panel data analysis using gauss 4 kuan-pin lin portland state university
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Panel Data Analysis Using GAUSS
4
Kuan-Pin LinPortland State University
Panel Data AnalysisHypothesis 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'
'0
'1
: ( , ) 0 ( )
: ( , ) 0 ( )
it it i it
i it
i it
y u e
H Cov u random effects
H Cov u fixed effects
x
x
x
Estimator Random Effects
E(ui|Xi) = 0
Fixed Effects
E(ui|Xi) =/= 0
GLS or RE-LS
(Random Effects)
Consistent and Efficient
Inconsistent
LSDV or FE-LS
(Fixed Effects)
Consistent
Inefficient
Consistent
Possibly Efficient
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
2
ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )
ˆ ˆ ˆ~ (# ), # # ( )
RE FE RE FE RE FE
FE FE RE
H Var Var
provided no intercept
β β β β β β
β β β
Random Effects vs. Fixed Effects
Alternative Hausman Test(Mundlak Approach)Estimate the random effects model with the group
means of time variant regressors:
F Test that = 0
' 'it it i ity e x β x γ
0 0: 0 : ( , ) 0i itH H Cov u γ x
Hypothesis Testing
Fixed Effects Model
Random Effects Model
' ' 2
' '
1 1 1
~ (0, )
, ,
1 1 1, ,
it it i it it it it it e
it it i it it i it it i
T T T
i it i it i itt t t
y u e y e e iid
where y y y e e e
y y e eT T T
x β x β
x x x
x x
' '
2
2 2
, ,
1
it it i it it it it
it it i it it i it it i
e
u e
y u e y e
where y y y e e e
T
x β x β
x x x
Heteroscedasticity
The Null Hypothesis
Based on the auxiliary regression
LM test statistic is NR2 ~ 2(K), N is total number of observation (i,t)s.
20 : ~ (0, )it eH e iid
2 '
' 2
ˆ
ˆ ˆ, ~ (0, )
it it it
it it it it v
e v
where e y v iid
x
x
Cross Sectional Correlation
The Null Hypothesis
Based on the estimated correlation coefficients
Breusch-Pagan LM Test (Breusch, 1980) As T ∞ (N fixed)
0 : ( , ) 0it jtH Cov e e t
2 2
ˆ ˆˆ , 1, 2,..., 1;
ˆ ˆit jtt
ij
it jtt t
e ei N j i
e e
12 2
1 1
( 1)ˆ ~
2
N N
BP iji j i
N NLM T
Cross Sectional Correlation
Bias adjusted Breusch-Pagan LM Test (Pesaran, et.al. 2008)
21
1 1
2
2 2 21 2
' 1
2
ˆ ˆ( )2(0,1) ,
ˆ( 1)
1ˆˆ [( ) ] ( )
ˆ[( ) ] [ ( ) ] 2 { [( )( )]}
( )
(3
N Nij ijAdj
BPi j i ij
ij ij i j
ij ij i j i j i j
i T i i i i
T KLM N as T then N
N N
where E T K traceT K
Var T K a trace a trace
T Ka
MM
MM MM MM
M I X X X X2
1 2 2
8)( 2) 24 1, , 8
( 2)( 2)( 4) ( )
T Ka a T K
T K T K T K T K
Time Serial Correlation
The Model and Null Hypothesis
LM Test Statistic
' 21
0
, , ~ (0, )
: 0it it it it it it it vy e e e v v iid
H
x
2
2'2 2 1
21 21'
2
1 1
'
ˆ ˆˆ ˆ
~ (1)ˆ ˆ1 1 ˆ
ˆ ˆ
N T
it iti tN T
iti t
it it it
e eNT NT
LMT T
e
where e y
ee
ee
x
Joint Hypothesis TestingRandom Effects and Time Serial Correlation
The Model
Joint Test for AR(1) and Random Effects
'1
22
2 2
,
, ,
1 , ~ (0, )
it it it it it it
it it i it it i it it i
eit v
u e
y e e e v
y y y e e e
v iidT
x
x x x
20 : 0, 0uH
Joint Hypothesis TestingRandom Effects and Time Serial Correlation
Based on OLS residuals :
2
22 2 2
0, 0
'1
4 2 ~ (2)2( 1)( 2)
ˆ ˆ ˆ ˆ'( ) '1,
ˆ ˆ ˆ ˆ' '
u
N T T
NTLM A AB TB
T T
A B
ε I i i ε ε ε
ε ε ε ε
ˆˆ ε y Xβ
Joint Hypothesis TestingRandom Effects and Time Serial Correlation
Marginal Tests for AR(1) & Random Effects
Robust Test for AR(1) & Random Effects
Joint Test Equivalence
2
2 2 22 2
00~ (1); ~ (1)
2( 1) 1u
NT A NT BLM LM
T T
2
2 2 2* 2 * 2
00
(2 ) ( / )~ (1); ~ (1)
2( 1)(1 2 / ) ( 1)(1 2 / )u
NT B A NT B A TLM LM
T T T T
2 2 2
* * 20 00, 0 0 0
~ (2)u u u
LM LM LM LM LM
Panel Data AnalysisExtensions
Seeming Unrelated Regression Allowing Cross-Equation Dependence Fixed Coefficients Model
Dynamic Panel Data Analysis Using FD Specification IV and GMM Methods
Spatial Panel Data Analysis Using Spatial Weights Matrix Spatial Lag and Spatial Error Models
References 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.