Stata 15 help for nlsur

[R] nlsur -- Estimation of nonlinear systems of equations

Syntax

Interactive version

nlsur (depvar_1=<sexp_1>) (depvar_2=<sexp_2>) ... [if] [in] [weight] [, options]

Programmed substitutable expression version

nlsur sexp_prog : depvar_1 depvar_2 ... [varlist] [if] [in] [weight] [, options]

Function evaluator program version

nlsur func_prog @ depvar_1 depvar_2 ... [varlist] [if] [in] [weight] , nequations(#) {parameters(namelist)|nparameters(#)} [options]

where

depvar_j is the dependent variable for equation j; <sexp>_j is the substitutable expression for equation j; sexp_prog is a substitutable expression program; and func_prog is a function evaluator program.

options Description ------------------------------------------------------------------------- Model fgnls use two-step FGNLS estimator; the default ifgnls use iterative FGNLS estimator nls use NLS estimator variables(varlist) variables in model initial(initial_values) initial values for parameters nequations(#) number of equations in model (function evaluator program version only) * 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

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

Reporting level(#) set confidence level; default is level(95) title(string) display string as title above the table of parameter estimates title2(string) display string as subtitle display_options control columns and column formats and line width

Optimization optimization_options control the optimization process; seldom used eps(#) specify # for convergence criteria; default is eps(1e-5) ifgnlsiterate(#) set maximum number of FGNLS iterations ifgnlseps(#) specify # for FGNLS convergence criterion; default is ifgnlseps(1e-10) delta(#) specify stepsize # for computing derivatives; default is delta(4e-7) noconstants no equations have constant terms hasconstants(namelist) use namelist as constant terms

coeflegend display legend instead of statistics ------------------------------------------------------------------------- * You must specify parameters(namelist), nparameters(#), or both. bootstrap, by, jackknife, rolling, and statsby are allowed; see prefix. Weights are not allowed with the bootstrap prefix. aweights are not allowed with the jackknife prefix. aweights, fweights, iweights, and pweights are allowed; see weight. coeflegend does not appear in the dialog box. See [R] nlsur postestimation for features available after estimation.

Menu

Statistics > Linear models and related > Multiple-equation models > Nonlinear seemingly unrelated regression

Description

nlsur fits a system of nonlinear equations by feasible generalized nonlinear least squares (FGNLS). With the interactive version of the command, you enter the system of equations directly on the command line or in the dialog box by using substitutable expressions. If you have a system that you use regularly, you can write a substitutable expression program and use the second syntax to avoid having to reenter the system 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 system of equations.

When you write a substitutable expression program or a function evaluator program, the first five letters of the name must be nlsur. sexp_prog and func_prog refer to the name of the program without the first five letters. For example, if you wrote a function evaluator program named nlsurregss, you would type nlsur regss @ ... to estimate the parameters.

Options

+-------+ ----+ Model +------------------------------------------------------------

fgnls requests the two-step FGNLS estimator; this is the default.

ifgnls requests the iterative FGNLS estimator. For the nonlinear systems estimator, this is equivalent to maximum likelihood estimation.

nls requests the nonlinear least-squares (NLS) estimator.

variables(varlist) specifies the variables in the system. nlsur ignores observations for which any of these variables has missing values. If you do not specify variables(), nlsur issues an error message 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 total number of parameters in the system, 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

nlsur ... , 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 matrix, the values must be in the same order in which the parameters are declared in your model. nlsur ignores the row and column names of the matrix.

nequations(#) specifies the number of equations in the system.

parameters(namelist) specifies the names of the parameters in the system. 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.

nparameters(#) specifies the number of parameters in the system. If you do not specify names with the parameters() option, nlsur 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.

sexp_options refer to any options allowed by your sexp_prog.

func_options refer to any options allowed by your func_prog.

+-----------+ ----+ 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.

+-----------+ ----+ Reporting +--------------------------------------------------------

level(#); see [R] estimation options.

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: noci, nopvalues, 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 to use for NLS at each round of FGNLS estimation. This option is different from ifgnlsiterate(), which controls the maximum rounds of FGNLS estimation to use when the ifgnls option is specified. log/nolog specifies whether to show the iteration log, and trace specifies that the iteration log should include the current parameter vector.

eps(#) specifies the convergence criterion for successive parameter estimates and for the residual sum of squares. The default is eps(1e-5) (.00001). eps() also specifies the convergence criterion for successive parameter estimates between rounds of iterative FGNLS estimation when ifgnls is specified.

ifgnlsiterate(#) specifies the maximum number of FGNLS iterations to perform. The default is the number set using set maxiter, which is 16,000 by default. To use this option, you must also specify the ifgnls option.

ifgnlseps(#) specifies the convergence criterion for successive estimates of the error covariance matrix during iterative FGNLS estimation. The default is ifgnlseps(1e-10). To use this option, you must also specify the ifgnls option.

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_i(x_i,b_1,b_2,...,b_i + d, b_[i+1],...) - f_i(x_i, b_1,b_2,...,b_i, b_[i+1],...)}/d, where d is delta*(b_i + delta). The default is delta(4e-7).

noconstants indicates that none of the equations in the system includes constant terms. This option is generally not needed, even if there are no constant terms in the system; though in rare cases without this option, nlsur may claim that there is one or more constant terms even if there are none.

hasconstants(namelist) indicates the parameters that are to be treated as constant terms in the system of equations. The number of elements of namelist must equal the number of equations in the system. The ith entry of namelist specifies the constant term in the ith equation. If an equation does not include a constant term, specify a period (.) instead of a parameter name. This option is seldom needed with the interactive and programmed substitutable expression versions, because in those cases nlsur can almost always find the constant terms automatically.

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

coeflegend; see [R] estimation options.

Remarks

Remarks are presented under the following headings:

Substitutable expressions Examples Substitutable expression programs Example Function evaluator programs

Substitutable expressions

You use substitutable expressions with the interactive and programmed substitutable expression versions of nlsur to define your system of equations. Substitutable expressions are just like any other mathematical expression in Stata, except that the parameters of your model are bound in braces.

You specify a substitutable expression for each equation in your system, and you must follow three rules:

1. Parameters of the model are bound in braces: {b0}, {param}, etc.

2. Initial values 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 system of equations

y1 = a1 + b1*x^g1

y2 = a2 + b2*x^g2

by iterative FGNLS, where a1, a2, b1, b2, g1, and g2 are parameters and 1 is a reasonable starting value for both g1 and g2, you would type

. nlsur (y1 = {a1} + {b1}*x^{g1=1}) (y2 = {a2} + {b2}*x^{g2=1}), ifgnls

2. nlsur makes imposing cross-equation parameter restrictions easy. Say that you want to fit a pair of exponential growth equations with the restriction that the constant terms in the two equations are equal,

y1 = a + b1*b2^x

y2 = a + c1*c2^x

where a, b1, b2, c1, and c2 are the parameters. To fit this model, you would type

. nlsur (y1 = {a} + {b1}*{b2}^x) (y2 = {a} + {c1}*{c2}^x)

Substitutable expression programs -- sexp_progs

If you intend to fit the same nonlinear system multiple times, then you can write a substitutable expression program and avoid having to reenter the equations every time. The first five letters of the program name must be nlsur, and the program is to accept a varlist, an if exp, and, optionally, weights. The program must return in r(n_eq) the number of equations in the system and in r(eq_1) through r(eq_m) the m equations in the system. You can add a title to the output by storing a string in r(title).

The outline of an nlsursexp_prog program is

program nlsursexp_prog, rclass version 15.1 syntax varlist [aw fw iw] [if] (obtain initial parameters if desired) return scalar n_eq = <neqn> return local eq_1 "<sexp>_1" ... return local eq_m "<sexp>_m" return local title "title" end

Example

Returning to the model

y1 = a1 + b1*x^g1

y2 = a2 + b2*x^g2

one way to obtain initial values is to let g1 = g2 = 1 and then fit a regression of y1 on x to obtain initial values for a1 and b1 and, similarly, fit a regression of y2 on x to obtain initial values for a2 and b2. The substitutable expression program is

program nlsurmyreg, rclass

version 15.1 syntax varlist(min=3 max=3) [aw fw iw] [if] local y1: word 1 of `varlist' local y2: word 2 of `varlist' local x : word 3 of `varlist'

// Obtain starting values assuming g1=g2=1 regress `y1' `x' [`weight'`exp'] `if' local a1 = _b[_cons] local b1 = _b[`x'] regress `y2' `x' [`weight'`exp'] `if' local a2 = _b[_cons] local b2 = _b[`x']

return scalar n_eq = 2 return local eq_1 "`y1' = {a1=`a1'} + {b1=`b1'}*`x'^{g1=1}" return local eq_2 "`y2' = {a2=`a2'} + {b2=`b2'}*`x'^{g2=1}"

end

To fit your model, you type

. nlsur myreg: y1 y2 x

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

The substitutable expression does not need to account for weights or the if exp, though we did need to use them in obtaining initial values.

Function evaluator programs -- func_progs

If your system of equations is particularly complex, then you may find that writing a substitutable expression for each equation is impractical. In those cases, you can write a function evaluator program instead. Whenever nlsur needs to evaluate the system of equations, it calls your program with a vector of parameters. Your program then fills in the dependent variables with the corresponding function values.

Function evaluator programs must accept a varlist, an if exp, and an option named at() that accepts the name of a matrix. They may optionally accept weights as well.

To illustrate the mechanics of a function evaluator program, we focus on a straightforward example. We want to fit the model

y1 = b0 + b1*x1 + b2*x2

y2 = c0 + c1*x2 + c2*x3

(Because this system is linear, we could in fact just use sureg.) Our function evaluator program is

program nlsursur

version 15.1 syntax varlist(min=5 max=5) [if], at(name) local y1: word 1 of `varlist' local y2: word 2 of `varlist' local x1: word 3 of `varlist' local x2: word 4 of `varlist' local x3: word 5 of `varlist'

// Retrieve parameters out of `at' tempname b0 b1 b2 c0 c1 c2 scalar `b0' = `at'[1,1] scalar `b1' = `at'[1,2] scalar `b2' = `at'[1,3] scalar `c0' = `at'[1,4] scalar `c1' = `at'[1,5] scalar `c2' = `at'[1,6]

// Fill in dependent variables quietly replace `y1' = `b0' + `b1'*`x1' + `b2'*`x2' `if' quietly replace `y2' = `c0' + `c1'*`x2' + `c2'*`x3' `if'

end

Our model has a total of five variables, so we made the syntax statement accept five variables. nlsur requires that our program accept an if clause and an option named at() by which nlsur passes a matrix holding the parameter values. The order in which you store the elements of the varlist will determine the order in which you specify the variables when calling nlsur.

Because we do not plan to use weighted estimation with this model, we did not make our syntax statement accept weights. If you do intend to perform weighted estimation, the syntax statement must accept them. When you replace the dependent variables with the values of the functions, you do not need to do anything with the weights, though if you use estimation or descriptive statistical commands when evaluating your functions, the weight expression must be passed onto those commands.

Our model has six parameters, so our program will receive a 1 x 6 row vector named `at'. We extract the six parameters from that vector and store them in temporary scalars. We could have referred directly to the elements of the `at' vector when evaluating the functions of our model, but storing the parameters in appropriately named scalars makes the process more transparent. The final part of our program computes the two equations.

There are several different ways we can call nlsur to fit our model.

1. This method uses the shortest syntax and initializes all the parameters to zero, which is the default:

. nlsur sur @ y1 y2 x1 x2 x3, nparameters(6) nequations(2)

Because we did not specify names for the parameters, they will be labeled b1 through b6 in the output.

2. Here we give names to the parameters and initialize the two constant terms to be 10:

. nlsur sur @ y1 y2 x1 x2 x3, parameters(b0 b1 b2 c0 c1 c2) initial(b0 10 c0 10) nequations(2)

3. When you use a function evaluator program, nlsur does not attempt to identify a constant term in each equation, so the R-squared statistics reported in the header of the output are uncentered. You use the hasconstants() option to indicate which parameter in each equation is a constant:

. nlsur sur @ y1 y2 x1 x2 x3, parameters(b0 b1 b2 c0 c2 c2) initial(b0 10 c0 10) nequations(2) hasconstant(b0 c0)

Now nlsur takes b0 to be the constant in the first equation and c0 in the second, and centered R-squared statistics will be reported.

Example

Setup . webuse petridish

Model growth of two bacteria populations over time; allow for correlation between error terms . nlsur (p1 = {b1}*{b2}^t) (p2 = {g1}*{g2}^t)

Stored results

nlsur stores the following in e():

Scalars e(N) number of observations e(k) number of parameters e(k_#) number of parameters for equation # e(k_eq) number of equation names in e(b) e(k_eq_model) number of equations in overall model test e(n_eq) number of equations e(mss_#) model sum of squares for equation # e(rss_#) RSS for equation # e(rmse_#) root mean squared error for equation # e(r2_#) R squared for equation # e(ll) Gaussian log likelihood (iflgs version only) e(N_clust) number of clusters e(rank) rank of e(V) e(converged) 1 if converged, 0 otherwise

Macros e(cmd) nlsur e(cmdline) command as typed e(method) fgnls, ifgnls, or nls e(depvar) names of dependent variables e(depvar_#) dependent variable for equation # 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(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(sexpprog) substitutable expression program e(sexp_#) substitutable expression for equation # e(params) names of all parameters e(params_#) parameters in equation # e(funcprog) function evaluator program e(rhs) contents of variables() e(constants) identifies constant terms e(properties) b V e(predict) program used to implement predict

Matrices e(b) coefficient vector e(init) initial values vector e(Sigma) error covariance matrix (Sigma hat matrix) e(V) variance-covariance matrix of the estimators

Functions e(sample) marks estimation sample


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