Econ 306
Panel Data
Repeated observations on the same set of crosssectional units
yit
X it
 yi1 
y 
 i2 
 . 
 . 
 
 . 
 . 
y 
 iT 
j
 X i11
 1
 X i2
 .
 .

 .
 .
 1
 X iT
2
X i1
2
X i2
.
.
.
.
X iT
2
K
X i1 
K
X i2 
. 
. 

. 
. 
K
...... X iT 
......
......
THE POOLED ESTIMATOR
yit  X it   e
eit ~ iid 0, 2 
EXTENSIONS OF THE SIMPLE MODEL
y it  X it   e it
eit   i  it
In the RANDOM EFFECTS MODEL αi is
uncorrelated with Xit
In the FIXED EFFECTS MODEL αi is
correlated with Xit
THE RANDOM EFFECTS MODEL
 ηit is uncorrelated with Xit

 αi is uncorrelated with Xit
 These othogonality conditions are sufficient
for OLS to be asymptotically unbiased
 OLS will produce consistent estimates of β
but the standard errors will be overstated
 OLS is not efficient compared to a Feasible
Generalised Least Squares estimator
 We first need an estimate of the covariance
matrix
E    0
E     2 IT
E  i   0
E  i j   0


E   i i    2


E  i jt   0


E  ei ei    2IT   2 ii 


  2   2 

2
  

.

.

2
  
 2
 2    2
......
......
.
 2   2
.
......
 2
 2




2
  

2
2 
    
This gives us the error-covariance for each
individual cross-section unit
  I n    E ee
  0 ...... 0 


 0  ...... 0 
. .

. .
 0


. .

 0 0 ......  


  E ee

1
2

1 
1

IT  

 
 T
 2
T 2   2
 
ii 
 
THE FIXED EFFECTS MODEL: THE 2
PERIOD CASE
y it  X it   Zi  eit
eit   i  it
Wit  X it Z it 
E Witeit   0
The independent variables are correlated with α
yi1  X i1   Z i  ei1
yi 2  X i 2   Z i  ei 2
yi 2  yi1   X 2i  X 1i   Z i  Z i   ei 2  ei1 
y  X  Z  e
Z i  0
 i  0
y  X  
EX    0
 OLS on the transformed variables yields
unbiased estimates of the coefficients of the
X variables
 With the fixed effects model we are able to
obtain consistent estimates of parameters of
interest even in the face of correlated
omitted effects when OLS on a cross-section
would fail to do so.
 With fixed effects estimators we cannot
generally recover estimates of any time
invariant explanatory variables since when
we difference to remove αi Zi drops out.
 Another way of looking at this is that the
fixed effects estimator is robust to the
omission of any relevant time-invariant
regressors such as unobservable individual
characteristics.
 When the random effects model is valid the
fixed effects estimator will still produce
consistent estimates of the identifiable
parameters although in this case the fixed
effects estimator is not as efficient as the
random effects estimator.
THE GENERAL FIXED EFFECTS
ESTIMATOR
yit  X it    i  it
 In this case the αi are treated as unknown
parameters and must be estimated. Note
however that we cannot obtain consistent
estimates.
 Typically T is small and n is large and
asymptotic theory relies on n getting
larger and larger.
 Here as n increases the number of αi to be
estimated grows at the same rate.
 We can however estimate the remaining
parameters consistently.
y  X  D  
D  I n  iT
Running this regression is actually equivalent
to:
1. running a regression of each of the y and X
variables on the dummy variables and then:
2. running a regression of the X residuals on
the y residuals
In practice the easiest way to implement a fixed
effects estimator with conventional software is
to include a different dummy variable for each
individual unit of observation.
This is often called the Least Squares Dummy
Variable method.
THE PERILS OF FIXED EFFECTS
ESTIMATION
 If you have measurement error in X then
the asymptotic bias in the estimates will be
far greater using a Fixed Effects estimator
than they would be if OLS was used.
RANDOM OR FIXED?
 The distinction between the two models is
whether the time invariant effects are
correlated with the regressors or not.
 When the Random Effects model is valid
but the Fixed Effects model is used the
Fixed Effects model still produces consistent
estimates of the identifiable parameters
although they are not the most efficient.
 Selecting the Random Effects model
imposes the restriction that the individual
effects are uncorrelated with the other
regressors and if this is not true then the
Random Effects estimates will be
inconsistent.
 Use a priori judgement?
 Rely on the Hausman Test?





ˆ
ˆ
H   RE   FE  FE   RE  ˆRE  ˆFE ~  2 K
where the null hypothesis is that the Random
effects model is correct.