Spatial Econometrics Regression

Generalized Nested Spatial Model (GSNM)

$$ y = \rho_{Lag} W y + X\beta + WX\gamma + \mu, \mu = \rho_{Err} W \mu + \xi $$

where $y$ is an $(N \times 1)$ vector of observations on a response variable taken at each of $N$ locations,$W$ is a spatial weight matrix, which is a non-negative matrix of size $N \times N$, $X$ is an $(N \times k)$ matrix of covariates, $\beta$ is a $(k \times 1)$ vector of parameters, $\mu$ is an $(N \times 1)$ spatially autocorrelated disturbance vector, $\xi$ is an $(N \times 1)$ vector of independent and identically distributed disturbances, $\rho_{Lag}$ and $\rho_{Err}$ is a scalar spatial parameter. $\gamma$ is a $(k’ \times 1)$ vector of parameters. $k’$ defines the subset of the intercept and covariates, often $k’ = k-1$ when using row standardised spatial weights and omitting the spatially lagged intercept.

This may be constrained to the double spatial coefficient model SAC/SARAR by setting ${\mathbf \gamma} = 0$, to the spatial Durbin (SDM) by setting $\rho_{\mathrm{Err}} = 0$, and to the error Durbin model (SDEM) by setting $\rho_{\mathrm{Lag}} = 0$. Imposing more conditions gives the spatial lag model (SLM) with ${\mathbf \gamma} = 0$ and $\rho_{\mathrm{Err}} = 0$, the SEM with ${\mathbf \gamma} = 0$ and $\rho_{\mathrm{Lag}} = 0$, and the SLX with $\rho_{\mathrm{Lag}} = 0$ and $\rho_{\mathrm{Err}} = 0$.

Maximum likelihood estimation in spatialreg

For models with single spatial coefficients (SEM and SDEM using errorsarlm(), SLM and SDM using lagsarlm()), the methods initially described by Ord are used. The following table shows the functions that can be used to estimate the models described above using maximum likelihood.

Spatial Econometrics Models

Review of spatial regression models