1
√NcNT
N (0, σz2σ2u).
Using these results and the fact that limN →∞T θ2T
as (N, T →∞),
= σ2v /σ2u , we can show that
τm Nnt (βw - β) =⇒ N
(0, -σv-Y
p1,2q1
(13)
Tm√N(βb - β) =⇒ N
0, σ2u
(m+1)2
p1,2
P2,1 / ,
(14)
τm√NT(βg - β) = τm√NT(βw
β)
1 σv2 (p1,2
P2,1)
T σU p1,2q1 (m + 1)2
τ m√N (b
b - β)
+θp(1∕√T );
(15)
τm Nττβ (βw
βbg)
σ2v (p1,2
P1,1)
σ2u p1,2q1(m+1)2
Tm NN (βb - β) + Op (1)
N 0,
σ4v (p1,2
Pι,ι)
σ2u (p1,2q1)2(m+1)2 ;
(16)
plimN,T→∞NT2m+2[V ar(βbw) -Var(βbg)] =
σ4v (p1,2
Pι,ι)
σ2u (p1,2q1)2(m + 1)2 .
(17)
Several remarks follow. First, not surprisingly, all of the within, between and
GLS estimators are superconsistent when the time-varying regressor xit contains
a time trend. Second, from (15), we can see that the two estimators β w andβ g
are Tm√NT-equivalent in the sense that (βw - βg) is op(1∕Tm√NT). This is
so because the second term in the right-hand side of (15) is Op(1∕y∕T). Nonethe-
less, from (16), we can see that (βw-βg) is Op(1∕Tm√NT2) and asymptotically
normal. These results indicate that the within and GLS estimators are equiva-
lent to each other by the order of Tm√NT, but not by the order of Tm√NT2.
Third, from (16) and (17), we can see that the Hausman statistic is asymptoti-
cally χ2 -distributed. Fourth, when the model is estimated without an intercept
term because ζ = 0, all of the results (14)-(17) are still valid with p1,2 replacing
p21,1).
(p1,2
Finally, (16) provides some intuition about the power property of the Haus-
man test. Observe that the asymptotic distribution of (βw
that of (βb - β). From this, we can conjecture that the Hausman statistic is for
testing consistency of the between estimator βbb , not exactly for testing the RE
assumption. In fact, the RE assumption (3) is not a necessary condition for the
asymptotic unbiasedness of βb . For example, if the effect is correlated with zi ,
ъ ʌ ,
-β g ) depends on
10
More intriguing information
1. MULTIPLE COMPARISONS WITH THE BEST: BAYESIAN PRECISION MEASURES OF EFFICIENCY RANKINGS2. Strengthening civil society from the outside? Donor driven consultation and participation processes in Poverty Reduction Strategies (PRSP): the Bolivian case
3. Kharaj and land proprietary right in the sixteenth century: An example of law and economics
4. Structural Influences on Participation Rates: A Canada-U.S. Comparison
5. The name is absent
6. Sector Switching: An Unexplored Dimension of Firm Dynamics in Developing Countries
7. Innovation Trajectories in Honduras’ Coffee Value Chain. Public and Private Influence on the Use of New Knowledge and Technology among Coffee Growers
8. The name is absent
9. Heavy Hero or Digital Dummy: multimodal player-avatar relations in FINAL FANTASY 7
10. The Mathematical Components of Engineering