Spatial Autocorrelation. Consider the regression model given in (9.1) with spatial autocorrelation defined in (9.36).

(a) Verify that the first-order conditions of maximization of the log-likelihood function given in

(9.39) yield (9.41).

(b) Show that for testing H0; λ = 0, the score ∂lnL/∂λ evaluated under the null, i.e., at λ = 0, is given by uWu/σ2.

(c) Show that the Information matrix with respect to σ2and λ, evaluated under the null of λ = 0, is given by

⎡

⎢⎢⎢⎣

n 2σ4 tr(W)

σ2 tr(W)

σ2 tr(W2) + tr(WW)

⎤

⎥⎥⎥⎦

(d) Conclude from parts

(b) and

(c) that the Lagrange Multiplier for H0; λ = 0 is given by LMλ

in (9.46). Hint: Use the fact that the diagonal elements of W are zero, hence tr(W) = 0.