Spatial Model Estimation

The Model

• e = u
• e = (I-rW)^-1u, or (I-rW)e = u ... Spatial AR(1)
• e = (I-qW)u, or (I-qW)^-1e = u ... Spatial MA(1)
• e = (I-rW)^-1(I-qW)u, or (I-qW)^-1(I-rW)e = u ... Spatial ARMA(1,1)

Log-Likelihood Function

f(u) = (1/(2ps²))^N/2 exp(-½ u'u/s²)

The joint likelihood of Y=[Y₁,Y₂,...,Y_N]' is f(Y) = f(u)|du/dY|, where |du/dY| is the Jacobian transformation. Then the log-likelihood of Y is written as

L = -N/2 ln(2p) -N/2 ln(s²) -½ u'u/s² + ln(|du/dY|)

By concentrating on the maximum likelihood estimate of s² = u'u/N, the concentrated log-likelihood function is

L^* = -N/2 [ln(2p)-ln(N)+1] - N/2 ln(u'u) + ln(|du/dY|)

The Jacobian term |du/dY| is defined by

• Spatial Lag: |(I-lW)|
• Spatial AR(1): |(I-rW)|
• Spatial MA(1): |(I-qW)^-1|=1/|(I-qW)|
• Spatial ARMA(1,1): |(I-qW)^-1(I-rW)| =|(I-rW)|/|(I-qW)|

Maximum Likelihood Estimation

L^* = -N/2 [ln(2p)-ln(N)+1] - N/2 ln(u^*'u^*)

where u^* = u/(|du/dY|^1/N) in which u is scaled by 1/(|du/dY|^1/N).

Maximizing the log-likelihood function is the same as minimizing the sum of squares of scaled errors: u^*'u^*. Since u^* depends on the parameters of error structure (r,q), in addition to the regression parameters (b,l,g), spatial model estimation is a nonlinear optimization problem which involves the evaluation of large Jacobian in the form of |I-aW| where a is l, r and/or q depending on model specification. The determinant of (I-aW) can be computed from the products of eigenvalues of (I-aW). That is, |I-aW| = (1-aw₁)(1-aw₂)...(1-aw_N) where w₁,w₂,...,w_N are the N eigenvalues of the spatial weights matrix W. Note that W is row-standardized but non-symmetry). In addition, the estimation requires that the parameter a associated with the spatial weights matrix W satisfies the range condition: 1/w_min < a < 1/w_max. Furthermore, for a row-standardized W, w_max = 1.

For a spatial MA(1) or ARMA(1,1), the computation of (I-qW)^-1 posts a challenging task since it is large in size and the result is not necessary sparse.