NLLS.RPF

NLLS.RPF is an example of non-linear least squares. It is adapted from Examples 6.7 and 6.8 from Martin, Hurn and Harris(2013).

The model being estimated is

\(C = \beta _0 + \beta _1 YD^{\beta _2 } \)

This has obvious guess values, since with \(\beta _2 = 1\), the other coefficients can be estimated by least squares. The NONLIN and FRML describe the parameter set and the model (given the chosen names for the parameters). The LINREG estimates the restricted model and copies the estimated coefficients into the parameters. NLLS then estimates the non-linear model:

nonlin beta0 beta1 beta2

frml nlconst cons = beta0+beta1*inc^beta2

linreg cons

# constant inc

compute beta0=%beta(1),beta1=%beta(2),beta2=1.0

nlls(frml=nlconst)

The example program then does tests for linearity (that is \(\beta_2=1\)) in two ways. First, with a "Wald" test which operates on the results of the NLLS:

test(title="Test of linearity")

# 3

# 1.0

A likelihood ratio test can be constructed easily since NLLS and LINREG both compute the log likelihood and puts it into %LOGL. Save the %LOGL from one model and use CDF (with degrees of freedom=1) to do the test:

compute loglunr=%logl

linreg cons

# constant inc

cdf(title="LR Test of linearity") chisqr 2.0*(loglunr-%logl) 1

Full Program

open data usdata.txt
calendar(q) 1960:1
data(format=prn,nolabels,org=columns) 1960:01 2009:04 cons inc
*
nonlin beta0 beta1 beta2
frml nlconst cons = beta0+beta1*inc^beta2
*
linreg cons
# constant inc
compute beta0=%beta(1),beta1=%beta(2),beta2=1.0
nlls(frml=nlconst)
*
* Wald test for beta2=1
*
test(title="Test of linearity")
# 3
# 1.0
*
* Likelihood ratio test
*
compute loglunr=%logl
*
linreg cons
# constant inc
*
cdf(title="LR Test of linearity") chisqr 2.0*(loglunr-%logl) 1
*
* LM test
* Generate the derivative with respect to beta2 at beta2=1. The other derivatives
* are just the two explanatory variables.
*
set dbeta2 = beta1*inc*log(inc)
*
* Generate an LM (outer-product-of-gradient) statistic
*
mcov(opgstat=lmstat) / %resids
# constant inc dbeta2
cdf(title="LM test of linearity") chisqr lmstat 1

Output

This is the preliminary least squares to get guess values:

Linear Regression - Estimation by Least Squares

Dependent Variable CONS

Quarterly Data From 1960:01 To 2009:04

Usable Observations 200

Degrees of Freedom 198

Centered R^2 0.9981108

R-Bar^2 0.9981013

Uncentered R^2 0.9996600

Mean of Dependent Variable 4906.7400000

Std Error of Dependent Variable 2304.4552354

Standard Error of Estimate 100.4149874

Sum of Squared Residuals 1996467.5986

Regression F(1,198) 104609.5461

Significance Level of F 0.0000000

Log Likelihood -1204.6450

Durbin-Watson Statistic 0.2516

Variable Coeff Std Error T-Stat Signif

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

1. Constant -228.5400745 17.3927175 -13.13999 0.00000000

2. INC 0.9498154 0.0029367 323.43399 0.00000000

And these are the estimates of the non-linear model:

Nonlinear Least Squares - Estimation by Gauss-Newton

Convergence in 39 Iterations. Final criterion was 0.0000000 <= 0.0000100

Dependent Variable CONS

Quarterly Data From 1960:01 To 2009:04

Usable Observations 200

Degrees of Freedom 197

Centered R^2 0.9987629

R-Bar^2 0.9987503

Uncentered R^2 0.9997774

Mean of Dependent Variable 4906.7400000

Std Error of Dependent Variable 2304.4552354

Standard Error of Estimate 81.4634087

Sum of Squared Residuals 1307348.5306

Log Likelihood -1162.3071

Durbin-Watson Statistic 0.4081

Variable Coeff Std Error T-Stat Signif

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

1. BETA0 299.01976924 0.59968767 498.62584 0.00000000

2. BETA1 0.28931296 0.00042143 686.49755 0.00000000

3. BETA2 1.12408729 0.00015266 7363.35287 0.00000000

This is the output from the Wald test:

Test of linearity

Chi-Squared(1)= 660702.158537 with Significance Level 0.00000000

This is the re-do of the linear model:

Linear Regression - Estimation by Least Squares

Dependent Variable CONS

Quarterly Data From 1960:01 To 2009:04

Usable Observations 200

Degrees of Freedom 198

Centered R^2 0.9981108

R-Bar^2 0.9981013

Uncentered R^2 0.9996600

Mean of Dependent Variable 4906.7400000

Std Error of Dependent Variable 2304.4552354

Standard Error of Estimate 100.4149874

Sum of Squared Residuals 1996467.5986

Regression F(1,198) 104609.5461

Significance Level of F 0.0000000

Log Likelihood -1204.6450

Durbin-Watson Statistic 0.2516

Variable Coeff Std Error T-Stat Signif

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

1. Constant -228.5400745 17.3927175 -13.13999 0.00000000

2. INC 0.9498154 0.0029367 323.43399 0.00000000

and this is the output from the likelihood ratio test for linearity. While quite different numerically from the Wald test, both are overwhelming rejections.

LR Test of linearity

Chi-Squared(1)= 84.675671 with Significance Level 0.00000000

LM test of linearity

Chi-Squared(1)= 53.859614 with Significance Level 0.00000000