J. Wesley Burnett*
There has been an explosion of research incorporating spatial dimensions into applied economic modeling, where the spatial aspects make a crucial difference in the analysis and policy implications for the particular problem or question. Most of the original spatial econometric analyses were conducted in the fields of regional economics and economic geography; however, these spatial models are now being used for a wide range of applied economics topics. Much of this research is being driven by the growing availability of geospatial data in the social sciences as well as the development of spatial modeling tools and methodological advances to analyze this data. In this same vein we are beginning to see an increase in spatial analysis incorporated into energy economics models. This makes sense given that energy resources, energy consumption, and energy production are defined over time and space, and therefore have spatial dimensions. Examples include patterns of energy use across space, spatial linkages between energy and the environment, spatial spillovers in regional energy consumption, spatial clustering in fossil fuel exploration activities, spatial structuring of electricity prices, etc.
Since spatial economic analysis is still an emerging field within economics, there may be some energy economists who are interested in incorporating spatial analysis into their own work but still do not quite understand the spatial modeling terminology and techniques. This monograph seeks to fill this gap by offering a basic overview and introduction to spatial economic modeling and terminology. This monograph is by no means a comprehensive overview of spatial economic analysis.
To begin, spatial data samples quite simply consist of observations that are associated with geographic points or regions. This data is generally collected through ground surveys, censuses, or by using remote sensing devices. In the social sciences, these samples are often analyzed through geographic information systems (GIS) or spatial econometrics. Within energy economics these spatial analysis tools can be used for several empirical applications. Pinske and Slade (2010) offer an example of drilling for petroleum. Following their example, a large discovery of a petroleum supply in a specific area will generally generate additional exploratory activity in that same area. The current Bakken play in North Dakota is an example. A question that an economist may ask is if the exploratory effort in the area is optimal in terms of costs versus benefits; in other words, do decision makers overreact in the exploration activity and create a bubble. To answer this question one would collect data on drilling activity in a specific region (e.g., the Bakken formation)–this data would be located in geographic space and time. An analysis of such data may require GIS or spatial econometric techniques.
2 Spatial Economics Terminology
Spatial economic analysis begins with the Tobler’s first law of geography which states: “everything is related to everything else, but near things are more related than distant things” (Tobler, 1970). This law is the basis for the concept of spatial effects. In a regression context, spatial effects pertain to spatial dependence (or spatial autocorrelation) and spatial heterogeneity. Spatial dependence is a cross-sectional dependence whereby near geographic locations influence each other and possess similar attributes. Spatial dependence can stem from interaction effects between neighbors such as externalities or spillovers, or from measurement error such as assuming that a data sample is independently distributed when in fact the underlying sample is characterized by a clustering of similar attributes. In contrast, random patterns within the underlying data sample exhibit no spatial autocorrelation. Spatial econometric techniques can be used to measure the strength of spatial autocorrelation and test the assumption of independently (or randomly) distributed data points across space. The structure of the dependence can relate to the location or distance in terms of geographic space as well as in more general economic terms (Anselin, Gallo, and Jayet, 2008). Spatial autocorrelation, on other hand, is defined as the correlation of a variable with itself through space. Spatial dependence is a property of a multivariate density function whereas spatial autocorrelation is a moment of this function (GeoDa Center for Spatial Analysis and Computation, 2011). In other words, spatial autocorrelation is a weaker form of spatial dependence. The two terms are often used interchangeably, but in practice only spatial autocorrelation can be estimated. Spatial autocorrelation can be positive whereby similar values occur near one another (e.g., clusters) or negative when dissimilar values occur near one another. In other words, positive spatial autocorrelation exists when values correlate well with neighboring values–this term simply operationalizes the first law of geography.
Spatial heterogeneity is a form of cross-sectional heterogeneity whereby relationships or model parameters vary within the sample data as one moves through space. Spatial regimes are a type of heterogeneity characterized by differing parameter values or functional forms through space; e.g., housing values in certain regions are structurally different from housing in other regions. Spatial heterogeneity can result in non-constant error variance (heteroskedasticity) across different regions (GeoDa Center for Spatial Analysis and Computation, 2011).
Many of the issues pertinent to spatial problems are similar to the issues faced in time series analysis. For example, in time series analysis, short lagged observations within a time series have a tendency to be more similar than observations at longer lag lengths–loosely speaking this is referred to as weak dependence. This concept is similar to first law of geography or spatial autocorrelation in spatial analysis–if the data points across space exhibit spatial autocorrelation then the observations are functionally related, in which case model parameters are not stable across locations (Irwin, 2000). This stability of the probability distribution within the series is similar to the notion of stationarity in time series analysis. Irwin and Geoghegan (2001) have argued that analyzing spatial data while ignoring potential spatial interactions or effects is analogous to analyzing time series data without knowing the chronological order of the observations. We will see in the spatial econometrics section how researchers control for spatial dependence.
3 Geographic Information Systems
Since geography or space is a major consideration in spatial econometrics then it is expected that researchers would make considerable use of GIS. Geographic information science is a multi-disciplinary academic field that is behind the development, use, and application of GIS. Spatial economics sometimes uses this geospatial data to augment or support economic theory and/or empirical modeling. GIS, in most economic applications, refers to a wide range of software packages that are used to input, store, manage, analyze, and map geospatial data. GIS is often used to identify observations by characteristics and location and then is used to perform statistical operations. The most common use of GIS in spatial economic analysis is in the study of land use, cost recreational demand modeling, and hedonic modeling of property price (Overman, 2010).
Despite a similar interest in spatial effects, there are fundamental differences in how researchers in geographic information science and spatial econometricians view the concept of space. Irwin and Geoghegan (2001) argue that the difference between the two camps is how to measure the spatial data “correctly” and how to measure the data “creatively.” The former is of interest to econometricians while the latter is of interest to (economic) geographers. In other words, (economic) geographers are interested in developing methods to create insights into understanding the driving forces behind spatial processes (Bateman et al., 2002). Most GIS packages provide little functionality for the types of statistical operations undertaken by spatial econometricians (Fotheringham, 1999). Generally, the creative use of space is the primary motive for researchers seeking to incorporate GIS analysis into empirical economic problems. In other words, the motivation for using GIS is to gather information on the spatial contexts that shape social and economic phenomena (Bateman et al., 2002). Put simply, a spatial context can play an important role in socio-economic processes, and so space can be important driver in our understanding of energy economics.
Two interesting papers in the energy economics literature have recently emerged. The first article uses a set of complicated GIS procedures to accurately model the potential diffusion of solar energy systems in a complex urban environment (Gennusa et al., 2011). The authors use GIS to examine an urban-wide project to install solar photovoltaics (PV) on rooftops. The planning of such a project requires reliable evaluation tools that can be used to assess the capacity of installation and determine the most suitable zones for installation. The GIS techniques afforded the authors a method to collect and process territorial information from technical and digital cartography. These GIS techniques allowed the authors to analyze various future scenarios of planning in order to determine the optimal usable surfaces of PVs that would exploit the most solar energy given environmental, architectural, and exposure constraints. This is a difficult task because it depends on the three-dimensional distribution of urban layout and the structural characteristics of the territory where the urban area is located (Gennusa et al., 2011). To overcome these difficulties the authors used a five-step procedure where they used a georeferenced technical map of the urban area (they use Palermo, Italy as an empirical example) to calculate the elevation of buildings, area of rooftops, angle of rooftops, optimal direction of rooftop face, shading factors, etc. The authors then took these structural characteristic measures and combined it with climatological data (amount of solar irradiation) over various months and times of day. Given this data the authors could analyze, at different times of the day, varying percentages of rooftop surfaces that would receive the most unshaded direct solar irradiation. Lastly, the authors analyzed the economic viability of launching an urban-wide PV development program in Palermo, taking into account the European Union’s (EU) sanction of 100 €/ton of CO2 emitted and varying levels of government incentivization of the program.
Another paper in the energy economics literature uses spatial modeling to analyze the viability of offshore wind energy in Denmark (Möller, 2011). The aim of the author’s work is to offer an analytical framework to address potential issues associated with proposed development sites (conflict free areas), risks and impacts, and costs. The author’s analytical framework consists of a spatially continuous, resource economic analysis model which uses raster-based spatial modeling in GIS. This analysis considers geographical factors, technology, costs, and planning information. The proposed offshore wind farms are constrained by (1) the visibility from shoreline (a negative externality) and (2) restrictions to marine activities and commerce–the author is able to model both constraints using the geographical analysis. The author identifies that this analysis is difficult because most planning proposals call for large scale developments to take advantage of economies of scale within offshore wind farms development. Development of these large scale wind farms requires movement at greater distances away from the coast and larger areas to achieve these economies. The author does however include scenarios with smaller developments which are able to be placed closer to shore. Taking all these factors into account, the first element of the analysis uses spatial inputs to provide a wind power density map for the calculation of wind farm electricity output. The second element calculates the levelized production costs. The third element uses spatial analysis to eliminate unfit areas within the proposed corridor of offshore wind development. Given each of these elements the author analyzes various scenarios of technological and development costs based upon the spatial constraints of developing offshore wind farms on a large scale. The author finds that there is an abundant source of economically feasible wind energy resources in Danish waters, but the feasibility is constrained by technological frontiers (e.g., transmission costs), natural conservation, marine hazards, and competing uses within Danish waters. Ultimately, the author finds that approximately 65 terawatt hours (TW·h), twice the current demand for final electricity in Denmark, could be produced at economically feasible production costs.
4 Spatial Econometrics
Spatial econometrics is a subfield of econometrics that focuses on the methodological concerns surrounding the explicit consideration of spatial effects in econometric models (Anselin, 1988). In a conventional, non-spatial, cross-sectional sample the data generating process (DGP) may take the form of (1) below, where a sample of independent observations, yi, are linearly related to the explanatory variables in a matrix, Xi.
This particular DGP assumes that the error term, εi, is normally distributed with mean zero and variance σ2. In a spatial context the observation i may represent regions or points in space. With the specification in equation (1) is it assumed that the observations at one location are independent from observations made at other locations or regions. Statistical independence of observations implies that
where E(·) denotes the expectation of the parameter. Spatial dependence (or autocorrelation), in contrast, reflects a situation in which the value of the observation at one location, say observation i, depends on the value of the observation at a neighboring location, say observation j. If we allow for observations i and j to be neighbors (e.g., regions with a shared border), then following Lesage and Pace (2009) the DGP may take the form
This situation implies a simultaneous DGP, where the value taken by yi depends on the value taken by yj and vice versa. One can easily see that if several more observations are spatially dependent then we would have to continue to add equations of a linear relationship to the system of equations in (3)-(4) above. If these additional observations are dependent on the first two equations then we would have to add those observations to the right hand side (RHS) of the first two equations. One could probably intuit that this system would get very complicated with a large number of parameters to estimate, and as we continue to add equations we would quickly run into a problem of over-parameterization. To overcome for this over-parameterization problem Ord (1975) proposed a parsimonious parameterization for the dependent variables. This proposed structure gives rise to a DGP called the spatial autoregressive (SAR) process. For ease of exposition it is listed in matrix notation below
where Wy on the RHS is called a spatial lag and In represents an identity matrix of dimensions (n x n). The spatial lag represents a linear combination of the values of the variable y constructed from observations that neighbor observation i (Lesage and Pace, 2009). The scalar parameter, ρ, denotes the spatial autocorrelation coefficient. This parameter represents the strength of the spatial dependence within observations, and as outlined in the Section 2 if it is positive then it implies that the observations near one another are more alike. The term W in the spatial lag denotes a square, spatial weighting matrix that consists of ones and zeros. A value of one is indicated if two observations are neighbors and zero otherwise; additionally, all the diagonal values are zero as an observation cannot be a neighbor to itself. This spatial weighting matrix can be row-standardized so that the values of each of its rows sum to one–this can make it easier to compare model coefficients with binary weights. If the weighting matrix is row-standardized, then loosely speaking, the spatial lag captures the average spatial effects of the neighbors’ values on the dependent variable. The DGP in (4) expresses the simultaneous nature of the spatial autoregressive process. This model is often estimated by a maximum likelihood procedure. LeSage (2011) offers a set of Matlab codes which estimates this model as well as numerous others.
Since the introduction of the SAR model in the 1970s the field of spatial econometrics has exploded–especially over the past two decades–and shows no signs of stopping. With increased interest has come far more complicated models including different DGPs (such as a spatial error process) and different estimation techniques including spatial panel data, spatial probit models, Bayesian spatial models, spatial GMM models, non-parametric spatial models, etc. These spatial econometric models can bring additional insights to energy economics, which as stated earlier often contains a spatial dimension. To give an example of spatial econometric models in energy economics we present here two recent papers in the energy economics literature.
The first paper uses spatial econometrics to estimate spatial patterns in electricity prices (Douglas and Popova, 2011). The authors take advantage of the changing spatial patterns of prices in wholesale electricity markets through the electricity transmission system to develop a spatial econometric model of locational marginal-cost prices (LMPs). Through this modeling technique the authors seeks to find more accurate estimates of price levels and volatility. The authors argue that adding spatial analysis to estimating electricity prices will not only improve the efficiency of estimation but also improve forecasting and interpolation of unknown prices. Specifically, the authors use data from the PJM Interconnection between 2002 and 2006 to estimate the spatial autocorrelation among prices in the twelve PJM regions. They find evidence for spatial influences in electricity price dynamics using a spatial panel data model, and demonstrate that spatial econometric modeling is a superior tool for the interpolation of unknown prices.
Another paper analyzes the industrial demand for coal in China at the provincial level (Cattaneo, Manera, and Scarpa, 2009). The authors use industrial demand because it accounts for over eighty percent of total final energy consumption in China, with power generation accounting for seventy-five percent of total industrial consumption. Specifically, the paper estimates and forecasts the demand of coal using a series of panel data estimation methods with the data disaggregated by provinces. To account for spatial dependence within the data the authors use a fixed effects estimation scheme with a spatial lag (as in equation (5) above) and a scheme with spatial error term. The fixed-effects spatial lag model results imply a strong interdependence of coal demand between provinces; and from these estimates the author forecast a four percent increase in coal demand in 2010.
Anselin, L. (1988). Spatial Econometrics: Methods and Models. Dordrecht, The Netherlands: Kluwer Academic Publishers.
Anselin, L., Florax, R., and Rey, S.J. Advances in Spatial Econometrics: Methodology, Tools and Applications. New York: Springer.
Anselin, L., Gallo, L.J., and Jayet, H. (2008). “Spatial Panel Econometrics.” In L. Matyas, and P. Sevestre, Eds. The Econometrics of Panel Data, pp. 625-660.
Arbia, G. and Baltagi, B.H., Eds. (2009) Spatial Econometrics: Methods and Applications. Heidelberg: Physica-Verlag.
Bateman, I.J., Jones, A.P., Lovett, A.A., Lake, I.R., and Day, B.H. (2002). ”Applying Geographical Information Systems (GIS) to Environmental and Resource Economics.” Environmental and Resource Economics, 22: 219-269.
Cattaneo, C., Manera, M., and Scarpa, E. (2009). “Industrial Coal Demand in China: a Provincial Analysis.” Retrieved from http://portale.unibocconi.it/wps/allegatiCTP/.
Combes, P.P., Mayer, T., and Thisse, J.F. (2008). Economic Geography: The Integration of Regions and Nations. Princeton, NJ: Princeton University Press.
Douglas, S.M., and Popova, J.N. (2011). “Econometric Estimation of Spatial Patterns in Electricity Prices.” Energy Journal, 32(2): 81-105.
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Gennusa, M.L., Lascari, G., Rizzo, G., Scaccianoce, G., and Sorrentino, G. (2011). “A Model of Predicting the Potential Diffusion of Solar Energy Systems in Complex Urban Environments.” Energy Policy, 39: 5335-5343.
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Irwin, E. G. and Geoghegan, J. (2001). “Theory, Data, Methods: Developing Spatially Explicit Economic Models of Land-Use Change.” Agriculture, Ecosystems and Environment, 85(1): 7–23.
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 For a more complete background we refer the reader to texts such as Economic Geography: The Integration of Regions and Nations by Combes, Mayer, and Thisse (2008); Spatial Econometrics: Methods and Applications by Arabia and Baltagi (2009); Introduction to Spatial Econometrics by LeSage and Pace (2009); GIS and Spatial Analysis for the Social Sciences: Coding, Mapping, and Modeling by Parker and Asenscio (2008); Advances in Spatial Econometrics: Methodology, Tools and Applications by Anselin, Florax, and Rey (2004); and the Handbook of Natural Resource and Energy Economics by Kneese and Sweeney (1993).
 The term GIS is however much broader than geospatial analysis software alone and is sometimes referred to operation systems that support activities such as traffic mapping or automated mapping (Bateman, et al., 2002).
 Historically this has been true that GIS packages provide little functionality for spatial econometric applications, but arguably the line between these two fields is becoming increasingly blurred by further advances in computational techniques.
 A raster data type is a digital image represented in a grid format.
 Those familiar with times series analysis will note the similarity of an autoregressive process (containing a lagged value of the independent variable on the RHS) with the spatial lag term. Note however that the spatial lag is not an actual lagged value as in time series, such as yt-1. The spatial lag is contemporaneous because of the nature of spatial dependence. This is different from the concept of dependence in time series which pertains to values of the series through time. Spatial dependence is a cross-sectional phenomenon, unless we introduce subsequent cross-sectional variables through time such as in panel data. This confuses some economists who are used to seeing the lagged term on the RHS.