NPREG.RPF

NPREG.RPF estimates a linear regression using weighted least squares with the spread series created from a non-parametric fit of the squared residuals to one of the explanatory variables (population). It's adapted from Pagan and Ullah(1999), pp 248-249. It demonstrates use of the NPREG instruction.

The code segment here first estimates the model by least squares, then does conventional weighted least squares using the square of population as the scedastic function. NPREG is then used to get a non-parametric fit of the squared residuals on population, creating VPOPNP as the fitted estimate; that is, instead of using a specific functional form in the population for the residual variance (the square) for weighted least squares, it uses a non-parametric estimate. Since the actual data series POP was used as the grid on NPREG, VPOPNP will also align with the data, so it can be used directly in LINREG as the SPREAD option.

linreg exptrav / resids

# constant income

set popsq = pop^2

linreg(spread=popsq) exptrav

set ressqr = resids^2

npreg(grid=input,type=gaussian) ressqr pop / pop vpopnp

linreg(spread=vpopnp,$

title="Semiparametric Weighted Least Squares") exptrav

# constant income

Full Program

open data travel.csv
data(format=prn,org=columns) 1 51 pop income exptrav
*
linreg exptrav / resids
# constant income
*
* Assumed scedastic function is pop^2
*
set popsq = pop^2
linreg(spread=popsq) exptrav
# constant income
*
* Nonparametric estimator
*
set ressqr = resids^2
npreg(grid=input,type=gaussian) ressqr pop / pop vpopnp
linreg(spread=vpopnp,title="Semiparametric Weighted Least Squares") exptrav
# constant income
*
* Parametric estimates of alternative scedastic functions
*
linreg ressqr
# popsq
prj vpopsq
linreg ressqr
# constant pop popsq
prj vpopquad
*
* The data set is already sorted by pop, so style=lines will work
*
scatter(style=lines,footer="Alternative Empirical Scedastic Functions",$
key=upleft,klabels=||"Quadratic Only","Full Polynomial","Non-Parametric"||) 3
# pop vpopsq
# pop vpopquad
# pop vpopnp
*
* This reveals that a big problem is one truly massive outlier
*
scatter(style=lines,footer="Non-Parametric Variance with Residuals",$
overlay=dots,ovsame) 2
# pop vpopnp
# pop ressqr

Output

Linear Regression - Estimation by Least Squares

Dependent Variable EXPTRAV

Usable Observations 51

Degrees of Freedom 49

Centered R^2 0.8836169

R-Bar^2 0.8812418

Uncentered R^2 0.9322972

Mean of Dependent Variable 4.3736470588

Std Error of Dependent Variable 5.2091982985

Standard Error of Estimate 1.7951583044

Sum of Squared Residuals 157.90707356

Regression F(1,49) 372.0234

Significance Level of F 0.0000000

Log Likelihood -101.1855

Durbin-Watson Statistic 2.6695

Variable Coeff Std Error T-Stat Signif

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

1. Constant 0.2664897296 0.3294409293 0.80892 0.42247443

2. INCOME 0.0675410389 0.0035017294 19.28791 0.00000000

Linear Regression - Estimation by Weighted Least Squares

Dependent Variable EXPTRAV

Usable Observations 51

Degrees of Freedom 49

Centered R^2 0.1577653

R-Bar^2 0.1405768

Uncentered R^2 0.6485201

Mean of Dependent Variable 1.1108773510

Std Error of Dependent Variable 0.9494764958

Standard Error of Estimate 0.8802129355

Sum of Squared Residuals 37.963965780

Log Likelihood -118.4384

Durbin-Watson Statistic 2.6422

Variable Coeff Std Error T-Stat Signif

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

1. Constant 0.5539567129 0.2248496450 2.46368 0.01730999

2. INCOME 0.0643900657 0.0138997026 4.63248 0.00002694

Linear Regression - Estimation by Semiparametric Weighted Least Squares

Dependent Variable EXPTRAV

Usable Observations 51

Degrees of Freedom 49

Centered R^2 0.8226464

R-Bar^2 0.8190269

Uncentered R^2 0.9305163

Mean of Dependent Variable 2.8608502722

Std Error of Dependent Variable 2.3189251803

Standard Error of Estimate 0.9864923217

Sum of Squared Residuals 47.685187933

Log Likelihood -84.3678

Durbin-Watson Statistic 2.6541

Variable Coeff Std Error T-Stat Signif

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

1. Constant 0.3010292386 0.2327686082 1.29326 0.20198673

2. INCOME 0.0652469911 0.0038782338 16.82389 0.00000000

Linear Regression - Estimation by Least Squares

Dependent Variable RESSQR

Usable Observations 51

Degrees of Freedom 50

Centered R^2 -0.0124249

R-Bar^2 -0.0124249

Uncentered R^2 0.0935548

Mean of Dependent Variable 3.0962171286

Std Error of Dependent Variable 9.1451526339

Standard Error of Estimate 9.2017910219

Sum of Squared Residuals 4233.6479005

Log Likelihood -185.0502

Durbin-Watson Statistic 1.5806

Variable Coeff Std Error T-Stat Signif

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

1. POPSQ 0.0250831946 0.0110416914 2.27168 0.02744221

Linear Regression - Estimation by Least Squares

Dependent Variable RESSQR

Usable Observations 51

Degrees of Freedom 48

Centered R^2 0.1189024

R-Bar^2 0.0821900

Uncentered R^2 0.2111349

Mean of Dependent Variable 3.0962171286

Std Error of Dependent Variable 9.1451526339

Standard Error of Estimate 8.7612755998

Sum of Squared Residuals 3684.4776065

Regression F(2,48) 3.2388

Significance Level of F 0.0479265

Log Likelihood -181.5074

Durbin-Watson Statistic 1.7840

Variable Coeff Std Error T-Stat Signif

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

1. Constant -1.377906335 2.240700176 -0.61494 0.54149582

2. POP 1.372387724 0.671473387 2.04385 0.04647697

3. POPSQ -0.041242004 0.030856975 -1.33655 0.18766950

Graphs