Economics Chapter 11 Homework Wald Statistic Require The Unrestricted Regression The

Chapter 11

Models for Panel Data

◼ Exercises

1. The pooled least squares estimator is

The fixed effects regression can be computed just by including the three dummy variables since the

sample sizes are quite small. The results are

The F statistic for testing the hypothesis that the constant terms are all the same is

In order to estimate the random effects model, we need some additional parameter estimates.

The group means are

y

x

In the group means regression using these three observations, we obtain

There is only one degree of freedom, so this is the candidate for estimation of

22

/.

u

T



 + 

In the least

squares dummy variable (fixed effects) regression, we have an estimate of

2



of 79.183/26 = 3.045.

One ought to proceed with some caution at this point, but it is difficult to place much faith in the group

means regression with but a single degree of freedom, so this is probably a preferable estimator in any

Chapter 11 Models for Panel Data 97

For the LM test, we return to the pooled ordinary least squares regression. The necessary quantities

2. There is no effect on the coefficients of the other variables. For the dummy variable coefficients,

with the full set of n dummy variables, each coefficient is

*

y

= mean residual for the ith group in the

3. (a) The pooled OLS estimator will be

1

11

nn

i i i i i i

−

==



=  

   

   

b X X X y

where Xi and yi have Ti

(b) We seek to establish consistency, not unbiasedness. As such, we will ignore the degrees of freedom

correction, −K, in (11-40). Use n(T − 1) as the denominator. Thus, the question is whether

2

1 1 . 2

()

nT

i t it i

ee

==

  − =

98 Greene • Econometric Analysis, Seventh Edition

The second term will converge to zero as the center matrix converges to a constant Q and the

vectors converge to zero as b converges to . (We use the Slutsky theorem.) The third term

4. To find plim(1/n)LM = plim [T/(2(T − 1))]{[i(teit)2]/[iteit2] − 1}2 we can concentrate on the sums

inside the curled brackets. First, i(teit)2 = nT 2{(1/n)i [(1/T)teit]2} and i teit2 = nT(1/(nT))iteit2.

5. The ordinary least squares regression results are

R2 = 0.92803, ee = 146.761, 40 observations.

Variable Coefficient Standard Error

X2 1.83915 0.1534

Constant 3.60568 2.555

Period 1 −3.57906 1.723

Period 6 −1.49474 1.714

Chapter 11 Models for Panel Data 99

Estimated covariance matrix for the slopes:

1 2

1 0.0062209

For testing the hypotheses that the sets of dummy variable coefficients are zero, we will require the

sums of squared residuals from the restrictions. These are

Regression Sum of Squares

All variables included 146.761

The F statistics are therefore,

(1) F[9,25] = [(318.503 − 146.761)/9]/[146.761/25] = 3.251

(3) F[12,25] = [(585.622 − 146.761)/12]/[146.761/25] = 6.23

6. The covariance matrix would be

2 2 2 2 2

1, 1 1, 2 2, 1 2, 2

1, 1 0

u v u v

i t i t i t i t

it



= = = = = = = =

= =  +  +   

7. The two separate regressions are as follows:

Sample 1 Sample 2

b = xy/xx 4/5 = 0.8 6/10 = 0.6

ee = yy − bxy 20 − 4(4/5) = 84/5 10 − 6(6/10) = 64/10

To carry out a Lagrange multiplier test of the hypothesis of equal variances, we require the separate

and common variance estimators based on the restricted slope estimator. This, in turn, is the pooled

least squares estimator. For the combined sample, we obtain

2

ˆ



100 Greene • Econometric Analysis, Seventh Edition

not 39, because we are computing maximum likelihood estimators. The individual estimators are

e1e1/20 = (y1y1 − 2b(x1y1) + b2(x1x1))/20 = (20 − 2(2/3)4 + (2/3)25)/20 = 0.84444

This has one degree of freedom for the single restriction. The critical value from the chi-squared table is

3.84, so we would reject the hypothesis.

In order to compute a two step GLS estimate, we can use either the original variance estimates based

on the separate least squares estimates or those obtained above in doing the LM test. Since both pairs

The estimated sampling variance is 1/[(1/

2

1

ˆ)

x1x1 + (1/

2

ˆ



)x2x2] = 0.02688. This implies an

asymptotic standard error of (0.02688)2 = 0.16395. To test the hypothesis that  = 1, we would referz =

(0.632 − 1)/ 0.16395 = −2.245 to a standard normal table. This is reasonably large, and at the usual

significance levels, would lead to rejection of the hypothesis.

The Wald test is based on the unrestricted variance estimates. Using b = .632, the variance estimators

while the pooled estimator would be

2

ˆ



= [yy − 2b(xy) + b2(xx)]/40 = 0.583784. The statistic is

given at the end of Example 16.3,

We reach the same conclusion as before.

To compute the maximum likelihood estimators, we begin our iterations from the two separate

ordinary least squares estimates of b which produce estimates

2

1

ˆ



= 0.84 and

2

ˆ



= 0.32.

The iterations are

Iteration

2

1

ˆ



2

ˆ







Chapter 11 Models for Panel Data 101

Now, to compute the likelihood ratio statistic for a likelihood ratio test of the hypothesis of equal

Finally, we allow for cross-sectional correlation of the disturbances. Our initial estimate of b is the

pooled least squares estimator, 2/3. The estimates of the two variances are 0.84444 and 0.32222 as

which is not particularly large. The LM test statistic given in (9-37) is 1.533, which is well under

the critical value of 3.84. Thus, we would not reject the hypothesis of zero cross-section correlation.

Nonetheless, we proceed. The estimator is shown in (9-35). The two step FGLS and iterated maximum

likelihood estimates appear below.

Iteration

2

1

ˆ



2

ˆ



12

ˆ



ˆ



1 0.8521955 0.3202177 0.1597994 0.5731058

3 0.8529155 0.3203725 0.1609873 0.5726805

8. If all of the regressor matrices are the same, the estimator in (9-35) reduces to

ˆ



= (XX)−1

1

n

i=



{(1/i2)/[

1

n

i=



(1/j2)]}Xyi =

1

n

i=



wi bi

9. The various least squares estimators of the parameters are

Sample 1 Sample 2 Sample 3 Pooled

a 11.6644 5.42213 1.41116 8.06392

b 0.926881 1.06410 1.46885 1.05413

102 Greene • Econometric Analysis, Seventh Edition

(Values of ee in parentheses above are based on the pooled slope estimator.) The FGLS estimator

and its estimated asymptotic covariance matrix are

Note that the FGLS estimator of the slope is closer to the 1.46885 of sample 3 (the highest of the

three OLS estimates). This is to be expected since the third group has the smallest residual variance.

The LM test statistic is based on the pooled regression,

Applications

As usual, the applications below require econometric software. The computations can be done with any

modern software package, so no specific program is recommended.

Application 11.1

+—————————————————-+

| Ordinary least squares regression |

| LHS=I Mean = 145.9582 |

| Standard deviation = 216.8753 |

| WTS=none Number of observs. = 200 |

| Model size Parameters = 3 |

+——–+————–+—————-+——–+——–+———-+

+——–+————–+—————-+——–+——–+———-+

Chapter 11 Models for Panel Data 103

+—————————————————-+

| Ordinary least squares regression |

| LHS=I Mean = 145.9582 |

| Standard deviation = 216.8753 |

+———————————————————————+

| Covariance matrix for the model is adjusted for data clustering. |

| Sample of 200 observations contained 10 clusters defined by |

+———————————————————————+

+——–+————–+—————-+——–+——–+———-+

The standard errors increase substantially. This is at least suggestive that there is

correlation across observations within the groups. A formal test would be based on one

of the panel models below. When the random effects model is fit by maximum likelihood,

–> REGRESS ; Lhs = I ; Rhs = F,C,one ; Panel ; Pds=20 ; Fixed $

+—————————————————-+

| Least Squares with Group Dummy Variables |

| Ordinary least squares regression |

| LHS=I Mean = 145.9583 |

104 Greene • Econometric Analysis, Seventh Edition

+——–+————–+—————-+——–+——–+———-+

+——————————————————————–+

| Test Statistics for the Classical Model |

+——————————————————————–+

| Model Log Likelihood Sum of Squares R-squared |

|(1) Constant term only -1359.15096 .9359943929D+07 .0000000 |

+——————————————————————–+

| Hypothesis Tests |

| Likelihood Ratio Test F Tests |

| Chi-squared d.f. Prob. F num. denom. P value |

|(2) vs (1) 285.604 9 .00000 66.932 9 190 .00000 |

+————————————+

| Listed Calculator Results |

The F statistic of 49.18 is far larger than the critical value, so the hypothesis of

equal constant terms is rejected.

–> REGRESS ; Lhs = I ; Rhs = F,C,one

; Panel ; Pds=20 ; Random $

+————————————————–+

| Random Effects Model: v(i,t) = e(i,t) + u(i) |

| Estimates: Var[e] = .278446D+04 |

+————————————————–+

+——–+————–+—————-+——–+——–+———-+

+——–+————–+—————-+——–+——–+———-+

Chapter 11 Models for Panel Data 105

1

+————–

1| 2.45500

–> CALC ; List ; Ctb(.95,2) $

+————————————+

| Listed Calculator Results |

+————————————+

Result = 5.991465

The Hausman statistic is quite small, which suggests that the random effects approach

is consistent with the data.

Application 11.2

create ; logc=log(cost/pfuel)

; logp1=log(pmtl/pfuel)

Namelist ; cd = logp1,logp2,logp3,logp4,logp5 $

create

; p11=.5* logp1^2

; p22=.5* logp2^2

Namelist ; tl = p11,p12,p13,p14,p15,p22,p23,p24,p25,p33,p34,p35,p44,p45,p55$

Namelist ; z = loadfctr,stage,points $

regress;lhs=logc;rhs=one,logq,logq2,cd,z $

+—————————————————-+

| Ordinary least squares regression |

| LHS=LOGC Mean = .7723984 |

+—————————————————-+

106 Greene • Econometric Analysis, Seventh Edition

+——–+————–+—————-+——–+——–+———-+

+——–+————–+—————-+——–+——–+———-+

Constant| 20.3856176 22.8643711 .892 .3735

LOGQ | .95227889 .01832119 51.977 .0000 -1.11237037

LOGQ2 | .06568531 .01060839 6.192 .0000 1.45687077

?

? Turns out the translog model cannot be computed with the firm

? dummy variables. I’ll use the Cobb Douglas form.

?

regress;lhs=logc;rhs= one,logq,logq2,cd ; panel ; pds=ti $

+—————————————————-+

| OLS Without Group Dummy Variables |

| Ordinary least squares regression |

| LHS=LOGC Mean = .7723984 |

| Standard deviation = 1.074424 |

+—————————————————-+

| Panel Data Analysis of LOGC [ONE way] |

| Unconditional ANOVA (No regressors) |

| Source Variation Deg. Free. Mean Square |

+—————————————————-+

+——–+————–+—————-+——–+——–+———-+

+——–+————–+—————-+——–+——–+———-+

LOGQ | .93708702 .01772733 52.861 .0000 -1.11237037

LOGQ2 | .07754607 .01211431 6.401 .0000 1.45687077

+—————————————————-+

| Least Squares with Group Dummy Variables |

| Ordinary least squares regression |

| LHS=LOGC Mean = .7723984 |

| Standard deviation = 1.074424 |

Chapter 11 Models for Panel Data 107

+—————————————————-+

| Panel:Groups Empty 0, Valid data 25 |

| Smallest 2, Largest 15 |

| Average group size 10.24 |

+—————————————————-+

+——–+————–+—————-+——–+——–+———-+

+——–+————–+—————-+——–+——–+———-+

LOGQ | .66448665 .03580894 18.556 .0000 -1.11237037

LOGQ2 | -.00955723 .01280811 -.746 .4563 1.45687077

+——————————————————————–+

| Test Statistics for the Classical Model |

+——————————————————————–+

| Model Log Likelihood Sum of Squares R-squared |

|(1) Constant term only -381.12407 .2943684435D+03 .0000000 |

+——————————————————————–+

| Hypothesis Tests |

| Likelihood Ratio Test F Tests |

| Chi-squared d.f. Prob. F num. denom. P value |

|(2) vs (1) 659.911 24 .00000 117.116 24 231 .00000 |

+——————————————————————–+

+————————————————–+

| Random Effects Model: v(i,t) = e(i,t) + u(i) |

| Estimates: Var[e] = .418468D-02 |

| Var[u] = .127110D-01 |

+————————————————–+

+——–+————–+—————-+——–+——–+———-+

+——–+————–+—————-+——–+——–+———-+

LOGQ | .79769706 .02494671 31.976 .0000 -1.11237037

LOGQ2 | .02011534 .01130089 1.780 .0751 1.45687077

+—————————————————-+

| OLS Without Group Dummy Variables |

| Ordinary least squares regression |

| LHS=LOGC Mean = .7723984 |

| Standard deviation = 1.074424 |

| WTS=none Number of observs. = 256 |

+—————————————————-+

| Panel Data Analysis of LOGC [ONE way] |

| Unconditional ANOVA (No regressors) |

| Source Variation Deg. Free. Mean Square |

| Between 272.013 24. 11.3339 |

+—————————————————-+

+——–+————–+—————-+——–+——–+———-+

+——–+————–+—————-+——–+——–+———-+

LOADFCTR| -.94688632 .18441823 -5.134 .0000 .54786115

STAGE | -.00021794 .402227D-04 -5.418 .0000 507.879666

POINTS | .00199712 .00031682 6.304 .0000 72.9843750

+—————————————————-+

| Least Squares with Group Dummy Variables |

| Ordinary least squares regression |

| LHS=LOGC Mean = .7723984 |

| Standard deviation = 1.074424 |

| WTS=none Number of observs. = 256 |

+—————————————————-+

| Panel:Groups Empty 0, Valid data 25 |

+—————————————————-+

+——–+————–+—————-+——–+——–+———-+

+——–+————–+—————-+——–+——–+———-+

LOADFCTR| -.89457348 .14242570 -6.281 .0000 .54786115

Chapter 11 Models for Panel Data 109

LOGP1 | 1.38217070 .72421015 1.909 .0575 .37999226

+——————————————————————–+

| Test Statistics for the Classical Model |

+——————————————————————–+

| Model Log Likelihood Sum of Squares R-squared |

|(1) Constant term only -381.12407 .2943684435D+03 .0000000 |

+——————————————————————–+

| Hypothesis Tests |

| Likelihood Ratio Test F Tests |

| Chi-squared d.f. Prob. F num. denom. P value |

|(2) vs (1) 659.911 24 .00000 117.116 24 231 .00000 |

|(4) vs (1) 1521.362 34 .00000 2470.054 34 221 .00000 |

|(4) vs (3) 344.355 24 .00000 26.140 24 221 .00000 |

+——————————————————————–+

+————————————————–+

| Random Effects Model: v(i,t) = e(i,t) + u(i) |

| Estimates: Var[e] = .349594D-02 |

| (High values of LM favor FEM/REM over CR model.) |

| Baltagi-Li form of LM Statistic = 170.10 |

+————————————————–+

+——–+————–+—————-+——–+——–+———-+

+——–+————–+—————-+——–+——–+———-+

LOADFCTR| -1.07921018 .13264921 -8.136 .0000 .54786115

STAGE | -.00016415 .672354D-04 -2.441 .0146 507.879666

POINTS | .00044792 .00035950 1.246 .2128 72.9843750