where B = In — XW and In is an identity matrix of dimension N. The model (2.1) can be
rewritten in matrix notation as
y = Xβ + u; (2.5)
where y is now of dimension NT £ 1, X is NT £ k, β is k £ 1 and u is NT £ 1: The
observations are ordered with t being the slow running index and i the fast running index,
i.e., y0 = (y11; : : : ; y1N; : : : ; yT1; : : : ; yTN): X is assumed to be of full column rank and its
elements are assumed to be asymptotically bounded in absolute value. Equation (2.2) can
be written in vector form as:
u = (lt In )μ + (It B-1)° (2.6)
where °0 = (° 1; ¢ ¢ ¢ ; °T), Lt is a vector of ones of dimension T, It is an identity matrix of
dimension T and denotes the Kronecker product. Under these assumptions, the variance-
covariance matrix for u can be written as
u = ¾2μ(Jτ In ) + ¾° (It (B0B)-1); (2.7)
where JT is a matrix of ones of dimension T. This variance-covariance matrix can be rewritten
as:
u = ¾° Jττ (TφIN + (B0B)-1) + Eτ (B0B)-1i = ¾°§u, (2.8)
where φ = ¾2 ¾°, Jτ = Jτ=T; Eτ = Iτ — Jτ and §u = JTτ (T≠In + (B0B)-1) + Eτ
(B0B)-1 : Using results in Wansbeek and Kapteyn (1983), §u-1 is given by
§-1 = Jt (T≠In + (B0B)-1)-1 + Et B0B: (2.9)
Also, j§uj = 7'<'√v + (B0B)-1j ∙ j(B0B)-1jτ-1: Under the assumption of normality, the log-
likelihood function for this model was derived by Anselin (1988, p.154) as
|
L= |
— NT ln2¼¾° — |
2ln j§uj — 2¾2 u0§u1u | |
|
= |
—N2T ln2¼¾° — 1 0 -1 2¾2u §u u; |
2 ln[∣T≠IN + (B0B)-1 j] + T 2 ' ln ∣B0Bj |
(2.10) |
with u = y — Xβ. Anselin (1988, p.154) derived the LM test for X = 0 in this model. Here,
we extend Anselin’s work by deriving the joint test for spatial error correlation as well as
random region effects.
The hypotheses under consideration in this paper are the following: