Econometric Analysis of Panel Data
• Panel Data Analysis: Extension
– Generalized Random Effects Model
• Seemingly Unrelated Regression
– Cross Section Correlation
• Parametric representation
• Spatial dependence defined by cross section
contiguity or distance
Panel Data Analysis: Extension
• Generalized Random Effects Model
yit  xit' β  ui  eit (t  1, 2,..., T ; i  1, 2,...N )
 yit  xit' β  xi' γ  ui  eit
 yit  x β   s 1 xis' γ s  ui  eit
'
it
T
yit  xit' βt  ui  eit
(βt  β  γ t , t  1, 2,..., T )

y t  Xt β t  ε t , ε t  u  et
E (εt | Xt )  0, Var (ε t | Xt )  Var (u | Xt )  Var (et | Xt )
Cov(εt , ε s | Xt , X s )  Var (u | Xt )
Panel Data Analysis : Extension
• Seemingly Unrelated Regression
y t  Xt β t  ε t
 y1   X1
y   0
 2  
  
  
yT   0
0
X2
0
0   β1   ε1 
0   β 2   ε 2 

   
   
XT  βT  εT 
y  Xβ  ε
E (ε | X)  0, Var (ε | X)  Σ
Panel Data Analysis : Extension
• Cross Section Correlation
y t  Xt β t  ε t
Cov(εt , ε s )  0
– Unobserved heterogeneity: fixed effects or
random effects
– OLS with robust inference
– GLS allowing time serial correlation
Panel Data Analysis : Extension
• Cross Section Correlation
– Parametric Representation
yit  t  l ( yit )  xit' β   it
 it  t  l ( it )  it
y t  t  l (y t )  Xt βt  εt
εt  t  l (εt )  υt
E (υt )  0, Var (υt )   v2I
Panel Data Analysis : Extension
• Spatial Lag Variables
l ( yit )   j 1 wij y jt  l (y t )  Wy t ,   t
N
l ( it )   j 1 wij jt  l (εt )  Wεt ,   t
N
• Spatial Weights
wii  0
wij  0, i  j

N
i 1
wij  1
Panel Data Analysis : Extension
• Spatial Lag Model
y t  Wy t  Xt β  εt
E (εt | Xt , W)  0, Var (εt | Xt , W)   2I
– OLS is biased and inconsistent
(I  W )y t  Xt β  εt
Cov(εt ,Wy t )   2W (I  W )1  0
– Unobserved heterogeneity: fixed effects or random
effects  it  ui  eit
– Observed heterogeneity
y t  Wy t  Xt β  WXt γ  εt
Panel Data Analysis : Extension
• Spatial Error Model
y t  Xt β  εt , εt  Wεt  υt
E (υt | Xt , W)  0, Var (υt | Xt , W)   2I
 (I  W )y t  (I  W ) Xt β  υt
– Unobserved heterogeneity
et  Wet  υt where  it  ui  eit
• Fixed effects
• Random effects εt  Wεt  υt
– Observed heterogeneity
y t  Xt β  WXt γ  εt , εt  Wεt  υt
Panel Data Analysis : Extension
• Spatial Panel Data Analysis
– Model specification could be a mixed structure of
spatial lag and spatial error model.
– Unobserved heterogeneity could be fixed effects or
random effects.
– OLS is biased and inconsistent; Consistent IV or
2SLS should be used, with robust inference.
– If normality assumption of the model is maintained,
efficient ML estimation could be used but with
computational complexity.
– Efficient GMM estimation is recommended.
Panel Data Analysis : Extension
• Panel Spatial Model Estimation
– IV / 2SLS / GMM
– Instrumental variables for the spatial lag
variable Wyt: [Xt, WXt, W2Xt,…]
– W is a predetermined spatial weights matrix
based on geographical contiguity or distance:
wii  0; wij  0, i  j; i 1 wij  1
N
Panel Data Analysis : Extension
• Space-Time Dynamic Model
y t  Wy t   y t 1  Xt β  εt
y t  Wy t  Wy t 1   y t 1  Xt β  εt
y t  Wy t  Wy t 1   y t 1  Xt β  WXt γ  εt
• Arellano-Bond estimator may be extended
to include cross section correlation in the
space-time dynamic models.
Example: U. S. Productivity
• The Model (Munnell [1988])
– One-way panel data model
ln( gspit )  0  1 ln(capit )   2 ln(hwyit )  3 ln( waterit )
  4 ln(utilit )  5 ln(empit )  6unempit  ui  eit
– 48 U.S. lower states
– 17 years from 1970 to 1986
– Variables: gsp (gross state output); cap (private
capital); 3 components of public capital (hwy, water,
util); emp (labor employment); unemp (unemployment
rate)
Example: U. S. Productivity
• Spatial Panel Data Model
– Cross Section Correlation
• Cross section dependence is defined by state
contiguity: if state i is adjacent with state j, then
wij=1; otherwise wij=0. The spatial weights matrix
W is then row-standardized with diagonal 0.
ln( gspit )  0  1 ln(capit )   2 ln(hwyit )  3 ln( waterit )
  4 ln(utilit )  5 ln(empit )  6unempit  7W ln( gspit )  ui  eit
• Pooled, fixed effects, random effects models are
all biased and inconsistent. IV or 2SLS methods
should be used with proper instruments.
Example: U. S. Productivity
• Spatial Panel Data Model
– Space-Time Dynamics
ln( gspit )   0  1 ln(capit )   2 ln(hwyit )  3 ln( waterit )
  4 ln(utilit )  5 ln(empit )   6unempit
  7W ln( gspit )  8W ln( gspit 1 )  9 ln( gspit 1 )  ui  eit