Empirical Example

Baltagi and Griffin (1983) considered the following gasoline demand equation:

log Car = a + в ilog 7Ф + e2log ppMO + в 3log Cf + U (12.40)

where Gas/Car is motor gasoline consumption per auto, Y/N is real income per capita, PMO/ PGdp is real motor gasoline price and Car/N denotes the stock of cars per capita. This panel consists of annual observations across eighteen OECD countries, covering the period 1960-1978. The data for this example are provided on the Springer web site as GASOLINE. DAT. Table 12.1
gives the Stata output for the Within estimator using xtreg, fe. This is the regression described in (12.5) and computed as in (12.9). The Within estimator gives a low price elasticity for gasoline demand of -.322. The F-statistic for the significance of the country effects described in (12.14) yields an observed value of 83.96. This is distributed under the null as an F(17,321) and is statistically significant. This F-statistic is printed by Stata below the fixed effects output. In EViews, one invokes the test for redundant effects after running the fixed effects regression.

Table 12.1 Fixed Effects Estimator – Gasoline Demand Data

Coef.

Std. Err.

T

P> |t|

[95% Conf. Interval]

log(Y/N)

0.6622498

0.073386

9.02

0.000

0.5178715

0.8066282

l°g(PMG/PGDP )

-0.3217025

0.0440992

-7.29

0.000

-0.4084626

-0.2349425

log(Car/N)

-0.6404829

0.0296788

-21.58

0.000

-0.6988725

-0.5820933

Constant

2.40267

0.2253094

10.66

0.000

1.959401

2.84594

sigma u

0.34841289

sigma e

0.09233034

Rho

0.93438173

(fraction of

variance due to u i)

Table 12.2 gives the Stata output for the Between estimator using xtreg, be. This is based on the regression given in (12.24). The Between estimator yields a high price elasticity of gasoline demand of -.964. These results were also verified using TSP.

Table 12.2 Between Estimator – Gasoline Demand Data

Coef.

Std. Err.

T

P> |t|

[95% Conf. Interval]

log(Y/N)

0.9675763

0.1556662

6.22

0.000

0.6337055

1.301447

log(PMG/PGDP )

-0.9635503

0.1329214

-7.25

0.000

-1.248638

-0.6784622

log(Car/N)

-0.795299

0.0824742

-9.64

0.000

-0.9721887

-0.6184094

Constant

2.54163

0.5267845

4.82

0.000

1.411789

3.67147

Table 12.3 gives the Stata output for the random effect model using xtreg, re. This is the Swamy and Arora (1972) estimator which yields a price elasticity of -.420. This is closer to the Within estimator than the Between estimator.

Table 12.3 Random Effects Estimator – Gasoline Demand Data

Coef.

Std. Err.

T

P> |t|

[95% Conf. Interval]

log(Y/N)

0.5549858

0.0591282

9.39

0.000

0.4390967

0.6708749

log(PMG/PGDP )

-0.4203893

0.0399781

-10.52

0.000

-0.498745

-0.3420336

log(Car/N)

-0.6068402

0.025515

-23.78

0.000

-0.6568487

-0.5568316

Constant

1.996699

0.184326

10.83

0.000

1.635427

2.357971

sigma u

0.19554468

sigma e

0.09233034

Rho

0.81769

(fraction of

variance due to u i)

Table 12.4 Gasoline Demand Data. One-way Error Component Results

ві

в2

вз

P

OLS

0.890

-0.892

-0.763

0

(0.036)*

(0.030)*

(0.019)*

WALHUS

0.545

-0.447

-0.605

0.75

(0.066)

(0.046)

(0.029)

AMEMIYA

0.602

-0.366

-0.621

0.93

(0.066)

(0.042)

(0.029)

SWAR

0.555

-0.402

-0.607

0.82

(0.059)

(0.042)

(0.026)

IMLE

0.588

-0.378

-0.616

0.91

(0.066)

(0.046)

(0.029)

* These are biased standard errors when the true model has error component disturbances (see Moulton, 1986). Source: Baltagi and Griffin (1983). Reproduced by permission of Elsevier Science Publishers B. V. (North-Holland).

Table 12.5 Gasoline Demand Data. Wallace and Hussain (1969) Estimator

Dependent Variable: GAS

Method: Panel EGLS (Cross-section random effects)

Sample: 1960 1978

Periods included: 19

Cross-sections included: 18

Total panel (balanced) observations: 342

Wallace and Hussain estimator of component variances

Coefficient

Std. Error

t-Statistic

Prob.

C

1.938318

0.201817

9.604333

0.0000

log(Y/N)

0.545202

0.065555

8.316682

0.0000

log(PMG/PGDP )

-0.447490

0.045763

-9.778438

0.0000

log(Car/N)

-0.605086

0.028838

-20.98191

0.0000

Effects Specification

S. D.

Rho

Cross-section random

0.196715

0.7508

Idiosyncratic random

0.113320

0.2492

Table 12.4 gives the parameter estimates for OLS and three feasible GLS estimates of the slope coefficients along with their standard errors, and the corresponding estimate of p defined in (12.16). These were obtained using EViews by invoking the random effects estimation on the individual effects and choosing the estimation method from the options menu. Breusch’s (1987) iterative maximum likelihood was computed using Stata(xtreg, mle) and TSP.

Table 12.5 gives the EViews output for the Wallace and Hussain (1969) random effects estima­tor, while Table 12.6 gives the EViews output for the Amemiya (1971) random effects estimator. Note that EViews calls the Amemiya estimator Wansbeek and Kapteyn (1989) since the latter paper generalizes this method to deal with unbalanced panels with missing observations, see Baltagi (2008) for details. Table 12.6 gives the Stata maximum likelihood output.

Table 12.6 Gasoline Demand Data. Wansbeek and Kapteyn (1989) Estimator

Dependent Variable: GAS

Method: Panel EGLS (Cross-section random effects)

Sample: 1960 1978

Periods included: 19

Cross-sections included: 18

Total panel (balanced) observations: 342

Wallace and Hussain estimator of component variances

Coefficient

Std. Error

t-Statistic

Prob.

C

2.188322

0.216372

10.11372

0.0000

log(Y/N)

0.601969

0.065876

9.137941

0.0000

log(PMG/PGDP )

-0.365500

0.041620

-8.781832

0.0000

log(Car/N)

-0.620725

0.027356

-22.69053

0.0000

Effects Specification

S. D.

Rho

Cross-section random

0.343826

0.9327

Idiosyncratic random

0.092330

0.0673

Table 12.7 Gasoline Demand Data. Random Effects Maximum Likelihood Estimator

. xtreg c y p car, mle

Random-effects ML regression

Number of obs =

342

Group variable (i): coun

Number of groups =

18

Random effects u i ~ Gaussian

Obs per group: min =

19

avg =

19.0

max =

19

LR chi2(3) =

609.75

Log likelihood = 282.47697

Prob > chi2 =

0.0000

c

Coef.

Std. Err.

z

P> |z|

[95% Conf. Interval]

log(Y/N)

.5881334

.0659581

8.92

0.000

.4588578

.717409

log(PMG/PGDP )

-.3780466

.0440663

-8.58

0.000

-.464415

-.2916782

log(Car/N)

-.6163722

.0272054

-22.66

0.000

-.6696938

-.5630506

cons

2.136168

.2156039

9.91

0.000

1.713593

2.558744

sigma u

.2922939

.0545496

.2027512

.4213821

sigma e

.0922537

.0036482

.0853734

.0996885

rho

.9094086

.0317608

.8303747

.9571561

Likelihood-ratio test of sigma u = 0: chibar2(01)= 463.97 Prob >= chibar2 = 0.000

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