Econometric Analysis of Panel Data
• Panel Data Analysis
– Random Effects
•
•
•
•
Assumptions
GLS Estimator
Panel-Robust Variance-Covariance Matrix
ML Estimator
– Hypothesis Testing
• Test for Random Effects
• Fixed Effects vs. Random Effects
Panel Data Analysis
• Random Effects Model
yit  xit' β  ui  eit (t  1, 2,..., Ti )

y i  Xi β  ui iTi  ei (i  1, 2,..., N )
– ui is random, independent of eit and xit.
– Define eit = ui + eit the error components.
Random Effects Model
• Assumptions
– Strict Exogeneity
E (eit | X)  0, E (ui | X)  0  E (e it | X)  0
• X includes a constant term, otherwise E(ui|X)=u.
– Homoschedasticity
Var (eit | X)   e2 , Var (ui | X)   u2 , Cov(ui , eit )  0
 Var (e it | X)   e2   e2   u2
– Constant Auto-covariance (within panels)
Var (εi | X)   e2ITi   u2iTi iT' i
Random Effects Model
• Assumptions
– Cross Section Independence
Var (ε i | X)  i   e2 ITi   u2 iTi iT' i
1 0
0 
2
Var (ε | X)  Ω  


0
0
0 
0 


N 
Random Effects Model
• Extensions
– Weak Exogeneity
E (e it | xi1 , xi 2 ,..., xiTi )  E (e it | Xi )  0
E (e it | xi1 , xi 2 ,..., xit )  0
E (e it | xit )  0
– Heteroscedasticity
2


 eit
2
2
Var (e it | Xi )   ui  Var (eit | Xi )   ui   2

 ei
Random Effects Model
• Extensions
– Serial Correlation
  eit 1  vit
yit  x β  ui  eit , eit  
 i eit 1  vit
'
it
– Spatial Correlation
yit  xit' β  e it , e it    wije jt  eit , eit  ui  vit
j
Model Estimation: GLS
• Model Representation
y i  Xi β  εi , εi  ui iTi  ei
E (εi | Xi )  0
Var (εi | Xi )  i   e2 ITi   u2 iTi iT' i
2
2




T

2
e
i u
  e Qi 
ITi  Qi 
2
e


1
where Qi  ITi  iTi iT' i
Ti


Model Estimation: GLS
• GLS
ˆβ  ( XΩ 1X) 1 XΩ 1y   N X  1X 
GLS
  i 1 i i i 
1
1
N
1
1
ˆ

Var (βGLS )  ( XΩ X)   i 1 Xi i1Xi 


2



1
1
e
where i  2 Qi  2
ITi  Qi 
2
e 
 e  Ti u


and 
1/2
i

 e2
1 

ITi  Qi
Qi 
2
2
 e 
 e  Ti u





1
X

 i 1 i i y i
N
Model Estimation: RE-OLS
• Partial Group Mean Deviations
yit  xit' β  e it  xit' β  (ui  eit )
yi  xi' β  (ui  ei )
 i  1 
 e2
 e2  Ti u2
yit  i yi  (xit'  i xi' )β  [(1  i )ui  (eit  i ei )]
yit  xit' β  e it
Model Estimation: RE-OLS
• Model Assumptions
E (e it | xi' )  0
Var (e it | xi' )  (1  i ) 2  u2  (1  2i / Ti  i 2 / Ti ) e2   e2
Cov(e it , e is | xi' )  (1  i ) 2  u2  (2i / Ti  i 2 / Ti ) e2  0, t  s
Note : i  1 
• OLS
 e2
 e2  Ti u2
N
βˆ OLS  ( X' X)1 X' y    i 1 Xi' Xi 


1

N
i 1
Xi y i
ˆ (βˆ )  ˆ 2 ( X' X) 1  ˆ 2  N X' X 
Var
OLS
e
e
 i 1 i i 
ˆ 2  εˆ ' εˆ / ( NT  K ), εˆ  y  Xβˆ
e
1
Model Estimation: RE-OLS
• Need a consistent estimator of :
ˆi  1 
ˆ e2
ˆ e2  Tiˆ u2
– Estimate the fixed effects model to obtain ˆ e
– Estimate the between model to obtain T ˆ u2  ˆ v2
– Or, estimate the pooled model to obtain ˆ e2  ˆ u2
– Based on the estimated large sample variances, it
is safe to obtain 0  ˆ  1
2
Model Estimation: RE-OLS
• Panel-Robust Variance-Covariance Matrix
– Consistent statistical inference for general
heteroscedasticity, time series and cross section
correlation.
ˆ (βˆ )  E[(βˆ  β)(βˆ  β) ']
Var
1
N
   i 1 X Xi    i 1 X εˆ εˆ Xi    i 1 Xi' Xi 

 


N
'
i
N
1
'
'
i i i
1
N
Ti
N
Ti
Ti
N
Ti
' 
' ˆ ˆ 
' 


  i 1  t 1 xit xit
x
x
e
e
x
x

   i 1  t 1  s 1 it is it is   i 1  t 1 it it 
εˆ i  y i  Xi βˆ , eˆit  yit  xit' βˆ
1
Model Estimation: ML
• Log-Likelihood Function
yit  xit' β  (ui  eit )  xit' β  e it
y i  Xi β  ε i
(t  1, 2,..., Ti )
(i  1, 2,..., N )
εi ~ iidn(0, i ), i   e2ITi   u2iTi iT' i

Ti
1
1
lli (β,  ,  | y i , Xi )   ln  2   ln i  εi i1εi
2
2
2
2
e
2
u
Model Estimation: ML
• ML Estimator
N
2
2
ˆ
ˆ
ˆ
(β,  e ,  u ) ML  arg max  i 1 lli (β,  e2 ,  u2 | y i , Xi )
where
Ti
1
1
ln  2   ln i  εi i1εi
2
2
2
Ti
1   e2  T  u2 
2
  ln  2 e   ln 

2
2   e2

2
2

1   Ti
Ti
'
2
'
u
  ( yit  xit β)  
 2   t 1 ( yit  xit β)  2
  e  Ti u2  t 1
 
2 e  
lli (β,  e2 ,  u2 | y i , Xi )  
Hypothesis Testing
To Pool or Not To Pool, Continued
• Test for Var(ui) = 0, that is
Cov(e it ,e is )  Cov(ui  eit ,ui  eis )  Cov(eit ,eis )
– If Ti=T for all i, the Lagrange-multiplier test statistic
(Breusch-Pagan, 1980) is:


2
2
N
T


2
'
ˆ
e
NT  eˆ ( I N  J T )eˆ 
NT   i 1  t 1 it

2
LM 

1


1
~

(1)


N
T
'


2
2 T  1 
eˆ eˆ
2 T  1   eˆit



i 1
t 1


βˆ 
'
where eˆit  yit   xit 1  
, J T  iT iT'
uˆ  Pooled
Hypothesis Testing
To Pool or Not To Pool, Continued
– For unbalanced panels, the modified BreuschPagan LM test for random effects (Baltagi-Li, 1990)
is:




2
2
Ti
 N

ˆ
T
e
 i1 i
  i 1  t 1 it

2
LM 

1
~

(1)
N
Ti


N
2
2  i 1 Ti (Ti  1)   i 1  t 1 eˆit




N
2

– Alternative one-side test:
LM ~ N (0,1) under H 0
P  Value : Prn ( z  LM )
Hypothesis Testing
To Pool or Not To Pool, Continued
• References
– Baltagi, B. H., and Q. Li, A Langrange Multiplier Test for the Error
Components Model with Incomplete Panels, Econometric Review, 9,
1990, 103-107.
– Breusch, T. and A. Pagan, “The LM Test and Its Applications to Model
Specification in Econometrics,” Review of Economic Studies, 47, 1980,
239-254.
Hypothesis Testing
Fixed Effects vs. Random Effects
H 0 : Cov(ui , xit' )  0 (random effects)
H1 : Cov(ui , xit' )  0 ( fixed effects)
Estimator
Random Effects
E(ui|Xi) = 0
Fixed Effects
E(ui|Xi) =/= 0
GLS or RE-OLS
(Random Effects)
Consistent and
Efficient
Inconsistent
LSDV or FE-OLS
(Fixed Effects)
Consistent
Inefficient
Consistent
Possibly Efficient
Hypothesis Testing
Fixed Effects vs. Random Effects
• Fixed effects estimator is consistent under H0
and H1; Random effects estimator is efficient
under H0, but it is inconsistent under H1.
• Hausman Test Statistic


'
1


H  βˆ RE  βˆ FE Var (βˆ RE )  Var (βˆ FE )  βˆ RE  βˆ FE
~  2 (# βˆ FE ), provided # βˆ FE  # βˆ RE (no intercept )
Hypothesis Testing
Fixed Effects vs. Random Effects
• Alternative (Asym. Eq.) Hausman Test
– Estimate any of the random effects models
( yit   yi )  (xit'   xi' )β  (xit'  xi' ) γ  eit
(or , random effects model : yit  xit' β  ( xit'  xi' ) γ  eit )
( yit   yi )  (xit'   xi' )β  xi' γ  eit
( yit   yi )  (xit'   xi' )β  xit' γ  eit
– F Test that g = 0
H0 : γ  0 
H 0 : Cov(ui , xit )  0
Hypothesis Testing
Fixed Effects vs. Random Effects
• Ahn-Low Test (1996)
– Based on the estimated errors (GLS residuals) of
the random effects model, estimate the following
regression:
eˆ    ( X  ˆ X )β  X γ  e
it
it
i
 NTR ~  (# γ )
2
2
i
it
Hypothesis Testing
Fixed Effects vs. Random Effects
• References
– Ahn, S.C., and S. Low, A Reformulation of the Hausman Test for
Regression Models with Pooled Cross-Section Time-Series Data,
Journal of Econometrics, 71, 1996, 309-319.
– Baltagi, B.H., and L. Liu, Alternative Ways of Obtaining Hausman’s Test
Using Artificial Regressions, Statistics and Probability Letters, 77, 2007,
1413-1417.
– Hausman, J.A., Specification Tests in Econometrics, Econometrica, 46,
1978, 1251-1271.
– Hausman, J.A. and W.E. Taylor, Panel Data and Unobservable Individual
Effects, Econometrics, 49, 1981, 1377-1398.
– Mundlak, Y., On the Pooling of Time Series and Cross-Section Data,
Econometrica, 46, 1978, 69-85.
Example: Investment Demand
• Grunfeld and Griliches [1960]
I it  i   Fit  g Cit  e it
– i = 10 firms: GM, CH, GE, WE, US, AF, DM, GY, UN,
IBM; t = 20 years: 1935-1954
– Iit = Gross investment
– Fit = Market value
– Cit = Value of the stock of plant and equipment