Section 11.4 Choosing a spatial regression model
Subsection 11.4.1 Basic spatial models
If the test is significant (as in this case), then we possibly need to think of a more suitable model to represent our data: a spatial regression model. Remember spatial dependence means that (more typically) there will be areas of spatial clustering for the residuals in our regression model. So our predicted line (or hyperplane) will systematically under-predict or over-predict in areas that are close to each other. That’s not good. We want a better model that does not display any spatial clustering in the residuals.
There are two traditional general ways of incorporating spatial dependence in a regression model, through what we called a spatial error model or by means of a spatially lagged model. The Moran test doesn’t provide information to help us choose, which is why [200] introduced the Lagrange multiplier tests, which we will examine shortly.
The difference between these two models is both technical and conceptual. The spatial error model treats the spatial autocorrelation as a nuisance that needs to be dealt with. It includes a spatial lag of the error in the model (
lambda in the R output). A spatial error model basically implies that the:
spatial dependence observed in our data does not reflect a truly spatial process, but merely the geographical clustering of the sources of the behaviour of interest. For example, citizens in adjoining neighbourhoods may favour the same (political) candidate not because they talk to their neighbours, but because citizens with similar incomes tend to cluster geographically, and income also predicts vote choice. Such spatial dependence can be termed attributional dependence [179].
In the example we are using, the spatial error model "evaluates the extent to which the clustering of homicide rates not explained by measured independent variables can be accounted for with reference to the clustering of error terms. In this sense, it captures the spatial influence of unmeasured independent variables" [187]. As [196] suggests, this spatial effect allows us to model "invisible or hardly measurable supra-regional characteristics, occurring more broadly than regional boundaries (e.g., cultural variables, soil fertility, weather, pollution, etc.)" (p. 218).
The spatially lagged model or spatially autoregressive model, on the other hand, incorporates spatial dependence by adding a "spatially lagged" variable \(y\) on the right-hand side of our regression equation (this will be denoted as
rho in the R output). Its distinctive characteristic is that it includes a spatially lagged measure of our outcome as an endogenous explanatory factor (a variable that measures the value of our outcome of interest, say crime rate, in the neighbouring areas, as defined by our spatial weight matrix). It is basically explicitly saying that the value of \(y\) in the neighbouring areas of observation \(n_i\) is an important predictor of \(y_i\) on each individual area \(n_i\text{.}\) This is one way of saying that the spatial dependence may be produced by a spatial process such as the diffusion of behaviour between neighbouring units:
If so the behaviour is likely to be highly social in nature, and understanding the interactions between interdependent units is critical to understanding the behaviour in question. For example, citizens may discuss politics across adjoining neighbours such that an increase in support for a candidate in one neighbourhood directly leads to an increase in support for the candidate in adjoining neighbourhoods [179].
So, in our example, what we are saying is that homicide rates in one county may be affecting the level of homicide in nearby counties (however you choose to define "nearby").
It is important to recognize that these models for spatial lag and spatial error processes are designed to yield indirect evidence of diffusion in cross-sectional data. However, any diffusion process ultimately requires "vectors of transmission," i.e., identifiable mechanisms through which events in a given place at a given time influence events in another place at a later time. The spatial lag model, as such, is not able to discover these mechanisms. Rather, it depicts a spatial imprint at a given instant of time that would be expected to emerge if the phenomenon under investigation were to be characterized by a diffusion process. The observation of spatial effects thus indicates that further inquiry into diffusion is warranted, whereas the failure to observe such effects implies that such inquiry is likely to be unfruitful (p. 567).
A third basic model is the one that allows for autocorrelation in the predictors (Durbin factor). Here we allow for exogenous interaction effects, where the dependent variable of a given area is influenced by independent explanatory variables of other areas. In this case we include in the model a spatial lag of the predictors. These, however, have been less commonly used in practice.
These three models only have one spatial component, but this could be combined. In the last 15 years, interest in models with more than one spatial component has grown. You could have models with two spatial components and even consider together the three ways of spatial interaction (in the error, in the predictors, and in the dependent variable). [203] classified the different models according to the spatial interactions included.
| Models | Features |
| Manski (GNS Model) | Includes spatial lag of dependent and explanatory variables, and autocorrelated errors |
| Kelejian/Prucha (SAC or SARAR Model) | Includes spatial lag of dependent model and autocorrelated errors, often called a spatial autoregressive-autoregressive model |
| Spatial Durbin Model (SDM) | Allows for spatial lag of dependent variable and explanatory variables |
| Spatial Durbin Error Model (SDEM) | Allows for lag of explanatory variables and autocorrelated errors |
| Spatial Lag Model (SAR) | The traditional spatial lag of the dependent variable model, or spatial autoregressive model |
| Spatial Error Model (SEM) | The traditional model with autocorrelated errors |
In practice, issues with overspecification mean you rarely find applications that allow for interaction at the three levels (the so-called Manski model or general nesting model (GNS)) simultaneously. But it is still useful for testing the right specification.
Subsection 11.4.2 Lagrange multipliers: the bottom-up approach to select a model
Moran’s I test statistic has high power against a range of spatial alternatives. However, it does not provide much help in terms of which alternative model would be most appropriate. Until a few years ago, the main solution was to look at a series of tests that were developed by [200] for this purpose: the Lagrange multipliers. The Lagrange multiplier test statistics do allow a distinction between spatial error models and spatial lag models, which was the key concern before interest developed in models with more than one type of spatial effect. The Lagrange multipliers also allow to evaluate if the Kelejian/Prucha (SAC) model is appropriate. This approach, starting with a baseline OLS model, tests residuals, runs Lagrange multipliers, and then selects the SAC or SEM model. It is referred to as the bottom-up or specific to general approach and was the dominant workflow until recently.
Both Lagrange multiplier tests (for the error and the lagged models,
LMerr and LMlag respectively), as well as their robust forms (RLMerr and RLMLag, also respectively) are included in the lm.LMtests() function. Again, a regression object and a spatial listw object must be passed as arguments. In addition, the tests must be specified as a character vector (the default is only LMerror), using the c( ) operator (concatenate), as illustrated below. Alternatively, one could ask to display all tests with test="all". This would also run a test for the so-called SARMA model, but Anselin advises against its use. For sparser reporting, we may wrap this function within a summary() function.
summary(lm.LMtests(fit3_90, rwm, test = c("LMerr","LMlag","RLMerr","RLMlag")))
## Lagrange multiplier diagnostics for spatial ## dependence ## data: ## model: lm(formula = eq1_90, data = ncovr) ## weights: rwm ## ## statistic parameter p.value ## LMerr 148.80 1 < 2e-16 *** ## LMlag 116.36 1 < 2e-16 *** ## RLMerr 35.41 1 2.7e-09 *** ## RLMlag 2.97 1 0.085 . ## --- ## Signif. codes: ## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
How do we interpret the Lagrange multipliers? The interpretation of these tests focuses on their significance level. First we look at the standard multipliers (
LMerr and LMlag). If both are below the .05 level, like here, this means we need to have a look at the robust version of these tests (Robust LM).
If the non-robust version is not significant, the mathematical properties of the robust tests may not hold, so we don’t look at them in those scenarios. It is fairly common to find that both the lag (
LMlag) and the error (LMerr) non-robust LM are significant. If only one of them were, then it is problem solved. We would choose a spatial lag or a spatial error model according to this (i.e., if the lag LM was significant and the error LM was not, we would run a spatial lag model or vice versa). Here we see that both are significant, thus, we need to inspect their robust versions.
When we look at the robust Lagrange multipliers (
RLMlag and RLMerr) here, we can see only the Lagrange for the error model is significant, which suggest we need to fit a spatial error model. But there may be occasions when both are as well significant. What do we do then? Luc Anselin (2008) proposes the following criteria:
When both LM test statistics reject the null hypothesis, proceed to the bottom part of the graph and consider the Robust forms of the test statistics. Typically, only one of them will be significant, or one will be orders of magnitude more significant than the other (e.g., p < 0.00000 compared to p < 0.03). In that case the decision is simple: estimate the spatial regression model matching the (most) significant "robust" statistic. In the rare instance that both would be highly significant, go with the model with the largest value for the test statistic. However, in this situation, some caution is needed, since there may be other sources of misspecification. One obvious action to take is to consider the results for different spatial weight and/or change the basic (i.e., not the spatial part) specification of the model. There are also rare instances where neither of the Robust LM test statistics are significant. In those cases, more serious specification problems are likely present and those should be addressed first (p. 199–200).
By other specification errors, Anselin refers to problems with some of the other assumptions of regression that you should be familiar with.
As noted, this way of selecting the "correct" model (the bottom-up approach), using the Lagrange multipliers, was the most popular until 15 years ago or so. It was attractive because it relied on observing the residuals in the non-spatial model, it was computationally very efficient, and when the correct model was the spatial error or the spatial lag model, simulation studies had shown it was the most effective approach [195]. Later we will explore other approaches. Also, since the spatial error and the spatial lag models are not nested, you could not compare them using likelihood ratio tests [193].
