Testing Panel Data Regression Models with Spatial Error Correlation

that λ = 0, see Breusch and Pagan (1980). This LM statistic should be asymptotically
distributed as Â₁² under H₀^b as N ! 1; for a given T . But this LM test has the problem
that the alternative hypothesis is assumed to be two-sided when we know that the variance
component cannot be negative. Honda (1985) suggested a uniformly most powerful test for
H₀^b based upon the square root of the G² term, i.e.,

nt I NT „

LMι — ∖ —-.------ G:

1 y 2(T - 1)

This should be asymptotically distributed as N(0,1) under H0 as N ! 1; for T fixed.
Moulton and Randolph(1989) showed that the asymptotic N(0,1) approximation for this
one sided LM test can be poor even in large samples. This occurs when the number of
regressors is large or the intra-class correlation of some of the regressors is high. They
suggest an alternative standardized LM (SLM) test statistic whose asymptotic critical values
are generally closer to the exact critical values than those of the LM test. This SLM test
statistic centers and scales the one sided LM statistic so that its mean is zero and its variance
is one:

where d1 — ——— and D1 — (Jτ   In) with U denoting the OLS residuals.  Using the

u~⁰u~

normality assumption and results on moments of quadratic forms in regression residuals (see

e.g. Evans and King, 1985), we get

E(d1) — tr(D1M)=s;                                                                  (2.14)
where s — NT - k and M — I_nt - X(X⁰X)"¹X⁰. Also.

var(d1) — 2fs tr(D1M)² - [tr(D1M)]²g=s²(s + 2):                                    (2.15)

Under H₀^b; SLM1 should be asymptotically distributed as N(0,1).

2.3 Marginal LM Test for Hθ: λ = 0 (assuming σⁱ_μ = 0)

Similarly, the second term in (2.11), call it LMh — NT H² ; is the basis for the LM test
statistic for testing H0: λ — 0 assuming there are no random regional effects, i.e., assuming
that σ²_μ — 0, see Anselin (1988). This LM statistic should be asymptotically distributed as

²₁ under H₀^c . Alternatively, this can be obtained as

/N ²T                                                                        ,    .

LM2 — V ~ H:                                               (2.16)

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