Imbens/Wooldridge, Lecture Notes 4, Summer ’07
What
’
s New in Econometrics
?
NBER
,
Summer 2007
Lecture 4
,
Monday
,
July 30th
,
3
.
15

4
.
15 pm
Nonlinear Panel Data Models
These notes summarize some recent, and perhaps notsorecent, advances in the estimation
of nonlinear panel data models. Research in the last 10 to 15 years has branched off in two
directions. In one, the focus has been on parameter estimation, possibly only up to a common
scale factor, in semiparametric models with unobserved effects (that can be arbitrarily
correlated with covariates.) Another branch has focused on estimating partial effects when
restrictions are made on the distribution of heterogeneity conditional on the history of the
covariates. These notes attempt to lay out the pros and cons of each approach, paying
particular attention to the tradeoff in assumptions and the quantities that can be estimated.
1
.
Basic Issues and Quantities of Interest
Most microeconomic panel data sets are best characterized as having few time periods and
(relatively) many cross section observations. Therefore, most of the discussion in these notes
assumes
T
is fixed in the asymptotic analysis while
N
is increasing. We assume random sample
in the cross section,
x
it
,
y
it
:
t
1,.
..,
T
.Take
y
it
to be a scalar for simplicity. If we are not
concerned about traditional (contemporaneous) endogeneity, then we are typically interested in
D
y
it

x
it
,
c
i
(1.1)
or some feature of this distribution, such as
E
y
it

x
it
,
c
i
, or a conditional median. In the case of
a mean, how do we summarize the partial effects? Let
m
t
x
t
,
c
be the mean function. If
x
tj
is
continuous, then
j
x
t
,
c
≡
∂
m
t
x
t
,
c
∂
x
tj
,
(1.2)
or look at discrete changes. How do we account for unobserved
c
i
?Ifwewanttoestimate
magnitudes of effects, we need to know enough about the distribution of
c
i
so that we can
insert meaningful values for
c
. For example, if
c
E
c
i
, then we can compute the
partial
effect at the average (PEA)
,
j
x
t
,
c
.
(1.3)
Of course, we need to estimate the function
m
t
and the mean of
c
i
. If we know more about the
distribution of
c
i
, we can insert different quantiles, for example, or a certain number of
standard deviations from the mean.
Alternatively, we can average the partial effects across the distribution of
c
i
:
1