# Spatial Econometrics

### Introduction to Spatial Econometrics

Traditional econometrics refers to the statistical analysis of economic data that aims to provide empirical assessment of the relationships between various economic phenomena (Geweke et al., 2008). Spatial econometrics extends traditional econometrics by considering the potential effects associated with the locations where data are collected (LeSage and Pace, 2009). Specifically, the potential effects associated with locations are regarded as spatial effects; these can be further divided into spatial dependence and spatial heterogeneity. The recent developments in spatial econometric modeling focus on spatial dependence and relatively little attention has been paid to spatial heterogeneity (Fotheringham, 2009). Some of the classic spatial econometric texts are now difficult to find or are out of print. A more recent text book is *Introduction to Spatial Econometrics** (*LeSage and Pace, 2009). The authors point out that spatial dependence and spatial heterogeneity violate the assumptions of the traditional econometrics models, such as independence and constancy. These violations can lead to biased coefficient estimates and incorrect inferences drawn from a model. In general, spatial dependence among locations could be captured by adding a spatial lag or spatial error component into a regression model. Spatial heterogeneity can be incorporated via what is known as a spatial regime model—a model that allows the estimated effects between independent and dependent variables to vary across different types of space (Anselin, 1990).

The spatial data-generating process underlying the regression model with a spatial lag effect is one where the value of the dependent variable in a certain location (observation) is related to the average value of the dependent variable in a set of neighboring locations (where neighboring is defined by the spatial weights matrix (W)∑ *β* ** *(*MCD Features*) + *W * **error* + *µ*. Chi and Zhu (2008) had relatively few explanatory variables in the explanatory analysis, and *W * **error* helps to capture the effects of the unknown factors that are not included in the regression. Note that *µ* indicates the independently and identically distributed errors.

As discussed previously, *W* is a spatial weight matrix where the spatial relationships among locations are specified.

As is perhaps apparent to the reader, the definition of the spatial weight matrix, *W*, is very important. In current spatial econometrics literature, *W* is regarded as a priori and has not been well connected to theories (Leenders, 2002), despite the argument that spatial matrices could go beyond the geographic associations (Beck et al., 2006). There are several different approaches to defining spatial relationships between two locations: (1) Spatial contiguity approach defines the first-order neighbors as sharing the same boundary or a vertex. The second-order neighbors are defined as the neighbors of the first-order neighbors. (2) Using the distance-based approach, researchers can define two locations/areas as neighbors if their geographic distance is within a certain threshold distance. (3) Researchers can assign the same number of neighbors to each location with the k-nearest neighbor approach. For example, using the 5-nearest neighbors to define the spatial weight matrix, each location will have 5 neighbors and these neighbors will have the top five shortest distances. While many approaches are available to define a spatial weight matrix, there is no agreement on which one is the best.

Both spatial lag and error models assume that the estimated relationships between the dependent and independent variables do not vary by locations. However, it is possible that the relationships may vary spatially, a concept of spatial heterogeneity. Baller and colleagues (2001) analyzed the county-level homicide rates and found that the homicide rates in the southern region of the U.S. were distinctive from the non-southern region. In order to capture the spatial heterogeneity, they implemented the spatial regime analysis with the following specifications: *Homicide Rates = *∑ *β* *** * *(*Structural Covariates*)* *+* *∑ *Ɵ *(*Structural Covariates ***** Southern Region Indicator *)* *+ *ɛ*. The spatial regime analysis accounts for the spatial heterogeneity and allows the spatially varying data-generating process (Anselin, 1990). It should be noted that the spatial regime modeling can be further combined with spatial lag and spatial error models (Curtis et al., 2012). Another analytic approach to spatial heterogeneity is the geographically weighted regression (GWR) (Fotheringham et al., 2002); however GWR is a localized perspective, rather than a spatial econometrics approach.

### Further Reading

The goal of this webpage is to give a brief introduction to spatial econometrics and we refer the audience to the following books and articles for more detailed discussions:

1. Anselin, L. 2002. Under the hood issues in the specification and interpretation of spatial regression models. *Agricultural Economics 27*(3): 247–267.

2. Arbia, G. and B.H. Baltagi. 2009. *Spatial Econometrics: Methods and Applications*. Physica-Verlag: Heidelberg, Germany.

3. LeSage, J.P. and R.K. Pace. 2009. *Introduction to Spatial Econometrics*. Boca Raton: Chapman & Hall/CRC.

4. Parent, O. and J.P. LeSage. 2010. Spatial econometric model averaging. In M.M. Fischer and A. Getis (Eds.), *Handbook of Applied Spatial Analysis: Software Tools, Methods and Applications*. New York: Springer Verlag, pp. 435–460.

5. Ward, M.D. and K.S. Gleditsch. 2008. *Spatial Regression M**odels*. Sage Publications, Inc.

### Software Programs for Spatial Econometrics Modeling

The following software programs are widely used to implement spatial econometrics models:

1. GeoDa: https://geodacenter.asu.edu/

2. Spdep package in R: http://cran.r-project.org/web/packages/spdep/spdep.pdf

3. Econometrics Toolbox in Matlab: http://www.spatial-econometrics.com/

The user’s manuals of these software programs can be found in these websites.

### References

Anselin, L. 1990. Spatial dependence and spatial structural instability in applied regression analysis. *Journal of Regional Science* 30(2): 185–207.

Baller, R. D., L. Anselin, S.F. Messner, G. Deane, and D.F.Hawkins. 2001. Structural covariates of U.S. county homicide rates: Incorporating spatial effects. *Criminology* 39(3): 561–590.

Beck, N., K.S. Gleditsch, and K. Beardsley. 2006. Space is more than geography: Using spatial econometrics in the study of political economy. *International Studies Quarterly* 50(1): 27–44.

Chi, G. and J. Zhu. 2008. Spatial regression models for demographic analysis. *Population Research and Policy Review* 27(1): 17–42

Curtis, K.J., P.R. Voss, and D.D.Long. 2012. Spatial variation in poverty-generating processes: Child poverty in the United States. *Social Science Research* 41(1): 146–159.

Fotheringham, A.S. 2009. The problem of spatial autocorrelation and local spatial statistics. *Geographical Analysis* 41: 398–403.

Fotheringham, A.S., C. Brunsdon, and M.E. Charlton. 2002. *Geographically Weighted Regression: The Analysis of Spatially Varying Relationships*. Wiley: Chichester.

Geweke, J., J. Horowitz, and H. Pesaran. 2008. Econometrics. In *The New Palgrave Dictionary of Economics*. Palgrave Macmillan: Basingstoke.

LeSage, J.P. and R.K. Pace. 2009. *Introduction to Spatial Econometrics*. Chapman & Hall/CRC.

Leenders, R.T.A.J. 2002. Modeling social influence through network autocorrelation: Constructing the weight matrix. *Social Networks *24: 21–47.

Voss, P.R., D.D. Long, R.B. Hammer, and S. Friedman. 2006. County child poverty rates in the U.S.: A spatial regression approach. *Population Research and Policy Review* 25(4): 369–391.