examine the cases in which this assumption is relaxed. The degree of the cor-
relation between the xit and zi may vary over time. As we demonstrate below,
the asymptotic properties of the panel data estimators and the Hausman test
statistic depend on how the correlation varies over time. The basic model we
consider here is given by
xit = Θ3,i + Ψ3,t + Πi zi /tm + eit, (30)
where the Θ3,i and Πi are individual-specific fixed parameters, the Ψ3,t are
the time-specific fixed effects, m is a non-negative real number. Observe that
because of the presence of the Θ3,i and Πi, the xit are not i.i.d. over different
i. The correlation between xit and zi decreases over time if m> 0. In contrast,
m = 0 implies that the correlation remains constant over time. We will not
report our detailed asymptotic results for model (30) with heterogeneous Θ3,i ,
because they are essentially the same as those we obtain for CASE 2.1. This is
so because the terms Θ3,i dominate and the terms Πizi /tm become irrelevant
in asymptotics. Thus, we set Θ3,i = 0 for all i. In addition, we set Ψ3,t = 0 for
all t. We do so because presence of the time effects is irrelevant for convergence
rates of panel data estimators and the Hausman statistic. For CASE 3, the
within and GLS estimators are always N———consistent regardless of the size
of m. Thus, we only report the asymptotic results for the between estimator
and the Hausman statistic.
For the cases in which the parameters Πi are the same for all i, it is easy
to show that the between estimator βb does not depend on Πi zi /tm . For such
cases, the terms Πizi∕tm do not play any important role in asymptotics. In
fact, when the parameters Πi are the same for all i, we obtain exactly the same
asymptotic results as those for CASE 2.2. This result is due to the fact that
the individual mean of the time-varying regressor xi becomes a linear function
of the time invariant regressor zi if the Πi are the same for all i. This particular
case does not seem to be of practical importance, because it assumes an overly
restrictive covariance structure of regressors. Thus we only consider the cases
in which the Πi are heterogeneous over different i.
We examine three possible cases: m ∈ ( 1, ∞], m = 1, and m ∈ [0,1 ). We
do so because, depending on the size of m, one (or both) of the two terms eit
and ∏iZi∕tm in xit becomes a dominating factor in determining the convergence
rates of the between estimator βb and the Hausman statistic HMTN.
CASE 3.1: Assume that m ∈ (2, ∞]. This is the case where the correlation
between xit and zi fades away quickly over time. Thus, one could expect that
the correlation between xit and zi (through the term ∏iZi∕tm) would not play
any important role in asymptotics. Indeed, a straightforward algebra, which
is not reported here, justifies this conjecture: The term eit in xit dominates
Πizi∕tm asymptotics, and thus, this is essentially the same case as CASE 2.2.9
9We can obtain this result using the fact that limτ→∞ √= Pt t-m = 0, if m > 1.
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