NONLINEAR.RPF

NONLINEAR.RPF estimates the same model with a non-linearity in the dependent variable using several different techniques. It is adapted from Greene, Econometric Analysis, 5th Edition, example 17.5, pages 498-499. A production function takes the form:

\begin{equation} \log \,y_i + \theta y_i = \beta _1 + \beta _2 \log \,K_i + \beta _3 \log \,L_i + \varepsilon _i ,\varepsilon _i \sim N(0,\sigma ^2 )\,i.i.d. \end{equation}

which would be a standard Cobb-Douglas production function except for the \(\theta y_i \) term. Because of the transformation of the dependent variable, there’s a Jacobian term that must be included in the likelihood. The model can be estimated directly by MAXIMIZE, or by NLLS with the JACOBIAN option.

The example first does a simple non-linear least squares estimate, which is a poor choice when \(\theta\) is unknown.

nonlin b1 b2 b3 theta
compute b1=b2=b3=theta=0.0
*
frml resid = log(y)+theta*y-b1-b2*logk-b3*logl
nlls(frml=resid)

Proper estimates require adjusting for the Jacobian—the derivative of \(\log \,y_i + \theta y_i \) with respect to \(y_i\), which is \(\left( {1/y + \theta } \right)\). To do maximum likelihood directly, we need to add the (unknown) residual variance to the parameter set. Also, because the FRML for maximum likelihood is the log likelihood, we need the log of the Jacobian in addition to the log Normal density. Note that this is using the RESID function defined above, which computes the residual for a given entry given the parameters:

nonlin b1 b2 b3 theta sigsq

frml loglike = log(1/y+theta)+%logdensity(sigsq,resid)

We need to initialize sigsq to a positive number (almost any number will work) or loglike won't be computable at the initial guesses.

compute sigsq=%seesq

maximize(method=bfgs) loglike

NLLS with the JACOBIAN option can also be used for this (since this is just non-linear least squares on a transformed variable). You provide a formula that computes the Jacobian itself, and it takes care of the adjustment. (This has the advantage compared with MAXIMIZE that the variance is concentrated out):

nonlin b1 b2 b3 theta

nlls(jacobian=(1/y+theta),frml=resid)

An alternative is to observe that, given \(\theta\), all the other coefficients can be estimated by a least squares regression. This uses FIND to search for the maximum likelihood \(\theta\). The regression part of the likelihood is concentrated into the single value %LOGL which is then adjusted by the sum of the log Jacobians (computed using SSTATS). FIND controls a loop much as DO does. The instructions between FIND and END are executed each time a new function evaluation is needed.

dec real concentrate

nonlin theta

find maximum concentrate

set yadjust = log(y)+theta*y

linreg(noprint) yadjust

# constant logk logl

sstats / log(1/y+theta)>>yterms

compute concentrate=yterms+%logl

end find

Full Program

open data tablef9-2[1].txt

data(format=prn,org=columns) 1 25 valueadd capital labor nfirm

set y = valueadd/nfirm

set k = capital/nfirm

set l = labor/nfirm

set logk = log(k)

set logl = log(l)

* Nonlinear least squares (which is a poor estimator if theta is

* unknown). Because the "dependent variable" is actually a combination

* of two terms, we just define the formula as the residual. If NLLS sees

* a FRML without an explicitly defined dependent variable, it will just

* minimize the sum of squares of the frml itself.

nonlin b1 b2 b3 theta

compute b1=b2=b3=theta=0.0

frml resid = log(y)+theta*y-b1-b2*logk-b3*logl

nlls(frml=resid)

*************************************************************************

* Maximum likelihood estimates. We need to add the variance to the list

* of parameters (if we're doing a joint estimate of all the parameters).

* The log likelihood formula itself uses the resid FRML defined above as

* a building block so we don't have to repeat it.

nonlin b1 b2 b3 theta sigsq

frml loglike = log(1/y+theta)+%logdensity(sigsq,resid)

* We need to initialize sigsq to a positive number or loglike won't be

* computable at the initial guesses.

compute sigsq=%seesq

maximize(method=bfgs) loglike

*************************************************************************

* This is an alternative way to do maximum likelihood. This uses NLLS

* with the JACOBIAN option to take care of those terms in the likelihood.

nonlin b1 b2 b3 theta

nlls(jacobian=(1/y+theta),frml=resid)

*************************************************************************

* This is the same maximum likelihood done by concentrating out the

* beta's and variance, since given theta, the likelihood is just least

* squares.

dec real concentrate

nonlin theta

find maximum concentrate

set yadjust = log(y)+theta*y

linreg(noprint) yadjust

# constant logk logl

sstats / log(1/y+theta)>>yterms

compute concentrate=yterms+%logl

end find

Output

Nonlinear Least Squares - Estimation by Gauss-Newton

Convergence in 1 Iterations. Final criterion was 0.0000000 <= 0.0000100

Usable Observations 25

Degrees of Freedom 21

Standard Error of Estimate 0.1909311795

Sum of Squared Residuals 0.7655490214

Log Likelihood 8.1020

Durbin-Watson Statistic 2.0367

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. B1 2.108924948 1.570598878 1.34275 0.17935226

2. B2 0.257900246 0.459701239 0.56102 0.57478589

3. B3 0.878387869 0.650499966 1.35033 0.17691102

4. THETA -0.031634242 0.251239073 -0.12591 0.89980086

MAXIMIZE - Estimation by BFGS

Convergence in 19 Iterations. Final criterion was 0.0000000 <= 0.0000100

Usable Observations 25

Function Value -8.9390

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. B1 2.9146396083 0.3439594778 8.47379 0.00000000

2. B2 0.3500318252 0.0883948446 3.95987 0.00007499

3. B3 1.0922472883 0.1306105504 8.36263 0.00000000

4. THETA 0.1066386111 0.0560053532 1.90408 0.05689991

5. SIGSQ 0.0427381564 0.0152962742 2.79402 0.00520566

Nonlinear Least Squares - Estimation by Gauss-Newton

Convergence in 1 Iterations. Final criterion was 0.0000059 <= 0.0000100

Usable Observations 25

Degrees of Freedom 21

Standard Error of Estimate 0.2255750331

Sum of Squared Residuals 1.0685660063

Log Likelihood -8.9390

Durbin-Watson Statistic 1.5469

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. B1 2.9148206024 1.0880209533 2.67901 0.00738399

2. B2 0.3500674182 0.2818202853 1.24217 0.21417553

3. B3 1.0922747920 0.4152567890 2.63036 0.00852945

4. THETA 0.1066652764 0.1773732820 0.60136 0.54759994

FIND Optimization - Estimation by Simplex

Function Value -8.9390

Variable Coeff

**********************************************

1. THETA 0.1066652764