Testing Panel Data Regression Models with Spatial Error Correlation



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 H
a: ¾l = λ = 0. Let μ = (¾°; ¾2)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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