15.1 Grunfeld’s Investment Data

15.2 Sets of Regression Equations

15.3 Seemingly Unrelated Regressions

15.4 The Fixed Effects Model

15.4 The Random Effects Model

The different types of panel data sets can be described as:

“long and narrow,” with “long” describing the time dimension and

“narrow” implying a relatively small number of cross sectional units;

“short and wide,” indicating that there are many individuals observed

over a relatively short period of time;

“long and wide,” indicating that both N and T are relatively large.

The data consist of T = 20 years of data (1935-1954) for N = 10 large firms.

Let yit = INVit and x2it = Vit and x3it = Kit

,it it itINV f V K

1 2 2 3 3it it it it it it ity x x e

, 1 2 , 3 , ,

, 1 2 , 3 , ,

1, ,20

1, ,20

GE t GE t GE t GE t

WE t WE t WE t WE t

INV V K e t

INV V K e t

1 2 2 3 3 1, 2; 1, ,20it it it ity x x e i t

, 1, 2, , 3, , ,

, 1, 2, , 3, , ,

1, ,20

1, ,20

GE t GE GE GE t GE GE t GE t

WE t WE WE WE t WE WE t WE t

INV V K e t

INV V K e t

1 2 2 3 3 1, 2; 1, ,20it i i it i it ity x x e i t

Assumption (15.5) says that the errors in both investment functions (i) have zero mean, (ii) are homoskedastic with constant variance, and (iii) are not correlated over time; autocorrelation does not exist. The two equations do have different error variances

2, , , ,

2, , , ,

0 var cov , 0

0 var cov , 0

GE t GE t GE GE t GE s

WE t WE t WE WE t WE s

E e e e e

E e e e e

2 2 and .GE WE

Let Di be a dummy variable equal to 1 for the Westinghouse

observations and 0 for the General Electric observations.

1, 1 2, 2 3, 3it GE i GE it i it GE it i it itINV D V D V K D K e

This assumption says that the error terms in the two equations, at the same point in time, are correlated. This kind of correlation is called a contemporaneous correlation.

, , ,cov ,GE t WE t GE WEe e

Econometric software includes commands for SUR (or SURE) that

carry out the following steps:

(i) Estimate the equations separately using least squares;

(ii)Use the least squares residuals from step (i) to estimate

;

(iii)Use the estimates from step (ii) to estimate the two equations jointly

within a generalized least squares framework.

2 2,, and GE WE GE WE

There are two situations where separate least squares estimation is

just as good as the SUR technique :

(i) when the equation errors are not contemporaneously correlated;

(ii)when the same explanatory variables appear in each equation.

If the explanatory variables in each equation are different, then a test

to see if the correlation between the errors is significantly different

from zero is of interest.

In this case

22,2

, 2 2

ˆ 207.58710.53139

ˆ ˆ 777.4463 104.3079GE WE

GE WEGE WE

r

20 20

, , , , ,1 1

1 1ˆ ˆ ˆ ˆ ˆ

3GE WE GE t WE t GE t WE tt tGE WE

e e e eTT K T K

3.GE WEK K

Testing for correlated errors for two equations:

LM = 10.628 > 3.84

Hence we reject the null hypothesis of no correlation between the

errors and conclude that there are potential efficiency gains from

estimating the two investment equations jointly using SUR.

0 ,: 0GE WEH

2 2, (1) 0 under .GE WELM Tr H

Testing for correlated errors for three equations:

0 12 13 23: 0H

2 2 2 212 13 23 (3)LM T r r r

Testing for correlated errors for M equations:

Under the null hypothesis that there are no contemporaneous

correlations, this LM statistic has a χ2-distribution with M(M–1)/2

degrees of freedom, in large samples.

12

2 1

M i

iji j

LM T r

Most econometric software will perform an F-test and/or a Wald χ2–test; in the context of SUR equations both tests are large sample approximate tests.

The F-statistic has J numerator degrees of freedom and (MTK) denominator degrees of freedom, where J is the number of hypotheses, M is the number of equations, and K is the total number of coefficients in the whole system, and T is the number of time series observations per equation. The χ2-statistic has J degrees of freedom.

0 1, 1, 2, 2, 3, 3,: , ,GE WE GE WE GE WEH

We cannot consistently estimate the 3×N×T parameters in (15.9) with only NT total observations.

1 2 2 3 3it it it it it it ity x x e

1 1 2 2 3 3, ,it i it it

All behavioral differences between individual firms and over time are

captured by the intercept. Individual intercepts are included to

“control” for these firm specific differences.

1 2 2 3 3it i it it ity x x e

This specification is sometimes called the least squares dummy

variable model, or the fixed effects model.

1 2 3

1 1 1 2 1 3, , , etc.

0 otherwise 0 otherwise 0 otherwisei i i

i i iD D D

11 1 12 2 1,10 10 2 2 3 3it i i i it it itINV D D D V K e

These N–1= 9 joint null hypotheses are tested using the usual F-test

statistic. In the restricted model all the intercept parameters are equal.

If we call their common value β1, then the restricted model is:

0 11 12 1

1 1

:

: the are not all equal

N

i

H

H

1 2 3it it it itINV V K e

We reject the null hypothesis that the intercept parameters for all

firms are equal. We conclude that there are differences in firm

intercepts, and that the data should not be pooled into a single model

with a common intercept parameter.

1749128 522855 948.99

522855 200 12

R U

U

SSE SSE JF

SSE NT K

1 2 2 3 3 1, ,it i it it ity x x e t T

1 2 2 3 31

1 T

it i it it itt

y x x eT

1 2 2 3 31 1 1 1

1 2 2 3 3

1 1 1 1T T T T

i it i it it itt t t t

i i i i

y y x x eT T T T

x x e

1 2 2 3 3

1 2 2 3 3

2 2 2 3 3 3

( )

( ) ( ) ( )

it i it it it

i i i i i

it i it i it i it i

y x x e

y x x e

y y x x x x e e

2 3it it it ity x x e

.1098 .3106

(se*) (.0116) (.0169)

itit itINV V K

2*ˆ 2e SSE NT

2 2 198 188 1.02625NT NT N

1 2 2 3 3i i i iy b b x b x

1 2 2 3 3 1, ,i i i ib y b x b x i N

1 1i iu

20, cov , 0, vari i j i uE u u u u

1 2 2 3 3

1 2 2 3 3

it i it it it

i it it it

y x x e

u x x e

Because the random effects regression error in (15.24) has two

components, one for the individual and one for the regression, the

random effects model is often called an error components model.

1 2 2 3 3

1 2 2 3 3

it it it it i

it it it

y x x e u

x x v

it i itv u e

0 0 0it i it i itE v E u e E u E e

2

2 2

var var

var var 2cov ,

v it i it

i it i it

u e

v u e

u e u e

There are several correlations that can be considered.

The correlation between two individuals, i and j, at the same

point in time, t. The covariance for this case is given by

cov , ( )

0 0 0 0 0

it jt it jt i it j jt

i j i jt it j it jt

v v E v v E u e u e

E u u E u e E e u E e e

The correlation between errors on the same individual (i) at

different points in time, t and s. The covariance for this case is

given by

2

2 2

cov , ( )

0 0 0

it is it is i it i is

i i is it i it is

u u

v v E v v E u e u e

E u E u e E e u E e e

The correlation between errors for different individuals in

different time periods. The covariance for this case is

cov , ( )

0 0 0 0 0

it js it js i it j js

i j i js it j it js

v v E v v E u e u e

E u u E u e E e u E e e

2

2 2

cov( , )corr( , )

var( ) var( )it is u

it isu eit is

v vv v

v v

1 2 2 3 3it it it ity x x e

1 2 2 3 3it it it ite y b b x b x

2

1 1

2

1 1

ˆ1

2 1 ˆ

N T

iti t

N T

iti t

eNT

LMT e

* * * * *1 1 2 2 3 3it it it it ity x x x v

* * * *1 2 2 2 3 3 3, 1 , ,it it i it it it i it it iy y y x x x x x x x

2 21 e

u eT

2 2

ˆ .1951ˆ 1 1 .7437

5 .1083 .0381ˆ ˆe

u eT

If the random error is correlated with any of the right-

hand side explanatory variables in a random effects model then the

least squares and GLS estimators of the parameters are biased and

inconsistent.

it i itv u e

1 2 2 3 31 1 1 1 1

1 2 2 3 3

1 1 1 1 1T T T T T

i it it it i itt t t t t

i i i i

y y x x u eT T T T T

x x u e

1 2 2 3 3 ( )it it it i ity x x u e

1 2 2 3 3

1 2 2 3 3

2 2 2 3 3 3

( )

( ) ( ) ( )

it it it i it

i i i i i

it i it i it i it i

y x x u e

y x x u e

y y x x x x e e

We expect to find

because Hausman proved that

, , , ,

1 2 1 22 2

, ,, ,se sevar var

FE k RE k FE k RE k

FE k RE kFE k RE k

b b b bt

b bb b

, ,var var 0.FE k RE kb b

, , , , , ,

, ,

var var var 2cov ,

var var

FE k RE k FE k RE k FE k RE k

FE k RE k

b b b b b b

b b

, , ,cov , var .FE k RE k RE kb b b

The test statistic to the coefficient of SOUTH is:

Using the standard 5% large sample critical value of 1.96, we reject the hypothesis that the estimators yield identical results. Our conclusion is that the random effects estimator is inconsistent, and we should use the fixed effects estimator, or we should attempt to improve the model specification.

, ,

1 2 1 22 2 2 2

, ,

.0163 (.0818) 2.3137

.0361 .0224se se

FE k RE k

FE k RE k

b bt

b b

Slide 15-50Principles of Econometrics, 3rd Edition

Slide 15-51Principles of Econometrics, 3rd Edition

Principles of Econometrics, 3rd Edition Slide 15-52

(15A.1)

(15A.2)

(15A.3)

1 2 2 3 3 ( )it it it i ity x x u e

2 2 2 3 3 3( ) ( ) ( )it i it i it i it iy y x x x x e e

2ˆ DVe

slopes

SSE

NT N K

Principles of Econometrics, 3rd Edition Slide 15-53

(15A.4)

(15A.5)

1 2 2 3 3 1, ,i i i i iy x x u e i N

1

22 2

2 21

22

var var var var var

1var

T

i i i i i itt

Te

u it ut

eu

u e u e u e T

Te

T T

T

Principles of Econometrics, 3rd Edition Slide 15-54

(15A.6)

(15A.7)

22 e BEu

BE

SSE

T N K

2 2

2 2 ˆˆ e e BE DV

u uBE slopes

SSE SSE

T T N K T NT N K