PROC IMPORT DATAFILE="/home/woodwc0/sasuser.v94/heterosk.xlsx"

OUT=woodwc0

DBMS=XLSX

REPLACE;

RUN;

data work.woodwc0;

SET work.woodwc0;

ODS graphics off;

PROC sort;

BY pop;

RUN;

PROC reg;

TITLE 'Engle Curve for Travel Expenditure';

MODEL exptrav = income;

RUN;

OUTPUT OUT=RESFILE RESIDUAL=E;

DATA TWO;

MERGE woodwc0 RESFILE;

BY POP;

LNSQRES = LOG(E**2);

LNPOP = LOG(POP);

RECIPOP= 1/POP;

DEXPTRAV = EXPTRAV/POP;

DINCOME = INCOME/POP;

RUN;

COMMENT 'Do not run all these tests in doing your assessment';

COMMENT 'Use discretion and run only the ones that are called for';

PROC REG;

TITLE 'Residual regression for Park test';

MODEL LNSQRES = LNPOP;

RUN;

PROC REG;

TITLE 'Estimation of transformed model';

MODEL DEXPTRAV = RECIPOP DINCOME /NOINT;

RUN;

OUTPUT OUT=RESFILE2 RESIDUAL=DE;

DATA THREE;

MERGE TWO RESFILE2;

BY POP;

LNSQDE = LOG(DE**2);

RUN;

PROC REG;

TITLE 'Residual regression for Park test on transformed equation';

MODEL LNSQDE = LNPOP;

RUN;

PROC REG;

TITLE 'Inherently per-capita model';

MODEL DEXPTRAV = DINCOME;

RUN;

OUTPUT OUT=RESFILE3 RESIDUAL=D2E;

DATA FOUR;

MERGE THREE RESFILE3;

BY POP;

LNSQD2E = LOG(D2E**2);

RUN;

PROC REG;

TITLE 'Residual regression for Park test on inherently per-capita equation';

MODEL LNSQD2E = LNPOP;

RUN;

PROC MODEL;

TITLE 'White Test for Heteroskedasticity';

PARMS B0 B1;

exptrav = B0 + B1*income;

FIT exptrav /WHITE;

RUN;

PROC MODEL;

TITLE 'Refitting with White Standard Errors';

PARMS B0 B1 B2;

exptrav = B0 + B1*income;

FIT exptrav /GMM KERNEL=(BART,1,0);

INSTRUMENTS income;

RUN;

Engle Curve for Travel Expenditure

The REG Procedure

Model: MODEL1

Dependent Variable: EXPTRAV EXPTRAV

Number of Observations Read / 51
Number of Observations Used / 51
Analysis of Variance
Source / DF / Sum of
Squares / Mean
Square / F Value / PrF
Model / 1 / 8313049205 / 8313049205 / 543.43 / <.0001
Error / 49 / 749572261 / 15297393
Corrected Total / 50 / 9062621467
Root MSE / 3911.18819 / R-Square / 0.9173
Dependent Mean / 11862 / Adj R-Sq / 0.9156
Coeff Var / 32.97331
Parameter Estimates
Variable / Label / DF / Parameter
Estimate / Standard
Error / tValue / Pr|t|
Intercept / Intercept / 1 / 1027.20093 / 718.30255 / 1.43 / 0.1591
INCOME / INCOME / 1 / 0.04541 / 0.00195 / 23.31 / <.0001

Residual regression for Park test

The REG Procedure

Model: MODEL1

Dependent Variable: LNSQRES

Number of Observations Read / 51
Number of Observations Used / 51
Analysis of Variance
Source / DF / Sum of
Squares / Mean
Square / F Value / PrF
Model / 1 / 51.51720 / 51.51720 / 6.71 / 0.0126
Error / 49 / 376.06639 / 7.67482
Corrected Total / 50 / 427.58359
Root MSE / 2.77035 / R-Square / 0.1205
Dependent Mean / 13.59789 / Adj R-Sq / 0.1025
Coeff Var / 20.37336
Parameter Estimates
Variable / DF / Parameter
Estimate / Standard
Error / tValue / Pr|t|
Intercept / 1 / -1.12446 / 5.69567 / -0.20 / 0.8443
LNPOP / 1 / 0.97397 / 0.37593 / 2.59 / 0.0126

Estimation of transformed model

The REG Procedure

Model: MODEL1

Dependent Variable: DEXPTRAV

Number of Observations Read / 51
Number of Observations Used / 51

Note:No intercept in model. R-Square is redefined.

Analysis of Variance
Source / DF / Sum of
Squares / Mean
Square / F Value / PrF
Model / 2 / 0.00031264 / 0.00015632 / 89.78 / <.0001
Error / 49 / 0.00008532 / 0.00000174
Uncorrected Total / 51 / 0.00039796
Root MSE / 0.00132 / R-Square / 0.7856
Dependent Mean / 0.00233 / Adj R-Sq / 0.7769
Coeff Var / 56.69934
Parameter Estimates
Variable / DF / Parameter
Estimate / Standard
Error / tValue / Pr|t|
RECIPOP / 1 / 1161.79589 / 406.37632 / 2.86 / 0.0062
DINCOME / 1 / 0.04738 / 0.00671 / 7.06 / <.0001

Residual regression for Park test on transformed equation

The REG Procedure

Model: MODEL1

Dependent Variable: LNSQDE

Number of Observations Read / 51
Number of Observations Used / 51
Analysis of Variance
Source / DF / Sum of
Squares / Mean
Square / F Value / PrF
Model / 1 / 84.54693 / 84.54693 / 11.17 / 0.0016
Error / 49 / 370.76519 / 7.56664
Corrected Total / 50 / 455.31212
Root MSE / 2.75075 / R-Square / 0.1857
Dependent Mean / -16.34038 / Adj R-Sq / 0.1691
Coeff Var / -16.83408
Parameter Estimates
Variable / DF / Parameter
Estimate / Standard
Error / tValue / Pr|t|
Intercept / 1 / 2.51998 / 5.65538 / 0.45 / 0.6579
LNPOP / 1 / -1.24773 / 0.37327 / -3.34 / 0.0016

Inherently per-capita model

The REG Procedure

Model: MODEL1

Dependent Variable: DEXPTRAV

Number of Observations Read / 51
Number of Observations Used / 51
Analysis of Variance
Source / DF / Sum of
Squares / Mean
Square / F Value / PrF
Model / 1 / 0.00002767 / 0.00002767 / 14.41 / 0.0004
Error / 49 / 0.00009407 / 0.00000192
Corrected Total / 50 / 0.00012174
Root MSE / 0.00139 / R-Square / 0.2273
Dependent Mean / 0.00233 / Adj R-Sq / 0.2115
Coeff Var / 59.53729
Parameter Estimates
Variable / DF / Parameter
Estimate / Standard
Error / tValue / Pr|t|
Intercept / 1 / -0.00192 / 0.00113 / -1.69 / 0.0976
DINCOME / 1 / 0.10889 / 0.02869 / 3.80 / 0.0004

White Test for Heteroskedasticity

The MODEL Procedure

Model Summary
Model Variables / 1
Parameters / 2
Equations / 1
Number of Statements / 1
Model Variables / exptrav
Parameters / B0 B1
Equations / exptrav
The Equation to Estimate is
exptrav = / F(B0(1), B1(income))
NOTE: At OLS Iteration 1 CONVERGE=0.001 Criteria Met.

White Test for Heteroskedasticity

The MODEL Procedure

OLS Estimation Summary

Data Set Options
DATA= / FOUR
Minimization Summary
Parameters Estimated / 2
Method / Gauss
Iterations / 1
Final Convergence Criteria
R / 0
PPC / 0
RPC(B0) / 10170305
Object / 0.953653
Trace(S) / 15297393
Objective Value / 14697495
Observations Processed
Read / 51
Solved / 51

White Test for Heteroskedasticity

The MODEL Procedure

NonlinearOLSSummaryofResidualErrors
Equation / DF Model / DF Error / SSE / MSE / Root MSE / R-Square / Adj R-Sq / Label
exptrav / 2 / 49 / 7.4957E8 / 15297393 / 3911.2 / 0.9173 / 0.9156 / EXPTRAV
Nonlinear OLS Parameter Estimates
Parameter / Estimate / Approx Std Err / tValue / Approx
Pr > |t|
B0 / 1027.201 / 718.3 / 1.43 / 0.1591
B1 / 0.04541 / 0.00195 / 23.31 / <.0001
Number of Observations / Statistics for System
Used / 51 / Objective / 14697495
Missing / 0 / Objective*N / 749572261
Heteroscedasticity Test
Equation / Test / Statistic / DF / PrChiSq / Variables
exptrav / White's Test / 2.10 / 2 / 0.3502 / Cross of all vars

Refitting with White Standard Errors

The MODEL Procedure

Model Summary
Model Variables / 1
Parameters / 3
Equations / 1
Number of Statements / 1
Model Variables / exptrav
Parameters / B0 B1 B2
Equations / exptrav
The Equation to Estimate is
exptrav = / F(B0(1), B1(income))
Instruments / 1 income
NOTE: At GMM Iteration 0 convergence assumed because OBJECTIVE=1.530624E-30 is almost zero (<1E-12).

Refitting with White Standard Errors

The MODEL Procedure

GMM Estimation Summary

Data Set Options
DATA= / FOUR
Minimization Summary
Parameters Estimated / 2
Kernel Used / BARTLETT
l(n) / 1
Method / Gauss
Iterations / 0
Final Convergence Criteria
R / 1
PPC / 0
RPC / .
Object / .
Trace(S) / 15297393
Objective Value / 1.53E-30
Observations Processed
Read / 51
Solved / 51

Refitting with White Standard Errors

The MODEL Procedure

NonlinearGMMSummaryofResidualErrors
Equation / DF Model / DF Error / SSE / MSE / Root MSE / R-Square / Adj R-Sq / Label
exptrav / 2 / 49 / 7.4957E8 / 15297393 / 3911.2 / 0.9173 / 0.9156 / EXPTRAV
Nonlinear GMM Parameter Estimates
Parameter / Estimate / Approx Std Err / tValue / Approx
Pr > |t|
B0 / 1027.201 / 621.6 / 1.65 / 0.1048
B1 / 0.04541 / 0.00253 / 17.94 / <.0001
Number of Observations / Statistics for System
Used / 51 / Objective / 1.531E-30
Missing / 0 / Objective*N / 7.806E-29
GMM Test Statistics
Test / DF / Statistic / Prob
Overidentifying Restrictions / 0 / 0.00 / .