Appendix A.1: Joint LM test
This appendix derives the joint LM test for spatial error correlation and random regional
effects. The null hypothesis is given by Ha: ¾l = λ = 0. Let μ = (¾°; ¾2,Λ)0. Note that
the part of the information matrix corresponding to β will be ignored in computing the LM
statistic, since the information matrix between the μ and β parameters will be block diagonal
and the first derivatives with respect to β evaluated at the restricted MLE will be zero. The
LM statistic is given by
LM = D~ μ J-1D μ; (A.1)
where D~μ = (@L/@0)(~) is a 3 £ 1 vector of partial derivatives with respect to each element
of μ, evaluated at the restricted MLE ~. Also, J = E[—@2L=@μ@μ0](~) is the information
matrix corresponding to 0, evaluated at the restricted MLE ~. Under the null hypothesis, the
variance-covariance matrix reduces to ¾°Itn and the restricted MLE of β is βoLs, so that
U = y — X0βOLS are the OLS residuals and ¾° = ~0~=NT.
Hartley and Rao(1967) or Hemmerle and Hartley(1973) give a useful general formula to obtain
~
D μ :
@L=@μr = — 1 tr[ -1(@ u/@0r)] + 2[u' -1 (@ u=@μr) -1u]; (A.2)
for r = 1; 2; 3. It is easy to check that @ u/@^° = IT (B0B)-1, @ u=9¾2 = JT IN
and @ u=@X = ¾°[IT (B0B)-1(W0B + B0W)(B0B)-1] using the fact that @(B0B)-1=@A =
(B0B)-1(W0B + B0W)(B0B)-1, see Anselin (1988, p.164).
Under H0a , we get
(A.3)
u1jHa = -2 IT IN ;
¾°
~jH jHa = IT IN ;
@^ 0
j0 jHa = JT IN ;
@^, 0
M
@bu jHa = ¾° IT (W0 + W ).
This uses the fact that B = IN under H0a. Using (A.2), we obtain
19
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