The SPATIALREG Procedure
Parameter Space of Autoregressive Parameters
For all models except linear regression models in PROC SPATIALREG, the autoregressive parameters and are often assumed to satisfy some assumptions to ensure consistency of the maximum likelihood estimator (Elhorst 2013). For SAR and SDM models, the Jacobian term involves the log-determinant of , and the parameter space of is often specified such that is nonsingular. For SEM, SMA, SDM, and SDMA models, the Jacobian term involves the log-determinant of , and the parameter space of is often specified such that is nonsingular. For SAC, SDAC, SARMA, and SDARMA models, the Jacobian term involves the log-determinants of both and . As a result, the parameter space of and is often specified such that both and are nonsingular.
In the SPATIALREG procedure, the parameter space of autoregressive parameters and depends on the spatial weights matrix that you choose. For , the parameter space of the autoregressive parameters and in PROC SPATIALREG is determined as follows:
For a symmetric , the nonsingularity condition requires and . Here and denote the smallest (that is, most negative) and largest real eigenvalues of , respectively.
If is symmetric and subsequently row-standardized, the nonsingularity condition requires and . Here denotes the smallest purely real eigenvalue of the row-standardized .
If is asymmetric and subsequently row-standardized, the nonsingularity condition requires and . Here denotes the smallest purely real eigenvalue of the row-standardized .
When Taylor approximation or Chebyshev approximation is used for SAR and SDM models, is required to be row-standardized. In these cases, the restriction on the autoregressive coefficient is .