LMχ
D(A)2
[(T - 1) + ¾4]b ,
(2.24)
where
1 ¾^2
D^(λ) = π U0[ 4 (Jt
2 ¾^1
(W 0 + W )) + ɪ (Et (W0 + W ))]^.
¾^
Here, ^° = ^(Et In)^=N(T — 1) and ^2 = U0(Jτ In)^=N are the maximum likelihood
estimates of ^° and ^2 under Hd, and U denotes the maximum likelihood residuals under the
null hypothesis H0d. See Appendix A.2 for more details.
Therefore, the one-sided test for zero spatial error dependence (assuming ¾μ ≥ 0) against an
alternative, say of λ > 0 is obtained from
^
LMX = l ( ) : ; (2.25)
√[(T —1) + ¾° ]b
and this test statistic should be asymptotically distributed as N(0; 1) under H0d for N ! 1
and T fixed.
We can also get the LR test for H0d, using the scoring method. Details are available upon
request from the authors. Under the null hypothesis, the LR test statistic will have the same
asymptotic distribution as its LM counterpart.
2.7 Conditional LM Test for H0e: ¾μ = 0 (assuming A may or may not be
= 0)
Similarly, if one uses LMi; given by (2.12), to test Hθ : ^μ = 0; one is implicitly assuming
that no spatial error correlation exists. This may lead to incorrect decisions especially when
A is significantly different from zero. To overcome this problem, this section derives a condi-
tional LM test for no random regional effects assuming the possible existence of spatial error
correlation. The null hypothesis for this model is Hθ : σ2μ = 0 (assuming A may or may not
be = 0).
This LM test statistic is derived in Appendix A.3 and is given by
_ _ _ ^ » ±1 ʌ
LMμ = D^'μ J^ Dμ,
(2.26)
where
T
dμ = - .r 2tr(B -B) +
2^
ɪ b0[Jτ (B0B)2]b;
2^
(2.27)
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