Spatial Modeling: SAR vs CAR

Question

I'm trying to understand why one would use a Conditional Autoregressive Model (CAR) vs a Spatial Autoregressive Model (SAR) to look at migratory decisions. I have been reading through some of the lit and need some help understanding this passage from:

Spatial Regression Models for Demographic Analysis, by Guangqing Chi & Jun Zhu (numbers and emphasis added to relate to how I'm understanding it so far)

"Spatial error models and sometimes spatial lag models are referred to as the simultaneous autoregressive model (SAR). Another popular class of models is the conditional autoregressive (CAR) models. The key distinction between the SAR and the CAR models is in the model specification (Cressie 1993). (1) SAR models explain the relations among response variables at all locations on the lattice simultaneously and the spatial effect is considered to be endogenous. (2) In contrast, CAR models specify the distribution of a response variable at one location by conditioning on the values of its neighbors in the neighborhood and the spatial effect of the neighbors is considered to be exogenous (Anselin 2003). (3) While CAR models are popular in the statistics literature and many other disciplines, SAR models are favored in spatial econometrics and spatial demography, possibly because interpretation of the spatial autocorrelation coefficient resembles that of standard linear regression and thus may seem more natural. (4) The relation between the two types of models is close, however, as SAR models (from a spatial error model) may be represented by possibly higher order CAR models (Cressie 1993)."

Assuming the level is census blocks, am I understanding this correctly to mean that:

1. SAR models consider the migration flows from all census blocks to matter, with closer census blocks being more important while
2. CAR models just look at direct neighbors
3. However, Demographers really just use SAR because habit and "it's easier to understand", and
4. I'm not entirely sure what they really mean by higher order?

Did that make sense? Any advice would be appreciated!

P.S. I did see this answer here:Spatial statistics models: CAR vs SAR, and I still need a bit more help

Answer 1

1, Yes, the SAR models consider migration flows from all census blocks as important, but do so in a decaying fashion. One useful trick is to expand the "spatial filter matrix" in the reduced form of the SAR using Leontief expansion.

Say, for starters, we have the SAR:

$$\begin{align} y &= \rho \mathbf{W}y + \mathbf{X}\beta + \epsilon \\ \epsilon &\sim \mathcal{N}(0,\sigma^2) \end{align}$$

Then, we can turn it into the reduced form to isolate $y$ on the left hand side:

$$\begin{align} y - \rho \mathbf{W}y &= \mathbf{X}\beta + \epsilon \\ (\mathbf{I} - \rho \mathbf{W})y &= X\beta + \epsilon \\ y &= (\mathbf{I} - \rho \mathbf{W})^{-1}(X\beta + \epsilon) \end{align} $$

The Leontief expansion works by taking matrices of the form $(I-A)^{-1}$ and transforming them into an infinite series: $$ (I - \rho \mathbf{W})^{-1} = (I + \rho \mathbf{W} + \rho^2\mathbf{W}^2 + \rho^3\mathbf{W}^3 + \dots )$$

So, this means that you can also spell the SAR model: $$ y = \sum_k^\infty \rho^k\mathbf{W}^k (\mathbf{X}\beta + \epsilon) $$ Which mirrors how back-substitution in a time series model results in an infinite sum.

For the very typical "standard" case of a row-standardised spatial weights matrix, $\mathbf{W}^k$ represents a measure of the "kth" order of spatial weight, or the observations that you can visit using $k$ steps from each site. For a simple lattice/grid setup, the $i$th row of $\mathbf{W}^2$ represents the cells you can reach in exactly two steps from $i$. The usual term to describe this kind of relationship is that these are the "higher order" neighbours of observation $i$.

So, these "higher order" terms are what measure the "global" structure of the model. The typical $\rho$ is somewhere between $-1$ and $1$, so these "higher order" terms get quite small quite quickly. But, when dependence is large, they decay more slowly.

2, Yes, the spillovers from the SAR do not arise in a CAR model, which can be stated as a typical multivariate normal model: $$\begin{align} y &= \mathbf{X}\beta + \epsilon \\ \epsilon &\sim \mathcal{N}(0, (\mathbf{I}-\delta\mathbf{W})\sigma^2) \end{align}$$

3, I can't speak to why demographers and econometricians generally use the SAR model over other forms, other than the fact that it's the "true" analogue to the autoregressive model common in time series. They both have very "surprising" behaviours in terms of the implied correlations (from Wall (2006), which I find to be very convincing).

4, I hope I've answered by way of the explanation in 1.