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

time-varying regressors with arbitrary autocovariance structure? Second, what
is the asymptotic distribution of the Hausman statistic when N, T → ∞? Is
the statistic HMNT χ² -distributed despite the equivalence result? Third, does
the Hausman test have power to detect violations of the RE assumption when
T is large? Our equivalence result implies that between variations in data
become less informative for the GLS estimation of β as T →∞. Then, the GLS
estimator of β may remain consistent even if the RE assumption is violated. If
this is the case, the power of the Hausman test might be inversely related to the
size of T . We will attempt to answer these questions in the following sections.

What makes it complex to investigate the asymptotic properties of the
within, GLS estimators and the Hausman statistic is that their convergence
rates crucially depend on data generating processes. The following subsection
considers several simple cases to illustrate this point.

2.2 Preliminary Results

This section considers several simple examples demonstrating that the con-
vergence rates of the within, GLS estimators and the Hausman statistic cru-
cially depend on whether or not data are cross-sectionally heteroskedastic, and
whether or not time-varying regressors contain time trends. For model (1), we
can easily show that

β^b_w - β = A^-_N¹_TaNT;                        (7)

β^b_b - β =(B_NT - C_NTH_N^-1C_N⁰ _T)^-1 [b_NT - C_NTH_N^-1c_NT],       (8)

where,

A0    0    00

NT = _i,t xitx_it ; BNT = _i xix_i ; CNT = _i xiz_i ; HN = _i ziz_i ;

aNT = Pi,_t eitvit; bNT = Pi è»(u» + vi); cnt = Pi e»(u» + Vi)∙

Using (7) and (8), we can also show that the GLS estimator is a convex combi-
nation of the within and between estimators:

β^b_g - β = [A_NT + Tθ²_T(B_NT - C_NTH_N^-1C_N⁰ _T)]^-1                (9)

×[ANT (β^b_w - β)+Tθ_T(BNT - CNT H_N^-1 C_N⁰ _T)(β^b_b - β)].

Using (7), (8) and (9), we can also obtain

β^b_w - β^b_g = [ANT + Tθ²_T (BNT - CNTH_N^-1CNT)]^-1
×Tθ²T(BNT-CNTH_N^-1CN⁰ T)[(β^bw-β)-(β^bb-β)]; (10)

Var(β^b_w) - V ar(β^b_g) = A^-_N¹_T - [A_NT + Tθ²_T (B_NT - C_NTH_N^-1C_N⁰ _T)]^-1. (11)

Equation (10) provides some insight into the convergence rate of the Hausman
test statistic. Note that (β^b_w-β^b_g) depends on both (β^b_w-β) and (β^b_b -β). Appar-
ently, the between estimator β_b exploits only N between-individual variations,

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