Panel Data Analysis
Using GAUSS
4
Kuan-Pin Lin
Portland State University
Panel Data Analysis
Hypothesis Testing

Panel Data Model Specification





Pool or Not To Pool
Random Effects vs. Fixed Effects
Heterscedasticity
Time Serial Correlation
Spatial Correlation
Fixed Effects vs. Random Effects

Hypothesis Testing
yit  xit'   ui  eit
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-LS
Consistent and
(Random Effects) Efficient
Inconsistent
LSDV or FE-LS
(Fixed Effects)
Consistent
Possibly Efficient
Consistent
Inefficient
Random Effects vs. Fixed 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 )
Random Effects vs. Fixed Effects

Alternative Hausman Test
(Mundlak Approach)


Estimate the random effects model with the group
means of time variant regressors:
yit  xit' β  xi' γ  eit
F Test that g = 0
H 0 : γ  0  H 0 : Cov(ui , xit )  0
Hypothesis Testing

Fixed Effects Model
yit  xit' β  ui  eit  yit  xit' β  eit
eit ~ iid (0,  e2 )
where yit  yit  yi , xit  xit  xi , eit  eit  ei
1 T
1 T '
1 T
'
yi   yit , xi   xit , ei   eit
T t 1
T t 1
T t 1

Random Effects Model
yit  xit' β  ui  eit  yit  xit' β  eit
where yit  yit   yi , xit  xit   xi , eit  eit   ei
  1
 e2
T  u2   e2
Heteroscedasticity

The Null Hypothesis
H 0 : eit ~ iid (0,  e2 )

Based on the auxiliary regression
eˆit2    xit' g  vit
where eˆ  y  x' ˆ , v ~ iid (0,  2 )
it

it
it
it
v
LM test statistic is NR2 ~ 2(K), N is total number
of observation (i,t)s.
Cross Sectional Correlation

The Null Hypothesis
H 0 : Cov(eit , e jt )  0 t

Based on the estimated correlation coefficients
ˆij 

 eˆ eˆ
 eˆ  eˆ
t it
2
t it
, i  1, 2,..., N  1; j  i
jt
t
2
jt
Breusch-Pagan LM Test (Breusch, 1980)

As T  ∞ (N fixed)
N ( N  1) 

 T   ˆ ~  

2


i 1 j i 1
N 1
LM BP
N
2
ij
2
Cross Sectional Correlation

Bias adjusted Breusch-Pagan LM Test (Pesaran,
et.al. 2008)
Adj
LM BP

N 1 N (T  K ) 
ˆij2  ˆ ij
2
 N (0,1) as T  , then N  


N ( N  1) i 1 j i 1
ˆ ij
1
trace(M i M j )
T K
 ij2  Var[(T  K ) ˆij2 ]  a1[trace(M i M j )2 ]  2a2{trace[(M i M j )(M i M j )]}
where ˆ ij  E[(T  K ) ˆij2 ] 
M i  IT  Xi ( Xi' Xi ) 1 Xi
2
 (T  K  8)(T  K  2)  24 
1
a2  3 
,
a

a

,T K 8
1
2

2
(T  K )
 (T  K  2)(T  K  2)(T  K  4) 
Time Serial Correlation

The Model and Null Hypothesis
yit  xit'   eit , eit   eit 1  vit , vit ~ iid (0,  v2 )
H0 :   0

LM Test Statistic
2
NT  eˆ 'eˆ 1 
LM 
 ˆ ' ˆ 
T 1  e e 
2
2
where eˆit  yit  xit' ˆ
 N T ˆ ˆ 
2   eit eit 1 
NT
 i 1 Nt  2T
 ~  2 (1)

T 1 
ˆ2 
e

it


 i 1 t 1

Joint Hypothesis Testing
Random Effects and Time Serial Correlation

The Model
yit  xit'   eit , eit   eit 1  vit
yit  yit   yi , xit  xit   xi , eit  eit   ei
  1

 e2
2
,
v
~
iid
(0,

it
v)
2
T  u2   e
Joint Test for AR(1) and Random Effects
H 0 :  u2  0,   0
Joint Hypothesis Testing
Random Effects and Time Serial Correlation

Based on OLS residuals εˆ  y  Xβˆ :
LM  0, 2 0
u
NT 2
 A2  4 AB  2TB 2  ~  2 (2)

2(T  1)(T  2)
εˆ '(I N  iT iT' )εˆ
εˆ ' εˆ 1
A
 1, B 
εˆ ' εˆ
εˆ ' εˆ
Joint Hypothesis Testing
Random Effects and Time Serial Correlation

Marginal Tests for AR(1) & Random Effects
LM  2 0
u

Robust Test for AR(1) & Random Effects
*
LM  2 0
u

NT A2
NT 2 B 2
2

~  (1); LM  0 
~  2 (1)
2(T  1)
T 1
NT (2 B  A)2
NT 2 ( B  A / T )2
2
*

~  (1); LM  0 
~  2 (1)
2(T  1)(1  2 / T )
(T  1)(1  2 / T )
Joint Test Equivalence
LM  0, 2 0  LM * 0  LM  2 0  LM * 2 0  LM  0 ~  2 (2)
u
u
u
Panel Data Analysis
Extensions

Seeming Unrelated Regression


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Dynamic Panel Data Analysis



Allowing Cross-Equation Dependence
Fixed Coefficients Model
Using FD Specification
IV and GMM Methods
Spatial Panel Data Analysis


Using Spatial Weights Matrix
Spatial Lag and Spatial Error Models
References
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Baltagi, B., Li, Q. (1995) Testing AR(1) against MA(1) disturbances in an
error component model. Journal of Econometrics, 68, 133-151.
Baltagi, B., Bresson, G., Pirotte, A. (2006) Joint LM test for
homoscedasticity in a one-way error component model. Journal of
Econometrics, 134, 401-417.
Bera, A.K., W. Sosa-Escudero and M. Yoon (2001), Tests for the error
component model in the presence of local misspecification, Journal of
Econometrics 101, 1–23.
Breusch, T.S. and A.R. Pagan (1980), The Lagrange multiplier test and its
applications to model specification in econometrics, Review of Economic
Studies 47, 239–253.
Pesaran, M.H. (2004), General diagnostic tests for cross-section
dependence in panels, Working Paper, Trinity College, Cambridge.
Pesaran, M.H., Ullah, A. and Yamagata, T. (2008), A bias-adjusted LM test
of error cross-section independence, The Econometrics Journal,11, 105–
127.