bNT = P1(Θ1,1 - Θι) (τmT Pt rm) Ui + Pι(el - 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 Tm will be
the dominant factors determining the asymptotic properties of BNT, CNT, 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 Tm . For this case,
the asymptotic properties of the three terms depend on (ei — e). This result
indicates that the asymptotic distribution of the between estimator βbb , 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 ei, respectively. Then, it is clear that the trend term Tm 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 Θ1,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:
|
plimN,τ →∞ T2m NTAnt = |
p1,2q1; |
|
1 1 plimN,τ →∞ T2m Nbnt = . ∙ 1 1 n plimN,τ →∞ Tm Ncnt = 1^ 1 и plimN →∞ n Hn = |
P1,2 — P1,1 (m +1)2 ; 0; σz2 , |
where qι = limτ→∞T P[(t∕T)m — T ∑t(t∕T)m]2 = R01[rm — 1∕(m + 1)]2dr.7
The first two equalities are obtained using the fact that limτ→∞ T Pt(t∕T)m
= R01 rm dr = 1∕(m + 1). In addition, we can also show that as (N,T →∞),
11
=⇒ N (0, p1,2q1σv2);
N Λ 2 P1,2 — P2
⇒ N I 0, σu / I -∣∖∙
u (m +1)
Tm √NTaNτ
ɪɪ b
Tm √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.
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