Section 11.1 Challenges for regression with spatial data
There are a number of issues when it comes to fitting regression with spatial data. We will highlight three: validity, spatial dependence, and spatial heterogeneity. One, validity, is common to regression in general, although with spatial data it acquires particular relevance. The other two (spatial dependence and spatial heterogeneity) are typical of spatial data.
The most important assumption of regression is that of validity. The data should be appropriate for the question that you are trying to answer. As noted by [188]:
Optimally, this means that the outcome measure should accurately reflect the phenomenon of interest, the model should include all relevant predictors, and the model should generalize to all cases to which it will be applied... Data used in empirical research rarely meet all (if any) of these criteria precisely. However, keeping these goals in mind can help you be precise about the types of questions you can and cannot answer reliably.
Although validity is a key assumption for regression, whether spatial or not, it is the case that the features of spatial data make it particularly challenging (as discussed in detail by [191]).
The second challenge is spatial dependence. A key assumption of regression analysis is the independence between the observations. That is, a regression model is formally assuming that what happens in area \(X_i\) is not in any way related (it is independent of) what happens in area \(X_{ii}\text{.}\) But if those two areas are adjacent or proximal in geographical space, we know that there is a good chance that this assumption may be violated. Values that are close together in space are unlikely to be independent. Crime levels, for example, tend to be similar in areas that are close to each other. The problem for regression is not so much autocorrelation in the response or dependent variable, but the remaining autocorrelation in the residuals. If the spatial autocorrelation of our outcome \(y\) is fully accounted by the spatial structure of the observed predictor variables in our model, we don’t really have a problem. On the other hand, when the residuals of your model display spatial autocorrelation, you are violating an assumption of the standard regression model, and you need to find a solution for this.
When positive spatial autocorrelation is observed, using a regression model that assumes independent errors will lead to an underestimation of the uncertainty around your parameters (or overestimation if the autocorrelation is negative). With positive spatial autocorrelation, you will be more likely to engage in a Type I error (concluding the coefficient for a given predictor is not zero when this decision is not justified). Essentially, we have less degrees of freedom, as [178] note for positive spatial autocorrelation (the most common case in crime analysis and social science):
Underlying the results... is the fact that ’less information’ about the parameter of interest... is contained in a dataset where observations are positively autocorrelated... The term ’effective’ sample size has been coined and used to measure the information content in a set of autocorrelated data. Compared with the case of N independent observations, if we have N autocorrelated data values then the effective sample size is less than N (how much less depends on how strongly autocorrelated values are). The effective sample size can be thought of as the equivalent number of independent observations available for estimating the parameter... It is this reduction in the information content of the data that increases the uncertainty of the parameter estimate... This arises because each observation contains what might call ’overlapping’ or ’duplicate’ information about other observations (p. 7).
In previous weeks we covered formal tests for spatial autocorrelation, which allow us to test whether this is a feature of our dependent (or predictor) variable. So before we fit a regression model with spatial data, we need to explore the issue of autocorrelation in the attributes of interest, but critically what we need to establish is whether there is spatial autocorrelation in the residuals. In this session, we will see that there are two basic ways of adjusting for spatial autocorrelation: through a spatial lag model or through a spatial error model. Although there are multiple types of spatial regression models (see [192]; [193]; [180]; and [194], specifically for a review of their relative performance), these are the two basic forms that you probably (particularly if your previous statistical training sits in the frequentist tradition) need to understand before you get into other types.
A third challenge for regression models with spatial data is spatial heterogeneity (also referred to as non-stationarity). We already discussed the concept of spatial homogeneity when introducing the study of point patterns. In that context we defined spatial homogeneity as the homogeneous intensity of the point pattern across the study surface. But spatial homogeneity is a more general concept that applies to other statistical properties. In the context of lattice or area level data, it may refer to the mean value of an attribute (such as crime rate) across areas, but it could also refer to how other variables are related to our outcome of interest across the study region. It could be the case, for example, that some predictors display spatial structure in how they affect the outcome. It could be, for example, that presence of licensed premises to sell alcohol have a different impact on the level of violent crime in different types of communities in our study region. If there is spatial heterogeneity in these relationships, we need to account for it. We could deal with this challenge through some form of data partition, spatial regime, or local regression model (where we allow the regression coefficient to be area specific) such as geographically weighted regression. We will discuss this third challenge to regression in greater detail in the next chapter.
Not so much a challenge but an important limitation is the presence of missing data. Unlike in standard data analysis where you have a sample of \(n\) observations and there are a number of approaches to impute missing data, in the context of spatial econometrics, you only have one realisation of the data generating mechanism. As such, if there is missing data for some of the areas, the models cannot be estimated; although some solutions have been proposed when the percentage of missing cases is very small (see [195]).
In general, the development of a spatial regression model requires taking into account these challenges. There are various aspects that building the model will require [196]: selecting the right model for the data at hand, the estimation method, and the model specification (variables to include).
