(a) Ho : A = ¾X = 0, and the alternative Hf is that at least one component is not zero.
(b) H : σ2μ = 0 (assuming no spatial correlation, i.e., A = 0), and the one-sided alternative
Hb is that σ2l > 0 (assuming A = 0).
(c) H0 : A = 0 (assuming no random effects, i.e., ¾2, = 0), and the two-sided alternative is
Hi : A = 0 (assuming σ2μ = 0).
(d) Hd : A = 0 (assuming the possible existence of random effects, i.e., ¾2, > 0), and the
two-sided alternative is Hd : A = 0 (assuming ¾2, > 0).
(e) Hf : ¾X = 0 (assuming the possible existence of spatial correlation, i.e., A may be zero
or different from zero), and the one-sided alternative is Hf : ¾2, > 0 (assuming that A
may be zero or different from zero).
In the next sections, we derive the corresponding LM tests for these hypotheses and we
compare their performance with the corresponding LR tests using Monte Carlo experiments.
2.1 Joint LM Test for Ha: A = ¾2 = 0
o
The joint LM test statistic for testing Hf : A = ¾X = 0 vs Hf is given by
LMj = _NL_G + nTh 2; (2.11)
j 2(T -1) b ; v j
where G = ~0( JTu ,uIN )u - 1, H = u'%W)u, b = tr(W + W0)2=2 = tr(W2 + W0W) and U
denotes the OLS residuals. The derivation of this LM test statistic is given in Appendix A.1.
It is important to note that the large sample distribution of the LM test statistics derived
in this paper are not formally established, but are likely to hold under similar sets of low
level assumptions developed in Kelejian and Prucha (2001) for the Moran I test statistic
and its close cousins the LM tests for spatial correlation. See also Pinkse (1998, 1999) for
general conditions under which Moran flavoured tests for spatial correlation have a limiting
normal distribution in the presence of nuisance parameters in six frequently encountered
spatial models. Section 2.4 shows that the one-sided version of this joint LM test should be
used because variance components cannot be negative
2.2 Marginal LM Test for Hob: ¾2x = 0 (assuming A = 0)
Note that the first term in (2.11), call it LMg = 2(NT1) G2; is the basis for the LM test statistic
for testing Hob : σ2x = 0 assuming there are no spatial error dependence effects, i.e., assuming