chapter 15
DESCRIPTION
Panel Data Models. Chapter 15. Prepared by Vera Tabakova, East Carolina University. Chapter 15: Panel Data Models. 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. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 15
Panel Data Models
Prepared by Vera Tabakova, East Carolina University
Chapter 15: Panel Data Models
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
Slide 15-2Principles of Econometrics, 3rd Edition
Chapter 15: Panel Data Models
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.
Slide 15-3Principles of Econometrics, 3rd Edition
15.1 Grunfeld’s Investment Data
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
Slide 15-4Principles of Econometrics, 3rd Edition
(15.1)
(15.2)
,it it itINV f V K
1 2 2 3 3it it it it it it ity x x e
15.2 Sets of Regression Equations
Slide 15-5Principles of Econometrics, 3rd Edition
(15.3a)
(15.3b)
, 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
15.2 Sets of Regression Equations
Slide 15-6Principles of Econometrics, 3rd Edition
(15.4a)
(15.4b)
, 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
15.2 Sets of Regression Equations
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
Slide 15-7Principles of Econometrics, 3rd Edition
(15.5)
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
15.2 Sets of Regression Equations
Slide 15-8Principles of Econometrics, 3rd Edition
15.2 Sets of Regression Equations
Let Di be a dummy variable equal to 1 for the Westinghouse
observations and 0 for the General Electric observations.
Slide 15-9Principles of Econometrics, 3rd Edition
(15.6)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
15.2 Sets of Regression Equations
Slide 15-10Principles of Econometrics, 3rd Edition
15.3 Seemingly Unrelated Regressions
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.
Slide 15-11Principles of Econometrics, 3rd Edition
(15.7) , , ,cov ,GE t WE t GE WEe e
15.3 Seemingly Unrelated Regressions
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.
Slide 15-12Principles of Econometrics, 3rd Edition
2 2,, and GE WE GE WE
15.3 Seemingly Unrelated Regressions
Slide 15-13Principles of Econometrics, 3rd Edition
15.3.1 Separate or Joint Estimation?
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.
Slide 15-14Principles of Econometrics, 3rd Edition
15.3.1 Separate or Joint Estimation?
In this case
Slide 15-15Principles of Econometrics, 3rd Edition
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
15.3.1 Separate or Joint Estimation?
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.
Slide 15-16Principles of Econometrics, 3rd Edition
0 ,: 0GE WEH
2 2, (1) 0 under .GE WELM Tr H
15.3.1 Separate or Joint Estimation?
Testing for correlated errors for three equations:
Slide 15-17Principles of Econometrics, 3rd Edition
0 12 13 23: 0H
2 2 2 212 13 23 (3)LM T r r r
15.3.1 Separate or Joint Estimation?
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.
Slide 15-18Principles of Econometrics, 3rd Edition
12
2 1
M i
iji j
LM T r
15.3.2 Testing Cross-Equation Hypotheses
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.
Slide 15-19Principles of Econometrics, 3rd Edition
(15.8)0 1, 1, 2, 2, 3, 3,: , ,GE WE GE WE GE WEH
15.4 The Fixed Effects Model
We cannot consistently estimate the 3×N×T parameters in (15.9) with only NT total observations.
Slide 15-20Principles of Econometrics, 3rd Edition
(15.9)
(15.10)
1 2 2 3 3it it it it it it ity x x e
1 1 2 2 3 3, ,it i it it
15.4 The Fixed Effects Model
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.
Slide 15-21Principles of Econometrics, 3rd Edition
(15.11)1 2 2 3 3it i it it ity x x e
15.4.1 A Dummy Variable Model
This specification is sometimes called the least squares dummy
variable model, or the fixed effects model.
Slide 15-22Principles of Econometrics, 3rd Edition
(15.12)
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
15.4.1 A Dummy Variable Model
Slide 15-23Principles of Econometrics, 3rd Edition
15.4.1 A Dummy Variable Model
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:
Slide 15-24Principles of Econometrics, 3rd Edition
(15.13)0 11 12 1
1 1
:
: the are not all equal
N
i
H
H
1 2 3it it it itINV V K e
15.4.1 A Dummy Variable Model
Slide 15-25Principles of Econometrics, 3rd Edition
15.4.1 A Dummy Variable Model
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.
Slide 15-26Principles of Econometrics, 3rd Edition
1749128 522855 948.99
522855 200 12
R U
U
SSE SSE JF
SSE NT K
15.4.2 The Fixed Effects Estimator
Slide 15-27Principles of Econometrics, 3rd Edition
(15.14)1 2 2 3 3 1, ,it i it it ity x x e t T
(15.15)
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
15.4.2 The Fixed Effects Estimator
Slide 15-28Principles of Econometrics, 3rd Edition
(15.16)
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
(15.17)2 3it it it ity x x e
15.4.2 The Fixed Effects Estimator
Slide 15-29Principles of Econometrics, 3rd Edition
15.4.2 The Fixed Effects Estimator
Slide 15-30Principles of Econometrics, 3rd Edition
(15.18) .1098 .3106
(se*) (.0116) (.0169)
itit itINV V K
2*ˆ 2e SSE NT
2 2 198 188 1.02625NT NT N
15.4.2 The Fixed Effects Estimator
Slide 15-31Principles of Econometrics, 3rd Edition
15.4.2 The Fixed Effects Estimator
Slide 15-32Principles of Econometrics, 3rd Edition
(15.19)
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
15.4.3 Fixed Effects Estimation Using a Microeconomic Panel
Slide 15-33Principles of Econometrics, 3rd Edition
15.4.3 Fixed Effects Estimation Using a Microeconomic Panel
Slide 15-34Principles of Econometrics, 3rd Edition
15.5 The Random Effects Model
Slide 15-35Principles of Econometrics, 3rd Edition
(15.20)
(15.22)
1 1i iu
(15.21) 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
15.5 The Random Effects Model
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.
Slide 15-36Principles of Econometrics, 3rd Edition
(15.23)
(15.24)
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
15.5.1 Error Term Assumptions
Slide 15-37Principles of Econometrics, 3rd Edition
(15.25)
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
15.5.1 Error Term Assumptions
Slide 15-38Principles of Econometrics, 3rd Edition
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
15.5.1 Error Term Assumptions
Slide 15-39Principles of Econometrics, 3rd Edition
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
(15.26)
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
15.5.1 Error Term Assumptions
Slide 15-40Principles of Econometrics, 3rd Edition
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
15.5.1 Error Term Assumptions
Slide 15-41Principles of Econometrics, 3rd Edition
(15.27)
2
2 2
cov( , )corr( , )
var( ) var( )it is u
it isu eit is
v vv v
v v
15.5.2 Testing for Random Effects
Slide 15-42Principles of Econometrics, 3rd Edition
(15.28)
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
15.5.3 Estimation of the Random Effects Model
Slide 15-43Principles of Econometrics, 3rd Edition
(15.29)
(15.30)
* * * * *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
(15.31)2 21 e
u eT
15.5.4 An Example Using the NLS Data
Slide 15-44Principles of Econometrics, 3rd Edition
2 2
ˆ .1951ˆ 1 1 .7437
5 .1083 .0381ˆ ˆe
u eT
15.5.5a Endogeneity in the Random Effects Model
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.
Slide 15-45Principles of Econometrics, 3rd Edition
it i itv u e
15.5.5b The Fixed Effects Estimator in a Random Effects Model
Slide 15-46Principles of Econometrics, 3rd Edition
(15.32)
(15.33)1 2 2 3 3
1 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
15.5.5b The Fixed Effects Estimator in a Random Effects Model
Slide 15-47Principles of Econometrics, 3rd Edition
(15.34)
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
15.5.5c A Hausman Test
We expect to find
because Hausman proved that
Slide 15-48Principles of Econometrics, 3rd Edition
(15.35) , , , ,
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
15.5.5c A Hausman Test
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.
Slide 15-49Principles of Econometrics, 3rd Edition
, ,
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
Keywords
Slide 15-50Principles of Econometrics, 3rd Edition
Balanced panel Breusch-Pagan test Cluster corrected standard errors Contemporaneous correlation Endogeneity Error components model Fixed effects estimator Fixed effects model Hausman test Heterogeneity Least squares dummy variable
model LM test Panel corrected standard errors Pooled panel data regression
Pooled regression Random effects estimator Random effects model Seemingly unrelated regressions Unbalanced panel
Chapter 15 Appendix
Slide 15-51Principles of Econometrics, 3rd Edition
Appendix 15A Estimation of Error Components
Appendix 15A Estimation of Error Components
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
Appendix 15A Estimation of Error Components
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
Appendix 15A Estimation of Error Components
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