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

inf N ≥_n aXN ≥ di and supN ≥_n aχN ≤ ⅛. We also use the following notation for
relevant sigma-fields: Fx_i = σ(xi1, ..., xiT); Fz_i = σ (zi); Fz = σ (Fz₁ , ..., Fz_N);
Fw_i = σ (Fx_i, Fz_i); and Fw = σ (Fw₁, ..., Fw_N) . The xit and zi are now k × 1
and g × 1 vectors, respectively.

As in Section 2, we assume that the regressors (x⁰_i1 , ..., x⁰_iT ,z_i⁰)⁰ are indepen-
dently distributed across different i. In addition, we make the following the
assumption about the composite error terms u_i and v_it :

Assumption 1 (about ui and vi_t): For some q>1,

(i) the u_i are independent over different i with sup_iE |u_i |^4q < ∞.

(ii) The vit are i.i.d. with mean zero and variance σ_v² across different i and t,
and are independent of xi_s, zi and ui, for all i, t, and s. Also, kvit k_4q ≡ κ_v
is finite.

Assumption 1(i) is a standard regularity condition for error-components models.
Assumption 1(ii) indicates that all of the regressors and individual effect are
strictly exogenous with respect to the error terms vit.¹⁴

We now make the assumptions about regressors. In Section 2, we have con-
sidered three different cases: CASEs 1, 2, and 3. Consistently with these cases,
we partition the k × 1 vector xit into three subvectors, x1,it, x2,it, and x3,it,
which are k1 × 1, k2 × 1, and k3 × 1, respectively. The vector x1_,it consists of
the regressors with deterministic trends. We may think of three different types
of trends: (i) cross-sectionally heterogeneous nonstochastic trends in mean (but
not in variance or covariances); (ii) cross-sectionally homogeneous nonstochas-
tic trends; and (iii) stochastic trends (trends in variance) such as unit-root time
series. In Section 2, we have considered the first two cases as CASEs 1.1 and
1.2, respectively. The latter case is materially similar to CASE 2.2, except that
the convergence rates of estimators and test statistics are different under these
two cases. Thus, we here only consider the case (i). We do not cover the cases
of stochastic trends (iii), leaving the analysis of such cases to future study.

The two subvectors x2_,it and x3_,it are random regressors with no trend in
mean. The partition of x2_,it and x3_,it is made based on their correlatedness
with zi . Specifically, we assume that the x2,it are not correlated with zi , while
the x3_,i_t are. In addition, in order to accommodate CASEs 2.1 and 2.2, we also
partition the subvector x2,it into x21,it and x22,it, which are k21 × 1 and k22 × 1,
respectively. Similarly to CASE 2.1, the regressor vector x_21,it is heterogeneous
over different i, as well as different t, with different means Θ_21,it . In contrast,
x21,it is homogeneous cross-sectionally with means Θ22,t for all i for given t.
We also incorporate CASEs 3.1, 3.2 and 3.3 into the model by partitioning x3_,i_t
into x31,it, x32,it, and x33,it, which are k31 × 1, k32 × 1, and k33 × 1, respectively,
depending on how fast their correlations with z_i decay over time. The more
detailed assumptions on the regressors xit and zi follow:

¹⁴As discussed in Section 2.1, this assumption rules out weakly exogenous or predetermined
regressors.

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