The presence of residual spatial autocorrelation need not bias the estimates of variance of regression coefficients, provided that the covariates themselves do not exhibit spatial autocorrelation
if covariates and the residual are autocorrelated, it is likely that the estimates of variance of regression coefficients will be biased downwards if attempts are not made to model the spatial processes.
Models
The spatial lag is usually taken as the mean of the values of the variable at neighbouring observations for row-standardised spatial weights, or the sum for binary spatial weights.
Spatial Lag Model (SLM) or Spatial Autoregressive (SAR) has spatially lagged values of the response variable \[
y = \rho_{\text{lag}}Wy + X\beta + \epsilon
\]
\(y\) : A \(n\times 1\) vector of observations on a response variable taken at each of \(n\) locations
\(\rho_{\text{lag}}\) : Scalar spatial parameter
\(W\) : A \(n\times n\) spatial weights matrix
\(X\) : A \(n\times k\) matrix of covariates usually including the intercept
\(\beta\) : A \(k\times 1\) vector of parameters
\(\epsilon\) : A \(n\times 1\) vector of iid disturbances
Spatial Error Model (SEM) takes account of the spatial dependence of the errors \[
\begin{align}
&y = X\beta + u \\
&\text{where} \;\; u = \rho_{\text{err}} Wu + \epsilon
\end{align}
\]
\(y\) : A \(n\;\times\;1\) vector of observations on a response variable taken at each of \(n\) locations
\(X\) : A \(n\;\times\;k\) matrix of covariates
\(\beta\) : A \(k\;\times\;1\) vector of parameters
\(\epsilon\) : A \(n\;\times\;1\) vector of iid disturbances
\(\rho_{\text{err}}\) : A scalar spatial parameter.
Spatial Durbin Model (SDM) adds spatially lagged covariates to the SLM model \[
y = \rho_{\text{lag}}Wy + X\beta + WX\gamma + \epsilon
\]
\(WX\) : Is the spatially weighted, lagged covariates term
\(\gamma\) : A \(k\;' \times 1\) vector of parameters where often \(k' = k - 1\) when using row-standardized spatial weights and ommitting the spatially lagged intercept.
The indirect spillover effect produced by the SDM model is both local and global with no prior restrictions on the magnitudes of these effects
The exogenous regressor of one location casts an effect on the dependent variable of the neighboring locations through feedback loops.
The feedback loops are a result of the Durbin term due to which a location, say \(i\) affects aneighboring location, say \(j\), which eventually has a consequence on \(i, j = 1,2, \ldots , n\)
An advantage of the spatial Durbin model is that it produces correct standard errors or t−values of the coefficient estimates also if the true data generating process is a spatial error model
The SDM model reduces to SEM under a non-linear common factor restriction. Thus, theoretically speaking, the choice between SDM and SEM can be settled by performing astatistical test for the common factor restriction.
Rao’s score test principle which is asymptotically equivalent to the Likelihood Ratio Test (LRT) but computationally much simpler
Use of the likelihood ratio test (LRT) is not attractive for testing a non-linear hypothesis, since LRT requires maximum likelihood estimation both under the null and the alternative hypotheses.
Others
Spatially Lagged X Model (SLX) omits the \(\rho_{\text{lag}}Wy\) term (from the SDM model?)
General Nested Model (GNM) or General Nested Spatial Model (GNS) adds (to the SDM model?) an additional spatial process in the residuals
Start from the General Nested Model (GNM) and eliminate unneeded terms
Venn Diagram of Econometric Spatial Models (source)
