 0 578309 10 gt 2 17441 0 0701569 EEES 2 17441 0 0662732 Sees 1 25435 2 17441 0 285261 13 gt 0 0 107306 0 236656 1 0 0 107306 0 0725174 15 0 0 107306 0 0658503 16 0 470042 0 107306 0 624976 res 0 507153 0 220348 18 0 507153 0 317646 fara 0 507153 0 188422 Figure 6 6 Saved Inclusive Values and Probabilities Chapter 6 NLOGIT Commands and Results 78 6 5 2 Computing Out of Sample Model Probabilities You can use an estimated model to compute list and or save all probabilities utilities elasticities and all descriptive statistics and crosstabulations for any specified set of observations whether they were used in estimating the model or not For example this feature will allow you to compute predicted probabilities for a control sample to a
138. ected estimator of the asymptotic covariance matrix based on the Hessian is Ve H Ei ELE The corrected asymptotic covariance matrix is Est Asy Var B V EL 8 8 Va Note that if there is exactly one observation per cluster then this is C C 1 times the sandwich robust estimator discussed above Also if you have fewer clusters than parameters then this matrix is singular it has rank equal to the minimum of C and JK the number of parameters This estimator is requested with Cluster specification where the specification is either a fixed number of observations per cluster or an identifier that distinguishes clusters such as an identification number This estimator can also be extended to stratified as well as clustered data using Stratum specification Chapter 4 The Basic Multinomial Logit Model 34 4 5 Output for the Multinomial Logit Model Initial ordinary least squares results are used for the starting values for this model For individual data J binary variables are implied by the model These are used in a least squares regression For the grouped data case a minimum chi squared generalized least squares estimate is obtained by the weighted regression of on the regressors with weights h nP Po 2 n may be 1 0 Note that the dependent variables in these regressions are the odds ratios The OLS estimates based on the individual data are inconsistent but the grouped d
139. ed sampling weights Data coded on one line Code C Variable number of choices Count variable r Choice names Perform IIA test on choices Use data scaling Cancel Figure 6 2a Main Page of Command Builder for Conditional Logit Model DISCRETE CHOICE Main Options Output Model type Discrete Choice 4 I Sequential estimation Conditional model Use one line setup Attribute labels Utility functions Attributes Interact with ASC GC ONE TTME lt HINC J Specify utility functions Hasir Tree Specification Optimization Hypothesis Tests Figure 6 2b Options Page of Command Builder for Conditional Logit Model A set of exactly J choice labels must be provided in the command These are used to label the choices in the output The number you provide is used to determine the number of choices there are in the model Therefore the set of the right number of labels is essential Use any descriptor of eight or fewer characters desired these do not have to be valid names just a set of labels separated in the list by commas The internal limit on J the number of choices is 100 Chapter 6 NLOGIT Commands and Results 52 There are K attributes Rhs variables measured for the choices The sections below will describe variations of this for different formulations and options The total number of parameters in the utility functions will include K for the Rhs variables and
140. eed to drop one of the cost coefficients because the variable cost varies by choices You can estimate a model with four separate coefficients for cost one in each utility function However it is not possible to do it by including cost in the Rh2 list as described above because this form will automatically drop the last term the one in the car utility function You could obtain this form albeit a bit clumsily by creating the four interaction terms yourself and including them on the right hand side We already have the alternative specific constants so the following would work CREATE cost_a gc aasc cost_t gc tasc cost_b gc basc cost_c ge casc NLOGIT 3 3 Rhs time cost_a cost_t cost_b cost_c Rh2 one income Having to create the interaction variables is going to be inconvenient The alternative method of specifying the model described in the next section will be much more convenient This method also allows you much greater flexibility in specifying utility functions HINT There are many different possible configurations of alternative specific constants ASCs and alternative specific variables In estimating a model it is not possible to determine a priori if a singularity will arise as a consequence of the specification You will have to discern this from the estimation results for the particular model The constant term one fits the hint above Recognizing this NLOGIT assumes that if your Rh
141. ent random effects SigmaE01 1 44001740 3 62060512 2398 6908 SigmaE02 1 70126558 2 89949978 58T 5574 STT rE Standard deviations of parameter distributions sdMGC 01138907 02143530 2931 5952 sdMTIME 08895746 02886146 3 082 0021 This is the same model once again now with Weibull distributed parameters SSS SSeS Random parameters in utility functions MGC 2 84950808 77609997 3 672 0002 MTTME 1 31927880 1 29537623 1 018 3085 ae ea oh tNonrandom parameters in utility functions A_AIR 10 1003905 1 80393746 5x599 0000 A_TRAIN 8 04274180 1 64404555 4 892 0000 A_BUS 7 04388422 1 83042969 3 848 0001 Heterogeneity in mean Parameter Variable MGC HIN 00573905 01801325 319 37500 MTTM HIN 00675433 00355598 1 899 0575 SSS Diagonal values in Cholesky matrix L WsMGC 24147776 37415604 645 5187 WsMTTME 00503652 68300577 007 9941 So Below diagonal values in L matrix V L Lt MTTM MGC 49268390 11909663 4 137 0000 ea S Standard deviations of latent random effects SigmaE01 99017999 6 82984761 145 8847 SigmaE02 2 08605480 3 40861849 612 5405 naei Standard deviations of parameter distributions sdMGC 24147776 37415604 645 5187 sdMTTIME 49270964 12066246 4 083 0000 The ASCs in the three models resemble one another but the coefficients on the attributes are vastly different and would seem to suggest very different models In fact that is not
142. enter the utility functions may be the same for all choices or may be defined differently for each utility function The estimator described in this chapter allows a large number of variations of this basic model In the discrete choice framework the observed dependent variable usually consists of an indicator of which among J alternatives was most preferred by the respondent All that is known about the others is that they were judged inferior to the one chosen But there are cases in which information is more complete and consists of a subjective ranking of all J alternatives by the individual NLOGIT allows specification of the model for estimation with ranks data In addition in some settings the sample data might consist of aggregates for the choices such as proportions market shares or frequency counts NLOGIT will accommodate these cases as well 8 2 Command for the Multinomial Logit Model The simplest form of the command for the discrete choice models is CLOGIT Lhs variable which indicates the choice made Choices a set of J names for the set of choices Rhs choice varying attributes in the utility functions Rh2 choice invariant characteristics With no qualifiers to indicate a different model such as RPL or MNP CLOGIT and NLOGIT are the same There are various ways to specify the utility functions i e the right hand sides of the equations that underlie the model and several different ways to spe
143. epaias sola seas cana mens R SEEE 11 o gt Bm NOC Mode irs EERON acai ly fo Makinada Logit Mode l kesise ena 14 2 3 Conditional Logit Model is ceccesiats ccsaccsseessctiacetsscesncsialsccsessdeuovedceecansscecuave ndedawwea teow ecceteensvieues 16 oo Nenad Lorn Mode lici ccimiunosninarceuiainntacaininnt Adio tae OAA EAO 17 2 7 Randon Parameicts Loci Models aranana manent cea 18 2e DAMIEN Propit Model siei E E E teed 20 Chapter 3 Model and Command Summary for Discrete Choice Models ccsssssseeee 22 aR ites cn ss Ue prey meepeeemeete Wome rer eer revere s renmen cr cee ener rece ertrensrrevrer sri trner rr vewnre Yeerer trary Vener 22 32 Model STONN ysis de ceus ei accsiasi pid rias Mavic E eels in eR ag ah Bays 22 ove Basie Disciete Choice Mod l Sirei a E EEA E EEEE 22 3A Multinomial Logit Models csissscs cass ssa veecscbsceanssensceruataedaveisevassscssdeesssidanatualseaereiteiwwvbadd ieosseetaxe 23 34 1 Multinomial LOZ ii siss2is sessstsescaseseiearsccadearasheoncsuasgnassncsebcursideduatant EEE ATT 23 AI Cond LAO usa 23 39 NLOGIT Extensions of Conditional Lonis iesnieoie tenien inkeer ia vie EEE NE bie teria 24 Du E e e N E E E E AAE 24 25 2 Random Parameters LOA aonni 24 3 5 3 Multinomial Probit ssseis sasstecesxorstnestanssesesunetadunasiciasasrsiaentursstqauateiepecaridesustseadeiarniaeive 25 30 Command SUMAT Y eirese ka a a ei 23 EET E E ee Lan a a E E E E TE E E E E 26 Chapter 4 The Basic Multinomial Logit Model
144. er this one will examine what we call again only for convenience the discrete choice model and also to differentiate the command the conditional logit model In this framework we observe the attributes of the choices rather than the characteristics of the individual A well known Chapter 4 The Basic Multinomial Logit Model 31 example is travel mode choice Samples of observations often consist of the attributes of the different modes and the choice actually made Usually no characteristics of the individuals are observed beyond their actual choice Models may also contain mixtures of the two types of choice determinants These are considered in the later chapters as well We emphasize these naming distinctions are meaningless in the modeling framework we just use them here only to organize the applicable parts of NLOGIT 4 2 The Multinomial Logit Model The general form of the multinomial logit model is exp B x gt exp B x m 1 Prob choice j J 09 J A possible J 1 unordered outcomes can occur In order to identify the parameters of the model we impose the normalization By 0 This model is typically employed for individual or grouped data in which the x variables are characteristics of the observed individual s not the choices For present purposes that is the main distinction between this and the discrete choice model described in Chapter 8 The characteristics are the same across all outcomes The d
145. ernatives in the choice set There may be up to 100 alternatives in the model a total of 25 branches throughout the tree 10 limbs and five trunks The model may contain one or more limbs Each limb may contain one or more branches and each branch may contain one or more twigs choices If there is only one trunk and one limb the model is by implication a two level model As for single level models choice sets may vary by individual However in order to construct a tree for such a setting a universal choice set as described in Chapter 5 is necessary The variable sized choice set is then indicated by setting up the full tree structure and indicating that certain choices are unavailable for the particular individual The command for fitting nested logit models is the same as described in Chapter 3 for one level models save for the addition of the tree definition in the command and optionally the specification of additional utility functions for choices made at higher levels in the tree The nested logit model is limited to four level models for full information maximum likelihood FIML estimation It also allows estimation of two and higher level models by sequential or two step estimation Utility functions can be specified for trunks the same as for limbs and branches though it is unlikely that there will be very many attributes at this level in a tree All options are available including logs Box Cox transformation fixed values start
146. es The predictions are obtained as the integer part of X jt Yi Add Crosstab to your model command For the same model this would produce Cross tabulation of actual vs predicted choices Row indicator is actual column is predicted Predicted total is F k j i Sum i 1l N P k j i Column totals may be subject to rounding error ATR TRAIN BUS CAR Total AIR 19 00000 13 00000 8 00000 18 00000 58 00000 TRAIN 12 00000 30 00000 9 00000 12 00000 63 00000 BUS 10 00000 8 00000 6 00000 6 00000 30 00000 CAR 17 00000 12 00000 7 00000 23 00000 59 00000 Total 58 00000 63 00000 30 00000 59 00000 210 00000 6 4 Marginal Effects and Elasticities In the discrete choice model the effect of a change in attribute k of alternative f on the probability that individual i would choose alternative m where m may or may not equal j is Sim klj OProbly m ox kij 1G m Pi PinBe You can request a listing of the effects of a specific attribute on a specified set of outcomes with Effects attribute list of outcomes The outcomes listing defines the variables f in the definition above The attribute is the kth A calculated marginal effect is then listed for all alternatives i e all m in the model You can request additional tables by separating additional specifications with slashes For example Effects gc car train ttme bus train HINT It may ge
147. es g and w For each observation we also observe which choice was made Suppose further that in the first three observations the choices made were two three and one respectively The data matrix would consist of 75 rows with 25 blocks of three rows Within each block there would be the set of attributes and a variable y which at each row takes the value one if the alternative is chosen and zero if not Thus within each block of J rows y will be one once and only once For the hypothetical case then we have Y Q W i 1 0 qi 1 Wi 1 1 Q2 1 W2 1 0 3 1 W3 1 i 2 0 qi 2 Wi 2 0 qz 2 W2 2 1 q3 2 W3 2 1 3 1 q 3 W1 3 0 q2 3 W2 3 0 q3 3 W3 3 and so on continuing to i 25 where gt marks the row of the respondent s actual choice When you read these data the data set is not treated any differently from any other panel Nobs would be the total number of rows in the data set in the hypothetical case 75 not 25 The separation of the data set into the above groupings would be done at the time your particular model is estimated NOTE Missing values are handled automatically by estimation programs in NLOGIT You should not reset the sample or use SKIP with the NLOGIT models Observations that have missing values are bypassed as a group Chapter 5 Data Setup for NLOGIT 42 Thus far it is assumed that the observed outcome is an indicator of which choice was made among a fixed set of up t
148. f Command Builder for RPL Model General options for NLOGIT s models are requested on the Output page shown in Figure 10 4 A separate page for model estimates may be opened by clicking Model Estimates in the lower right of the Output page See Figure 10 5 NLOGIT Multnomial Probit HEV RPL Main Options Output I Display covariance matrix Display crosstabs Display descriptive statistics Display predictions Display marginal effects l r r 7 Keep choice probabilities as variable l I Keep utilities as variable l I Keep individual parameters Figure 10 4 Output Page of Command Builder for RPL Model Chapter 10 The Random Parameters Logit Model 137 Model Estimates Keep ancillary parameters Keep model results for table as F Use cluster robust C Cluster Cancel Figure 10 5 Model Estimates Page of Command Builder for RPL Model 10 4 Heteroscedasticity and Heterogeneity in the Variances The random parameters model allows heterogeneity in the variances as well as in the means in the distributions of the random parameters The model is expanded to Oik Ok EXp hr If y equals 0 this returns the homoscedastic model The implied form of the RPL model is Biz B 6 Z Oir B 6 z o explor hr vir Request the heteroscedasticity model with Hfr list of variables in hr The variables in hr may be any variables but they must be choice invariant Only the last value in J
149. fect of the attribute Mean St Dev Choice AIR 2798 2044 Choice TRAIN 2420 305 Choice BUS 1342 0648 5 Choice CAR 6559 22159 These effects are always extremely small 73 They are multiplied by 100 in the output to make sure that some significant digits are shown in the tables The effects are computed by averaging the individual specific results so the report contains the average partial effects Since the mean is computed over a sample of observations we also report the standard deviation of the estimates Chapter 6 NLOGIT Commands and Results 74 NOTE The standard deviations are not the asymptotic standard errors for the estimators of the marginal effects In principle that could be computed using the delta method However the estimates computed by NLOGIT are average partial effects They are computed for each individual in the sample then averaged Computing an appropriate standard error for that statistic is difficult to impossible owing to its extreme nonlinearity and due to the fact that all observations in the average are correlated they use the same estimated parameter vector Nonetheless it may be tempting to use the standard deviations for tests of hypotheses that the marginal effects are zero We advise against this There is no meaning that could be attached to an elasticity or marginal effect being zero these are complicated functions of all parameters in the model The hypothesis tha
150. g Q v by T The model components may be restricted and varied in several ways e A variety of distributions may be chosen for the random parameters and they need not be the same for all parameters e The observed heterogeneity Az is optional You may specify that a coefficient is randomly distributed around a fixed mean Thus 6 may be set to a zero vector for some or all random coefficients e may be set equal to zero for some coefficients This may change the way a coefficient enters the model If o 0 and 8 0 then the coefficient is a nonrandom fixed parameter But including it in B allows you to force a coefficient to be positive This device also allows you to form a hierarchical model with nonrandom coefficients e Any coefficient in the model may be fixed at a specific value e The heteroscedasticity may apply to some or all or none of the random parameters e Different variables may be placed in the heterogeneous means Az or the heteroscedastic variances Q of any of the random parameters e The variables that enter the heteroscedasticity of the error components may be different e The model with both heteroscedasticity and cross parameter correlation is not estimable There is no way to make the covariance heterogeneous A number of additional features are listed in the sections to follow 10 3 Command for the Random Parameters Logit Models The command for the mixed logit model is as follows RPLOGIT
151. geneity terms are assumed to be independently and identically distributed with identical type 1 extreme value distribution the CDF is F e exp exp g Based on this specification the choice probabilities Prob choice j Prob U gt U Vg j exp B x PBR los J Da exp B X eno where i indexes the observation or individual and f and m index the choices The IID assumptions made about s are quite stringent and lead to the Independence from Irrelevant Alternatives or ITA implications that characterize the model Much perhaps all of the research on forms of this model consists of development of alternative functional forms and stochastic specifications that avoid this feature The observed data consist of the Rhs vectors X and the outcome or choice y We also consider a number of variants This chapter will examine what we call for the present the multinomial logit model In this model it is assumed that the Rhs variables consist of a set of individual specific characteristics such as age education marital status etc These are the same for all choices so the choice subscript on x in the formula above is dropped The observation setting is the individual s choice among a set of alternatives where it is assumed that the determinant of the choice is the characteristics of the individual An example might be a model of choice of occupation The remaining chapters of this manual aft
152. h2 list of variables lists choice invariant characteristic variables Model alternative way to specify utility functions followed by definitions of utility functions Fix list lists names of and values for coefficients that are to be fixed Uset list of alternatives list of values or list of values alternative method of specifying starting values or fixed coefficients Lambda value specifies coefficient to use for Box Cox transformation Attr list of names names for attributes used in one line entry format Chapter 3 Model and Command Summary for Discrete Choice Models 27 Output Control List and Retain Variables and Results Prob variable name keeps predicted probabilities from estimated model as variable Keep variable name keeps predicted values from estimated model as variable Used by PROBIT and BLOGIT only Utility name keeps predicted utilities as variable List lists predicted probabilities and predicted outcomes with model results Parameters retains additional parameters as matrices With RPL and LCM keeps matrices of individual specific parameter means WTP list of specifications retains computations of willingness to pay Covariance Matrices Printve displays estimated covariance matrix with model output Robust computes robust sandwich estimator for asymptotic covariance matrix Cluster specification computes robust cluster corrected asymptotic covariance matrix Displa
153. he mean Chapter 10 The Random Parameters Logit Model 133 The difference here is the parentheses in the first as opposed to the brackets in the second The second of these forces the applicable row of A to contain zeros instead of free parameters There are also some variations on this specification that allow some flexibility in the construction of A First an alternative equivalent form of name type is name type This requests that if there are RPL variables RPL list these not appear in the mean for this parameter This puts a row of zeros in the A matrix For example RPL income Fen gce n ttme nl specifies that income does not appear in the mean of the ttme parameter This form may be extended to exclude and include specific variables from the RPL list in the mean of a particular parameter The specification is name type pattern where the pattern consists of ones and zeros which indicate which variables in the list are included ones and excluded zeros There must be the same number of items in the pattern as there are in the list For example the specification RPL age sex income Fen ge n ttme nl 101 invt nl 011 inve nl 000 includes all three variables in the mean of gc excludes sex from the mean of ttme excludes age from the men of invt and excludes all three variables from the mean of invc All parameters may be specified independently and there is no restriction on how th
154. he within branch cross elasticities are not equal as would be imposed by the IID assumptions because we used Pwt to weight the observations Chapter 9 The Nested Logit Model 119 Derivative times 100 averaged over observations Attribute is GC in choice CAR Effects on probabilities of all choices in the model indicates direct Derivative effect of the attribute Decomposition of Effect if Nest Total Effect Trunk Limb Branch Choice Mean St Dev Limb TRAVEL Branch FLY Choice AIR 000 000 119 000 oll 9 083 Branch GROUND Choice TRAIN 000 000 Do7 S239 218 147 Choice BUS 000 000 014 e223 209 130 Choice CAR 000 000 T05 2 39 9 504 140 Chapter 10 The Random Parameters Logit Model 120 Chapter 10 The Random Parameters Logit Model 10 1 Introduction The random parameters logit RPL model also referred to as the mixed logit model is the most general model form in NLOGIT in terms of the variety of model specifications it can accommodate and in terms of the range of behavior that it can model This chapter will develop the numerous different specifications of the model that can be accommodated NLOGIT offers an extensive set of specifications within the mixed logit structure This model is gaining great popularity in applications Capabilities provided by the estimator include i choosing from among a large number of analytical distributions for each random parameter ii accounting
155. her Econometric Software Australia Recent developments especially the random parameters logit in its cross section and panel data variants have also benefited from the suggestions of Kenneth Train of UC Berkeley Version 4 0 has also been greatly improved by the enthusiastic collaboration of John Rose Econometric Software Australia We note the recently published work Applied Choice Analysis A Primer Hensher D Rose J and Greene W Cambridge University Press 2005 is a wide ranging introduction to discrete choice modeling that contains numerous applications developed with Versions 3 0 and 4 0 of NLOGIT This book should provide a useful companion to the documentation for NLOGIT William H Greene Econometric Software Inc 15 Gloria Place Plainview NY 11803 January 2007 NLOGIT Student Reference Guide Table of Contents vi Table of Contents Table of COMUNE Si ss cae ca case cece sc tm cate re erence vi Chapter 1 Introduction to NLOGIT is icccncccsssiscesssrsasennstsscrasscaastanaessaccsecsuatasraascenaiaserssneauacsnanaes 9 1 1 Diseteie Choice Madeline Will ALOG on inwsciesetnaiaeineaseia A 9 MEIC arad MOLE surani ven Lad nies Goan yay ccnassliy E nase pela TALEE 10 Chapter 2 Discrete Choloe MOGOIS wisissscccsssssccssecasascancnaccsscasacouacsnancescoserdiadsaviacconasssarsesroraven 11 2 TABOO BON asana nance anne inoesaks oi i ene aac 11 2 Randan Vility GES secre yuna sceaivcus xt juan cesses veUa yank aces cas gece dudk e
156. hoice j Prob Uj gt U Vg 4j exp B x e ee J 0 J Dad exp B X At this point we make a purely semantic distinction between two cases of the model When the observed data consist of individual choices and only data on the characteristics of the individual identification of the model parameters will require that the parameter vectors differ across the utility functions as they do above The study on labor market decisions by Schmidt and Strauss 1975 is a classic example For the moment we will call this the multinomial logit model When the data also include attributes of the choices that differ across the alternatives then the forms of the utility functions can change slightly and the coefficients can be generic that is the same across alternatives Again only for the present we will call this the conditional logit model It will emerge that the multinomial logit is a special case of the conditional logit model though the reverse is not true The conditional logit model is defined in Section 2 5 The general form of the multinomial logit model is exp B x Prob choice j j7 _ _ 2 0 amp XPBX f Osc A possible J 1 unordered outcomes can occur In order to identify the parameters of the model we impose the normalization By 0 This model is typically employed for individual or grouped data in which the x variables are characteristics of the observed individual s not the choices
157. hood function 199 6825 Number of parameters 6 Info Criterion AIC 1 95888 Finite Sampl AIC 1 96085 Info Criterion BIC 2 05451 Info Criterion HQIC 1 99754 R2 1 LogL LogL Log L fncn R sqrd RsqAdj Constants only 283 7588 29630 28953 Chi squared 3 168 15262 Prob chi squared gt value 00000 Response data are given as ind choice Number of obs 210 skipped 0 bad obs Notes No coefficients gt P i j 1 J i Constants only gt P i j uses ASCs only N j N if fixed choice set N j total sample frequency for j N total sample frequency These 2 models are simple MNL models R sqrd 1 LogL model logL other RsqAdj 1 nJ nJ nparm 1 R sqrd nJ sum over i choice set sizes Variable Coefficient Standard Error b St Er P Z gt z AA 6 41353627 1 10452186 5 807 0000 AT 3 69564345 52116476 7 091 0000 AB 2 96221779 54485066 5 437 0000 BC 01702110 00471351 3 611 0003 BTA 10758045 01791733 6 004 0000 BTG 08939996 01419339 6 299 0000 Chapter 6 NLOGIT Commands and Results 6 3 Standard Model Results Estimation results for the model commands consist of the initial display of diagnostic followed by notes about the model then the estimated coefficients without the tree structure or the initial echo of the model specification NLOGIT Rhs invc invt gc Rh2 one hinc produces the following results Nor
158. iSqrd 4 51 9631 Pr C gt c 000000 Variable Coefficient Standard Error b St Er P Z gt z INVC 04641792 02108920 22T SOTT INVT 00963276 00271137 3 593 0004 GC 04116251 01984102 2 075 0380 TIME 07938809 00991501 8 007 0000 81 Chapter 6 NLOGIT Commands and Results 82 In order to compute the coefficients in the restricted model it is necessary to drop those observations that choose the omitted choice s In the example above 59 observations were skipped They are marked as bad data because with car excluded no choice is made for those observations As a consequence the log likelihood functions are not comparable The Hausman statistic is used to carry out the test In the preceding example the large value suggests that the IIA restriction should be rejected Note that you can carry out several tests with different subsets of the choices without refitting the benchmark model Thus in the example above you could follow with a third model in which Ias bus instead of car There is a possibility that restricting the choice set can lead to a singularity It is possible that when you drop one or more alternatives some attribute will be constant among the remaining choices Thus you might induce the case in which there is a regressor which is constant across the choices In this case NLOGIT will issue a diagnostic about a singular Hessian it is Hausman and McFadden 1984 suggest estimating the mo
159. iagonal For convenience at this point we will simply gather the parameters choice specific or not under the subscript k The notation is a bit more cumbersome for the lognormally distributed parameters We will return to that in the technical details We can go a step further and allow the random parameters to be correlated All that is needed to obtain this additional generality is to allow T to be a triangular matrix with nonzero elements below the main diagonal Then the full covariance matrix of the random coefficients is IT The standard case of uncorrelated coefficients has I diag o 02 0 If the coefficients are freely correlated I is a full unrestricted lower triangular matrix and X will have nonzero off diagonal elements It will be convenient to aggregate this one step further We may gather the entire parameter vector for the model in this formulation simply by specifying that for the nonrandom parameters in the model the corresponding rows in A and T are zero We will also define the data and parameter vector so that any choice specific aspects are handled by appropriate placements of zeros in the applicable parameter vector An additional extension of the model allows the distribution of the random parameters to be heteroscedastic As stated above the variance of v is taken to be a constant The model is made heteroscedastic by assuming instead that Var va oj exp hr A convenient way to par
160. ibute that multiplies the indicated parameter Choice Parameter Row 1 GC EB A_AIR A_TRAIN A_BUS AIR 1 GC E Constant none none TRAIN 1 GC E none Constant none BUS 1 GC E none none Constant CAR 1 GC E none none none Normal exit from iterations Exit status 0 Discrete choice multinomial logit model Maximum Likelihood Estimates Dependent variable Choice Weighting variable None Number of observations 117 Iterations completed 6 Log likelihood function 62 58418 Number of parameters 3 Info Criterion AIC TL2 T10 Finite Sample AIC de T2297 Info Criterion BIC 1 19192 Info Criterion HQIC 1 14985 R2 1 LogL LogL Log L fnecn R sqrd RsqAdj Chapter 6 NLOGIT Commands and Results Chi squared 2 Constants only 81 0939 Prob chi squared gt value s00000 Response data are given as ind choice Number of obs 210 skipped 93 bad obs 22825 20794 37 01953 Restricted choice set Excluded choices are TRAIN BUS Variable Coefficient Standard Error b St Er P 2 gt z GC 01320101 00694790 1 900 0574 TIME 07141256 01604643 4 450 0000 A_AIR 3 96116758 98004184 4 042 0001 A_TRAIN 000000 peress Fixed Parameter A_BUS OOOOOO ise Sears Fixed Parameter 69 There are 210 individuals in the data set but this model was fit to a restricted choice set which reduced the data set to n 210 93 117 useable o
161. ic may identify a particular observation or value In the listing below we use the conventions lt AAAAAAAA gt indicates a variable name that will appear in the diagnostic lt nnnnnnnnnnnn gt indicates an integer value often an observation number that is given lt XXXXXXXXXXXX gt indicates a specific value that may be invalid such as a time that is negative The listing below contains the diagnostics and in some cases additional points that may help you to find and or fix the problem The actual diagnostic you will see in your output window is shown in the Courier font such as appears in diagnostic 82 above We note it should be extremely rare but occasionally an error message will occur for reasons that are not really related to the computation in progress We cannot give an example if we knew where it was we would remove the source before it occurred You will always know exactly what command produces a diagnostic an echo of that command will appear directly above the error message in the output window So if an absolutely unfathomable error message shows up try simplifying the command that precedes it to its bare essentials and by building it up reveal the source of the problem Finally there are the program crashes Obviously we hope that these never occur but they do The usual ones are division by zero and exponent overflow Once again we cannot give specific warnings about these since if we could
162. ict zero outcomes in one or more of the cells Even in a model with very high t ratios and great statistical significance it takes a very well developed model to make predictions in all cells The List specification produces a listing such as the following Observation Observed Y Predicted Y Residual MaxPr i Prob Y y 1 2 0000 00000 0000 42905 1443 2 00000 00000 0000 2538 2938 3 00000 00000 0000 2866 2866 4 5 0000 3 0000 0000 2532 1088 5 4 0000 3 0000 0000 22535 2452 6 4 0000 3 0000 0000 2584 2503 7 4 0000 4 0000 0000 2568 2568 8 5 0000 00000 0000 2354 1440 9 00000 4 0000 0000 2596 2045 10 1 0000 00000 0000 2554 s1027 In the listing the MaxPr i is the probability attached to the outcome with the largest predicted probability the outcome is shown as the Predicted Y The last column shows the predicted probability for the observed outcome Residuals are not computed there is no significance to the reported zero The results kept for further use are Matrices b and varb An additional matrix named b_logit is created which is J 1 xK This matrix contains the parameters arranged so that Bj is the jth row The first row is zero This matrix can be used to obtain fitted probabilities as discussed below Scalars kreg nreg logl and exitcode Labels for WALD are constructed from the outcome and variable numbers For example if there are three outcomes and Rhs one x1 x2
163. iduals that made that choice The full set of results from the model is as follows Discrete choice multinomial logit model Sample proportions are marginal not conditional Choices marked with are excluded for the IIA test Choice prop Weight IIA AIR 27619 1 000 TRAIN 30000 1 000 BUS 14286 1 000 CAR 28095 1 000 Model Specification Table entry is the attribute that multiplies the indicated parameter Choice Parameter Row 1 GC E A_AIR AIR_HIN1 A_TRAIN Row 2 TRA_HIN2 A_BUS BUS_HIN3 AIR 1 GC E Constant HINC none 2 none none none TRAIN 1 GC E none none Constant 2 HINC none none BUS 1 GC E none none none 2 none Constant HINC CAR 1 GC E none none none 2 none none none Normal exit from iterations Exit status 0 Discrete choice multinomial logit model Dependent variable Choice Number of observations 210 Log likelihood function 13 9 5252 Number of parameters 8 Info Criterion AIC 1 88119 R2 1 LogL LogL Log L fncen R sqrd RsqAdj Constants only 283 7588 33209 32350 Chi squared 5 188 46723 Prob chi squared gt value 00000 Response data are given as ind choice Number of obs 210 skipped 0 bad obs Chapter 8 The Multinomial Logit Model Constants only only N j N N j total N total These 2 models gt P i 3 Notes No coefficients gt P i j 1 J i uses ASCs if
164. ikelihood function 178 3248 Number of parameters 12 Info Criterion AIC 1 81262 Finite Sample AIC 1 82016 Info Criterion BIC 2 00388 Info Criterion HQIC 1 88994 Restricted log likelihood 291 1218 McFadden Pseudo R squared 3874565 Chi squared 225 5941 Degrees of freedom 12 Prob ChiSqd gt value 0000000 R2 1 LogL LogL Log L fncen R sqrd RsqAdj No coefficients 291 1218 38746 37556 Constants only 283 7588 37156 35936 At start values 199 9766 10827 09096 Response data are given as ind choice Chapter 10 The Random Parameters Logit Model Notes No coefficients gt P i j 1 J i Constants only gt P i j uses ASCs These 2 models are simple MNL models R sqrd RsqAdj 1 nJ nJ nparm 1 R sqrd only N j N if fixed choice set N j total sample frequency for j N total sample frequency 1 LogL model logL other nJ sum over i choice set sizes Hessian Random Parms Error Comps Logit Model Replications for simulated probs 25 Halton sequences used for simulations RPL model has correlated parameters Number of obs 210 skipped 0 bad obs was not PD Using BHHH estimator Variable Coefficient Standard Error b St Er P Z gt z Ranen SEN Random parameters in utility functions GC 03344523 02505267 1 335 1819 TIME 23084818 08682355 2 659 0078 eeSssasea4 tNonrandom parameters in utilit
165. imation I Conditional model J Use one line setup Attribute labels m Utility functions Attributes gt _ gt Interact with ASC TTME ONE INVC ea HINC GC gt gt gt gt I Specify utility functions ia Box Cox fo Tree Specification Optimization Hypothesis Tests Figure 8 1 Command Builder for Multinomial Logit Model Chapter 8 The Multinomial Logit Model 97 8 3 Results for the Multinomial Logit Model Results for the multinomial logit model will consist of the standard model results and any additional descriptive output you have requested The application below will display the full set of available results Results kept by this estimator are Matrices band varb coefficient vector and asymptotic covariance matrix Scalars logl log likelihood function nreg N the number of observational units kreg the number of Rhs variables Last Model b_variable the labels kept for the WALD command In the Last Model groups of coefficients for variables that are integrated with constants get labels choice_variable as in trai_gco Note that the names are truncated up to four characters for the choice and three for the attribute The alternative specific constants are a_choice with names truncated to no more than six characters For example the sum of the three estimated choice specific constants could be analyzed as follows WALD Fnl a_air a_train a_bus
166. imply by changing the square brackets above to parentheses as in Effects attribute list of outcomes The first set of results above would become as shown in the following table Chapter 6 NLOGIT Commands and Results 75 Elasticity Averaged over observations Attribute is GC in choice AIR Effects on probabilities of all choices in model Direct Elasticity effect of the attribute Mean St Dev X Choice AIR 2 6002 8212 Choice TRAIN 1 1293 9295 Choice BUS al 1293 92 95 Choice CAR 1 1293 9295 Elasticity Averaged over observations Attribute is GC in choice TRAIN Effects on probabilities of all choices in model Direct Elasticity effect of the attribute Mean St Dev Choice AIR 1 2046 8221 x Choice TRAIN 3 5259 2 1605 Choice BUS 1 2046 8221 Choice CAR 1 2046 8221 Elasticity Averaged over observations Attribute is GC in choice BUS Effects on probabilities of all choices in model Direct Elasticity effect of the attribute Mean St Dev Choice AIR 15695 42859 Choice TRAIN DGIS 2859 Choice BUS 3 6181 1 4924 Choice CAR 5695 2859 Elasticity Averaged over observations Attribute is GC in choice CAR Effects on probabilities of all choices in model Direct Elasticity effect of the attribute Mean St Dev Choice AIR 8688 5119 Choice TRAIN 8688 5119 Choice BUS 8688 poe wee x Choice CAR 2 5979 1 5604 The forc
167. in EQC list value is not a correlation Error Unrecognized alt name in EQC list value Error List needs more than 1 name in EQC list value Error A name is repeated in EQC list value Your model forces a free parameter equal to a fixed one Covariance heterogeneity model needs nonconstant variables Covariance heterogeneity model not available with HEV model Covariance heterogeneity model is only for 2 level models Covariance heterogeneity model needs 2 or more branches At least one variance in the HEV model must be fixed In NLOGIT in the heteroscedastic extreme value you have specified the model so that all the variances are free But for identification one of them must be fixed ultiple observation RPL MNP data must be individual ismatch of indivs and number implied by groups WARNING Halton method is limited to 25 random parameters Not used ODEL followed by a colon was expected not found Fl io Expected equation specs of form U after MOD Unidentified name found in lt string is given gt This occurs during translation of Model U specifications U list must define only choices branches or limbs An equals sign was not found where expected in utility function definition Mismatched or in parameter value specification Could not interpret string expected to find number Expected to find IVSET defn at
168. ing values trunk specific constants interaction terms and so on Utility functions for the trunks may include up to 10 variables including the set of constant terms if used Since the command structure and options for the nested logit model are the same as those for the one level model we will present in this chapter only the parts of the command setup that are specific to nested models All users of this program should read Chapters 2 6 before proceeding Chapter 9 The Nested Logit Model 104 9 2 Mathematical Specification of the Model Individuals are assumed to choose one of the alternatives at the lowest level of the tree Thus they also choose a branch a limb and a trunk We denote by jlb r the choice of alternative j in branch b in limb in trunk r The number of alternatives in the branch limb trunk Npr can vary in every branch limb and trunk and the number of branches in the rth limb trunk Nj is likely to vary across limbs and trunks as well No assumption of equal choice set sizes is made at any point in the following Note that for ease of presentation we have dropped the observation subscript The choice probability defined in Chapter 8 is now redefined to be the conditional probability of alternative j in branch b limb and trunk r jlb Lr exp B X par _ exp B X ip P jlb L e o XP Xqu 7 exp J where J is the inclusive value for branch b in limb trunk r Jp log Xgin eXp B Xoi
169. ins pasa cut esa ada wea pas Scar Hat vaio Sa aa vd bdh Lele ota 87 41 Observations Used Tor the Si lan Oiis i scessysivess vnevvessasevesssnerveztenanvcis easevbes EN 87 T2 Vetoes Used forthe SMU SMO enis a N 87 PAS Choices Simulated reris ere terere narea iene OTER aenea 87 TAA Oher NEOGIT puns asiron pa N 87 TAS Observations Used for the SMUN reccann 87 Fic PUP CAINE E E vena heaved E bead yay eee ae ska dis poet pans ed ns ee Leta 88 Chapter 8 The Multinomial Logit Model cccccceseeseeeeeeeeeeeeeeeeeeesnseseeneeeeeeeeeeeneeenenees 94 Ta AN i spi te sdk oe cida tak ets Sa etal hc bia cra peeping ca ak whey bana geek eprint ald ai 94 8 2 Command Tor the Multinomial Logit Model ssc cissess secs sins inaire ei nieni Eene rE 95 8 3 Res lts Tor ihe Mulinormial Logit MOC cactus cantscntioncsauantananinaatansatto ie 97 SA A Pp CAIN soecerrceisrrieorcerre eraren EEE en NEEE SE N Er EESE NEEE E AT E ENN EF eo Marginal Pets ennen R A 101 Chapter 9 The Nested Logit Model ccccccsssseeeeeeeeeeeeeeeneeeeeeeeeeeeeeeseeseeeceeeneeeeeeeeeesenees 103 D7 Ti OGG OI senao S EN 103 9 2 Mathematical Specification of the Model sci sscesscvcgevecscaccysiieinasdscseeaivin deowacsetueisvseinanisbedocivinke 104 oo Commands 100 TML Satine nant saena O AA 105 cM BET SEMI esse x E pens sng ncox etc siseg cer an wl Huns vets twee pe ends ta as ces eee cea 105 Ta Tree eS an cists avin ance sien Gat screeds 106 O32 UUU PUNCHING ein etrr e E E E
170. ion that logs have appeared in the equation The preceding for example produces the following estimation results Chapter 6 NLOGIT Commands and Results 64 Variable Coefficient Standard Error b St Er P Z gt z H 4 BA 5929844379 21339917 pe AP SS ae 0055 BCOST 2 630222154 45170772 5 823 0000 BC 9545367356 24330526 3 923 0001 BB 9785661344 22951885 4 264 0000 You may also use the Box Cox transformation to transform variables Indicate this with Bcx x where x is the variable which must be positive The transformation is Bex x Q 1 A which is Log x if equals 0 and is x 1 not x if A equals 1 The Bcx function may appear any number of times in the model specification In general if a variable is transformed with this function it should be transformed every time it appears in the model Not doing so is analogous to including both levels and logs of a variable which while not invalid is usually avoided The default value of the transformation parameter A is 1 0 The same value is used in all transformations You may specify a different value by including the specification Lambda value in your NLOGIT command Lambda is treated as a fixed value during estimation not an estimated parameter Thus no standard error is computed for lambda since you provide the fixed value and the standard errors for the other estimates are not adjusted for the
171. is feature is used Do note however if you exclude an RPL variable from all parameters the model becomes inestimable 10 3 5 Correlated Parameters The model specified thus far assumes that the random parameters are uncorrelated Use Correlation to allow free correlation among the parameters In this case estimates of the below diagonal elements of I will be obtained with the other parameters of the model No restrictions may be imposed on these new parameters After these are presented the elements of X IT are given An example appears below Some ambiguity in the results will be unavoidable when this feature is used with other modifications of the model such as mixed distributions and heteroscedasticity The most favorable case for use of this feature would be a sparse model B B Tv We would note many perhaps most of the received applications of the mixed logit model are of this form it is much less restrictive than its bare appearance would suggest Chapter 10 The Random Parameters Logit Model 134 In the model developed thus far the covariance matrix for the random components for the simple distributions normal uniform triangle is Var B lx z T In the uncorrelated case I is a diagonal matrix and the variance of B is simply OL When the parameters are correlated then the diagonal element of X is y y where y is the kth row of Ir The model results will show the elements of I and the implied stan
172. is set to zero We normalize the first element subscript 0 to zero For convenience this normalization is left implicit in what follows It is automatically imposed by the software To allow the remaining random effects to be freely correlated we write the Jx1 vector of nonzero as as Q Ty where T is a lower triangular matrix to be estimated and v is a standard normally distributed mean vector 0 covariance matrix I vector Chapter 2 Discrete Choice Models 16 2 5 Conditional Logit Model If the utility functions are conditioned on observed individual choice invariant characteristics Z as well as the attributes of the choices x then we write U choice j for individual i Uj B xj Y Zi j j 1 Ji For this model which uses a different part of NLOGIT we number the alternatives 1 J rather than 0 There is no substantive significance to this it is purely for convenience in the context of the model development for the program commands The random individual specific terms j1 2 6i7 are once again assumed to be independently distributed across the utilities each with the same type 1 extreme value distribution F e exp exp Under these assumptions the probability that individual t chooses alternative j is Prob U gt U for all q j It has been shown that for independent type 1 extreme value distributions as above this probability is exp B x 12 Prob
173. ith the selection model for z as stated above the bivariate probability for y and z is a mixture of a logit and a probit model The log likelihood can be obtained but it is not in closed form and must be computed by approximation We do so with simulation There are several formulations for extensions of the binary choice models to panel data setting These include e Fixed effects Prob y 1 F B X a correlated with Xy e Random effects Prob y 1 Prob B X x u gt 0 u uncorrelated with X e Random parameters Prob y 1 F B xi B i h Bli with mean vector B and covariance matrix X e Latent class Prob y Ilclass j F B Xi G 8 z Prob class j where Zz is a set of observed characteristics of the individual Other variations include simultaneous equations models and semiparametric formulations Chapter 2 Discrete Choice Models 14 2 4 Multinomial Logit Model The canonical random utility model is as follows U alternative 0 BoXo i U alternative 1 B x n U alternative J By xi u Observed y choice j if U alternative j gt U alternative q V q The disturbances in this framework individual heterogeneity terms are assumed to be independently and identically distributed with identical type lextreme value distribution the CDF is F g exp exp Based on this specification the choice probabilities are Prob c
174. ives affected Change type Value ME AIR Scale base by value 14250 ME TRAIN Scale base by value 1 250 The simulator located 209 observations for this scenario Simulated Probabilities shares for this scenario Chapter 7 Simulating Probabilities in Discrete Choice Models 91 Choice Base Scenario Scenario Base SShare Number Share Number ChgShare ChgNumber AIR 27619 58 16 417 34 11 202 24 TRAIN 30 000 63 23 178 49 6 822 14 BUS 14 286 30 18 796 39 4 510 9 CAR 28 095 59 41 609 87 13 514 28 Total 100 000 210 100 000 209 000 You may also compare the effects of different scenarios For example rather than assume that ttme for both air and train changed you might compare the two scenarios To do a pairwise comparison of scenarios separate them with amp in the command For example NLOGIT Lhs mode Rhs one gc ttme Choices air train bus car Simulation Scenario ttme air 1 25 amp ttme train 1 25 produces the separate results then the pairwise comparison Simulations of Probability Model Model Discrete Choic One Level Model Simulated choice set may be a subset of the choices Number of individuals is the probability times the number of observations in the simulated sample Column totals may be affected by rounding error The model used was simulated with 210 observations Specification of scenario 1 is
175. k r Ji log Xgiz exp B Xoi At the next level up the tree we define the conditional probability of choosing a particular branch in limb J trunk r exp y Tred pur exp y Tor our P bilr yy expa Y sr T ied digi exp where In is the inclusive value for limb 7 in trunk r In log Ysy exp Q ysir Tsis The probability of choosing limb in trunk r is exp 8 Z OnLn exp 8 Z On L1 Pdlr i gt exP sZ Cal ip exp H Chapter 2 Discrete Choice Models 18 where H is the inclusive value for trunk r H log X exp 8 Zs Oss Finally the probability of choosing a particular limb is exp h 6 H Ne gt exp 6 h 6 H By the laws of probability the unconditional probability of the observed choice made by an individual is P j b Lr P jlb Lr x P bllr x P r x P r This is the contribution of an individual observation to the likelihood function for the sample The nested logit aspect of the model arises when any of the Ty Or On or Q differ from 1 0 If all of these deep parameters are set equal to 1 0 the unconditional probability reduces to exp B X ipis t A Y pi 8 z Oh P j b Lr ae gt 2 D a exp B x Y 5Z 6h which is the probability for a one level conditional multinomial logit model 2 7 Random Parameters Logit Models In its most general form we write the multinomial logit probability a
176. l specifications that are documented in the LIMDEP Econometric Modeling Guide for these specific models The specifications below are those that may appear in the NLOGIT command or the conditional logit extensions described above General Model Specification and Data Setup Data on Dependent Variable Ranks indicates that data are in the form of ranks possibly ties at last place Shares indicates that data are in the form of proportions or shares Frequencies indicates that data are in the form of frequencies or counts Checkdata checks validity of the data before estimation Wts weighting variable uses a weighting variable Noscale is not used here Scale list of variables values for scaling loop specifies scaling of certain variables during iterations Pds specification used by RPL LCM ECM MNP and by binary choice models to indicate a panel data set Indicates multiple choice situations for individuals Specification of the Dependent Variable Lhs list of variables used by all models to name the dependent variable Second Lhs variable indicates variable choice set size Third Lhs variable indicates specific choices in a universal choice set First variable is a set of utilities if MCS is used MCS requests data generated by Monte Carlo simulation Choices list lists names for alternatives Specification of Utility Functions Rhs list of variables lists choice varying attribute variables R
177. l with uncorrelated disturbances are equal This parametric test is likely to be a more powerful test than the McFadden Hausman test first because it is based on the Neyman Pearson methodology and second because it will always use the entire sample You could do it as follows CALC Ran seed for generator MNPROBIT _ 5 specify the choices and utility functions Cor 0 CALC lu logl CALC Ran same seed for generator MNPROBIT specify the choices and utility functions Sdv 1 Cor 0 CALC Ir logl List Irstat 2 lu Ir We applied this procedure in passing in the preceding section The log likelihoods for the three models estimated were Most restrictive oj 1l Pm 0 Log likelihood 195 5496 Restrictive oj 1 Log likelihood 194 9204 Unrestricted Log likelihood 189 5252 In principle a test of the first assumption as the null hypothesis against the alternative of the second is sufficient to reject ITA We found the chi squared to be 10 132 with two degrees of freedom The critical value is 5 99 so the hypothesis is rejected A test of the third model against the null of the first produced a chi squared of 12 048 with five degrees of freedom The critical value is 11 07 so once again the hypothesis is rejected Which test should be preferred is uncertain Under the null hypothesis the estimated parameters in the second model are more precisely estimated so this may favor it We are u
178. lternatives specification enclose any choices to be excluded in parentheses For example the specification Choices air train bus car produces in part the results are omitted the following display in the model output Discrete choice multinomial logit model Response data are given as ind choice Number of obs 210 skipped 93 bad obs Restricted choice set Excluded choices are TRAIN BUS Chapter 6 NLOGIT Commands and Results 55 Note that as in the IIA test this procedure results in exclusion of some bad observations that is the ones that selected the excluded choices The model specified is fit to the data set that is based on the remaining choices We note a caution at this point that applies equally here and in the IIA test It is possible that there are attributes that do not vary within the retained choice set If these remain in the model it will not be possible to fit it Consider for example a six choice model with choices air train bus car as driver car as passenger motorcycle Now suppose that one of the attributes is terminal time It will happen that the last three choices always have terminal time equal to zero So while it may be no trouble to fit a six choice model which includes terminal time the same model specified with Choices air train bus car_d car_p mc will not be estimable as terminal time will always be zero for all choices for all individuals
179. mal exit from iterations Exit status 0 Discrete choice multinomial logit model aximum Likelihood Estimates Dependent variable Choice Weighting variable None umber of observations 210 Iterations completed 5 Log likelihood function 246 1098 umber of parameters 9 Info Criterion AIC 2 42962 Finite Sample AIC 2 43390 Info Criterion BIC 2 57306 Info Criterion HQIC 2 48761 R2 1 LogL LogL Log L fncen R sqrd RsqAdj Constants only 283 7588 13268 12011 Chi squared 6 7529796 Prob chi squared gt value 00000 Response data are given as ind choice Number of obs 210 skipped 0 bad obs Notes No coefficients gt P i j 1 J i Constants only gt P i j uses ASCs only N j N if fixed choice set N j total sample frequency for j N total sample frequency These 2 models are simple MNL models R sqrd 1 LogL model logL other RsqAdj 1 nJ nJ nparm 1 R sqrd Lhs mode Choices air train bus car nJ sum over i choice set sizes Variable Coefficient Standard Error b St Er P Z gt z INVC 04612501 01664864 2 770 0056 INVT 00838543 00214019 3 918 0001 GC 03633292 01477727 2 459 0139 A_AIR 1 31602481 72323155 1 820 0688 AIR_HIN1 00648950 01079433 601 5477 A_TRAIN 2 10710471 43179879 4 880 0000 TRA_HIN2 05058498 01206873 4 191 0000 A_BUS 86502331 50318615 1719 0856 BUS_HIN3 03316081 01299
180. n Section 5 2 data for NLOGIT will generally consist of a record row of data for each alternative in the choice set for each individual Thus the data file contains 210 observations or 840 records The variables in the data set are as follows Original Data mode 0 1 for four alternatives air train bus car this variable equals one for the choice made labeled choice below ttme terminal waiting time invc invehicle cost for all stages invt invehicle time for all stages gc generalized cost measure Invc Invt x value of time chair dummy variable for chosen mode is air hinc household income in thousands psize traveling party size Chapter 5 Data Setup for NLOGIT 48 Transformed variables aasc choice specific dummy for air generated internally tasc choice specific dummy for train basc choice specific dummy for bus casc choice specific dummy for car hinca hinc x aasc psizea psize x aasc The table below lists the first 10 observations in the data set In the terms used here each observation is a block of four rows The mode chosen in each block is boldfaced mode choice ttme invc invt gc chair hinc psize aasc tasc basc casc hinca psizea obs Air 0 69 59 100 70 0 35 1 1 0 0 QO 35 1 i 1 Train 0 34 3d 3T2 71 0 35 1 0 1 0 0 0 0 Bus 0 35 25 417 70 0 35 1 0 0 i 0 0 0 Car 1 0 10 180 30 0 35 1 0 0 0 1 0 0 Air 0 64 58 68 68 0 30 2 cr 0 0 0 30 2 1 2 Train 0 44 3A 354 84 0 30 2
181. n bus car Rhs invc invt gc ttme Rh2 one Prlist Chapter 6 NLOGIT Commands and Results 79 Discrete choice multinomial logit model Dependent variable Choice Number of observations 200 Log likelihood function 174 8393 Number of parameters 7 Info Criterion AIC 1 81839 Finite Sampl AIC 1 82131 Info Criterion BIC 1 93383 Info Criterion HQIC 1 86511 R2 1 LogL LogL Log L fncn R sqrd RsqAdj Constants only 267 3168 34595 33823 Chi squared 4 184 95510 Prob chi squared gt value 00000 Response data are given as ind choice Number of obs 200 skipped 0 bad obs Notes No coefficients gt P i j 1 J i Constants only gt P i j uses ASCs only N j N if fixed choice set N j total sample frequency for j N total sample frequency These 2 models are simple MNL models R sqrd 1 LogL model logL other RsqAdj 1 nJ nJ nparm 1 R sqrd nJ sum over i choice set sizes Variable Coefficient Standard Error b St Er P Z gt z INVC 08826012 01987417 4 441 0000 INVT 01344131 00256769 52235 0000 GC 07053307 01778244 3 966 0001 TTME 10176138 01117372 9 107 0000 A_AIR 533347705 92158644 5 787 0000 A_TRAIN 4 44686822 52777949 8 426 0000 A_BUS 3 69334154 52916432 6 980 0000 Discrete Choice One Level Model odel Simulation Using Previous Estimates umber of observations 10
182. naware of any other evidence on the question Chapter 12 Diagnostics and Error Messages 167 Chapter 12 Diagnostics and Error Messages 12 1 Introduction The following is a complete list of diagnostics that will be issued by NLOGIT Altogether there are over 1 000 specific conditions that are picked up by the command translation and computation programs in LIMDEP and NLOGIT Those listed here are specific to NLOGIT The full set of diagnostics is given in Chapter R18 of the LIMDEP Reference Guide Nearly all of the error messages listed below identify problems in commands that you have provided for the command translator to parse and then to pass on to the computation programs Most diagnostics are self explanatory and will be obvious For example 82 LHS variable in list is not in the variable names table states that your Lhs variable in a model command does not exist No doubt this is due to a typographical error the name is misspelled Other diagnostics are more complicated and in many cases it is not quite possible to be precise about the error Thus in many cases a diagnostic will say something like the following string contains an unidentified name and a part of your command will be listed the implication is that the error is somewhere in the listed string Finally some diagnostics are based on information that is specific to a variable or an observation at the point at which it occurs In that case the diagnost
183. nce Matrices Printve displays estimated asymptotic covariance matrix normally not shown x Choice uses choice based sampling sandwich with weighting estimated matrix This is specified in the Choices list specification for NUOGIT Cluster name cluster form of corrected covariance estimator Robust sandwich estimator or robust VC for TSCS and some discrete choice Optimization Controls for Nonlinear Optimization Start list gives starting values for a nonlinear model Tlg value sets convergence value for gradient TIf value sets convergence value for function TIb value sets convergence value for parameters Alg name algorithm Newton s method is best BFGS is occasionally needed Maxit n maximum iterations Chapter 6 NLOGIT Commands and Results 53 Output n technical output Lpt n Laguerre quadrature number of points to use Hpt n Hermite quadrature number of points to use Set keeps current setting of optimization parameters as permanent Predictions and Residuals List displays a list of estimated probabilities with model results 7 Keep name keeps fitted values as a new or replacement variable in data set Several other similar specifications are used with NLOGIT s Res name keeps residuals as a new or replacement variable Prob name saves probabilities as a new or replacement variable j Fill fills missing values outside estimating sample f
184. nd the Choice Solaires 53 bo Resrictins Me Choice S6l ican tos 54 6 2 4 Specifying the Utility Functions with Rhs and Rh2 oe ee eeeeeeeeeereeeneeeneeenees 55 Gas Puldi he Ulty Fomio Gabor rey emerrer cement tere error A te tener 59 hoo lan ard Model Fe Wiesner tans cena ia ader ar eb aane ea aia 67 0al Retained Resti rarse kn nces nuce Tana eaten ERANA cee tlugseutnian teed ov Metered seen 69 maL RODUS Standard Erroi osian Rename 70 6 3 3 Descnpuve Statistics for ANGIVE Sisirin narrans 71 6 4 Marginal Ettects and Elasticities s2ssicaseoyscsivcsssccesactiat sedaresteiwovsesctensssedevaveetsecaivesesum besateossrieexe 72 fo Predicted Probabili nes and Inclusive Y aeS asinansonsaniidnneian a aAA 76 6 3 1 In Sample Predicted Probabilities and Inclusive Values ccicss ccscsascccssesssecovesnsceeevs 76 6 5 2 Computing Out of Sample Model Probabilities ojo cearie on tirienn 78 GO Testina Hype SOS rasire sri arenen EEEE EAE EAEN SEE EEs assume relies 80 6 6 1 Testing the Assumption of Independence from Irrelevant Alternatives ITA 80 6 6 2 Lagrange Multiplier Wald and Likelihood Ratio TestSs sssissssisisisscsssssiseesssvsseess 83 Chapter 7 Simulating Probabilities in Discrete Choice Models ssssssssssssseeeeeeees 84 PATIO BICHON scann E cetascdn a eae sae eed 84 Peas Hey ile ale est clas siana reteeer trey an trn oem herremn A 85 7 3 Multiple Attribute Specifications and Multiple Scemarios naisiin 86 Fale a A NS aise e
185. ndom the probability that the analyst will observe a purchase is Prob purchase Prob U purchase gt U not purchase Prob B x gt BoXo o Prob o lt Bi x BoXo F Bi x Bo Xo where F z is the CDF of the random variable amp The model is completed and an estimator generally maximum likelihood is implied by an assumption about this probability distribution For example if and are assumed to be normally distributed then the difference is also and the familiar probit model emerges The sections to follow will outline the models described in this manual in the context of this random utility model The different models derive from different assumptions about the utility functions and the distributions of their random components 2 3 Binary Choice Models Continuing the example in the previous section the choice of alternative 1 purchase reveals that U gt Uo or that E E amp 0 lt Bi X Bo xo Let 1 o and B x represent the difference on the right hand side of the inequality x is the union of the two sets of covariates and B is constructed from the two parameter vectors with zeros in the appropriate locations if necessary Then a binary choice model applies to the probability that lt B x Two of the parametric model formulations in NLOGIT for binary choice models are the probit model based on the normal distribution f ly CED i o x an
186. ndow Help Elm Data Description Time Series Linear Models Nonlinear Regression Binary Choice Censoring and Truncation Count Data Duration Models H E Variables Frontiers Namelists Discrete Choice Matrices Numerical Analysis Nested Logit Scalars i i MLogit E Strings Ordered G Procedures Multnomial Probit HEY RPL Sy Output Ga nee Gen Maximum Entropy Output Window b b b b b d d Figure 6 1 Command Builders for NLOGIT Models The Main and Options pages of the command builder for the conditional logit model are shown in Figures 6 2a and 6 2b Some features of the models and the ECM model are not provided by the command builders Most of the features of these models are much easier to specify in the editor or command mode of entry The model and the choice set are set up on the Main page The Rhs variables attributes and Rh2 variables characteristics are defined on the Options page Note in the two windows on the Options page the Rhs variables of the model are defined in the left window and the Rh2 variables are specified in the right window Chapter 6 NLOGIT Commands and Results 51 DISCRETE CHOICE Main Options Output Choice variable Choice variable MODE z Data type Individual choice z I Use ordinary weights Individual choice Choice set Proportion Fixed number Aa id 5 ir Train Bus Car an Use choice bas
187. nds and Results 6 1 Introduction This chapter will describe the common features of the NLOGIT models The specification of models for NLOGIT follows the general pattern for model commands in LIMDEP The different models such as nested logit and multivariate probit are requested by modifying the basic command NLOGIT is built around estimation of the parameters of the random utility model for discrete choice U choice j for individual i Uj B Xy amp 7 1 Ji in which individual 7 makes choice j if U is the largest among the J utilities in the choice set The parameters in the model are the weights in the utility functions and the deeper parameters of the distribution of the random terms In some cases the taste parameters in the utility functions might vary across individuals and in most cases they will vary across choices The latter is simple to accommodate just by merging all parameters into one grand B and redefining x with some zeros in the appropriate places But for the former case we will be interested in a lower level parameterization that involves what are sometimes labeled the hyperparameters Thus it might be the extreme case as in the random parameters logit model that B f z A T B v where A T B are lower level parameters z is observed data and v is a set of latent unobserved variables The parameters of the random terms will generally be few in number usually consisting of a small numbe
188. ne limb travel Gravel 1 0 The other two parameters are free and unrestricted You can modify the specification of these parameters in two ways e You may specify that they are equal to each other e You may specify that they are fixed values instead of free parameters to estimate To use these features add the specification Ivset specification Note once again the presence of a colon in this specification For purposes of this specification ts os and os are treated the same To force parameters to be equal put the names of the branches and or limbs together in parentheses in the Ivset specification For the example given above to force the two ts to be equal in the estimated model use Ivset public private For a second example consider this larger tree Commute TRUNK Private Public LIMBS Drive Land Water BRANCHES So A Plane Helicopter Car_Drv Car_Ride Train Bus Ferry Raft TWIGS Chapter 9 The Nested Logit Model 109 We would define this with Tree private fly plane helicptr drive car_ride car_drv public land train bus water ferry raft There are six IV parameters Ti for each of fly drive land and water and c for private and public If it were desired to force Oprivate Opublic Tflylprivate Uland public gt and Twaterl public for some reason to equal Opublic you could use Ivset private public water fly land Note once again separate specifications are separated
189. nerate quite a lot of output if your model is large but you can request an analysis of all alternatives by using the wildcard attribute Chapter 6 NLOGIT Commands and Results The tables below are produced by NLOGIT Lhs mode Choices air train bus car Rhs invc invt gc Rh2 one hinc Effects gc Derivative times 100 Attribute is GC averaged over observations in choice AIR Effects on probabilities of all choices in model Direct Derivative effect of the attribute Mean St Dev Choice AIR 6042 82397 Choice TRAIN 2007 ELSZ Choice BUS s1237 0798 Choice CAR 2798 2044 Derivative times 100 Attribute is GC averaged over observations in choice TRAIN Effects on probabilities of all choices in model Direct Derivative effect of the attribute Mean St Dev Choice AIR 2007 oLE32 4 Choice TRAIN 6180 2612 Choice BUS 1754 ASI Choice CAR 2420 21305 Derivative times 100 Attribute is GC averaged over observations in choice BUS Effects on probabilities of all choices in model Direct Derivative effect of the attribute Mean St Dev Choice AIR 1237 0798 Choice TRAIN 1754 sha Tt Choice BUS 4332 1431 Choice CAR 1342 0648 Derivative times 100 Attribute is GC averaged over observations in choice CAR Effects on probabilities of all choices in model Direct Derivative ef
190. ng our estimates and taking the square root produces an estimate of the population standard deviation of 0 014864085 The result for the normal distribution is 01723103 We emphasize we are implicitly averaging over incomes in these computations the results are close to but not exactly equal to the analytical results The results for the lognormal distribution correctly interpreted are quite similar to those for the normal distribution The structural parameters however are quite different A similar characterization applies to the other distributions that are obtained as transformations of the underlying random terms In most cases it is not possible to obtain closed form results for the overall means and variances the lognormal distribution is a convenient special case The program will report its estimates of the structural parameters but it is not generally possible to disentangle the reduced form to report the actual mean and standard deviation in spite of the labeling of the estimates in the program output Random parameter distributions that depend on the uniform distribution present another ambiguity in the interpretation of the results For the uniform distribution we estimate the spread of the distribution not the standard deviation or the variance Suppose we now change the earlier model to Fen mge u By this construction B B Zi OVi Vi U 1 1 the values of B are distributed uniformly between
191. nly be used for 1 level nonnested models
192. not vary across the choices such as age marital status income etc will simply fall out of the probability Consider an example with a constant one attribute and one characteristic expla B cost B income Prob choice j pla B cost B i Ei exp a B cost B income exp a B income exp B cost 7 Dy _ exp a B income exp B cost exp a B income exp B cost 7 exp a B income X m exp B cost exp B cost Dy _ exp B cost With a generic coefficient the choice invariant characteristic falls out of the model This includes the constant term one A model which contains such a characteristic with a generic coefficient is not estimable This carries over to all of the more elaborate models such as the HEV nested logit and MNP models as well The solution to this complication is to create choice specific constant terms and if need be interact the invariant characteristic with the constant term This is what appears in the last eight columns in the example above Here it produces a hybrid model which can have both types of variables in the utility functions exp B cost a y Income Prob choice j vx pice aes DE expBicost a Income There remains an indeterminacy in the model after it is expanded in this fashion Suppose the same constant say 9 is added to each y The resulting model is exp B cost a y 8 Income Prob choice j gt exp B
193. ns that one must be careful in interpreting the estimated coefficients even in simple cases in which there is no heterogeneity in the means or variances It is possible to learn about these empirically however it is often not possible to state a priori what the population means are for most of the distributions The problem becomes yet more complicated as additional features such as heterogeneity in the means and heteroscedasticity are added to the model Some practical aspects of the specifications are as follows If you will be mixing distributions the specification of correlated parameters while allowable produces ambiguous results The nature of the correlation is difficult to define However the program will have no unusual difficulty estimating a model in which correlated parameters have different distributions One particular case worth noting is a mixture of normal and lognormal parameters In such a model the reported correlation will be between the normally distributed parameter and the log of the lognormally distributed parameter This is probably not a useful result e Researchers often find that the long thick tail of the lognormal distribution produces an implausible distribution of parameters Type c is the same as not including the parameter in the Fcn list which is how this usually should be done But sometimes for convenience this might be preferred Variable name c specifies a free mean and zero variance of
194. o 100 choices Numerous variations on this are possible e Data on the observed outcome may be in the form of frequencies market shares or ranks e The number of choices may differ across observations e The choice set may be extremely large The preceding described the base case model for a fixed number of choices using individual level data There are several alternative formulations that might apply to the data set you are using 5 3 Fixed and Variable Numbers of Choices When every individual in the sample chooses from the same choice set and all alternatives are available to all individuals then the data set will appear as in the first example above and will consist of n sets of J observations You indicate this case with a command such as NLOGIT Lhs y or CLOGIT Choices a list of J names for the choices Or se the rest of the command For convenience in what follows we will use the generic model name NLOGIT in the command The specific verbs CLOGIT RLPOGIT etc will be used in the specific chapters where the model itself is developed For example NLOGIT Lhs mode Choices air train bus car the rest of the command The list of choices is crucial as it tells the program how many choices constitute an observation Otherwise for example there is no way to tell if 12 rows of data are three observations on a four choice setting or four observations on a three choice setting We now conside
195. oduced by this model command are as follows Chapter 6 NLOGIT Commands and Results Variable Coefficient Standard Error b St Er P Z gt z BA 1 55491032 37580063 4 138 0000 BCOST 02020918 00434927 4 647 0000 BTIME 08680295 OLT2Z223 7 mt SD 0000 BC 3 65316491 46378035 Eo 0000 BB 3 91982604 45611114 8 594 0000 One point that you might find useful to note The order of the parameters in this list is determined by moving through the model definition from beginning to end Each time a new parameter name is encountered it is added to the list Looking at the model command above you can now see how the order in the displayed output arose The last example in the preceding subsection which has four separate coefficients on a cost variable gc could be specified using NLOGIT Lhs mode Choices air train bus car Model U air be inve bt invt aa cha hinc cga ge U train be inve bt invt at cht hinc cgt ge U bus be inve bt invt ab chb hinc cgb ge U car be inve bt invt The estimates are Variable Coefficient Standard Error b St Er P Z gt z BC 04386562 01712959 224561 0104 BT 00815115 00241976 3 369 0008 AA 1 43 749 3591 83837138 1 640 eLO41 CHA 00703267 01078793 652 5145 CGA 03762100 01676624 2 244 0248 AT 2 53156832 60800716 4 164 0000 CHT 05096641 01214303 4 197 0000
196. oefficient income or price coefficient The random parameters logit model will compute and retain person specific WTP measures Use WTP name name where names are either variable names if Rhs is used or parameter names if utility functions are specified directly In general the WTP calculation will have an attribute level coefficient in the numerator and a cost or income measure in the denominator Parameters can be random or nonrandom This will create two matrices wtp_i and sdwtp_i These are computed the same way that beta_i and sdbeta_i are computed where wtp_i contains estimates of the conditional expectation of WTP and sdwtp_i contains estimates of the conditional standard deviation These matrices can be examined and analyzed in precisely the same way that beta_i was used earlier You may compute more than one WTP variable by adding additional ratios in the command separated by commas For example WTP time income space price To illustrate we use the Weibull model once again with a small modification RPLOGIT Lhs mode Choices air train bus car Rhs mgc mttme hinca Rh2 one s ECM air car train bus WTP mttme hinca Fen mgc w mttme w Correlated Parameters Halton Pds 3 Pts 200 The willingness to pay is computed as the ratio of the terminal time in minutes to the income variable hinca this equals income for the air alternative and zero otherwise The basic coefficient estimates a
197. of obs 210 skipped 0 bad obs Variable Coefficient Standard Error b St Er P Z gt z GC 01578374 00438279 3 601 0003 TTME 09709052 01043509 9 304 0000 A_AIR 5 77635901 65591873 8 807 0000 A_TRAIN 3 92300113 44199360 8 876 0000 A_BUS 3 21073472 44965283 7 140 0000 140 Results from the random parameters logit model display the standard pattern an initial box containing diagnostic statistics followed by an indication of the size R and type random or Halton of the simulation then the output for the model In this model there are likely to be many different components of the probability function such as in the earlier example As shown in the sample output below the results will contain the lowest level structural parameters first the constant terms in the random parameters in the utility functions then the nonrandom parameters and finally the parameters of the underlying distribution The final parameters shown are the scale factors for the underlying random terms in the parameters The leading character matches your specification in the Fen part of your command The s to follow indicates this is a diagonal element of I Finally up to five characters of the original name are appended Random Parms Error Comps Logit Model Maximum Likelihood Estimates Dependent variable MODE Weighting variable None Number of observations 210 Iterations completed 36 Log l
198. oftware product To the maximum extent permitted by applicable law Econometric Software disclaims all other warranties and conditions either express or implied including but not limited to implied warranties of merchantability fitness for a particular purpose title and non infringement with respect to the software product This limited warranty gives you specific legal rights You may have others which vary from state to state and jurisdiction to jurisdiction Limitation of Liability Under no circumstances will Econometric Software be liable to you or any other person for any indirect special incidental or consequential damages whatsoever including without limitation damages for loss of business profits business interruption computer failure or malfunction loss of business information or any other pecuniary loss arising out of the use or inability to use the software product even if Econometric Software has been advised of the possibility of such damages In any case Econometric Software s entire liability under any provision of this agreement shall not exceed the amount paid to Econometric Software for the software product Some states or jurisdictions do not allow the exclusion or limitation of liability for incidental or consequential damages so the above limitation may not apply to you Preface NLOGIT is a major suite of programs for the estimation of discrete choice models It is built on the original DISCRETE CHOICE comm
199. on of the data which allows some heteroscedasticity across individuals The correlations pj are restricted to 1 lt pj lt 1 but they are otherwise unrestricted save for a necessary normalization The correlations in the last row of the correlation matrix must be fixed at zero The standard deviations are unrestricted with the exception of a normalization two standard deviations are fixed at 1 0 NLOGIT fixes the last two This model may also be fit with panel data In this case the utility function is modified as follows Uit B Xjir Eje Vit where indexes the periods or replications There are two formulations for v Random effects Vjix Vir the same in all periods First order autoregressive vj Oj Viti Ajit Chapter 2 Discrete Choice Models 21 It is assumed that you have a total of T observations choice situations for person i Two situations might lend themselves to this treatment If the individual is faced with a set of choice situations that are similar and occur close together in time then the random effects formulation is likely to be appropriate However if the choice situations are fairly far apart in time or if habits or knowledge accumulation are likely to influence the latter choices then the autoregressive model might be the better one You can also add a form of individual heterogeneity to the disturbance covariance matrix The model extension is Var e exp y h x where amp
200. or MNP model 1053 Scaling option is not available with Ranks data may not be used with HEV Nested models are not available with Cannot keep cond probs or IVs with Choice based sampling not useable in Ea E i iise ES These diagnostics are produced by problems setting up the scaling option for mixed data sets 1054 Scaling option is not available with one line data setup Ranks data may not be used with one line data setup Choice set may not be variable with one line data setup One line data setup requires RHS and or RH2 spec Nested models are not available with one line data setup Chapter 12 Diagnostics and Error Messages 171 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 Cannot keep probabilities or IVs with one line data setup Did not find closing paren in SCALE list spec The list of variables to be scaled has an error Only 40 or fewer variables may be scaled You are attempting to scale the LHS variable The list of values given for SCALE grid is bad Grid must Lo Hi N or Lo Hi N N2 Check spec Grid must have Low gt 0 and High gt low Check s Number of grid points must be 2 3 up to 20 Unidentified name in IIA list Procedure omitted More than 5 names in IIA list Limit is 5 Size variables only available with Nested MNL
201. or fitted values Hypothesis Tests and Restrictions Wald spec Wald test of linear restrictions in any model s CML spec constrained maximum likelihood Test spec Wald test of linear restrictions same as Wald spec r Rst list specifies equality and fixed value restrictions Maxit 0 Start the restricted values specifies Lagrange multiplier test 6 2 2 Specifying the Choice Variable and the Choice Set Every model fit by NLOGIT must include a specification for the choice variable and a definition of the choice set The basic formulation would appear as Lhs the dependent or choice variable Choices the names of the choices in the model In general your dependent variable is the name of a variable which indicates by a one or zero whether a particular alternative is selected or it gives the proportion or frequency of individuals sampled that selected a particular alternative When they are enumerated the Choices list gives names and possibly sampling weights for the set of alternatives All command builders begin with these two specifications The discrete choice and nested logit models allow the full set of variants discussed earlier while the other command builders expect the simple form with a fixed choice set The Main page of the conditional logit command builder shown in Figure 6 3 illustrates A similar Main page is used for the nested logit command builder The command builder allows you to specify the choice v
202. ost gc btime ttme U train be bcost gc btime ttme U bus bcost gc btime ttme U car be bcost gc Effects ge car Chapter 9 The Nested Logit Model 113 This lists the effects on all four probabilities of changes in attribute generalized cost gc of choice car Partial effects average over observations dinP alt j br b lmb 1 tr r dx k alt J br B lmb L tr R B LR P L R P R defined likewis n N 1 if n N 0 else for n j b 1 r and N J B L R lasticity x k D 3j B L R arginal effect F is decomposed into the 4 parts in the tables mH tu D k J B L R delta k F delta k coefficient on x k in U J B L R F r R 1 L b B j J P J BLR r R 1 L b B P B LR P J BLR t B LR r R 1 L P LIR P B LR P J BLR t B LR s L R r R P R P LIR P B IR P J BIR t B LR s L R f R J BLR Prob choice J branch B limb L trunk R P JBLR D P J BLR P B LR P L R P R D Elasticity averaged over observations Attribute is GC in choice CAR Effects on probabilities of all choices in the model indicates direct Elasticity effect of the attribute Decomposition of Effect if Nest Trunk Limb Branch Choice Trunk Trunk 1 Limb TRAVEL Branch PUBLIC Choice BUS 000 000 857 0 00 Choice TRAIN 000 000 LBD 000 Branch PRIVATE Choice
203. owever the software may not be used on the primary computer by another person while the secondary computer is in use For a multi user site license the specific terms of the site license agreement apply for scope of use and installation Limited Warranty Econometric Software warrants that the software product will perform substantially in accordance with the documentation for a period of ninety 90 days from the date of the original purchase To make a warranty claim you must notify Econometric Software in writing within ninety 90 days from the date of the original purchase and return the defective software to Econometric Software If the software does not perform substantially in accordance with the documentation the entire liability and your exclusive remedy shall be limited to at Econometric Software s option the replacement of the software product or refund of the license fee paid to Econometric Software for the software product Proof of purchase from an authorized source is required This limited warranty is void if failure of the software product has resulted from accident abuse or misapplication Some states and jurisdictions do not allow limitations on the duration of an implied warranty so the above limitation may not apply to you To the extent permissible any implied warranties on the software product are limited to ninety 90 days Econometric Software does not warrant the performance or results you may obtain by using the s
204. ppear in the utility functions for all the choices and in all periods or choice situations Denote this implied conditional distribution as p y la B x he E where a is the set of ASCs With these in hand we will form p Biy x z hr he E as follows First we will have to eliminate E from the conditional distribution of y The unconditional distribution is p y B x he ply IB x he E p E dE Note that the marginal distribution is actually known it is the M variate standard normal distribution Nonetheless it will be more convenient to carry it through in generic form below We now obtain the conditional density of B using Bayes theorem B i p y B x he E p E dE p B z hr p B ly x Z he hr f p y 1 2 he hhr E p E dE f P Y B x he E p E dE p B z hr f f p y B x he E p E dE p B z hr dB Note that it is the joint density p B y x z hr hf that appears in the fraction the product of the conditional density times the marginal density Proceeding we are interested in forming the conditional expectation E Bjly x z hr hf Since the preceding gives the conditional density the conditional expectation is formed in the usual manner J Bfe PO B x he B p B aE p B 12 hr AB f fe PO 1Bix he E p B dE p B z hr dB f Ja BPO 1B x he E p p B z hr dE AB h h PIB x she E p E p B 12 hr aE 4B E B ly
205. r Moreover we may also have information about individual specific error components E m specifically in the form of he the observed heterogeneity in the variation of the error components The following details a method of forming a conditional estimator E B all data on individual i By using Bayes Theorem we can form the joint distribution of B and y yi1 Vi2 Vir as follows Denote the unconditional marginal distribution of B lz hr as p Biz hr This distribution is implied by whatever is assumed about v in the general model B B Az T Q v Chapter 10 The Random Parameters Logit Model 145 where if there is heteroscedasticity o exp hr Elements of B might also be functions of the exponent of this expression for the lognormal and Weibull distributions We can also form the conditional distribution of y lB x he E based on the assumptions about v and E En En Eim in the conditional multinomial logit model Prob y j t 1 T J exp a B X ji gt a a exp y he E it Site Eyres ti J 7 pee kee BX ji Eitan exp y he E The conditional distribution is defined by the multinomial logit probabilities for the outcomes that have been assumed throughout We are looking ahead a bit here and treating the panel data case here rather than developing it separately later Note as well that x denotes the collection of data on attributes and characteristics that a
206. r of scaling parameters as in the heteroscedastic logit model but they might be quite numerous again in the random parameters model In all cases the main function of the routines is estimation of the structural parameters then use of the estimated model for analysis of individual and aggregate behavior 6 2 NLOGIT Commands The essential command for the set of discrete choice models in NLOGIT is the same for all with the exception of the model name Model Lhs variable which indicates the choice made Choices a set of J names for the set of choices Rhs choice varying attributes in the utility functions Rh2 choice invariant variables including one for ASCs The various models are as follows where either of the two forms given may be used Model Command Alternative Command Form Conditional Logit CLOGIT NLOGIT Nested Logit NLOGIT NLOGIT Tree Random Parameters Logit RPLOGIT NLOGIT RPL Multinomial Probit MNPROBIT NLOGIT MNP Chapter 6 NLOGIT Commands and Results 50 The description to follow in the rest of this chapter applies equally to all models For convenience we will use the generic NUOGIT command in most of the discussion while you can use the specific model names in your estimation commands The commands builders for these models can be found in Model Discrete Choice There are several model options as shown in Figure 6 1 ai Limdep Untitled Project 1 File Edit Insert Project SMAN Run Tools Wi
207. r the random utility model first in which the number of choices is not constant from one observation to the next Two possible arrangements that might occur are as follows e There is a universal choice set from which individuals make their choices But not all choices are available to all individuals Consider for example the choice of travel mode among air train bus car If respondents are observed at many different locations one or more of the choices for example train might be unavailable to some of them and those might vary from person to person e Individuals each choose among a set of J alternatives However there is no universal choice set defined as such Consider for example the choice of which shopping center to shop at If observations are taken in many different cities we will observe numerous different choice sets but there is no well defined universal choice set Either case can be accommodated For both cases you will provide a second Lhs variable which gives the number of choices for each observation The command is NLOGIT Lhs y nij Specification of the utility functions the rest of the command Chapter 5 Data Setup for NLOGIT 43 Note that the Choices list is not defined in this command since in this case the second one above there is no clearly defined choice set Nothing else need be changed NLOGIT does all of the accounting internally In this case it is simply assumed
208. r valued variable taking the values 0 1 J This model may also be fit with proportions data In that case you will provide the names of J 1 Lhs variables that will be strictly between zero and one and will sum to one at every observation The MLOGIT command is the same as LOGIT The program inspects the command Lhs and the data and determines internally whether BLOGIT or MLOGIT is appropriate Note on proportions data if you want to fit a binary logit model with proportions data you will supply a single proportions variable not two What would be the second one is just one minus the first If you want to fit a multinomial logit model with proportions data with three or more outcomes you must provide the full set of proportions Thus you would never supply two Lhs variables in a LOGIT BLOGIT or MLOGIT command 3 4 2 Conditional Logit The command for the conditional model and the commands in the sections to follow are variants of the NLOGIT command This is a full class of estimators based on the conditional logit form There are several forms of the essential command for fitting the conditional logit model with NLOGIT The simpler one is CLOGIT Lhs dependent variable Choices the names of the J alternatives Rhs list of choice specific attributes Rh2 list of choice invariant individual characteristics As discussed in Chapter 5 the data for this estimator consist of a set of J observations one for each alterna
209. rce the attribute to take this value in all cases or value to multiply observed values by the value or value to add value to the observed values or value to divide the attribute by the specified value or value to subtract value from the observed values The following example Choices air train bus car Simulation air car Scenario gc car 1 5 specifies a simulation over two choices in a four choice model The scenario is enacted by changing the gc attribute for car only by multiplying whatever value is found in the original sample by 1 5 7 3 Multiple Attribute Specifications and Multiple Scenarios The simulation may specify that more than one attribute is to change The multiple settings may provide for changes in different alternatives The specification is Scenario attribute name 1 list of alternatives action magnitude of action attribute name 2 list of alternatives action magnitude of action repeated up to a maximum of 20 attributes specifications The different change specifications are separated by slashes To continue the earlier example we might specify Choices air train bus car Simulation air train car Scenario gc car 1 5 ttme air train 1 25 You may also provide more than one full scenario for the simulation In this case each scenario is compared to the base case then the scenarios are compared to each other
210. re E Random parameters in utility functions MGC 3 10624315 69007784 4 501 0000 MTTME 1 22068334 1 01138340 1 207 Ry ara a SSessasao4 tNonrandom parameters in utility functions HINCA 02916424 02180170 1 4338 1810 A_AIR 8 30401343 1 72839556 4 804 0000 A_TRAIN 7 44116617 1 45457284 5 116 0000 A_BUS 6 50022714 1 60098502 4 060 0000 Chapter 10 The Random Parameters Logit Model 156 As before the structural parameters do not suggest what the implied parameters will look like For these data the estimated WTP values for the first 10 individuals copied from wtp_i are 8 048 7 11862 6 41581 8 01403 8 31522 5 56074 4 42096 1 58768 2 36362 3 1795 The overall average computed by averaging the 70 values in the matrix is 5 23031 This is in minute 10 8 Applications The preceding sections contain numerous examples of the mixed logit model The applications below show a few of the most basic procedures This is a basic formulation with two random parameters and heterogeneity in the means as a function of household income RPLOGIT Lhs mode Choices air train bus car Rhs gc ttme Rh2 one RPL hinc Fen gc n ttme n Effects ge air Discrete choice and multinomial logit models Start values obtained using MNL model Maximum Likelihood Estimates Dependent variable Choice Number of observations 210 Log likelihood function 199 9766 Info Criterion AIC 1 95216 Fini
211. relevant part of the output would appear as follows Covariance matrix for the model is adjusted for data clustering Sample of 210 observations contained 70 clusters defined by 3 observations fixed number in each cluster Variable Coefficient Standard Error b St Er P Z gt z GC 01578375 00543575 2 904 0037 TTME 09709052 01366784 7 104 0000 A_AIR 5 77635888 74564933 7 747 0000 A_TRAIN 3 92300124 47890812 8 192 0000 A_BUS 3 21073471 48991386 6 554 0000 Chapter 6 NLOGIT Commands and Results 71 6 3 3 Descriptive Statistics for Alternatives You may request a set of descriptive statistics for your model by adding Describe to the model command For each alternative a table is given which lists the nonzero terms in the utility function and the means and standard deviations for the variables that appear in the utility function Values are given for all observations and for the individuals that chose that alternative For the example shown above the following tables would be produced NLOGIT Lhs mode Choices air train bus car Rhs invc invt gc Rh2 one hinc Show Model Describe Descriptive Statistics for Alternative AIR Utility Function 58 0 observs Coefficient All 210 0 obs that chose AIR Name Value Variable Mean Std Dev Mean Std Dev INVC 0461 INVC 85 252 27 409 97 569 31 4733 INVT 0084 INVT T33 710 48 521 124 828 50 288 GC
212. ristics that produce heterogeneity in the variances of the error components The model specification will dictate which parameters are random and which are not how the heteroscedasticity if any is parameterized the distributions of the random terms and how the error components enter the model The probabilities defined above are conditioned on the random terms v and the error components E The unconditional probabilities are obtained by integrating vz and Eim out of the conditional probabilities P Ey P jlv E This is a multiple integral which does not exist in closed form The integral is approximated by sampling nrep draws from the assumed populations and averaging See Bhat 1996 and Revelt and Train 1998 and Greene 2003 for discussion Parameters are estimated by maximizing the simulated log likelihood exp a BX jn iid jm On eXp y he E m jm m im r Ji baie exp a Bi X gi T pe Mae exp y he Em 1 betys Yi oe SDT with respect to B A T Q 6 y where R the number of replications B B Az TON the rth draw on B v the rth multivariate draw for individual i Eim the rth univariate normal draw on the underlying effect for individual i Chapter 10 The Random Parameters Logit Model 124 Note that the multivariate draw v is actually K independent draws The heteroscedasticity is induced first by multiplying by Q then the correlation is induced by multiplyin
213. rogeneity in the mean so what appears there may be of the form 0 B 8 z We have also written the scaling parameter in each form as o however you may also specify heterogeneity in the variances so what appears there may be of the form o o exp h The list above suggests the variety of different distributions that may be used Any distribution may be used for any parameter The normal distribution will be the usual choice However you may wish to restrict a particular coefficient in the model to be positive The lognormal distribution is the obvious choice though there are several other possibilities The normal lognormal exponential Erlang and Weibull distributions all have infinite ranges If you wish to restrict the range of variation of a parameter then the triangular dome or uniform can be used The lognormal distribution has an infinite tail in the positive direction and is anchored at zero while the Erlang and Weibull models as specified have infinite range from B oE v to It is important to note that the means and variances of the distributions are not always simple functions when the parameters are not linear functions of the underlying random variables For all but the Weibull distributions shown above the mean of v is zero which centers the distributions at B For the lognormal and Weibull models the mean depends on the parameters This is also true of the modified distributions shown below This mea
214. roportions data and your data contain zeros or ones certain kinds of observations cannot be distinguished from erroneous individual data and they may be flagged as such For example in a frequency data set the observation 0 0 1 1 0 0 is a valid observation but for individual data it looks like a badly coded observation In order to avoid this kind of ambiguity if you have frequency data containing zeros add Frequencies to your NLOGIT command You may use this in any event to be sure that the data are always recognized correctly If you have proportions data instead you may use Shares to be sure that the data are correctly marked Again this will only be relevant if your data contain zeros and or ones Data are checked for validity and consistency An unrecognizable mixture of the three types will cause an error For example a mixture of frequency and proportions data cannot be properly analyzed For the ranks data an error will occur if the set of ranks is miscoded or incomplete or if ties are detected at any ranks other than the lowest 5 5 Data for the Applications The documentation of the NLOGIT program in the chapters to follow includes numerous applications based on the data set CLOGIT DAT that is distributed with NLOGIT These data are a survey of the transport mode chosen by a sample of 210 travelers between Sydney and Melbourne about 500 miles and other points in nonmetropolitan New South Wales As discussed i
215. s Bx By t 8 z o exp hr v x when T I Q diag 1 On A parameters that enter the heterogeneous means of the distributions of the random parameters B Az the heterogeneous means Mx exp hr heterogeneity in the variances of the distributions of the random parameters parameters in the variance heterogeneity of the random parameters Oik 0 heterogeneous standard deviations in the distributions of the random parameters Oz in a homoscedastic model Zi observed variables that measure the heterogeneity in the means of the random parameters hr observed variables that measure the heterogeneity in the variances of the random parameters Chapter 10 The Random Parameters Logit Model 123 Correlated Parameters with Heterogeneous Means Bx By 8 z 3o Tzs vis when T I and Q diag oj 0 T lower triangular matrix with ones on the diagonal that allows correlation across random parameters when T I Individual Error Components Eim the individual specific underlying random error components m 1 M Eim N 0 1 dim 1 if Eim appears in utility for alternative j and 0 otherwise Omn scale factor for error component m Yim Xp Yn he heterogeneity in the variances of the error components Aim OmYim Standard deviations of random error components Ym parameters in the heteroscedastic variances of the error components he individual choice invariant characte
216. s exp a 02 of Bix J f ae exp a 0 2 Qf ByXX P jlv soy where U j i a 052 Of B x he i ji j 1 J alternatives in individual i s choice set Qj is an alternative specific constant which may be fixed or random ay 0 0 is a vector of nonrandom fixed coefficients 0 0 b is a vector of nonrandom fixed coefficients B is a coefficient vector that is randomly distributed across individuals v enters Bi Z is a set of choice invariant individual characteristics such as age or income f i 4S a vector of M individual and choice varying attributes of choices multiplied by 9 X ji is a vector of L individual and choice varying attributes of choices multiplied by B Chapter 2 Discrete Choice Models 19 The term mixed logit is often used in the literature for this model The choice specific constants a and the elements of Bj are distributed randomly across individuals such that for each random coefficient px any not necessarily all of a or Bix the coefficient on attribute xj k 1 K Piki Of OF Biki Pie OWI Okki or Pjki Oj Or Byer EXP Pje K Wi Okri The vector w which does not include one is a set of choice invariant characteristics that produce individual heterogeneity in the means of the randomly distributed coefficients p is the constant term and 6 is a vector of deep coefficients which produce an indi
217. s You can also keep as variables the fitted probabilities and the branch limb and trunk inclusive values The predicted probabilities are PG b Lr The inclusive values for the branches are repeated for each choice row of data within the branches The inclusive values for the limbs are likewise repeated for every alternative in the limb and similarly for trunks An example appears below The command specifications are Prob name to retain predicted probabilities as a variable Ivb name to retain the branch level inclusive values as a variable Ivl name to retain the limb level inclusive values as a variable Ivt name to retain the trunk level inclusive values as a variable Normally in this setting the unconditional probability PG b r is the one of interest However for some purpose you might want instead the conditional probabilities at the twig level PG b r You can request to have this retained as a variable with Cprob name to retain estimated conditional probabilities Lastly the utility values at the twig level of the tree are UGlb Lr B Xjnir These are the values that you define in your Model specification You may request to retain these for later use with Utility name of the variable If you have not defined a utility function for an alternative the value returned for that row of data is 0 0 not missing 999 Utility values may be further processed like any other variable You may fin
218. s an estimator of B is subject to interpretation The vector B is a draw from a distribution that has an unconditional mean E pilz hr B Az and a conditional mean iF f B p y B x he E p B E z hr dE dB E B ly X Z he hr f f PLY B x he E p B E z hr dE dB What we are computing here are estimates of the means of these distributions In principle these are conditioned on the particular data sets associated with individual i not individual i themselves as such To underscore the point note that the computations would produce the same predictions for two individuals say i and i if they have the same measured data even though they would have different draws from the underlying population v E and v E So the mean computed here is an estimate of the center of this distribution not a formal estimator of B as such We can take this a step further and examine the unconditional and conditional distributions The variance of the unconditional distribution is Var B lz hr TO T for a particular element of B the variance is Var Bial exp hr x t r For the conditional distribution no such expression exists For a particular element of B f Bi PCY 1B x he E p B E z hr dE dB Var B ly X Z he hr aP a cht e aa R Ge e e Se ea f PCy 1B x he E p B E z hr dE d 2 I f B p y B x he E p B E z hr dE dB I f p y B
219. s are computed ITA list IIA is not testable here since it is not imposed Cprob name Conditional and unconditional probabilities are the same Ranks This estimator may not be based on ranks data Scale Data scaling is only for the nested logit model The command builder may also be used for this model by selecting Model Discrete Choice Multinomial Probit HEV RPL The choice set and utility functions for the model are defined on the Main page and the MNP format of the model is selected on the Options page See Figures 11 1 and 11 2 for the setup of the model shown in the application below Chapter 11 The Multinomial Probit Model 163 NLOGIT Multnomial Probit HEV RPL Main Options Output Choice variable MODE v Data type Individual hd Choice names ir Train Bus Ca F Data coded on one line Code i _ m Utility functions r Phs Rh2 Attributes Rhs gt Characteristics Rh2 TTME GC NLOGIT Multnomial Probit HEV RPL Main Options Qutput Model Multinomial Probit v MNP model V Number of points for simulated probabilities E I Specify correlation structure Specify standard deviation pattern I Covariance heterogeneity Variables 2 Optimization Hypothesis Tests Figure 11 2 Options Page of Command Builder for the MNP model Chapter 11 The Multinomial Probit Model 164 11 3 An Application The m
220. s list includes one you are requesting a set of alternative specific constants As such when the Rhs list includes one NLOGIT will create a full set of J 1 choice specific constants One of them must be dropped to avoid what amounts to the dummy variable trap HINT You need not have choice specific dummy variables in your data set The Rh2 setup described here allows you to produce these variables as part of the model specification Chapter 6 NLOGIT Commands and Results 59 The remaining columns of the utility functions in the example above are produced with Rh2 one hinc You should note in addition how the variables are expanded as a set in constructing the utility functions Command Builders You can specify utility functions in this format in any of the command builders as shown in Figure 6 4 The two windows allow you to select variables from the list at the right and assemble the Rhs list at the left or the Rh2 list in the center DISCRETE CHOICE Main Options Output Model type Discrete Choice z Sequential estimation I Conditional model I Use one line setup Attribute labels Utility functions Attributes Interact with ASC GC ONE TTME Eal HINC wi esi Specify utility functions I Box Cox Tree Specification Optimization Hypothesis Tests Run Cancel Figure 6 4 Specifying Utility Functions in the Command Builder 6 2 5 Building the Utility Functions
221. sing the same parameter name For example nothing precludes Model U air car bus train lt ba bc 0 bt gt lt ba bc bcpub bcepub gt ge This forces two of the slope coefficients to equal the alternative specific constants Expanded this specification would be equivalent to Model U air ba ba gc U car bc be gc U bus bepub gc U train bt bepub gc Ff Logs and the Box Cox Transformation Variables may be specified in logarithms This will be useful when you are using aggregate data and you wish to include e g market size in a choice To indicate that you wish to use logs use Log variable name instead of just variable name in the utility definition The syntax Rhs Log x as described above is not available This option may only be used when you are explicitly defining the utility functions Thus the model above might have been NLOGIT Lhs mode Choices air train bus car Model U air ba bcost Log gce U car be beost Log ge U bus bb bcost Log ge U train beost Log gc When a variable appears in more than one utility function you should take logs each time it appears Although this is not mandatory if you do not your model will contain a mix of levels and logs which is probably not what you want Also it will be necessary for you to be aware in your results when you have used this transformation The model results will not contain any indicat
222. ssess how well the model predicts outcomes for observations outside the estimation sample For this feature use the following steps Step 1 Set up the full model for estimation and estimate the model parameters Step 2 Reset the sample to specify the observations for which you wish to simulate the model Step 3 Use the identical NUOGIT command but add the specification Prlist to the command The sample that you specify at Step 2 may contain as many observations as you wish it may be just one individual or it may be an altogether different set of data NOTE The observations in the new sample must be consistent with the specification of the model The usual data checking is done to ensure this WARNING You must not change the specification of the model between Steps 1 and 3 The coefficient vector produced by Step 1 is used for the simulation at Step 3 But it is not possible to check whether the coefficient vector used at Step 3 is actually the correct one for the model command used at Step 3 It will be if your model commands at Steps 1 and 3 are identical The following sequence fits the model in the preceding examples using the first 200 observations 800 data rows then simulates the probabilities for the remaining 10 observations in the full sample SAMPLE 3 1 800 NLOGIT Lhs mode Choices air train bus car Rhs invc invt gc ttme Rh2 one SAMPLE 801 840 NLOGIT Lhs mode Choices air trai
223. sticity averaged over observations Attribute is GC in choice AIR Effects on probabilities of all choices in model Direct Elasticity effect of the attribute Mean St Dev x Choice AIR 8019 3834 Choice TRAIN 3198 3370 Choice BUS 3198 3370 Choice CAR 3198 3370 The difference between the two commands is the use of air for derivatives and air for elasticities The full set of tables one for each alternative is requested with alternative or alternative Note that for this model the elasticities take only two values the own value when j equals m and the cross elasticity when j is not equal to m The fact that the cross elasticities are all the same is one of the undesirable consequences of the ITA property of this model Chapter 9 The Nested Logit Model 103 Chapter 9 The Nested Logit Model 9 1 Introduction The nested logit model is an extension of the multinomial model presented in Chapter 8 The models described here are based on variations of a four level tree structure such as the following ROOT root TRUNKS trunk1 trunk2 LIMBS limb1 limb2 limb3 limb4 he ww cw we chee oi BRANCHES branch1 branch2 branch3 branch4 branch5 branch branch7 branch8 patatas asp Ey ALTS al a2 a3 a4 a5 a6 a7 a8 a9 a10 ali a12 a13 a14 a15 a6 Individuals are assumed to make a choice among NALT J alternatives alts in a choice set The twigs in the tree are the elemental alt
224. structural parameters for the Weibull model which are completely different The Weibull distribution which involves the exponent of B Az T Q v looks quite different from the normal These are the basic MNL estimates with both parameters fixed Variable Coefficient Standard Error b St Er P Z gt z MGC 01578374 00438279 3 601 0003 MTTME 09709052 01043509 9 304 0000 A_AIR 5 77635901 65591873 8 807 0000 A_TRAIN 3 92300113 44199360 8 876 0000 A_BUS 3 21073472 44965283 7 140 0000 This is the same model with two correlated normally distributed random parameters with heterogeneous means There are also two random error components in the model SSeS S Random parameters in utility functions MGC 03170589 01949180 1 627 1038 MTTME 13551247 02907461 4 661 0000 SSS Senos tNonrandom parameters in utility functions A_AIR 10 1292509 1 85832856 5 451 0000 A_TRAIN 8 20598683 1 73422590 4 732 0000 A_BUS 7 19813304 1 91386320 3 761 0002 Chapter 10 The Random Parameters Logit Model 150 Heterogeneity in mean Parameter Variable MGC HIN 450634D 05 00044082 010 9918 MTTM HIN 00078233 00056928 1 374 1694 aia Diagonal values in Cholesky matrix L NsMGC 01138907 02143530 z931 lt 5952 NsMTTME 06637718 07932744 837 4027 San Hes Below diagonal values in L matrix V L Lt MTTM MGC 05922416 09092201 651 5148 Saas Standard deviations of lat
225. summarize the results comparing them to the original base case Chapter 7 Simulating Probabilities in Discrete Choice Models 85 Steps 3 6 may be repeated as many times as desired once a model has been estimated The model is not reestimated the existing model is used to compute the simulation results The simulation produces an output table that compares absolute frequencies and shares for each alternative in the full or a restricted choice set to the base case in which the predicted shares are the means of the sample predictions from the model absent the changes specified in the scenario In addition this feature provides a capability for implementing simulation scenario analysis when one is using mixtures of data for example stated preference and revealed preference This option allows you to combine the two types of data in a simulation An example is shown in the case study below 7 2 Essential Subcommands NLOGIT s models are all built around the specification which indicates the choice set being modeled Choices the full list of alternatives in the model This simulation program is used to compute simulated probabilities assuming that the individuals in the sample being simulated are choosing among some or all of these alternatives The first subcommand for the simulation is Simulation a list of names of alternatives The list of names must be some or all of the names in the Choices list If they are to be all of them
226. t a variable is not influential in the determination of the choices should be tested at the coefficient level As noted in the tables the marginal effects are computed by averaging the individual sample observations An alternative way to compute these is to use the sample means of the data and compute the effects for this one hypothetical observation Request this with Means For the first table above the results would be as follows Derivative times 100 Computed at sample means Attribute is GC in choice AIR Effects on probabilities of all choices in model Direct Derivative effect of the attribute Mean St Dev A Choice AIR 7263 0000 Choice TRAIN 3010 0000 Choice BUS 1434 0000 Choice CAR 2819 0000 Note that the changes are substantial The literature is divided on this computation Current practice seems to favor the first approach Rather than see the partial effects you may want to see elasticities Nim klj T OlogProbly m Ologx kj XAKY Pim x imCklj 1G m Pi XK Bx Notice that this is not a function of Pim The implication is that all the cross elasticities are identical This will be obvious in the results below This aspect of the model is specific to the basic multinomial logit model As will emerge in the chapters to follow the IIA property which produces this result is absent from every other model in NLOGIT You may request elasticities instead of partial effects s
227. te Sample AIC 1 95356 Info Criterion BIC 2 03185 Info Criterion HQIC 1 98438 R2 1 LogL LogL Log L fnen R sqrd RsqAdj Constants only 283 7588 29526 28504 Chi squared 2 167 56429 Prob chi squared gt value 00000 Response data are given as ind choice Number of obs 210 skipped 0 bad obs Notes No coefficients gt P i j 1 J i Constants only gt P i j uses ASCs only N j N if fixed choice set N j total sample frequency for j N total sample frequency These 2 models are simple MNL models R sqrd 1 LogL model logL other RsqAdj 1 nJ nJ nparm 1 R sqrd nJ sum over i choice set sizes Chapter 10 The Random Parameters Logit Model Variable Coefficient Standard Error b St Er P Z gt z H GC 301578374 00438279 3 601 0003 TTME 09709052 01043509 9 304 0000 A_AIR 5 77635901 65591873 8 807 0000 A_TRAIN 3 92300113 44199360 8 876 0000 A_BUS 3 21073472 44965283 7 140 0000 Random Parameters Logit Model Dependent variable MODE Number of observations 210 Log likelihood function 182 9290 Number of parameters 9 Info Criterion AIC 1 82789 Finite Sample AIC 1 83218 Info Criterion BIC 1 97134 Info Criterion HQIC 1 88589 Restricted log likelihood 291 1218 McFadden Pseudo R squared 3716412 Chi squared 216 3857 Degrees of freedom 9 Prob ChiSqd gt value 000
228. test of the IIA assumption The second part of this section considers more conventional hypothesis tests about the coefficients in the model 6 6 1 Testing the Assumption of Independence from Irrelevant Alternatives IIA Hausman and McFadden 1984 have proposed a specification test for this model to test the inherent assumption of the independence from irrelevant alternatives IIA IIA is a consequence of the initial assumption that the stochastic terms in the utility functions are independent and extreme value distributed Discussion may be found in standard texts on qualitative choice modeling such as Hensher Rose and Greene 2005 and Greene 2011 The procedure is first to estimate the model with all choices The alternative specification is the model with a smaller set of choices Thus the model is estimated with this restricted set of alternatives and the same model specification The set of observations is reduced to those in which one of the smaller set of choices is made The test statistic is 1 q b baJ V V b bil where w and r indicate unrestricted and restricted smaller choice set models and V is an estimated variance matrix for the estimates To use NLOGIT to carry out this test it is necessary to estimate both models In the second it is necessary to drop the outcomes indicated This is done with the Ias list specification The list gives the names of the outcomes to be dropped Chapter 6 N
229. that each individual has his or her own choice set For example one such data set might appear as follows Y Q W Nij i 1 0 G11 Wi1 1 3 q2 1 W2 1 3 0 93 1 W3 1 3 i 2 0 1 2 Wi 2 4 0 q2 2 W2 2 4 1 3 2 W3 2 4 0 4 2 W4 2 4 i 3 1 q 3 W1 3 2 0 q2 3 W2 3 2 The model command might be NLOGIT Lhs y nij Rhs q w Notice once again that the command does not contain a definition of the choice set such as Choices list specification For the case of a universal choice set suppose that the data set were instead Y Q W Nij Altij i 1 0 qd Wi 3 1 Air gt 1 2 1 W2 1 3 2 Train 0 3 1 W3 1 3 4 Car i 2 0 i 2 Wi 2 4 1 Air 0 2 2 Wo 2 4 2 Train 1 3 2 W3 2 4 3 Bus 0 a 2 Wa 2 4 4 Car i 3 gt 1 qd 3 W1 3 2 3 Bus 0 2 3 W2 3 2 4 Car The specific choice identifier when it is needed is provided as a third Lhs variable For this case the choice set would have to be defined For example NLOGIT Lhs y nij altij Choices air train bus car Rhs q w Once again in this setting every individual is assumed to choose from a set of four alternatives though the altij variable indicates that some of these choices are unavailable to some individuals Chapter 5 Data Setup for NLOGIT 44 Do note that if you are not defining a universal choice set NLOGIT simply uses the largest number of choices for any individual in the sample to determine J As such
230. then you may use Simulation or just Simulation NOTE Simulation on a subset of alternatives in the full choice set is done by analyzing the full set of data while in process pretending simulating that alternatives not in the simulation list are not available to these individuals even if they are physically in the data set and actually available Note this is just for the purposes of the simulation You must not change the sample settings in any way to produce this effect yourself It is handled completely internally by this program simply by using a set of switches on for included off for excluded for the choice set while numerical results are computed The second specification you will provide is the name of the attribute that is being set or changed and the names of the alternatives in which this attribute is changing This is the scenario The base case for a single changing attribute is Scenario attribute name list of alternatives whose attribute levels will change action magnitude of action Chapter 7 Simulating Probabilities in Discrete Choice Models 86 If you wish to include in the scenario all the alternatives that are defined in the simulation simply use the wildcard character as the list Note that this all items in list refers back to your Simulation list not to the Choices list The actions in the scenario specification are as follows specific value to fo
231. this point Expected to find a list of names in parens in IVSET IVSET list Unidentified name appears in list You have given a spec for an IV parm that is fixed at 1 You have specified an IV parameter more than once Chapter 12 Diagnostics and Error Messages 173 1095 Count variable lt nnnnnn gt at row lt nnnnnn gt equals lt nnnn gt The peculiar value for the count variable has thrown off the counter that keeps track of where the estimator is in the data set 1096 Choice variable lt AAAAAAAA gt at row lt nnnnn gt Choice lt nnnnn gt The most likely cause is a coding error Check for bad data 1097 Obs lt nnnnnn gt Choice set contains lt nnnn gt lt nnnn gt times The choice variable for individual data has more than one 1 0 in it NLOGIT cannot determine which alternative is chosen 1098 Obs lt nnnnnn gt alt lt nnn gt is not an integer nor a proportion 1099 Obs lt nnnnnn gt responses should sum to 1 0 Sum is lt xxxxxx gt 1100 Cannot classify obs lt nnnnnn gt as IND PROPs or FREQs Your data appear to be a mix of individual and frequency data This occurs when an individual s Lhs variable data include zeros It then becomes difficult to determine what kind of data you have You can settle the question by including Frequencies in your command if that is appropriate 1101 of parms in lt list gt greater than choices in U list 1102 RANK data can o
232. tions that were not in the sample Observations which contain missing data are bypassed as usual You can also compute a vector of probabilities for a specific observation for example the sample means by using the matrix b_logit The following suggests how this might be done using the group means NAMELIST x the Rhs variables MATRIX xb Mean x MATRIX pvec b_logit xb pvec Expn pvec pvec lt 1 pvec gt pvec Chapter 5 Data Setup for NLOGIT 41 Chapter 5 Data Setup for NLOGIT 5 1 Introduction In general the data for the models described in Parts III and IV will be arranged in a format that is set up to work well with the specific NLOGIT estimators In almost all cases the data used for all models that you fit with NLOGIT will be set up as if they were a panel That is each individual observation will have a set of observations with one line of data for each choice in the choice set Thus in the analogy to a panel the group is a person and the group size would be the number of choices You will use this arrangement in nearly all cases This chapter will explain the various aspects of setting up the data for the NLOGIT models 5 2 Basic Data Setup for NLOGIT In the base case the data are arranged as follows where we use a specific set of values for the problem to illustrate Suppose you observe 25 individuals Each individual in the sample faces three choices and there are two attribut
233. tive The observation resembles a group in a panel data set The command just given assumes that every individual in the sample chooses from the same size choice set J The choice sets may have different numbers of choices in which case the command is changed to Lhs dependent variable choice set size variable The second Lhs variable is structured exactly the same as a Pds variable for a panel data estimator In the second form of the model command the utility functions are specified directly symbolically The Rhs and Rh2 specifications can be replaced with Model specification of the utility functions This is discussed in Chapter 6 Chapter 3 Model and Command Summary for Discrete Choice Models 24 The CLOGIT command is the same as DISCRETE CHOICE It is also the same as NLOGIT when the only information given in the command is that specified above that is when none of the specifications that invoke the model extensions that are described in the sections to follow are provided 3 5 NLOGIT Extensions of Conditional Logit 3 5 1 Nested The nested logit model is the default form of the NUOGIT command Request the nested logit model with NLOGIT Tree specification of the tree structure Choices the names of the J alternatives Rhs list of choice specific attributes 3 5 2 Random Parameters Logit The random parameters logit model mixed logit model is requested by specifying a conditional logit model and
234. ts to signify interaction with choice specific constants Thus lt b1 b2 bL gt indicates L interactions with choice specific dummy variables L may be any number up to the number of alternatives Use a zero in any location in which the variable does not appear in the corresponding equation For example Model U air ba bcost gc U car bc bcost ge U bus beost gc U train bt beost gc could be specified as Model U air car bus train lt ba bc 0 bt gt bcost gc NOTE Within a lt gt construction the correspondence between positions in the list is with the U list list not with the original Choices list Note the considerable savings in notation The same device may also be used in interactions with attributes For example s Model U air ba beprv ge U car be beprv ge U bus bepub gc U train bt bepub ge Chapter 6 NLOGIT Commands and Results 63 There are two cost coefficients but the variable gc is common This entire model can be collapsed into the single specification Model U air car bus train lt ba bc 0 bt gt lt beprv beprv bepub bcpub gt ge Parameters inside the brackets need not all be different if you wish to impose equality constraints Equality Constraints There is no requirement that parameters be unique across any specification Equality constraints may be imposed anywhere in the model just by u
235. ultinomial probit model based on the clogit data is estimated with the command MNPROBIT _ Lhs mode Choices air train bus car Rhs gc ttme Rh2 one hinc Effects gc air Pts 20 This is the model that was fit as an MNL model in Chapter 8 We have now relaxed the equal variances assumption and replaced the extreme value distribution with a multivariate normal distribution The probabilities are computed with 20 replications which is fairly small we do this for purposes of a simple illustration Results are shown below The MNL model is fit first to obtain the starting values for the iterations The results for the MNP model are given next The two sets of results are merged in the display below Discrete choice multinomial logit model Dependent variable MODE Log likelihood function 189 5252 Info Criterion AIC 1 88119 Finite Sample AIC 1 88460 Info Criterion BIC 2 00870 Info Criterion HQIC 1 93274 R2 1 LogL LogL Log L fnen R sqrd RsqAdj Constants only 283 7588 33209 31802 Chi squared 5 188 46723 Prob chi squared gt value 00000 Response data are given as ind choice Number of obs 210 skipped 0 bad obs Multinomial Probit Model Log likelihood function 189 8452 Info Criterion AIC 1 93186 Finite Sample AIC 1 94070 Info Criterion BIC 2 13906 Info Criterion HQIC 2 01562 Restricted log likelihood 291 1218 McFadden Pseudo R squared 347
236. variance But there is an indirect method If you add Parameters to your RPLOGIT command then NLOGIT creates two matrices from the model results The matrix beta_i contains for each random parameter column and each individual row an estimate of 6 EIB lall information about individual i Chapter 10 The Random Parameters Logit Model 131 The information about individual i includes their choices so this is not quite the same as the estimator that we are using above E Bj lz But since the average of conditional means gives the unconditional mean the average of the estimates contained in beta_i provides an estimator of the unconditional population mean that we are estimating above Figure 10 1 below shows the first 10 rows of this 70x1 matrix as created by the model command that generated the Weibull results above ii Matrix BETA I O X Matrix soBETA I E BR 70 1 Cel 0 0274657 70 1 Cel 0 0101905 0 0274657 0 0101905 0 0158211 0 00785383 0 024132 0 0120969 0 0271134 0 0108668 0 0168981 0 00880909 0 015507 0 00729046 0 015860 0 00725378 0 0180881 0 0120018 0 031572 0 0136248 0 0139569 0 00803417 Figure 10 1 Estimated Conditional Means and Standard Deviations We can estimate the overall mean by averaging the elements in beta_i This produces MATRIX ebi list 1 70 beta_i 1 1 01984 which is the now familiar result Estimating the population variance is a bit more complicated
237. ve Command Conditional Logit CLOGIT NLOGIT Nested Logit NLOGIT NLOGIT Tree Random Parameters Logit RPLOGIT NLOGIT RPL Multinomial Probit MNPROBIT NLOGIT MNP The model is viewed as a random utility model in which the utility functions are functions of attributes x x The model is then fit to describe the choice among J alternatives C Cy This may be a very simple model such as the basic multinomial logit model MNL of Chapter 8 or as complicated as a four level nested logit model as described in Chapter 9 In any event the model is ultimately viewed in terms of these attributes and choices If desired Reset the sample to any desired setup that is consistent with the model This may be all or a subset of the data used to fit the model or a set of individuals that were not used in fitting the model or any mixture of the two Specify which of the choices possibly but not necessarily all are to be used as the choice set for the simulation The simulation is then produced to predict choice among this possibly reduced set of choices Probabilities for the full choice set are reallocated but not necessarily proportionally This would only occur in the MNL model which satisfies IIA Specify how the attributes that enter the utility functions will change for example that a particular price is to rise by 25 Simulate the model by computing the probabilities and predicting the outcomes for the specified sample and
238. ve specific constants are not shown The values for the random parameters are B and o For the normally distributed variables these are the means and standard deviations For the other distributions they are only the structural parameters To see the similarity however note for the coefficient on mgc in the Rayleigh model exp 3 3585415 is about 0 035 which resembles the value for the normal distribution Accounting for o would likely bring them yet closer Sapa tMultinomial logit with nonrandom parameters GC 01578374 00438279 3 601 0003 TIM 09709052 01043509 9 304 0000 af i a an E oe Normal Random parameters in utility functions GC 02179446 00691475 34 192 0016 TIM 14140119 01958762 7 219 0000 Derived standard deviations of parameter distributions NsMGC 00867259 01372168 632 5274 NsMTTME 07424180 01489564 4 984 0000 at Sasa ES Weibull Random parameters in utility functions GC 03546490 02059812 Lar e2 0851 TTME 23934417 03347361 7 150 0000 Derived standard deviations of parameter distributions WsMGC 00683164 00915062 747 4553 WsMTTME 05604114 01237808 4 527 0000 a a Rayleigh Random parameters in utility functions MGC 3 39385419 1 36800576 2 r452 0142 MTTME 1 26324106 21598047 5 849 0000 Derived standard deviations of parameter distributions RsMGC 33488248 89249718 375 T075 RsMTTME 47230483 10945909 4 315 0000 Chapter 10 The
239. vidual specific mean The random term vj is normally distributed or distributed with some other distribution with mean 0 and standard deviation 1 so is the standard deviation of the marginal distribution of pj The vjxis are individual and choice specific unobserved random disturbances the source of the heterogeneity Thus as stated above in the population 2 oi Or Bizi Normal or Lognormal pj k Wi Oj Other distributions may be specified For the full vector of K random coefficients in the model we may write pi p Aw Tv where T is a diagonal matrix which contains ox on its diagonal A nondiagonal T allows the random parameters to be correlated Then the full covariance matrix of the random coefficients is amp IT The standard case of uncorrelated coefficients has diag o 02 0 If the coefficients are freely correlated I is a full unrestricted lower triangular matrix and X will have nonzero off diagonal elements An additional level of flexibility is obtained by allowing the distributions of the random parameters to be heteroscedastic Ok On x exp y x hj This is now built into the model by specifying pi p Aw T Q v where Q diag on and now is a lower triangular matrix of constants with ones on the diagonal Finally autocorrelation can also be incorporated by allowing the random components of the random parameters to obey an autoregressive process Vkit Tki Vkit 1
240. well as the ancillary tools including MATRIX etc 3 3 Basic Discrete Choice Models The binomial probit and logit models and the ordered probit and logit models are the primary model frameworks for single equation single decision discrete choice models The ordered choice and the bivariate and multivariate probit models are multivariate extensions of the simple probit model There are five binary choice models probit logit complementary log log Gompertz and Burr The ones that interest us here are the binary probit and logit models The probit model is requested with PROBIT Lhs dependent variable Rhs independent variables The binary logit model may be invoked with BLOGIT Lhs dependent variable Rhs independent variables In earlier versions you would use the LOGIT command which is still useable LOGIT is the same as BLOGIT when the data on the dependent variable are either binary zeros and ones or proportions strictly between zero and one Chapter 3 Model and Command Summary for Discrete Choice Models 23 3 4 Multinomial Logit Models The multinomial logit model is an early restrictive version of the conditional logit model which itself is the gateway model to the main model extensions described in Section 3 5 3 4 1 Multinomial Logit The multinomial logit model is invoked with MLOGIT Lhs dependent variable Rhs independent variables Data for the MLOGIT model consist of an intege
241. y functions A_AIR 15 2077878 5 00957004 3 036 0024 A_TRAIN 12 7374035 4 54631279 2 802 0051 A_BUS 11 4866808 4 49644235 2 4559 0106 Heterogeneity in mean Parameter Variable GC HI 00048503 00052537 923 23959 TIME HIN 00098231 00095140 1 032 3018 Saas Diagonal values in Cholesky matrix L NsGC 01920669 02520301 762 4460 NsTIME 04635102 04963601 934 3504 SRSA ad Below diagonal values in L matrix V L Lt TIME GC 14938411 06697675 2 230 0257 DERAS Standard deviations of latent random effects SigmaE01 1 47749388 1 42144790 1 039 2986 SigmaE02 1 65809550 1 69694056 977 3285 a Standard deviations of parameter distributions sdGC 01920669 02520301 162 4460 sdTIME 15640981 06299625 2 483 0130 Random Effects Logit Model Appearance of Latent Random Effects in Utilities Alternative E01 E02 AIR TRAIN BUS CAR 141 Chapter 10 The Random Parameters Logit Model 142 Parameter Matrix for Heterogeneity in Means Matrix Delta has 2 rows and 1 columns HINC GC 00049 TTME 00098 Correlation Matrix for Random Parameters GC TTME GC 1 00000 95508 TTME 95508 1 00000 Covariance Matrix for Random Parameters GC TIME GC 00037 00287 TTME 00287 02446 Cholesky Matrix for Random Parameters GC TIME GC 01921 0000000D 00 TTME 14938 04635 Note two important points about th
242. y of Estimation Results Show displays model specification and tree structure Describe lists descriptive statistics for attributes by alternative Odds includes odds ratios in estimation results Used only by BLOGIT Crosstab includes crosstabulation of predicted and actual outcomes Table name adds model results to stored tables Marginal Effects Effects specification displays estimated marginal effects Used by NLOGIT Marginal Effects displays estimated marginal effects Used by PROBIT BLOGIT BVPROBIT MVPROBIT OLOGIT OPROBIT Means computes marginal effects using data means Uses average partial effects if this is not specified Pwt uses probability weights to compute average partial effects Hypothesis Testing Wald specification computes Wald test statistic for specified linear restrictions Test specification same as Wald specification IAS list of choices used with CLOGIT to test ITA assumption Chapter 3 Model and Command Summary for Discrete Choice Models 28 Optimization Iterations Controls Alg algorithm specifies optimization method Maxit value specifies maximum iterations Tlg value tolerance for convergence on gradient Tlb value tolerance for convergence on change in parameters Tlg value tolerance for convergence on change in function Set keeps settings of tolerance values Output value displays technical output during iterations Starting Values
243. y the terms and conditions of this agreement Copyright Trademark and Intellectual Property This software product is copyrighted by and all rights are reserved by Econometric Software Inc No part of this software product either the software or the documentation may be reproduced distributed downloaded stored in a retrieval system transmitted in any form or by any means sold or transferred without prior written permission of Econometric Software You may not modify adapt translate or change the software product You may not reverse engineer decompile dissemble or otherwise attempt to discover the source code of the software product LIMDEP and NLOGIT are trademarks of Econometric Software Inc The software product is licensed not sold Your possession installation and use of the software product does not transfer to you any title and intellectual property rights nor does this license grant you any rights in connection with software product trademarks Use of the Software Product You have only the non exclusive right to use this software product A single user license is registered to one specific individual and is not intended for access by multiple users on one machine or for installation on a network or in a computer laboratory For a single user license only the registered single user may install the software on a primary stand alone computer and one home or portable secondary computer for his or her exclusive use H

NLOGIT Student User`s Manual - NYU Stern School of Business

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