Imbens, Lecture Notes 20, ARE213 Spring ’06
1
ARE213
Econometrics
Spring 2006 UC Berkeley Department of Agricultural and Resource Economics
Panel Data I: Random Effects
Panel
or
longitudinal
data sets consists of repeated observations for the same units, firms,
individuals or other economic agents. Typically the observations are at different points in
time.
Let
Y
it
denote the outcome for unit
i
in period
t
, and
X
it
a vector of explanatory
variables. The index
i
denotes the unit and runs from 1 to
N
, and the index
t
denotes time
and runs from 1 to
T
. Typically
T
is relatively small (as small as two), and
N
is relatively
large.
As a result, when we try to approximate sampling distributions for estimators, we
typically approximate them using large
N
asymptotics, keeping
T
fixed. (Recently there has
been some work looking at asymptotic approximations where
T/N
→
α
for some 0
< α <
1.)
Here we will mainly look at
balanced
panels, where
T
is the same for each unit.
An
unbalanced
panel has potentially different numbers of observations for each unit. This may
arise because of units dropping out of the sample, or more directly for firms going out of
business.
The key issue with panel data is that
Y
it
and
Y
is
tend to be correlated even conditional
on the covariates
X
it
and
X
is
. Let us look at this in a linear model setting:
Y
it
=
X
it
β
+
c
i
+
ε
it
.
The presence of
c
i
, the unobserved individual effect, creates this correlation even with the
ε
it
uncorrelated over time and units.
The two main approaches to dealing with such issues are
fixed effects
and
random effects
.
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 Spring '06
 IMBENS
 Variance, random effects, Xit

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