J12 = E h - @@@ ■'i = 2teh((¾Γ2JT + ¾-2Et) InK2(Jt In)] (A.19)
- °-' Ц- —
_ NT
= 2¾4,
J13 = Eh - a¾2Lλi = 24((¾1^2JT + ¾-2ET) In)
(A.20)
(A.21)
((¾-2Jr + --2Et ) In )(-° It (W + W 0))i = 0,
@2L
J23 =
E [-@ 2 ∂λ J = 2 tr ¾12(JT In )((¾ι2tJτ + ¾°ET ) In )
n
2
(¾° It (W + W 0))i = -° tr h 14J Jt (W + W 0)i = 0,
2 ¾1
where the result that J13 = J23 = 0 follows from the fact that the diagonal elements of W is
0 and J33 uses the fact that tr(W2) = tr(W02), and b = tr(W 2 + W0W).
Therefore, the information matrix evaluated under H0d is given by
|
Γ Nl 1 i τ-1∖ 2 (¾1 + ¾° ) |
NT 2<^1 |
0 | |
|
ʌ |
NT |
NT 2 |
∩ |
|
jμ = |
2¾4 |
2<^1 |
0 |
|
0 |
0 |
4 (T - 1 + ¾4)b _ | |
|
Therefore, |
LMχ = D0x J- Dλ
= D(A)2
[(T - 1) + ¾4]b,
(A.22)
as described in (2.24).
Appendix A.3: Conditional LM test for ¾2 = 0 (assuming ʌ = 0)
This appendix derives the conditional LM tests for zero random regional effects assuming that
spatial error correlation exists. We give the detailed derivation of the score and information
matrix for testing H0e : -2n = 0 (assuming A 6= 0).
Under H0e,
0 = -° It (B0B)-1,
(A.23)
(A.24)
-1 = ^12 It (B0B),
2
23