Testing Panel Data Regression Models with Spatial Error Correlation

Using (A.2), one obtains

∂L ∂σ²	I Hd =	—2tr[ u ¹(It (B⁰B)-¹)]+ 2[U⁰ U¹ (It (B^,B)~¹) U¹ u]
	=	—v tr[( v₂ Jt + v₂ Eτ ) In ] + v[u⁰[( V4 Jτ + v4 Eτ ) In ]u] (A.12) ² ^1 ¾° ² ¾ 1 ¾ °
∂L ^dσμ	=	N(T — 1) N , 1 „0, ¹ - 1 ʌʌ , 1 ʌ,, 1 . _n vv₂ vv₂ ⁺ u^u⁽ V4 ^jt ⁱn⁾^u⁺ vu ⁽~^et ⁱn^)u = ⁰, ²¾° ²¾1 ² ¾1 ² ¾ °
∂L ^dσμ	jH₀^d =	^—2^tr[ U ¹⁽^jt ⁱn^{)] +} 2^[u0 U ¹⁽^jt ⁱn) U 1^u] (A.13)
		nt 1 _r„,_z^ „
	=	^— TTT ⁺ TΓ4 ^[^u⁽^JT ¹N^)u] = ⁰; 2¾ 1 2¾ 1
∂L Ж	jHo^d =	—2tr[ U ¹(¾2It (W + W⁰))] + ∣[u⁰ U ¹(¾2It (W + W⁰)) U ¹u]
	=	-¹2 [u⁰(Eτ (W + W ⁰))u] + ⅛ [u⁰(J⁰τ (W + W ⁰))^] = D λ, (A.14) ²¾° ²¾1

where ¾° = ^⁰(Eτ In)u∕N(T — 1) and ¾2 = ^⁰(Jτ In)u∕N are the maximum likelihood

estimates of ¾° and σ², and u is the maximum likelihood residual under the null hypothesis
Hd
H0 .

Therefore, the score vector under Hd is given by

(A.15)

D _λ

Using (A.5), the elements of the information matrix are given by

^J11 = ^e [ ^— _g(¾L)₂ ] =2 ^tr[(((^σι²^jr ⁺^σJ²Eτ ) ⁱn ))] (A.¹6)

_ N _l 1 T — 1 ʌ

= у M ⁺ ~°^~ ɔ;

J22 = E [ — @@22 ] =2 tr[(((¾Γ² Jτ + ¾-² Eτ ) In )( Jt In ))2] (A.17)

~∖~μ∕

_ NT²

= l¾f ’

^j33 = ^E [ ^— @\₂ ] =2^tr[(((^σ1 ²^jT ^{+ σ}!7²ET) ⁱn)

= ¾4 tr ' ( ¾₁ ¹J/ + ¾^-4Eτ ) (2W² + 2W ⁰W

• (¾°It (W + W⁰)))2] (A.18)

: ( ¾° + (T — 1))b,

⁴¾1 /