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 Ha: ¾_l = 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/@0_r)] ₊ 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 H₀^a , we get

u^1jHa =  ^-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 H₀^a. Using (A.2), we obtain

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