J23 = ôtrh-2 (JT IN)(IT (W + W0
2 l¾°
= 122 tr[JT (W + W0)] = O;
2¾2
where the result that J13 = J23 = 0 follows from the fact that the diagonal elements of W
are 0 and J33 uses the fact that tr(W2) = tr(W02) and b = tr(W 2 + W0W).
Therefore, the information matrix evaluated under H0a is given by
11 0
NT
Jμ = 2¾°
1T 0
(A.6)
0 0 2b¾°
0 0 N
using (A.1), we get
LMJ
2¾~4
NT 2¾~4
0;—^G;NTH (—° )
; 2¾2 ; NTTf
NT
T
T -1
1
T-1
0
1
T -1
1
T-1
0
0
0
N
2b¾°
0
Nil-
2¾° G
NTH
√^G2 + N2Th 2:
(A.7)
2(T - 1) b
where G = u (JT, IN)u — 1 and H = u (IT-W)u as descibed in (2.11).
uu uu
Appendix A.2: Conditional LM test for λ = O (given ¾2 > O)
In this appendix we derive the conditional LM test which tests for no spatial error correlation
given the existence of random regional effects. The null hypothesis is given by Hd: λ = 0
(assuming ¾2 > 0). Under the null hyphothesis, the variance-covariance matrix reduces to
0 = ¾2Jτ IN + ¾°INT. It is the familiar form of the one-way error component model, see
Baltagi(1995), with -1 = (¾2)-1(JT IN) + (¾°)-1(Et IN), where ¾2 = T¾2 + ¾°.
Under the null hypothesis Hd : λ = 0 (assuming ¾2 > 0), we get
u-1jH0d
@ ul
@^ jHd
@ ul
@^ jHd
@ ul
@\ jHd
( — JT +—2 Et ¢ IN ;
¾1 ¾°
IT IN ;
JT IN ;
¾° IT (W + W 0):
(A.8)
(A.9)
(A.10)
(A.11)
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