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

bNT = P₁(Θ1,₁ - Θι) (τ^mT Pt rm) Ui + P_ι(e_l - e)ui

+ Pi(Θι,i - Θ1) TmmT Pt rm) Vi + Pi(ei — e)Vi.

From these three equations, it is obvious that the terms including T^m will be
the dominant factors determining the asymptotic properties of B_NT, C_NT, and
bNT. However, if the parameters Θ1_,i are constant over different individuals so
that Θ1_,i - Θ1 = 0, none of BNT, CNT, and bNT depends on T^m . For this case,
the asymptotic properties of the three terms depend on (e_i — e). This result
indicates that the asymptotic distribution of the between estimator β^b_b , which is
a function of BNT, CNT, and bNT, will depend on whether the parameters Θ1_,i
are cross-sectionally heteroskedastic or homoskedastic. Somewhat interestingly,
however, the distinction between these two cases becomes unimportant when
the model has no intercept term (ζ = 0) and is estimated with this restriction.
For such a case, Bnt, Cnt and bNτ depend on Xi instead of ei. With Xi, the
terms (Θι,i — Θι) and (e — e) in Bnt, Cnt, and bNτ are replaced by Θι,i
and e_i, respectively. Then, it is clear that the trend term T^m remains as a
dominating factor whether or not the Θ1,i are heterogenous.

We now consider the asymptotic distributions of the within, between, GLS
estimators and the Hausman statistic under the two alternative assumptions
about the parameters Θ_1,i.

CASE 1.1: Assume that the parameters Θ1,i are heterogeneous over differ-
ent individuals; that is, p1,2 - p₁²_,1 6= 0. For this case, we can easily show:

where qι = limτ→∞T P[(t∕T)^m — T ∑_t(t∕T)^m]2 = R01[r^m — 1∕(m + 1)]²dr.7
The first two equalities are obtained using the fact that limτ→∞ T P_t(t∕T)^m
= ^R₀¹ r^m dr = 1∕(m + 1). In addition, we can also show that as (N,T →∞),

11

Tm √NTaN^τ

ɪɪ _b

T^m √N ^nt

⁷We can obtain these results using the fact that under given assumptions, √= Pt eit and
√= Pt Vit are i.i.d. over different i with N(0, σ²) and N(0, σ2), respectively, for any T.

<<   6   7   8   9   10   11   12   >>

More intriguing information