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

¹

∖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

¹

NλJ^{h (w}i

i

— w) (Wi

— w)' Dt

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

i

× [(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:

N X I¹,i,^NTI1 ,i,NT i	^→p Ξ1;	(48)
N X I2,i,NTI2,i,NT i	^→p Ξ2;	(49)
I3,N T ^→p 0; and		(50)
NX I1,i,NTI2,i,NT, nX I1,i,NTI3,NT ,	nX I2,i,NTI3,NT→p 0. i	(51)

Proof of (48): Write

-¹ X I¹^,i^,NTI1 ,i,NT
i

38

More intriguing information

1. An Efficient Circulant MIMO Equalizer for CDMA Downlink: Algorithm and VLSI Architecture
2. Dual Track Reforms: With and Without Losers
3. Neural Network Modelling of Constrained Spatial Interaction Flows
4. The name is absent
5. Influence of Mucilage Viscosity On The Globule Structure And Stability Of Certain Starch Emulsions
6. Spousal Labor Market Effects from Government Health Insurance: Evidence from a Veterans Affairs Expansion
7. An Economic Analysis of Fresh Fruit and Vegetable Consumption: Implications for Overweight and Obesity among Higher- and Lower-Income Consumers
8. Improving the Impact of Market Reform on Agricultural Productivity in Africa: How Institutional Design Makes a Difference
9. Gender and aquaculture: sharing the benefits equitably
10. Sector Switching: An Unexplored Dimension of Firm Dynamics in Developing Countries