Stata 15 help for nl

[R] nl -- Nonlinear least-squares estimation

Syntax

Interactive version

nl (depvar=<sexp>) [if] [in] [weight] [, options]

Programmed substitutable expression version

nl sexp_prog : depvar [varlist] [if] [in] [weight] [, options]

Function evaluator program version

nl func_prog @ depvar [varlist] [if] [in] [weight], {parameters(namelist)|nparameters(#)} [options]

where

depvar is the dependent variable; <sexp> is a substitutable expression; sexp_prog is a substitutable expression program; and func_prog is a function evaluator program.

options Description ------------------------------------------------------------------------- Model variables(varlist) variables in model initial(initial_values) initial values for parameters * parameters(namelist) parameters in model (function evaluator program version only) * nparameters(#) number of parameters in model (function evaluator program version only) sexp_options options for substitutable expression program func_options options for function evaluator program

Model 2 lnlsq(#) use log least-squares where ln(depvar - #) is assumed to be normally distributed noconstant the model has no constant term; seldom used hasconstant(name) use name as constant term; seldom used

SE/Robust vce(vcetype) vcetype may be gnr, robust, cluster clustvar, bootstrap, jackknife, hac kernel, hc2, or hc3

Reporting level(#) set confidence level; default is level(95) leave create variables containing derivative of E(y) title(string) display string as title above the table of parameter estimates title2(string) display string as subtitle display_options control column formats and line width

Optimization optimization_options control the optimization process; seldom used eps(#) specify # for convergence criterion; default is eps(1e-5) delta(#) specify # for computing derivatives; default is delta(4e-7)

coeflegend 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 prefix. Weights are not allowed with the bootstrap prefix. aweights are not allowed with the jackknife prefix. vce(), leave, and weights are not allowed with the svy prefix. aweights, fweights, and iweights are allowed; see weight. coeflegend does not appear in the dialog box. See [R] nl postestimation for features available after estimation.

Menu

Statistics > Linear models and related > Nonlinear least-squares estimation

Description

nl fits 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 nl 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, 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 expressions. 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 +----------------------------------------------------------

lnlsq(#) fits the model by using log least-squares, which we define as least squares with shifted lognormal errors. In other words, ln( depvar-#) is assumed to be normally distributed. Sums of squares and deviance are adjusted to the same scale as depvar.

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 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 to the general form for combining weighted matrices to form the variance estimate. There are three kernels available for nl:

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 sigma_j^2 = {n/(n-k)} u_j^2 as an estimate of the variance of the jth observation, where u_j is the calculated residual and n/(n-k) is included to improve the overall estimate's small-sample properties.

vce(hc2) instead uses u_j^2/(1-h_jj) as the observation's variance estimate, where h_jj 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_j^2/(1-h_jj)^2 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] 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 nl to issue an error message with return code 110.

leave may not be specified with vce(cluster clustvar) or the svy prefix.

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(%fmt), sformat(%fmt), and nolstretch; see [R] estimation 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 derivatives. The derivative for parameter b_i is computed as {f(X,b_1,b_2,...,b_i + d, b_[i+1],...) - f(X, b_1,b_2,...,b_i,b_[i+1],...)}/d, where d is delta(b_i + delta). The default is delta(4e-7).

The following option is available with nl but is not shown in the dialog box:

coeflegend; see [R] estimation options.

Remarks

Remarks are presented under the following headings:

Substitutable expressions Examples Some commonly used models Substitutable expression programs Example Function evaluator programs Example

Substitutable expressions

Using a substitutable expression is the easiest way to define your nonlinear function. Substitutable expressions are just like any other mathematical expression in Stata, except that the parameters of your model are bound in braces. There are three rules to follow:

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: {b1=1.267}, {gamma=3}, etc. If you do not specify an initial value, that parameter is initialized to zero. The initial() option overrides initial values provided in substitutable expressions.

3. Linear combinations can be included using the notation {eqname:varlist}:

{xb:mpg price weight} is equivalent to {xb_mpg}*mpg + {xb_price}*price + {xb_weight}*weight

Examples

1. To fit the model

y = alpha + beta*x^gamma

where alpha, beta, and gamma are parameters and a starting value for gamma is one, you would type

. nl (y = {alpha} + {beta}*x^{gamma=1})

2. To regress y on a constant and the reciprocal of x you could do

. nl (y = {b0} + {b1} / x), initial(b0 2 b1 3)

which obviates the need to generate a new variable equal to 1/x before calling regress. Here b0 is initialized to two and b1 is initialized to three.

Some commonly used models

The following models are used so often that they are built into nl.

Exponential regression with one asymptote:

exp3 y = b0 + b1*b2^x exp2 y = b1*b2^x exp2a y = b1*(1-b2^x)

Logistic function (symmetric sigmoid shape)(*):

log4 y = b0 + b1/(1 + exp(-b2*(x-b3))) log3 y = b1/(1 + exp(-b2*(x-b3)))

Gompertz function (asymmetric sigmoid shape):

gom4 y = b0 + b1*exp(-exp(-b2*(x-b3))) gom3 y = b1*exp(-exp(-b2*(x-b3)))

(*) not to be confused with logistic regression

To use any of these, you type

. nl model : depvar indepvar

For example,

. nl exp3: y x . nl gom3: response dosage

Initial values are chosen automatically, though you can override the defaults by using the initial() option.

Substitutable expression programs -- sexp_progs

If you use the same nonlinear function repeatedly, then you can write a substitutable expression program so that you do not have to retype the expression every time. The first two letters of the program name must by nl. The nlsexp_prog is to accept a varlist, an if exp, and, optionally, weights. The program will then return a substitutable expression in the r-class macro r(eq) and, optionally, a title in r(title).

The outline of an nlsexp_prog program is

program nlsexp_prog, rclass version 15.1 syntax varlist [aw fw iw] if (obtain initial parameters if desired) return local eq "<sexp>" return local title "title" end

Example

Returning to the model

y = alpha + beta*x^gamma

one way to obtain initial values is to let gamma = 1 and then run a regression of x on y to obtain alpha and beta. The substitutable expression program is

program nlmyreg, rclass version 15.1 syntax varlist(min=2 max=2) [aw fw iw] if local lhs: word 1 of `varlist' local rhs: word 2 of `varlist' regress `lhs' `rhs' [`weight'`exp'] `if' tempname a b scalar `a' = _b[_cons] scalar `b' = _b[`rhs'] return local eq "`lhs' = {alpha=`=`a''}+{beta=`=`b''}*`rhs'^{gamma=1}" return local title "`lhs' = alpha+beta*`rhs'^gamma" end

To fit your model, you type

. nl myreg: y x

(There is a space between nl and myreg, even though the program is named nlmyreg.)

The substitutable expression does not need to account for weights or the if exp, though you do need to use them in obtaining initial values. Also, the substitutable expression is not bound in parentheses, unlike when typing it in interactively.

Function evaluator programs -- func_progs

If your function is particularly complex, then you may find that writing one substitutable expression is impractical. In those cases, you can write a function evaluator program instead. Whenever nl needs to evaluate your function, it calls your program with a vector of parameters. Your program then fills in the dependent variable with function values.

Function evaluator programs must accept a varlist, an if exp, and an option named at() that accepts the name of a matrix. It may optionally accept weights as well. Unlike substitutable expression programs, a function evaluator program is not declared to be r-class. The outline of a nlfunc_prog program is

program nlfunc_prog version 15.1 syntax varlist [aw fw iw] if, at(name) local lhs: word 1 of `varlist' local rhs: subinstr local varlist "`lhs'" "", word (evaluate the function at matrix) replace `lhs' = <the function values> `if' end

When evaluating your function, remember to restrict the estimation sample by using `if'. Also, remember to include the weights if using commands such as summarize or regress if you intend to do weighted estimation.

Example

The CES production function can be written

ln Q = b0 - 1/rho*ln{delta*K^-rho + (1-delta)*L^-rho}

where Q denotes output and b0, rho, and delta are parameters to be estimated. The function evaluator program is

program nlces version 15.1 syntax varlist(min=3 max=3) [aw fw iw] 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 b0 rho delta scalar `b0' = `at'[1,1] scalar `rho' = `at'[1,2] scalar `delta' = `at'[1,3] // Some temporary variables tempvar kterm lterm generate double `kterm' = `delta'*`capital'^(-1*`rho') `if' generate double `lterm' = (1-`delta')*`labor'^(-1*`rho') `if' // Now fill in dependent variable replace `logout' = `b0' - 1/`rho'*ln(`kterm'+`lterm') `if' end

If your variables are logout, capital, and labor, then any of the following methods can be used to estimate the parameters:

1. This method uses b0 = 0 as an initial value by default:

. nl ces @ logout capital labor, parameters(b0 rho delta) initial(rho 1 delta 0.5)

2. This method initializes b0 to 2, rho to 1, and delta to 0.5. Because we do not give parameter names, nl names them b1, b2, and b3:

. nl ces @ logout capital labor, nparameters(3) initial(b1 2 b2 1 b3 0.5)

3. This method sets up a vector holding the initial values:

. matrix ivals = (2, 1, 0.5) . nl ces @ logout capital labor, parameters(b0 rho delta) initial(ivals)

or

. nl ces @ logout capital labor, nparameters(3) initial(ivals)

Example

Setup . webuse production

Fit CES production function with initial values rho=1 and delta=.5 . nl (lnoutput = {b0} - 1/{rho=1}*ln({delta=0.5}*capital^(-1*{rho}) + (1-{delta})*labor^(-1*{rho})))

Stored results

nl stores the following in e():

Scalars e(N) number of observations e(k) number of parameters e(k_eq_model) number of equations in overall model test; always 0 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(ll) 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(lnlsq) value of lnlsq if specified e(log_t) 1 if lnlsq specified, 0 otherwise e(gm_2) square of geometric mean of (y-k) if lnlsq, 1 otherwise e(cj) position of constant in e(b) or 0 if no constant e(delta) relative change used to compute derivatives e(rank) rank of e(V) e(ic) number of iterations e(converged) 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_kernel) 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) b V e(predict) program used to implement predict e(marginsok) predictions allowed by margins e(marginsnotok) predictions disallowed by margins

Matrices e(b) coefficient vector e(init) initial values vector e(V) variance-covariance matrix of the estimators e(V_modelbased) model-based variance

Functions e(sample) marks estimation sample

References

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.


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