Large-N and Large-T Properties of Panel Data Estimators and the Hausman Test

plimN,T→∞NT^2m+2 [V ar(β^bw) - V ar(β^bg)] =

So far, we have considered several simple cases to demonstrate how the
convergence rates of the popular panel data estimators and the Hausman test are
sensitive to data generating processes. For these simple cases, all of the relevant
asymptotics can be obtained in a straightforward manner. In the following
sections, we will show that the main results obtained from this section apply
to more general cases in which regressors are serially dependent with arbitrary
covariance structures.

3 Conditional α - Mixing

In the asymptotic analysis of the general model (1) with large T, some technical
difficultes arise when some of the time varying regressors xit are correlated with
the time invariant regressors zi . For such cases, the temporal dependence of
the time-varying regressors may persist through their correlations with the time
invariant regressors; that is, the time series of xit may not be ergodic in time.
Thus, for general asymptotic results, we need to study the probability limits
of the random variables containing time-averages of such non-ergodic regressors
(i.e., BNT and bNT in Section 2.2). In this section, we discuss the assumptions
that can facilitate derivations of the (joint) limits of such random variables as
(N, T →∞) simultaneously.

Consider CASE 3.3 with m = 0. Observe that the time series of xit is
not ergodic, because of the presence of the time invariant random component
Πizi in xit. In addition, cov(xit,xi,t+l)=E(xit - E(xit))(xi,t+l - E(xi,t+l))
= Π_i²σ_z² 9 0asl →∞. Thus, the termporal dependence of xit does not
decay. Despite these problems, we were able to obtain handy asymptotic results
based on the two strong assumptions: E (xi_t | zi) = Πizi, and the conditional
terms e_it = x_it - E (x_it | z_i) are i.i.d. over time. This example illustrates that
under some certain conditions imposed on non-ergodic time-varying regressors,
we can analyze the asymptotic properties of sample averages of panel data. In
fact, our major findings from CASE 3 remain unaltered even if we alternatively
assume that E(xi_t | zi) is an arbitrary nonlinear function of zi, and/or the ei_t
are autocorrelated, so long as the eit satisfy the conditions we discuss in detail
below. Formally, we consider a mixing model that is defined conditionally on the
sigma field generated by time invariant regressors zi , which we call a conditional

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