Testing Panel Data Regression Models with Spatial Error Correlation

2.5 LR Test for Ha: λ = ¾2 = 0

We also compute the Likelihood ratio (LR) test for Ha: A = ¾2_l = 0.  Estimation of the

unrestricted log-likelihood function is obtained using the method of scoring. The details of
the estimation procedure are available upon request from the authors. Let A°, A, b and b
denote the unrestricted maximum likelihood estimators and let Bb = IN—AW and u = y—X⁰b,
then the unrestricted maximum log-likelihood estimator function is given by

NT         1                                    1

Lu =       ln2¼¾° — ~n InljTAIN + (BA⁰IA)-¹ j] + (T — 1) ln ∖B∖ —   ._l ti§-¹^;    (2.21)

^{2            2                                        2}¾°

see Anselin (1988), where §^b is obtained from (2.8) with B^b replacing B and A^b replacing A: But
under the null hypothesis Ha, the variance-covariance matrix reduces to U = _u = ¾°IτN
and the restricted maximum likelihood estimator of β is ~~ols , so that ~ = y — X^,∕~ols
are the OLS residuals and ¾° = U⁰U=ΝT. Therefore, the restricted maximum log-likelihood
function under H_a^a is given by

NT        1

Lr = —in ln 2¼¾° — ..₂ u ~∙                                               (2.22)

2            2¾_°

Hence, the likelihood ratio test statistic for H_a^a : A = ¾²_p = 0 is given by

LRJ = 2(Lu — Lr);                                                            (2.23)

and this should be asymptotically distributed as a mixture of ² given in (2.20) under the
null hypothesis.

2.6 Conditional LM Test for Had: λ = 0 (assuming ¾² > 0)

When one uses LM₂; given by (2.16), to test H_a^c : A = 0; one implicitly assumes that the
random region effects do not exist. This may lead to incorrect decisions especially when ¾²
is large. To overcome this problem, this section derives a conditional LM test for spatially
uncorrelated disturbances assuming the possible existence of random regional effects. The
null hypothesis for this model is H_a^d : A = 0 (assuming ¾²_p > 0). Under the null hyphothesis,
the variance-covariance matrix reduces to a = ¾²_pJT IN + ¾²_°INT . It is the familiar form
of the one-way error component model, see Baltagi(1995), with ^-1 = (¾2)^-1(Jτ IN) +
(¾°)^-1(Et   IN), where ¾² = Tσⁱ_μ + ¾°; and Et = It — Jt. Using derivations analogous

to those for the joint LM-test, see Appendix A.1, we obtain the following LM test for H_a^d vs
^H1^d,

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