nl - Stata
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1. Title stata com nl Nonlinear least squares estimation Syntax Menu Description Options Remarks and examples Stored results Methods and formulas Acknowledgments References Also see Syntax Interactive version nl depvar lt sexp gt if in weight options Programmed substitutable expression version nl sexp_prog depvar varlist lif in weight le options Function evaluator program version nl func_prog depvar varlist lif in weight 5 parameters namelist nparameters options where depvar is the dependent variable lt sexp gt is a substitutable expression sexp_prog is a substitutable expression program and func_prog is a function evaluator program 2 nl Nonlinear least squares estimation options Description Model variables varlist variables in model initial Cinitial_values parameters namelist nparameters sexp options func_options Model 2 1nlsq noconstant hasconstant name SE Robust vce vcetype Reporting level leave title string title2 string display_options Optimization optimization_options eps delta coeflegend initial values for parameters parameters in model function evaluator program version only number of parameters in model function evaluator program version only options for substitutable expression program options for function evaluator program use log least s2. See R nlcom and U 13 5 Accessing coefficients and standard errors for more information K nl s output closely mimics that of regress see R regress for more information The R sums of squares and similar statistics are calculated in the same way that regress calculates them If no constant term is specified the usual caveats apply to the interpretation of the R statistic see the comments and references in Goldstein 1992 Unlike regress nl does not report a model F statistic because a test of the joint significance of all the parameters except the constant term may not be relevant in a nonlinear model Substitutable expression programs If you fit the same model often or if you want to write an estimator that will operate on whatever variables you specify then you will want to write a substitutable expression program That program will return a macro containing a substitutable expression that nl can then evaluate and it may optionally calculate initial values as well The name of the program must begin with the letters n1 To illustrate suppose that you use the CES production function often in your work Instead of typing in the formula each time you can write a program like this program nlces rclass version 13 syntax varlist min 3 max 3 if local logout word 1 of varlist local capital word 2 of varlist local labor word 3 of varlist Initial value for bO given delta 0 5 and rho 1 t
3. gom4 yi Bo Bi exp exp b2 xi 6s amp gom3 Yi Bi exp exp b2 x 6s amp Lognormal errors A nonlinear model with errors that are independent and identically distributed normal may be written as yi f x b ui u N 0 07 3 fori 1 n If the y are thought to have a k shifted lognormal instead of a normal distribution that is In y k N 77 and the systematic part f x 3 of the original model is still thought appropriate for y the model becomes In y k i vi In f xi 8 k vi vi N 0 7 4 This model is fit if Inlsq k is specified If model 4 is correct the variance of y k is proportional to f x 3 kY Probably the most common case is k 0 sometimes called proportional errors because the standard error of y is proportional to its expectation f x 8 Assuming that the value of k is known 4 is just another nonlinear model in and it may be fit as usual However we may wish to compare the fit of 3 with that of 4 using the residual sum of squares RSS or the deviance D D 2 x log likelihood from each model To do so we must allow for the change in scale introduced by the log transformation Assuming then the y to be normally distributed Atkinson 1985 85 87 184 by considering the Jacobian O In y amp Oy showed that multiplying both sides of 4 by the geometric mean of yi k y gives residuals
4. labor 1 rho obs 100 Iteration 0 residual SS 29 38631 Iteration 1 residual SS 29 36637 Iteration 2 residual SS 29 36583 Iteration 3 residual SS 29 36581 Iteration 4 residual SS 29 36581 Iteration 5 residual SS 29 36581 Iteration 6 residual SS 29 36581 Iteration 7 residual SS 29 36581 Source ss df MS Number of obs 100 Model 91 1449924 2 45 5724962 R squared 0 7563 Residual 29 3658055 97 302740263 Adj R squared 0 7513 Root MSE 5502184 Total 120 510798 99 1 21728079 Res dev 161 2538 lnoutput Coef Std Err t P gt ltl 95 Conf Interval b0 3 792158 099682 38 04 0 000 3 594316 3 989999 rho 1 386993 472584 2 93 0 004 4490443 2 324941 delta 4823616 0519791 9 28 0 000 3791975 5855258 Parameter bO taken as constant term in model amp ANOVA table nl will attempt to find a constant term in the model and if one is found mention it at the bottom of the output nl found bO to be a constant because the partial derivative InQ ObO has a coefficient of variation less than eps in the estimation sample The elasticity of substitution for the CES production function is 0 1 1 p and having fit the model we can use nlcom to estimate it nlcom 1 1 _b rho nl_i 1 1 _b rho lnoutput Coef Std Err z P gt lz 95 Conf Interval _nl_i 4189372 0829424 5 05 0 000 256373 5815014 8 nl Nonlinear least squares estimation
5. its initial value another parameter name its initial value and so on For example to initialize alpha to 1 23 and delta to 4 57 you would type nl initial alpha 1 23 delta 4 57 Initial values declared using this option override any that are declared within substitutable expres sions If you specify a parameter that does not appear in your model nl exits with error code 480 If you specify a matrix the values must be in the same order that the parameters are declared in your model nl ignores the row and column names of the matrix parameters namelist specifies the names of the parameters in the model The names of the parameters must adhere to the naming conventions of Stata s variables see U 11 3 Naming conventions If you specify both parameters and nparameters the number of names in the former must match the number specified in the latter if not nl issues an error message with return code 198 nparameters specifies the number of parameters in the model If you do not specify names with the parameters option nl names them b1 b2 b If you specify both parameters and nparameters the number of names in the former must match the number specified in the latter if not nl issues an error message with return code 198 sexp_options refer to any options allowed by your sexp_prog func_options refer to any options allowed by your func_prog Model 2 1nlsq fits the model by using log least sq
6. 08 Oxford Numerical Algorithms Group 1990 Nonlinear Estimation New York Springer Royston P 1992 sg7 Centile estimation command Stata Technical Bulletin 8 12 15 Reprinted in Stata Technical Bulletin Reprints vol 2 pp 122 125 College Station TX Stata Press 1993 sg1 4 Standard nonlinear curve fits Stata Technical Bulletin 11 17 Reprinted in Stata Technical Bulletin Reprints vol 2 p 121 College Station TX Stata Press 20 nl Nonlinear least squares estimation Also see R nl postestimation Postestimation tools for nl R gmm Generalized method of moments estimation R ml Maximum likelihood estimation R mlexp Maximum likelihood estimation of user specified expressions R nleom Nonlinear combinations of estimators R nlsur Estimation of nonlinear systems of equations R regress Linear regression SVY svy estimation Estimation commands for survey data U 20 Estimation and postestimation commands
7. Taking a second order Taylor series expansion centered at Bg yields SSR B SSR Bo 2 Bp B Bo 5 8 Bo H Bo 8 Bo 8 where g Bo denotes the k x 1 gradient of SSR 3 evaluated at Bg and H G denotes the k x k Hessian of SSR evaluated at Bg Letting X denote the N x k matrix of derivatives of f x 3 with respect to 3 the gradient g 3 is g 8 2X Du 9 X and u are obviously functions of 6 though for notational simplicity that dependence is not shown explicitly The m n element of the Hessian can be written as i l OBmOBn where di is the ith diagonal element of D As discussed in Davidson and MacKinnon 2004 chap 6 the first term inside the brackets of 10 has expectation zero so the Hessian can be approximated as H B 2X DX 11 Differentiating the Taylor series expansion of SSR 3 shown in 8 yields the first order condition for a minimum g 8o H 60 8 Bo 9 which suggests the iterative procedure Bj pe aH B g B 12 where a is a step size parameter chosen at each iteration to improve convergence Using 9 and 11 we can write 12 as B41 b a X DX X Du 13 where X and u are evaluated at 8j Apart from the scalar a the second term on the right hand side of 13 can be computed via a weighted regression of the columns of X on the errors nl computes the derivatives numerically and then calls regress At each iteration a is set to one and a candidate
8. choose between the two versions of the logistic model 6 whereas for the exponential model 7 the fit using 1lnlsq 0 is much better a deviance difference of 7 28 The reciprocal transformation has introduced heteroskedasticity into y which is countered by the proportional errors property of the lognormal distribution implicit in 1nlsq 0 The deviances are not comparable between the logistic and exponential models because the change of scale has not been allowed for although in principle it could be Other uses Even if you are fitting linear regression models you may find that nl can save you some typing Because you specify the parameters of your model explicitly you can impose constraints on them directly gt Example 2 In example 2 of R cnsreg we showed how to fit the model mpg o Piprice foweight S3displ S4gear_ratio sforeign Belength u subject to the constraints b b2 B3 Bo Ba Bs Bo 20 12 nl Nonlinear least squares estimation An alternative way is to use nl use http www stata press com data ri3 auto clear 1978 Automobile Data nl mpg b0 bi price bi weight bi displ gt b0 20 gear_ratio b0 20 foreign bi length obs 74 Iteration 0 residual SS 1578 522 Iteration 1 residual SS 1578 522 Source ss df MS Number of obs 74 Model 34429 4777 2 17214 7389 R squared 0 9562 Residual 1578 52226 72 21 9239203 Adj R squared 0 95
9. estimation options leave leaves behind after estimation a set of new variables with the same names as the estimated parameters containing the derivatives of E y with respect to the parameters If the dataset contains an existing variable with the same name as a parameter then using leave causes n1 to issue an error message with return code 110 leave may not be specified with vce cluster clustvar or the svy prefix nl Nonlinear least squares estimation 5 title string specifies an optional title that will be displayed just above the table of parameter estimates title2 string specifies an optional subtitle that will be displayed between the title specified in title and the table of parameter estimates If title2 is specified but title is not title2 has the same effect as title display_options cformat fmt pformat 4 fmt sformat fmt and nolstretch see R es timation options Optimization optimization_options iterate no log trace iterate specifies the maximum number of iterations log nolog specifies whether to show the iteration log and trace specifies that the iteration log should include the current parameter vector These options are seldom used eps specifies the convergence criterion for successive parameter estimates and for the residual sum of squares The default is eps 1e 5 delta specifies the relative change in a parameter to be used in computing the numeric deriv
10. expression version of n1 with a function evaluator program or vice versa Stata issues an error message Verify that you are using the syntax appropriate for the program you have 14 nl Nonlinear least squares estimation General comments on fitting nonlinear models Achieving convergence is often problematic For example a unique minimum of the sum of squares function may not exist Much literature exists on different algorithms that have been used on strategies for obtaining good initial parameter values and on tricks for parameterizing the model to make its behavior as linear like as possible Selected references are Kennedy and Gentle 1980 chap 10 for computational matters and Ross 1990 and Ratkowsky 1983 for all three aspects Ratkowsky s book is particularly clear and approachable with useful discussion on the meaning and practical implications of intrinsic and parameter effects nonlinearity An excellent text on nonlinear estimation is Gallant 1987 Also see Davidson and MacKinnon 1993 and 2004 To enhance the success of nl pay attention to the form of the model fit along the lines of Ratkowsky and Ross For example Ratkowsky 1983 49 59 analyzes three possible three parameter yield density models for plant growth a rs Bz 1 E y 4 a Bri yz a paf All three models give similar fits However he shows that the second formulation is dramatically more linear like than the other two and therefo
11. it more convenient to pass a column vector In those cases you could type Matrix myvals 0 1 0 5 nl ces2 lnoutput capital labor nparameters 3 initial myvals 16 nl Nonlinear least squares estimation In summary a function evaluator program receives a list of variables the first of which is the dependent variable that you are to replace with the values of your nonlinear function Additionally it must accept an if exp as well as an option named at that will contain the vector of parameters at which n1 wants the function evaluated You are then free to do whatever is necessary to evaluate your function and replace the dependent variable If you wish to use weights your function evaluator program s syntax statement must accept them If your program consists only of for example generate statements you need not do anything with the weights passed to your program However if in calculating the nonlinear function you use commands such as summarize or regress then you will want to use the weights with those commands As with substitutable expression programs nl will pass to it any options specified that n1 does not accept providing you with a way to pass more information to your function Q Technical note Before version 9 of Stata the nl command used a different syntax which required you to write an nlfcn program and it did not have a syntax for interactive use other than the seven functions that were built in Th
12. number of iterations e converge 1 if converged 0 otherwise Macros e cmd nl e cmdline command as typed e depvar name of dependent variable e wtype weight type e wexp weight expression e title title in estimation output e title_2 secondary title in estimation output e clustvar name of cluster variable e hac_kernel1 HAC kernel e hac_lag HAC lag e vce vcetype specified in vce e vcetype title used to label Std Err e type 1 interactively entered expression 2 substitutable expression program 3 function evaluator program e sexp substitutable expression e params names of parameters e funcprog function evaluator program e rhs contents of variables e properties bv e predict program used to implement predict e marginsnotok predictions disallowed by margins Matrices e b coefficient vector e init initial values vector e V variance covariance matrix of the estimators Functions marks estimation sample 18 nl Nonlinear least squares estimation Methods and formulas The derivation here is based on Davidson and MacKinnon 2004 chap 6 Let denote the k x 1 vector of parameters and write the regression function using matrix notation as y f x 3 u so that the objective function can be written as ssR 8 y f x B D y f x B The D matrix contains the weights and is defined in R regress if no weights are specified then D is the N x N identity matrix
13. on the same scale as those of y The geometric mean is given by y e 5 ln y k which is a constant for a given dataset The residual deviance for 3 and for 4 may be expressed as D A 1 In 27 bn 5 nl Nonlinear least squares estimation 11 where B is the maximum likelihood estimate MLE of 3 for each model and nG is the RSS from 3 or that from 4 multiplied by 7 Because 3 and 4 are models with different error structures but the same functional form the arithmetic difference in their RSS or deviances is not easily tested for statistical significance However if the deviance difference is large gt 4 say we would naturally prefer the model with the smaller deviance Of course the residuals for each model should be examined for departures from assumptions nonconstant variance nonnormality serial correlations etc in the usual way Alternatively consider modeling E y 1 C Ae 6 E 1 yi E y C Ae 7 where C A and B are parameters to be estimated Using the data y x 0 04 5 0 06 12 0 08 25 0 1 35 0 15 42 0 2 48 0 25 60 0 3 75 and 0 5 120 Danuso 1991 fitting the models yields Model C A B RSS Deviance 6 1 781 25 74 0 03926 0 001640 51 95 6 with 1Inlsq 0 1 799 25 45 0 04051 0 001431 53 18 7 1 781 25 74 0 03926 8 197 24 70 7 with 1nlsq 0 1 799 27 45 0 04051 3 651 17 42 There is little to
14. to the general form for combining weighted matrices to form the variance estimate There are three kernels available for n1 nwest gallant anderson specifies the number of lags If is not specified N 2 is assumed vce hac kernel is not allowed if weights are specified vce hc2 and vce hc3 specify alternative bias corrections for the robust variance calculation vce hc2 and vce hc3 may not be specified with the svy prefix By default vce robust uses gt n n k u as an estimate of the variance of the jth observation where uy is the calculated residual and n n k is included to improve the overall estimate s small sample properties vce hc2 instead uses u 1 h as the observation s variance estimate where hjj is the jth diagonal element of the hat projection matrix This produces an unbiased estimate of the covariance matrix if the model is homoskedastic vce hc2 tends to produce slightly more conservative confidence intervals than vce robust vce hc3 uses u 1 hjj as suggested by Davidson and MacKinnon 1993 and 2004 who report that this often produces better results when the model is heteroskedastic vce hc3 produces confidence intervals that tend to be even more conservative See in particular Davidson and MacKinnon 2004 239 who advocate the use of vce hc2 or vce hc3 instead of the plain robust estimator for nonlinear least squares Reporting level see R
15. value 8 is computed by 13 If SSR G54 lt SSR G then B 4 and the iteration is complete Otherwise is halved a new B 41 is calculated and the process is repeated Convergence is declared when a 8j41 m lt 8jm 7 for all m 1 k nl uses T 1078 and by default 1075 though you can specify an alternative value of with the eps option nl Nonlinear least squares estimation 19 As derived for example in Davidson and MacKinnon 2004 chap 6 an expedient way to obtain the covariance matrix is to compute u and the columns of X at the final estimate and then regress that u on X The covariance matrix of the estimated parameters of that regression serves nN as an estimate of Var If that regression employs a robust covariance matrix estimator then the covariance matrix for the parameters of the nonlinear regression will also be robust All other statistics are calculated analogously to those in linear regression except that the nonlinear function f x 8 plays the role of the linear function x3 See R regress This command supports estimation with survey data For details on VCEs with survey data see SVY variance estimation Acknowledgments The original version of nl was written by Patrick Royston 1992 of the MRC Clinical Trials Unit London and coauthor of the Stata Press book Flexible Parametric Survival Analysis Using Stata Beyond the Cox Model Francesco Danuso s m
16. 49 Root MSE 4 682299 Total 36008 74 486 594595 Res dev 436 4562 mpg Coef Std Err t P gt t 95 Conf Interval b0 26 52229 1 375178 19 29 0 000 23 78092 29 26365 b1 000923 0001534 6 02 0 000 0012288 0006172 The point estimates and standard errors for 39 and 6 are identical to those reported in example 2 of R cnsreg To get the estimate for 64 we can use nlcom nlcom _b b0 20 _nl_i _b b0 20 mpg Coef Std Err z P gt iz 95 Conf Interval _nl_i 1 326114 0687589 19 29 0 000 1 191349 1 460879 The advantage to using nl is that we do not need to use the constraint command six times lt nl is also a useful tool when doing exploratory data analysis For example you may want to run a regression of y on a function of x though you have not decided whether to use sqrt x or In x You can use n1 to run both regressions without having first to generate two new variables nl y b0 b1 1n x nl y b0 b1 sqrt x Poi 2008 shows the advantages of using n1 when marginal effects of transformed variables are desired as well Weights Weights are specified in the usual way analytic and frequency weights as well as iweights are supported see U 20 23 Weighted estimation Use of analytic weights implies that the y have different variances Therefore model 3 may be rewritten as Yi f Xi B ui ui N 0 07 wi 3a where w are positive weights assumed to be know
17. U 17 Ado files a Sometimes you may want to pass additional options to the substitutable expression program You can modify the syntax statement of your program to accept whatever options you wish Then when you call nl with the syntax nl func_prog varlist options any options that are not recognized by n1 see the table of options at the beginning of this entry are passed on to your function evaluator program The only other restriction is that your program cannot accept an option named at because nl uses that option with function evaluator programs 10 nl Nonlinear least squares estimation Built in functions Some functions are used so often that nl has them built in so that you do not need to write them yourself nl automatically chooses initial values for the parameters though you can use the initial option to override them Three alternatives are provided for exponential regression with one asymptote exp3 Yi Pot b1b3 tea exp2 Yi PBs i exp2a Yi b 1 BS i For instance typing nl exp3 ras dvl fits the three parameter exponential model parameters 6o 61 and 62 using y ras and x dvl Two alternatives are provided for the logistic function symmetric sigmoid shape not to be confused with logistic regression log4 Yi bo Bj exp Balti B H i log3 yi b i exp b2 zi 8s Finally two alternatives are provided for the Gompertz function asymmetric sigmoid shape
18. a tives The derivative for parameter 3 is computed as f X 31 G2 8 d Bi4i F X G1 b2 Bi Bi41 d where d is 8i The default is delta 4e 7 The following options are available with n1 but are not shown in the dialog box coeflegend see R estimation options Remarks and examples stata com Remarks are presented under the following headings Substitutable expressions Substitutable expression programs Built in functions Lognormal errors Other uses Weights Potential errors General comments on fitting nonlinear models Function evaluator programs nl fits an arbitrary nonlinear function by least squares The interactive version allows you to enter the function directly on the command line or dialog box using substitutable expressions You can write a substitutable expression program for functions that you fit frequently to save yourself time Finally function evaluator programs give you the most flexibility in defining your nonlinear function though they are more complicated to use The next section explains the substitutable expressions that are used to define the regression function and the section thereafter explains how to write substitutable expression program files so that you do not need to type in commonly used functions over and over Later sections highlight other features of n1 The final section discusses function evaluator programs If you find substitutable expressions adeq
19. am are immaterial however The rest of the program computes the nonlinear function using some temporary variables to hold intermediate results The final line of the program then replaces the dependent variable with the values of the function Notice the use of if to restrict attention to the estimation sample nl makes a copy of your dependent variable so that when the command is finished your data are left unchanged To use the program and fit your model you type use http www stata press com data r13 production clear nl ces2 lnoutput capital labor parameters b0 rho delta gt initial bO 0 rho 1 delta 0 5 The output is again identical to that shown in example 1 The order in which the parameters were specified in the parameters option is the same in which they are retrieved from the at matrix in the program To initialize them you simply list the parameter name a space the initial value and so on If you use the nparameters option instead of the parameters option the parameters are named b1 b2 bk where k is the number of parameters Thus you could have typed nl ces2 lnoutput capital labor nparameters 3 initial bi 0 b2 1 b3 0 5 With that syntax the parameters called bO rho and delta in the program will be labeled b1 b2 and b3 respectively In programming situations or if there are many parameters instead of listing the parameter names and initial values in the initial option you may find
20. an be rewritten as Bo nQ In 0 5K 0 5271 2 so a good initial value for So is the mean of the right hand side of 2 ignoring Lines 7 10 of the function evaluator program calculate that mean and store it in a local macro Notice the use of if in the summarize statement so that the mean is calculated only for the estimation sample nl Nonlinear least squares estimation 9 The final part of the program returns two macros The macro title is optional and defines a short description of the model that will be displayed in the output immediately above the table of parameter estimates The macro eq is required and defines the substitutable expression that n1 will use If the expression is short you can define it all at once However because the expression used here is somewhat lengthy defining local macros and then building up the final expression from them is easier To verify that there are no errors in your program you can call it directly and then use return list use http www stata press com data r13 production nlces lnoutput capital labor output omitted return list macros r title CES ftn ln Q lnoutput K capital L labor r eq lnoutput b0 3 711606264663641 1 rho 1 ln delt gt a 0 5 capital 1 rho 1 delta labor 1 rho The macro r eq contains the same substitutable expression that we specified at the command line in the preceding example except for
21. bly robust to the inability of your nonlinear function to be evaluated at some parameter values nl does assume that your function can be evaluated at the initial values of the parameters If your function cannot be evaluated at the initial values an error message is issued with return code 480 Recall that if you do not specify an initial value for a parameter then n1 initializes it to zero Many nonlinear functions cannot be evaluated when some parameters are zero so in those cases specifying alternative initial values is crucial Thereafter as nl changes the parameter values it monitors your function for unexpected missing values If these are detected n1 backs up That is n1 finds a point between the previous known to be good parameter vector and the new known to be bad vector at which the function can be evaluated and continues its iterations from that point nl requires that once a parameter vector is found where the predictions can be calculated small changes to the parameter vector be made to calculate numeric derivatives If a boundary is encountered at this point an error message is issued with return code 481 When specifying 1nlsq an attempt to take logarithms of y k when y lt k results in an error message with return code 482 If iterate iterations are performed and estimates still have not converged results are presented with a warning and the return code is set to 430 If you use the programmed substitutable
22. e old syntax of n1 still works and you can still use those nlfcn programs If nl does not see a colon an at sign or a set of parentheses surrounding the equation in your command it assumes that the old syntax is being used The current version of nl uses scalars and matrices to store intermediate calculations instead of local and global macros as the old version did so the current version produces more accurate results In practice however any discrepancies are likely to be small Q nl Nonlinear least squares estimation Stored results nl stores the following in e e sample Scalars e N number of observations e k number of parameters e k_eq_model number of equations in overall model test always O e df_m model degrees of freedom e df_r residual degrees of freedom e df_t total degrees of freedom e mss model sum of squares e rss residual sum of squares e tss total sum of squares e mms model mean square e msr residual mean square e 11 log likelihood assuming i i d normal errors e r2 R squared e r2_a adjusted R squared e rmse root mean squared error e dev residual deviance e N_clust number of clusters e 1nlsq value of 1nlsq if specified e log_t 1 if 1nlsq specified O otherwise e gm_2 square of geometric mean of y k if 1nlsq 1 otherwise e cj position of constant in e b or O if no constant e delta relative change used to compute derivatives e rank rank of e V e ic
23. empvar y generate double y logout 1n 0 5 capital 1 0 5 labor 1 summarize y if meanonly local bOval r mean Terms for substitutable expression local capterm delta 0 5 capital 1 rho local labterm 1 delta labor 1 rho local term2 1 rho 1 ln capterm labterm Return substitutable expression and title return local eq logout b0 bOval term2 return local title CES ftn ln Q logout K capital L labor end The program accepts three variables for log output capital and labor and it accepts an if exp qualifier to restrict the estimation sample All programs that you write to use with nl must accept an if exp qualifier because when nl calls the program it passes a binary variable that marks the estimation sample the variable equals one if the observation is in the sample and zero otherwise When calculating initial values you will want to restrict your computations to the estimation sample and you can do so by using if with any commands that accept if exp qualifiers Even if your program does not calculate initial values or otherwise use the if qualifier the syntax statement must still allow it See P syntax for more information on the syntax command and the use of if As in the previous example reasonable initial values for 6 and p are 0 5 and 1 respectively Conditional on those values 1 c
24. enu driven nonlinear regression program 1991 provided the inspiration References Atkinson A C 1985 Plots Transformations and Regression An Introduction to Graphical Methods of Diagnostic Regression Analysis Oxford Oxford University Press Canette I 2011 A tip to debug your nl nlsur function evaluator program The Stata Blog Not Elsewhere Classified http blog stata com 201 1 12 05 a tip to debug your nlnlsur function evaluator program Danuso F 1991 sgl Nonlinear regression command Stata Technical Bulletin 1 17 19 Reprinted in Stata Technical Bulletin Reprints vol 1 pp 96 98 College Station TX Stata Press Davidson R and J G MacKinnon 1993 Estimation and Inference in Econometrics New York Oxford University Press 2004 Econometric Theory and Methods New York Oxford University Press Gallant A R 1987 Nonlinear Statistical Models New York Wiley Goldstein R 1992 srd7 Adjusted summary statistics for logarithmic regressions Stata Technical Bulletin 5 17 21 Reprinted in Stata Technical Bulletin Reprints vol 1 pp 178 183 College Station TX Stata Press Kennedy W J Jr and J E Gentle 1980 Statistical Computing New York Dekker Poi B P 2008 Stata tip 58 nl is not just for nonlinear models Stata Journal 8 139 141 Ratkowsky D A 1983 Nonlinear Regression Modeling A Unified Practical Approach New York Dekker Ross G J S 1987 MLP User Manual Release 3
25. its an arbitrary nonlinear regression function by least squares With the interactive version of the command you enter the function directly on the command line or in the dialog box by using a substitutable expression If you have a function that you use regularly you can write a substitutable expression program and use the second syntax to avoid having to reenter the function every time The function evaluator program version gives you the most flexibility in exchange for increased complexity with this version your program is given a vector of parameters and a variable list and your program computes the regression function When you write a substitutable expression program or function evaluator program the first two letters of the name must be nl sexp_prog and func_prog refer to the name of the program without the first two letters For example if you wrote a function evaluator program named nlregss you would type nl regss to estimate the parameters Options Model variables varlist specifies the variables in the model nl ignores observations for which any of these variables have missing values If you do not specify variables then n1 issues an error message with return code 480 if the estimation sample contains any missing values initial initial_values specifies the initial values to begin the estimation You can specify a 1 x k matrix where k is the number of parameters in the model or you can specify a parameter name
26. n and normalized such that their sum equals the number of observations The residual deviance for 3a is D B 1 m 278 n X In w 5a nl Nonlinear least squares estimation 13 compare with 5 where A 12 rio RSS X wi ys fxi B Defining and fitting a model equivalent to 4 when weights have been specified as in 3a is not straightforward and has not been attempted Thus deviances using and not using the 1nlsq option may not be strictly comparable when analytic weights other than O and 1 are used You do not need to modify your substitutable expression in any way to use weights If however you write a substitutable expression program then you should account for weights when obtaining initial values When n1 calls your program it passes whatever weight expression if any was specified by the user Here is an outline of a substitutable expression program that accepts weights program nl name rclass version 13 syntax varlist aw fw iw if Obtain initial values allowing weights Use the syntax weight exp For example summarize varname weight exp if regress depvar varlist weight exp if Return substitutable expression return local eq substitutable expression return local title description of estimator end For details on how the syntax command processes weight expressions see P syntax Potential errors nl is reasona
27. parameter appears in your function multiple times you need only specify an initial value only once or never if you wish to set the initial value to zero If you do specify more than one initial value for the same parameter nl will use the last value given Parameter names must follow the same conventions as variable names See U 11 3 Naming conventions Frequently even nonlinear functions contain linear combinations of variables As an example suppose that you wish to fit the function Yi Bo 1 _ greene ete amp nl allows you to declare a linear combination of variables by using the shorthand notation nl y b0 1 1 exp 1 xb x1 x2 x3 In the syntax xb x1 x2 x3 you are telling n1 that you are declaring a linear combination named xb that is a function of three variables x1 x2 and x3 nl will create three parameters named xb_x1 xb_x2 and xb_x3 and initialize them to zero Instead of typing the previous command you could have typed nl y b0 1 1 exp 1 xb_xi x1 xb_x2 x2 xb_x3 x3 and yielded the same result You can refer to the parameters created by n1 in the linear combination later in the function though you must declare the linear combination first if you intend to do that When creating linear combinations n1 ensures that the parameter names it chooses are unique and have not yet been used in the function In general there are three rules to follow when defining substitutable ex
28. pressions 1 Parameters of the model are bound in braces b0 param etc 2 Initial values for parameters are given by including an equal sign and the initial value inside the braces b0 1 param 3 571 etc 3 Linear combinations of variables can be included using the notation eqname varlist for example xb mpg price weight score w x z etc Parameters of linear combinations are initialized to zero nl Nonlinear least squares estimation 7 If you specify initial values by using the initial option they override whatever initial values are given within the substitutable expression Substitutable expressions are so named because once values are assigned to the parameters the resulting expression can be handled by generate and replace gt Example 1 We wish to fit the CES production function 1 InQ bo E 4 1 6 L y amp 1 where InQ is the log of output for firm 7 K and L are firm it s capital and labor usage respectively and is a regression error term Because p appears in the denominator of a fraction zero is not a feasible initial value for a CES production function p 1 is a reasonable choice Setting 6 0 5 implies that labor and capital have equal impacts on output which is also a reasonable choice for an initial value We type use http www stata press com data r13 production nl lnoutput b0 1 rho 1 1n delta 0 5 capital 1 rho gt 1 delta
29. quares where In depvar is assumed to be normally distributed the model has no constant term seldom used use name as constant term seldom used vcetype may be gnr robust cluster clustvar bootstrap jacknife hac kernel hc2 or hc3 set confidence level default is level 95 create variables containing derivative of E y display string as title above the table of parameter estimates display string as subtitle control column formats and line width control the optimization process seldom used specify for convergence criterion default is eps 1e 5 specify for computing derivatives default is delta 4e 7 display legend instead of statistics For function evaluator program version you must specify parameters namelist or nparameters or both bootstrap by jackknife rolling statsby and svy are allowed see U 11 1 10 Prefix commands Weights are not allowed with the bootstrap prefix see R bootstrap aweights are not allowed with the jackknife prefix see R jackknife vce leave and weights are not allowed with the svy prefix see SVY svy aweights fweights and iweights are allowed see U 11 1 6 weight coeflegend does not appear in the dialog box See U 20 Estimation and postestimation commands for more capabilities of estimation commands Menu Statistics gt Linear models and related gt Nonlinear least squares nl Nonlinear least squares estimation 3 Description nl f
30. re has better convergence properties In addition the parameter estimates are virtually unbiased and normally distributed and the asymptotic approximation to the standard errors correlations and confidence intervals is much more accurate than for the other models Even within a given model the way the parameters are expressed for example or ef affects the degree of linearity and convergence behavior Function evaluator programs Occasionally a nonlinear function may be so complex that writing a substitutable expression for it is impractical For example there could be many parameters in the model Alternatively if you are implementing a two step estimator writing a substitutable expression may be altogether impossible Function evaluator programs can be used in these situations nl will pass to your function evaluator program a list of variables a weight expression a variable marking the estimation sample and a vector of parameters Your program is to replace the dependent variable which is the first variable in the variables list with the values of the nonlinear function evaluated at those parameters As with substitutable expression programs the first two letters of the name must be nl To focus on the mechanics of the function evaluator program again let s compare the CES production function to the previous examples The function evaluator program is nl Nonlinear least squares estimation 15 program nlces2 ver
31. sion 13 syntax varlist min 3 max 3 if at name local logout word 1 of varlist local capital word 2 of varlist local labor word 3 of varlist Retrieve parameters out of at matrix tempname bO rho delta scalar bO at 1 1 scalar rho at 1 2 scalar delta at 1 3 tempvar kterm lterm generate double kterm delta capital 1 rho if generate double lterm 1 delta labor 1 rho if Fill in dependent variable replace logout bO 1 rho ln kterm lterm if end Unlike the previous nlces program this one is not declared to be r class The syntax statement again accepts three variables one for log output one for capital and one for labor An if exp is again required because nl will pass a binary variable marking the estimation sample All function evaluator programs must accept an option named at that takes a name as an argument that is how nl passes the parameter vector to your program The next part of the program retrieves the output labor and capital variables from the variables list It then breaks up the temporary matrix at and retrieves the parameters bO rho and delta Pay careful attention to the order in which the parameters refer to the columns of the at matrix because that will affect the syntax you use with nl The temporary names you use inside this progr
32. the initial value for bO In short an n1 substitutable expression program should return in r eq the same substitutable expression you would type at the command line The only difference is that when writing a substitutable expression program you do not bind the entire expression inside parentheses Having written the program you can use it by typing nl ces lnoutput capital labor There is a space between nl and ces The output is identical to that shown in example 1 save for the title defined in the function evaluator program that appears immediately above the table of parameter estimates Q Technical note You will want to store nlces as an ado file called nlces ado The alternative is to type the code into Stata interactively or to place the code in a do file While those alternatives are adequate for occasional use if you save the program as an ado file you can use the function anytime you use Stata without having to redefine the program When n1 attempts to execute nlces if the program is not in Stata s memory Stata will search the disk s for an ado file of the same name and if found automatically load it All you have to do is name the file with the ado suffix and then place it in a directory where Stata will find it You should put the file in the directory Stata reserves for user written ado files which depending on your operating system is c ado personal Windows ado personal Unix or ado personal Mac See
33. uares which we define as least squares with shifted lognormal errors In other words In depvar is assumed to be normally distributed Sums of squares and deviance are adjusted to the same scale as depvar 4 nl Nonlinear least squares estimation noconstant indicates that the function does not include a constant term This option is generally not needed even if there is no constant term in the model unless the coefficient of variation over observations of the partial derivative of the function with respect to a parameter is less than eps and that parameter is not a constant term hasconstant name indicates that parameter name be treated as the constant term in the model and that nl should not use its default algorithm to find a constant term As with noconstant this option is seldom used SE Robust vce vcetype specifies the type of standard error reported which includes types that are derived from asymptotic theory gnr that are robust to some kinds of misspecification robust that allow for intragroup correlation cluster clustvar and that use bootstrap or jackknife methods bootstrap jackknife see R vce_option vce gnr the default uses the conventionally derived variance estimator for nonlinear models fit using Gauss Newton regression nl also allows the following vce hac kernel specifies that a heteroskedasticity and autocorrelation consistent HAC variance estimate be used HAC refers
34. uate to define your nonlinear function then you can skip that section entirely Function evaluator programs are generally needed only for complicated problems such as multistep estimators The program receives a vector of parameters at which it is to compute the function and a variable into which the results are to be placed 6 nl Nonlinear least squares estimation Substitutable expressions You define the nonlinear function to be fit by nl by using a substitutable expression Substitutable expressions are just like any other mathematical expressions involving scalars and variables such as those you would use with Stata s generate command except that the parameters to be estimated are bound in braces See U 13 2 Operators and U 13 3 Functions for more information on expressions For example suppose that you wish to fit the function Yi Bol e Pim i where 6o and are the parameters to be estimated and is an error term You would simply type nl y bO 1 exp 1 b1i x You must enclose the entire equation in parentheses Because bO and b1 are enclosed in braces n1 knows that they are parameters in the model nl will initialize bO and b1 to zero by default To request that n1 initialize bO to 1 and b1 to 0 25 you would type nl y b0 1 1 exp 1 b1 0 25 x That is inside the braces denoting a parameter you put the parameter name followed by an equal sign and the initial value If a
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