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

Then, under Assumption 11, as (N, T →∞), the followings hold.

(a) NN Pi Dt^wi^w⁰DT →p ^ξ

(b) sup_{N t}sup₁≤_i≤N E \\D_Twik4 < M, for some constant M < ∞.

(^c) √N Pi DT^wi⅛ ⇒ N (≡^{λ, σ}U≡0 .

(d) √N Pi DTW^iVi →p 0.

Proof

Part (a)

To find the joint limit of

N X DTwJiwJ⁰DT =
i

∖UD (^wji
i

— w) (wi — w)' DT,

we define

E^w = Ex^Ex'^EFi^χ3,i^,zi)' ;

E*W = (Ex⁰₁,Ex'₂,EF_zx'₃,z').

With this notation, we write

NλJ^{h (w}i

— w) (Wi

— w)' Dt

= NXDt [(wui — E*_iw^i) + (E*_iWi — E*wj) + (E*w — wu)]

× [(w7i — E*ιvi) + (EiWi — E*w) + (E*w — w)]' Dt

-1X

N
i

(I1,i,N T+I2,i,NT+I3,N T)(I1,i,N T+I2,i,N T+I3,N T)^' , say.

(47)

We complete the proof by showing the following:

Proof of (48): Write

-¹ X I¹^,i^,NTI1 ,i,NT
i