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

Using the fact that tr (A ® B) = tr (A) tr (B) and the Cauchy-Schwarz inequal-
ity, we have

T12 XX tr(

Gx,T (xit xi) (^χis xi) Gx,T
®Gx,T (xit xi) (xis xi) Gx,T

⅛∑ E©tr £Gx,T (Xit

xi) (xis

Xi)0 Gχ,τ]}

T X ∣∣Gχ,τ (xit
t

xi⁾^k

Similarly,

T12 XXtr(

Gx,T (xit xi) (xis xi) Gx,T

®Gx,T (xis xi) (xit xi) Gx,T

T X ∣G∙,T (xi.
t

- xi )k²

and

¹ XX tr f Gx,T (xit — xi) (xit — ^cci)° GX,T
T² t^l s' × ®^Gx,T (xis ^- xi) (xis ^- xi) Gx,T

= (T X

tr Gx,T (xit

- xi) (xit

T X ∣∣Gχ,T (xit
t

- xi)k²

Thus, the right hand side of (84) is less than or equal to

3M1 κ⁴_v sup sup E
i,T 1≤t≤T

T X ∣∣Gχ,T (xit
t

— xi)k²

Note that

sup sup E
i,T 1≤t≤T

T X ∣∣Gχ,t (xit — xi)k²

≤ sup sup E
i,T 1≤t≤T

T X l∣Gχ,Txitk²

_≤ M sup_i,T sup₁_≤_t_≤_T E ∣D1Tx1,it ∣⁴ + sup_i,T sup₁_≤_t_≤_T E ∣x2,it ∣⁴

+sup_i,Tsup₁_≤_t_≤_TE∣x3,it∣⁴