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



NT(β (βg β)

Nttw - β)

+   1 σ2v    (P2,2 - P2,1)

√T σU (q2,2 - q2,ι + σ2)

+op √Jf) ;

N( (βb - β)


(24)


√NT2 (βw — βg )


-V    (P2,2 - p2,ι)

σU (q2,2 - q2,ι + σ2)


n (βb - β) + op(1)


n 0  σV    (p2,2 - p2,1)

(25)


, -U (q2,2 - q2,ι + σ^)2

σ4    (p2 2 - p22 1 )

plimN,τ→∞NT2[Var(βw) - Var(βg)] = -v ,     , 2   ,1 2λ2 .      (26)

w            g      σ2u (q2,2 - q22,1 + σe2)2

Note that (22) - (26) are essentially the same as (13) - (17) except that they do
not include the time trend
Tm.

CASE 2.2: Now, we consider the case in which the Θ2,i are constant over
different
i; that is, p2,2 - p22,1 =0. If the model contains no intercept term
(ζ =0) and it is estimated with this restriction, all of the results (22) - (26) are
still valid with
p2,2 replacing (p2,2 - p22,1). However, if the model contains an
intercept term, the assumption of cross-sectional homoskedasticity affects the
convergence rates of the between estimator and the Hausman statistic, while
it does not to the within and GLS estimators. To see this, we assume that
Θ
2,i =0, for all i, without loss of generality. Then, we can show

N (βb - β)→ N μ0, -u ) ;


(27)


-—^ ^   ^         —2       2        /n ^

NT3(β(βw - βg)   =  --v7-------2r—. ʌNr (βb -β) + op(i)

w g             -2u (q22,2 - q222,1 + -e2)   T b

= N (0, -v 7-------2- . J ;          (28)

-2u (q22,2 - q222,1 + -e2)2

-4 -2

plimN,τ→∞NT3[Var(βw) - Var(βg)] = -27-------2---:—2τ2,     (29)

w            g     -2u(q22,2 - q222,1 +-e2)2

as (N, T → ∞). Observe that the between estimator is ypN-consistent as in
CASE 1.2.

CASE 3: So far, we have considered the cases in which the time-varying
regressor x
it and the time invariant regressor zi are uncorrelated. We now

13



More intriguing information

1. The name is absent
2. The name is absent
3. Concerns for Equity and the Optimal Co-Payments for Publicly Provided Health Care
4. The name is absent
5. The name is absent
6. The name is absent
7. The name is absent
8. The name is absent
9. The name is absent
10. Valuing Access to our Public Lands: A Unique Public Good Pricing Experiment