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

while the within estimator β_w is computed based on N(T - 1) within-individual

variations. Accordingly, (β_b - β) converges to a zero vector in probability much
slower than (β_w - β) does. Thus, we can conjecture that the convergence rate
of (β^b_w - β^b_g) will depend on that of (β^b_b - β), not (β^b_w - β). Indeed, we below

In this subsection, we only consider a simple model which has a single time-
varying regressor (x_it) and a single time-invariant regressor (z_i). Accordingly,
all of the terms defined in (7)-(11) are scalars. We consider asymptotics under
the RE assumption (3). To save space, this section only considers the estimators
of β and the Hausman test. The asymptotic distributions of the estimators of
γ will be discussed in Section 4. Throughout the examples below, we assume
that the zi are i.i.d. over different i with N (0, σ²_z). In addition, we introduce
a notation e_it to denote a white noise component in the time-varying regressor
xit. We assume that the eit are i.i.d. over different i and t with N(0, σ_e²), and
are uncorrelated with the zi .

CASE 1: We here consider a case in which the time-varying regressor x_it
contains a time trend of order m. Specifically, we assume:

xit = Θ1_,it^m + eit.                              (12)

We assume the parameters Θ₁,_i are fixed with finite IimN→∞ -N Pi Θ₁,_i =
IimN→∞Θι ≡ p1,1 and IimN→∞-N Pi Θ^ i ≡ p1,2. We can allow them to be
random without changing our results, but at the cost of analytical complexity.⁶
We consider two possible cases: one in which the parameters Θ1,i are hetero-
geneous, and the other in which they are constant over different individuals.
Allowing the Θ1,i to be different across different individuals, we allow the xit
be cross-sectionally heteroskedastic. In contrast, if the Θ1,i are constant over
different i, the xit become cross-sectionally homogeneous. As we show below,
the convergence rates of the between estimator and Hausman test statistic are
different in the two cases. Furthermore, whether or not the model is estimated
with an overall intercept could matter for convergence rates.

To be more specific, consider the three terms B_NT, C_NT, and b_NT defined
below (8). A straightforward algebra reveals that with rt ≡ t/T ,

BNT = Pi(Θ1,i - Θ1)² TΓ^mTT Pt rm 2

+2 Pi(Θ1,i — Θι) Tm^mT Pt rm (ei - e) + Pi(ei - e)²;

CNT = Pi(Θ1,i - Θ1) Tm^mT Pt rm) (zi - z)+Pi(ei - e)(zi - z);

⁶We can consider a more general case: for example, xi_t = ait^m + Θi + Θt + bizi + ei_t .
However, the same asymptotic results apply to this general model. This is so because the
trend term (t^m) dominates asymptotics.

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