# Spatial autocorrelation versus spatial stationarity

Let's assume we have points in two-dimensional space, and we wish to measure the effects of attributes $X$ on attribute $y$. The typical linear regression model is of course $$y= X\beta + \epsilon$$

There are two problems here: the first is that the $\epsilon$ terms may be spatially correlated (violating the independent and identical errors assumption), and the second is that the regression slope may vary throughout space. The first issue can be dealt with by incorporating spatial lag terms into the model, as in

$$y=\rho W y + X\beta + \epsilon$$

We can even incorporate spatially autoregressive omitted variables (spatial fixed effects) with the spatial Durbin model described in the text by LeSage and Pace

$$y=\rho W y + X\beta + WX\lambda + \epsilon$$

where $\rho$ is the strength of spatial correlation controlled by the weights matrix $W$. Clearly the form of the spatial lag will depend on assumptions about the form of the spatial correlation.

The second problem has been addressed using "geographically-weighted regression" (GWR), a technique I am not as familiar with, but which is explained by Brunsdon, et al. (1998). As far as I can tell, it involves fitting an array of regression models to weighted sub-regions, thus getting an estimate of each $\beta_i$ that changes based on its space, $$\hat{\beta}_i = (X^TW_iX)^{-1}X^T W_i y$$ where $W$ is another spatial weights matrix, not necessarily different from the one above.

My question: Is the first method (spatial autoregression) not sufficient to yield an unbiased estimate of the average marginal effect of $X$ on $y$? GWR seems like it is over-fitting: of course the $\beta$ changes in space, but if we want to know the average expected effect of a treatment without regard for its spatial position, what could GWR contribute?

Here's my attempt at an initial answer:

1. If I want to know the premium for an additional bedroom in a specific neighborhood, it seems that GWR would be my best option.
2. If I want to know the unbiased global average premium for an additional bedroom, I should use spatial autoregressive techniques.

Would love to hear other perspectives.

• As currently worded your question is in the abstract until at the end you refer to bedrooms. This perhaps suggests that $y$ is housing rents and $X$ includes various attributes of housing including number of bedrooms, but it would be helpful to clarify this in the question. – Adam Bailey Jan 31 '13 at 16:29
• I wish for an abstract answer, though my particular application is indeed home prices. – gregmacfarlane Jan 31 '13 at 19:32
• Have you thought of looking up Panel data econometrics for modeling ideas? Your specific example at the end looks like a hedonic price-index model in a panel-data setting with individual effects (or with varying coefficients), and errors that possibly are cross-sectionally correlated, while in abstract, panel-data methods provide both the "space" dimension and the "time" dimension. – Alecos Papadopoulos Apr 7 '14 at 2:30

## 2 Answers

I think you are answering properly your own set of questions.

Housing market research is normally tackled by using non-parametric models.

For your second question, I agree in the use of SAR models, and I will go with the Durbin for two reasons: First, the Durbin model produces unbiased coefficient estimates. Second, it is able to produce spillover effects that in relation to their correspondent direct effect may be different for each explanatory variable.

Hope this helps!

The problem is not with spatial Durbin estimation itself. It could be estimated by maximum likelihood and you can calculate the partial effects. The problem occurs when the space effect is not stationary in dgp, so that you cannot properly model its effect this way. GWR does many regressions over your space, thus provides you a vector of coefficients over your space. Statistical inferences over those coefficients is not straightforward but it shows well on a map as an exploratory tool. So, for finding out the premium of an additional bedroom in a specific neighborhood, your best bet would probably be running a separate spatial regression on that neighborhood. For finding premium of an additional bedroom globally, use spatial regression as well, but also do mind that the coefficients are not linear in parameters with such regressions; for that reason, the premiums are defined at specific values such as the mean.