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

Observe that for any conformable matrices P and Q, we have

(P + Q)^-1 - P^-1 = -P-¹QP^-1 + (P + Q)^-1 QP-¹QP^-1.

Using this fact, we write

(A₁ + θ²_τF₁)^-1 - A^-1 = -θTA^-1F₁A^-1 + θTR₁,         (100)

where R₁ = (A₁ + θ²_τF₁)-¹ F₁A^-1F₁A^-1. Define

Q = (A1 + θTF1 )^-1 √NT (A2 + θTF2) - A^-1 √NTA2.

Then,

Q

= (A1 + θTF1)^-1 √NT (A2 + θTF2) - A^-1√NT (A2 + θTF2)

+A^-1 √NT θT F₂

= {(A1 + θTF1)^-1 - A^-11 √NT {A2 + θTF2} + A^-1 √NTθTF₂

= -A^-1 (θTF1) A^-1 √NT {A2 + θTF2} + A^-1 √NTθTF2

+θT R1 √NT {A2 + θT F2}

= -θT√NT ∣A^-1F₁ A^-1A₂ - A^-1 F₂^] - θT√NTr₂,            (101)
where R2 = A^-1F1A^-1F2 - R1 {A2 + θTF2} .

In view of (100) and (101) , we now can rewrite the Hausman statistic as

HMnt = Q [σ^A-¹ - σ² (A1 + θTF1)-¹]-1 Q

= θτ√NT [A^-1F1A^-1A2 - A^-1F2 + θTR2] '

× [σV A-¹F1A-¹ - σ2 θTR1] -¹

×θτ√NT [A^-1F1A^-1A2 - A^-1F2 + θTR2] ;

or equivalently,

HMnt

× [σ2 G1 J'  ' ^F ■ D J ' ) G1 + σ2 ^θT (⅛ ^r1^g⅛) ] "ɪ

Jχ,TDχ,τF1Dχ,τ J^-T) G1Gχ,τA2
G1 Jx^-T Dχ,τ F2 + θT G^-1_τ R2

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