and
|
2 TNN NN tr[(W0B + B'W) + (B0B)-1] |
rτι _ -^- -^- -I ɪ tr[2b02b] | ||
|
Jμ = |
T t tr[((W 0B + Bb0W ) + (jb0 B?)-1¢ 2 ] |
° rr> . ^ ^. 2T2 tr[W 0B + BfW ] |
7 ; (2.28) |
|
2¾4 tr[(ib0ib)2] J |
T
2¾°
N ¾° g h
24 2
¾°g ¾° c ¾° d
h ¾° d Te
(2.29)
where g = tr[(W0B + Bb0W)(B0B) 1], h = tr[B0B], c = tr
. ^ - - , ^. ^ .ï-
tr[W0Bb + Bb0W] and e = tr[(Bb0Bb)2]. Therefore,
2
((W 0B + B0W )(B0B)-1)
d=
LMμ = (D)μ)2^2¾°^TNσ°ec - N¾°d2 - Tσ°g2e + 2¾°ghd - ¾°h2c)
(≡°c - ¾°g2^ (2.30)
where Dμ and Jμ are evaluated at the maximum likelihood estimates under the null hypoth-
esis H0e. However, LMμ ignores the fact that the variance component cannot be negative.
Therefore, the one-sided version of this LM test is given by
DμJ(2¾,=T)(N¾°c - ¾°g2)
(2.31)
LMμ = V _
∙ξ=TN¾°ec - N¾°d2 - T¾°g2e + 2¾°ghd - ¾°h2c
and this should be asymptotically distributed as N(0; 1) under He as N ! 1 for T fixed.
3 MONTE CARLO RESULTS
The experimental design for the Monte Carlo simulations is based on the format extensively
used in earlier studies in the spatial regression model by Anselin and Rey (1991) and Anselin
and Florax (1995) and in the panel data model by Nerlove (1971).
The model is set as follows :
yit = ® + x'itβ + uit; i = 1,∙ ∙ ∙ N; t = 1,∙ ∙ ∙ ; T; (3.1)
where ® = 5 and β = 0:5. xit is generated by a similar method to that of Nerlove (1971). In
fact, xit = 0:1t + 0:5xi;t-1 +zit, where zit is uniformly distributed over the interval [-0:5; 0:5].
The initial values xio are chosen as (5 + 1Ozio). For the disturbances, uit = μi + "it, "it =
λ pN=1 Wij "it + ° it with μi » IIN (0, ¾μ) and °it » IIN (0,¾° ): The matrix W is either a