4 Conclusion

Maximum likelihood estimation based on the matrix exponential spatial specification (MESS) introduced here was shown to be computationally superior to most spatial estimators, requiring O() operations (the same as OLS) conditional upon formation of the spatial weight matrix and as low as O(log()) for the formation of the spatial weight matrix. When used in conjunction with common approaches to specifying spatial influences, the MESS results in a situation where the log-determinant term in the spatial likelihood function vanishes. The matrix exponential spatial specification provides an unusual situation (in spatial problems) where non-linear least-squares and maximum likelihood methods yield the same estimates. A simplification of the log-likelihood stemming from use of the matrix exponential spatial specification produces a situation where a unique closed-form solution for the estimates exists. This unique closed form solution greatly accelerates computation.
As an illustration of the speed gained through the use of these techniques, it took only 3.36 seconds using Matlab to compute a spatial autoregression involving 57,647 observations. We also demonstrated that the estimates for the parameters from the MESS model were almost identical to those from a spatial autoregressive (SAR) model and the inferences were identical, while the MESS model’s computational speed was over 1000 times faster than the more traditional SAR model. , the Fortran code ran 6 to 7 times faster than the Matlab code for the 57,647 observation census dataset. In other experiments (not reported here) we found that the MESS model when specified with spatially lagged explanatory variables produces estimates and inferences similar to those from the spatial Durbin model introduced in Anselin (1988).
This speed, along with the simpler MESS log-likelihood, facilitates maximization over a host of spatial parameter settings that can be used to vary the nature and extent of spatial influences in the model. The application by Bell and Bockstael (2000) provides a compelling motivation for this type of exploration. It may also enable Bayesian model selection criterion to be used in place of traditional likelihood ratio tests which would allow the rejection regions to vary with the sample size. This may have the potential to produce more parsimonious global model specifications because the rejection regions would narrow with larger sample sizes. Recent literature in the area of ‘Bayesian model averaging’ suggests that another potential role for Bayesian methods may be to produce a single posterior model that averages over alternative specifications associated with alternative spatial parameter settings. This would greatly facilitate reporting of results that are not conditional on a particular setting for decay in spatial weights, or number of neighbors employed. A final point is that the simpler MESS log-likelihood may make it an easier model to use in theoretical derivations needed to produce Bayesian and other spatial econometric extensions.
The computational advantages of the MESS should prove useful in solving a number of problems that arise in application of spatial econometric analysis. First, the MESS should provide an easily calculated benchmark against which to gauge the performance of other spatial estimators. In other words, MESS can serve as a more sophisticated null hypothesis than the typical assumption of spatial independence. In addition, since MESS provides a unique optimal estimate, it could help identify when another more complex model has become trapped in a local optima.
Second, the computational efficiency for large problems means that the MESS model can serve as a global description for very large data sets. Such global descriptions can help identify smaller regions where it may be of interest to apply more computationally costly techniques for analysis. Policy decisions often require global descriptions, so a collection of regional descriptions based on smaller subsets of the data set may not serve the desired purpose. This could become particularly important with the pending release of the year 2000 Census that will contain nearly 250,000 observations at the block-group microlevel.
A third area of applications opened up by this approach is computation of diagnostic statistics that have traditionally been problematical in the maximum likelihood spatial estimation setting. We provided a brief demonstration of the application of these statistics, but the potential of these diagnostics in large sample problems represents a relatively unexplored area for future research.
Another area where the MESS method could be useful is Monte Carlo experiments. Research examining the performance characteristics of alternative spatial estimation methodologies has been limited to relatively small data sets because of the computational burdens. Since both simulation and estimation proceed rapidly in the case of the MESS, this should facilitate Monte Carlo experiments based on larger, more realistic data sets.
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Table 2: Spatial and Aspatial Regression Models
Variables Aspatial Model Spatial Model
Intercept 1.224 -0.151
ln(Land Area) -0.085 -0.003
ln(Land Area) -0.017
Deviance 9,379.1 1,218.8
ln(Population) 0.115 0.022
ln(Population) 0.030
Deviance 1,358.6 366.6
ln(Per Capita Income) 1.084 0.677
ln(Per Capita Income) -0.463
Deviance 43,355.2 29,764.3
ln(Age) -0.127 -0.138
ln(Age) 0.127
Deviance 1,175.2 2,289.5
(# of neighbors) 30
Deviance (=29) 2.72
(geometric decay) 0.90
Deviance (=0.95) 612.82
Deviance (=0.85) 1240.52
(autoregressive parameter) -1.673
Deviance () 64,450.6
57,647 57,647
5 12
Maximum Log-likelihood -266,505.2 -228,850.4

Figure 1: Scaled Log-likelihood vs. Number of Neighbors across Differing
Figure 2: US Census Tract Locations with Smallest (O) and Largest () Leverage Observations Identified
Figure 3: US Census Tract Locations with Largest (O) and Smallest () Delete-1 Identified