Econometric Analysis of Panel Data
• Hypothesis Testing
– Specification Tests
• Fixed Effects vs. Random Effects
– Heteroscedasticity
– Autocorrelation
• Serial Autocorrelation
• Spatial Autocorrelation
• More on Autocorrelation
Hypothesis Testing
• Heteroscedasticity
2


e

yit  xit' β  ui  eit , Var (ui )   u2i , Var (eit )   2it

 ei
• Serial Correlation
  vit 1  eit
yit  x β  ui  vit , vit  
 i vit 1  eit
• Spatial Correlation
'
it
yit  xit' β  ui  vit , vit    wij v jt  eit
j
Hypothesis Testing
• Heteroscedasticity
yit  xit' β   it ,  it  ui  eit
Var (ui )   u2i   u2 hu (fi' u ), i  1,..., N (e.g., fi'  xi' )
Var (eit )   e2it   e2 he ( z it'  e ), i  1,..., N , t  1,..., T (e.g., z it'  xit' )
  u2i   e2   u2 hu (fi' u )   e2
 2
Var ( it )    u   e2it   u2   e2 he (z it'  e )
 2
2
2
'
'
 ui   eit   u hu (fi  u )  he ( z it e )
Note : h(.)  0, h(0)  1, h' (0)  0,  u2  0,  e2  0
A special case of  e2i is  e2i   e2 he ( zi' e ), i  1,..., N
then  2   u2   e2i   u2   e2 he ( zi' e )
Hypothesis Testing
Test for Homoscedasticity
• If u2=0 (constant effects or pooled model), then
yit  xit' β  ui  eit
Var (eit )   e2it   e2 he (z it'  e ) or  e2 he ( zi' e )
• LM Test (Breusch and Pagan, 1980)
1
LM  2 0 ( e  0)  G ' Z ( Z ' Z ) 1 Z 'G ~  2 (#  e )
u
2
G  eˆ 2 / ˆ e2  1, ˆ e2  eˆ 'eˆ / NT , eˆ  y  Xβˆ  uˆ
Z  [z it' , i  1,..., N , t  1,..., T ]
ESS
~  2 (#  e )
2
(2) Regress eˆ 2 on  0  Z e , obtain NTR 2 ~  2 (#  e )
(1) Regress [eˆ 2 / ˆ e2  1] on Z , obtain
H0 : e  0
Hypothesis Testing
Test for Homoscedasticity
• If u2>0 (random effects), then
yit  xit' β  ui  eit
Var (ui )   u2i   u2 hu (fi'u )
Var (eit )   e2it   e2 he (z it'  e ) or  e2 he ( zi' e )
• LM Tests (Baltagi, Bresson, and Pirott, 2006)
H 0 : e  0 | u  0
H 0 : u  0 | e  0
H 0 :  u  0,  e  0
Hypothesis Testing
Test for Homoscedasticity
• Marginal LM Test H 0 :  e  0 |  u  0,  u2  0
Regress ˆit2 on  0  z it'  e , obtain NTR 2 ~  2 (#  e )
where ˆ  uˆ  eˆ  y  x' ˆ
it
i
it
it
it
(e.g., z it'  xit' , not including constant )
• See, Montes-Rojas and Sosa-Escudero (2011)
Hypothesis Testing
Test for Homoscedasticity
2
• Marginal LM Test H 0 :  u  0 |  e  0,  u  0
Regress ˆi2 on  0  fi' u , obtain NR 2 ~  2 (#  u )
T
1
where ˆi   ˆit , ˆit  uˆi  eˆit  yit  xit' ˆ
T t 1
(e.g ., fi'  xi' )
• Joint LM Test
H 0 :  u  0,  e  0 |  u2  0
– Sum of the above two marginal test statistics
(approximately)
– See, Montes-Rojas and Sosa-Escudero (2011)
Hypothesis Testing
Testing for Homoscedasticity
• References
– Batagi, B.H., G. Bresson, and A. Priotte, Joint LM Test for
Homoscedasticity in a One-Way Error Component
Model, Journal of Econometrics, 134, 2006, 401-417.
– Breusch, T. and A. Pagan, “A Simple Test of
Heteroscedasticity and Random Coefficient Variations,”
Econometrica, 47, 1979, 1287-1294.
– Montes-Rojas, G. and W. Sosa-Escudero, Robust Tests for
Heteroscedasticity in the One-Way Error Components
Model, Journal of Econometrics, 2011, forthcoming.
Hypothesis Testing
• Serial Correlation AR(1) in a Random Effects Model
yit  xit' β  ui  vit , ui ~ iid (0,  u2 )
vit   vit 1  eit ~ iid (0,  e2 )
• LM Test for Serial Correlation and Random Effects
H 0 :  u2  0,   0
H 0 :   0 (assuming  u2  0 or pooled model)
H 0 :   0 (assuming  u2  0 or random effects model)
Hypothesis Testing
Test for Serial Correlation
• LM Test Statistics: Notations
Based on OLS residuals of the restricted model (i.e. pooled
model with no serial correlation)
2


eˆit 



'
eˆ ( I N  J T )eˆ
i 1  t 1
 1
A

1

N T
eˆ 'eˆ
2
ˆ
e
 it
N
T
i 1 t 1
N
eˆ eˆ 1
B ' 
eˆ eˆ
'
T
 eˆ eˆ
i 1 t  2
N T
it it 1
2
ˆ
e
 it
i 1 t 1
Hypothesis Testing
Test for Serial Correlation
• Marginal LM Test Statistic for a Pooled Model
NT
LM  0 (  0) 
A2 ~  2 (1)
2(T  1)
2
u
– See Breusch and Pagan (1980)
• Marginal LM Test Statistic for Serial Correlation
NT 2 2
LM  2 0 (   0) 
B ~  2 (1)
u
(T  1)
– See Breusch and Godfrey (1981)
Hypothesis Testing
Test for Serial Correlation
• Robust LM Test Statistic
NT
2
LM  0 (  0) 
 2B  A ~  2 (1)
2(T  1)(1  2 / T )
*
2
u
2
NT
2
LM * 2 0 (   0) 
B

A
/
T
~  2 (1)


u
(T  1)(1  2 / T )
• See Baltagi and Li (1995)
Hypothesis Testing
Test for Serial Correlation
• Joint LM Test Statistic for Pooled Model with Serial
Correlation
NT
LM (  0,   0) 
A2  4 AB  2TB 2  ~  2 (2)

2(T  1)(T  2)
2
u
LM ( u2  0,   0)  LM  0 ( u2  0)  LM  2 0 (   0)
u
 LM * 0 ( u2  0)  LM  2 0 (   0)
u
 LM  0 ( u2  0)  LM * 2 0 (   0)
u
• See Baltagi and Li (1995)
Hypothesis Testing
Test for Serial Correlation
• LM Test Statistic for a Fixed Effects Model
NT 2 eˆ 'eˆ 1
2
LM fixed effects (   0) 
~

(1)
'
T  1 eˆ eˆ
eˆ  y  Xβˆ  residuals of mean deviation regression
– See Baltagi, Econometric Analysis of Panel Data (2008)
Hypothesis Testing
Test for Serial Correlation
• References
– Breusch, T. and A. Pagan, “A Simple Test of Heteroscedasticity and
Random Coefficient Variations,” Econometrica, 47, 1979, 1287-1294.
– Breusch, T. and A. Pagan, “The LM Test and Its Applications to Model
Specification in Econometrics,” Review of Economic Studies, 47, 1980,
239-254.
– Breusch, T. and L.G. Godfrey, A Review of Recent Work on Testing for
Autocorrelation in Dynamic Simultaneous Models, in D.A. Currie, R.
Nobay and D. Peel (eds.), Macroeconomic Analysis, Essays in
Macroeconomics and Economics (Croom Helm, London), 63-100.
– Baltagi, B.H. and Q. Li, Testing AR(1) Against MA(1) Disturbances in an
Error Components Model, Journal of Econometrics, 68, 1995, 133-151.
Autocorrelation
• AR(1)
yit  xit' β  ui  eit
eit   eit 1  it (t  1, 2,..., Ti ; i  1, 2,..., N )
• Assumptions
E (it | xit' )  0
Var (it | xit' )   2  Var (eit | xit' )   e2  (1   2 ) 2
Cov(it , eit 1 )  0, Cov(eit , ui )  0, Cov (eit , xit' )  0
Cov(ui , xit' )  0 (random effects only )
Autocorrelation
• AR(1) Model Estimation (Paris-Winsten)
– Begin with =0, estimate the model
yit  xit' β  ui  eit  eˆit  yit  xit' βˆ  uˆi
eˆ eˆ


ˆ 
  eˆ
N
Ti
t  2 it it 1
Ti
2
i 1
t 1 it
i 1
N
– Transform variables according
zit*  zit  ˆ zit 1 , t  1
zi*1  1  ˆ 2 zi1
Autocorrelation
– Estimate the transformed model
yit*  x*'it β  ui*  eit*  eˆit  yit  xit' βˆ  uˆi
eˆ eˆ


ˆ 
  eˆ
N
Ti
t  2 it it 1
Ti
2
i 1
t 1 it
i 1
N
– Iterate until
 βˆ 
 
 ̂ 
converges
Autocorrelation
• Notational Complexity with time lags in unbalanced panel data
(Unbalanced unequal space panel data)
i
t
zit
zit-1
z*it
1
1
z11
.
(1-2)1/2 z11
i
t
zit
1
2
z12
z11
z12 -z11
1
1
z11
1
3
.
.
.
1
2
z12
1
4
z14
.
(1-2)1/2 z14
1
4
z14
1
5
z15
z14
z15 -z14
1
5
z15
2
1
z21
.
(1-2)1/2 z21
2
1
z21
2
2
.
.
.
2
4
z24
2
3
.
.
.
2
5
z25
2
4
z24
.
(1-2)1/2 z24
3
3
z33
2
5
z25
z24
z25 -z24
3
1
.
.
.
3
2
.
.
.
3
3
z33
.
(1-2)1/2 z33
3
4
.
z33
.
3
5
.
.
.
Autocorrelation
• Hypothesis Testing
– Modified Durbin-Watson Test Statistic (Bhargava,
Franzini, Narendranathan, 1982)
d1
eˆ



 
N
Ti
i 1
t 1
N
i 1
it
Ti
 eˆit 1 
2
2
ˆ
e
t 1 it
– LBI Test Statistic (Baltagi-Wu, 1999)
• For unbalanced unequal spaced panel data
Example: Investment Demand
• Grunfeld and Griliches [1960]
I it  i   Fit   Cit   it
 it   it 1  eit ~ iid (0,  e2 )
– 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