Tài liệu The theory and practice of spatial econometrics

The Theory and Practice of Spatial Econometrics James P. LeSage Department of Economics University of Toledo February, 1999 Preface This text provides an introduction to spatial econometric theory along with numerous applied illustrations of the models and methods described. The applications utilize a set of MATLAB functions that implement a host of spatial econometric estimation methods. The intended audience is faculty, students and practitioners involved in modeling spatial data sets. The MATLAB functions described in this book have been used in my own research as well as teaching both undergraduate and graduate econometrics courses. They are available on the Internet at http://www.econ.utoledo.edu along with the data sets and examples from the text. The theory and applied illustrations of conventional spatial econometric models represent about half of the content in this text, with the other half devoted to Bayesian alternatives. Conventional maximum likelihood estimation for a class of spatial econometric models is discussed in one chapter, followed by a chapter that introduces a Bayesian approach for this same set of models. It is well-known that Bayesian methods implemented with a diffuse prior simply reproduce maximum likelihood results, and we illustrate this point. However, the main motivation for introducing Bayesian methods is to extend the conventional models. Comparative illustrations demonstrate how Bayesian methods can solve problems that confront the conventional models. Recent advances in Bayesian estimation that rely on Markov Chain Monte Carlo (MCMC) methods make it easy to estimate these models. This approach to estimation has been implemented in the spatial econometric function library described in the text, so estimation using the Bayesian models require a single additional line in your computer program. Some of the Bayesian methods have been introduced in the regional science literature, or presented at conferences. Space and time constraints prohibit any discussion of implementation details in these forums. This text describes the implementation details, which I believe greatly enhance understanding and allow users to make intelligent use of these methods in applied settings. Audiences have been amazed (and perhaps skeptical) when I tell them it takes only 10 seconds to generate a sample of 1,000 MCMC draws from a sequence of conditional distributions needed to estimate the Bayesian models. Implementation approaches that achieve this type of speed are described here in the hope that other researchers can apply these ideas in their own work. I have often been asked about Monte Carlo evidence for Bayesian spatial i ii econometric methods. Large and small sample properties of estimation procedures are frequentist notions that make no sense in a Bayesian setting. The best support for the efficacy of Bayesian methods is their ability to provide solutions to applied problems. Hopefully, the ease of using these methods will encourage readers to experiment with these methods and compare the Bayesian results to those from more conventional estimation methods. Implementation details are also provided for maximum likelihood methods that draw on the sparse matrix functionality of MATLAB and produce rapid solutions to large applied problems with a minimum of computer memory. I believe the MATLAB functions for maximum likelihood estimation of conventional models presented here represent fast and efficient routines that are easier to use than any available alternatives. Talking to colleagues at conferences has convinced me that a simple software interface is needed so practitioners can estimate and compare a host of alternative spatial econometric model specifications. An example in Chapter 5 produces estimates for ten different spatial autoregressive models, including maximum likelihood, robust Bayesian, and a robust Bayesian tobit model. Estimation, printing and plotting of results for all these models is accomplished with a 39 line program. Many researchers ignore sample truncation or limited dependent variables because they face problems adapting existing spatial econometric software to these types of sample data. This text describes the theory behind robust Bayesian logit/probit and tobit versions of spatial autoregressive models and geographically weighted regression models. It also provides implementation details and software functions to estimate these models. Toolboxes are the name given by the MathWorks to related sets of MATLAB functions aimed at solving a particular class of problems. Toolboxes of functions useful in signal processing, optimization, statistics, finance and a host of other areas are available from the MathWorks as add-ons to the standard MATLAB software distribution. I use the term Econometrics Toolbox to refer to my public domain collection of function libraries available at the internet address given above. The MATLAB spatial econometrics functions used to implement the spatial econometric models discussed in this text rely on many of the functions in the Econometrics Toolbox. The spatial econometric functions constitute a “library” within the broader set of econometric functions. To use the spatial econometrics function library you need to download and install the entire set of Econometrics Toolbox functions. The spatial econometrics function library is part of the Econometrics Toolbox and will be available for use along with more traditional econometrics functions. The collection of around 500 econometrics functions and demonstration programs are organized into libraries, with approximately 40 spatial econometrics library functions described in this text. A manual is available for the Econometrics Toolbox in Acrobat PDF and postscript on the internet site, but this text should provide all the information needed to use the spatial econometrics library. A consistent design was implemented that provides documentation, example programs, and functions to produce printed as well as graphical presentation of iii estimation results for all of the econometric and spatial econometric functions. This was accomplished using the “structure variables” introduced in MATLAB Version 5. Information from estimation procedures is encapsulated into a single variable that contains “fields” for individual parameters and statistics related to the econometric results. A thoughtful design by the MathWorks allows these structure variables to contain scalar, vector, matrix, string, and even multidimensional matrices as fields. This allows the econometric functions to return a single structure that contains all estimation results. These structures can be passed to other functions that intelligently decipher the information and provide a printed or graphical presentation of the results. The Econometrics Toolbox along with the spatial econometrics library functions should allow faculty to use MATLAB in undergraduate and graduate level courses with absolutely no programming on the part of students or faculty. Practitioners should be able to apply the methods described in this text to problems involving large spatial data samples using an input program with less than 50 lines. Researchers should be able to modify or extend the existing functions in the spatial econometrics library. They can also draw on the utility routines and other econometric functions in the Econometrics Toolbox to implement and test new spatial econometric approaches. I have returned from conferences and implemented methods from papers that were presented in an hour or two by drawing on the resources of the Econometrics Toolbox. This text has another goal, applied modeling strategies and data analysis. Given the ability to easily implement a host of alternative models and produce estimates rapidly, attention naturally turns to which models best summarize a particular spatial data sample. Much of the discussion in this text involves these issues. My experience has been that researchers tend to specialize, one group is devoted to developing new econometric procedures, and another group focuses on applied problems that involve using existing methods. This text should have something to offer both groups. If those developing new spatial econometric procedures are serious about their methods, they should take the time to craft a generally useful MATLAB function that others can use in applied research. The spatial econometrics function library provides an illustration of this approach and can be easily extended to include new functions. It would also be helpful if users who produce generally useful functions that extend the spatial econometrics library would submit them for inclusion. This would have the added benefit of introducing these new research methods to faculty and their students. There are obviously omissions, bugs and perhaps programming errors in the Econometrics Toolbox and the spatial econometrics library functions. This would likely be the case with any such endeavor. I would be grateful if users would notify me via e-mail at [email protected] when they encounter problems. The toolbox is constantly undergoing revision and new functions are being added. If you’re using these functions, update to the latest version every few months and you’ll enjoy speed improvements along with the benefits of new iv methods. Instructions for downloading and installing these functions are in an Appendix to this text along with a listing of the functions in the library and a brief description of each. Numerous people have helped in my spatial econometric research efforts and the production of this text. John Geweke explained the mysteries of MCMC estimation when I was a visiting scholar at the Minneapolis FED. He shared his FORTRAN code and examples without which MCMC estimation might still be a mystery. Luc Anselin with his encylopedic knowledge of the field was kind enough to point out errors in my early work on MCMC estimation of the Bayesian models and set me on the right track. He has always been encouraging and quick to point out that he explored Bayesian spatial econometric methods in 1980. Kelley Pace shared his sparse matrix MATLAB code and some research papers that ultimately lead to the fast and efficient approach used in MCMC estimation of the Bayesian models. Dan McMillen has been encouraging about my work on Bayesian spatial autoregressive probit models. His research in the area of limited dependent variable versions of these models provided the insight for the Bayesian logit/probit and tobit spatial autoregressive methods in this text. Another paper he presented suggested the logit and probit versions of the geographically weighted regression models discussed in the text. Art Getis with his common sense approach to spatial statistics encouraged me to write this text so skeptics would see that the methods really work. Two colleagues of mine, Mike Dowd and Dave Black were brave enough to use the Econometrics Toolbox during its infancy and tell me about strange problems they encountered. Their feedback was helpful in making improvements that all users will benefit from. In addition, Mike Dowd the local LaTeX guru provided some helpful macros for formatting and indexing the examples in this text. Mike Magura, another colleague and co-author in the area of spatial econometrics read early versions of my text materials and made valuable comments. Last but certainly not least, my wife Mary Ellen Taylor provided help and encouragement in ways too numerous to mention. I think she has a Bayesian outlook on life that convinces me there is merit in these methods. Contents 1 Introduction 1.1 Spatial econometrics . . . . . . . . . 1.2 Spatial dependence . . . . . . . . . . 1.3 Spatial heterogeneity . . . . . . . . . 1.4 Quantifying location in our models . 1.4.1 Quantifying spatial contiguity 1.4.2 Quantifying spatial position . 1.4.3 Spatial lags . . . . . . . . . . 1.5 Chapter Summary . . . . . . . . . . 2 The 2.1 2.2 2.3 2.4 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 7 10 11 14 17 20 MATLAB spatial econometrics library Structure variables in MATLAB . . . . . . Constructing estimation functions . . . . . Using the results structure . . . . . . . . . . Sparse matrices in MATLAB . . . . . . . . Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 22 24 28 35 42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 45 47 57 63 64 66 71 76 78 82 83 85 87 89 92 97 . . . . . . . . . . . . . . . . . . . . . . . . 3 Spatial autoregressive models 3.1 The first-order spatial AR model . . . . . 3.1.1 Computational details . . . . . . . 3.1.2 Applied examples . . . . . . . . . . 3.2 The mixed autoregressive-regressive model 3.2.1 Computational details . . . . . . . 3.2.2 Applied examples . . . . . . . . . . 3.3 The spatial autoregressive error model . . 3.3.1 Computational details . . . . . . . 3.3.2 Applied examples . . . . . . . . . . 3.4 The spatial Durbin model . . . . . . . . . 3.4.1 Computational details . . . . . . . 3.4.2 Applied examples . . . . . . . . . . 3.5 The general spatial model . . . . . . . . . 3.5.1 Computational details . . . . . . . 3.5.2 Applied examples . . . . . . . . . . 3.6 Chapter Summary . . . . . . . . . . . . . v . . . . . . . . . . . . . . . . CONTENTS vi 4 Bayesian Spatial autoregressive models 4.1 The Bayesian regression model . . . . . . . . . . 4.1.1 The heteroscedastic Bayesian linear model 4.2 The Bayesian FAR model . . . . . . . . . . . . . 4.2.1 Constructing a function far g() . . . . . . 4.2.2 Using the function far g() . . . . . . . . . 4.3 Monitoring convergence of the sampler . . . . . . 4.3.1 Autocorrelation estimates . . . . . . . . . 4.3.2 Raftery-Lewis diagnostics . . . . . . . . . 4.3.3 Geweke diagnostics . . . . . . . . . . . . . 4.3.4 Other tests for convergence . . . . . . . . 4.4 Other Bayesian spatial autoregressive models . . 4.4.1 Applied examples . . . . . . . . . . . . . . 4.5 An applied exercise . . . . . . . . . . . . . . . . . 4.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 99 102 107 113 118 124 126 127 129 132 134 138 142 147 5 Limited dependent variable models 5.1 Introduction . . . . . . . . . . . . . . . . . . 5.2 The Gibbs sampler . . . . . . . . . . . . . . 5.3 Heteroscedastic models . . . . . . . . . . . . 5.4 Implementing probit models . . . . . . . . . 5.5 Comparing EM and Bayesian probit models 5.6 Implementing tobit models . . . . . . . . . 5.7 An applied example . . . . . . . . . . . . . 5.8 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 150 153 155 156 160 164 168 180 6 Locally linear spatial models 6.1 Spatial expansion . . . . . . . . . . . . . . 6.1.1 Implementing spatial expansion . . 6.1.2 Applied examples . . . . . . . . . . 6.2 DARP models . . . . . . . . . . . . . . . . 6.3 Non-parametric locally linear models . . . 6.3.1 Implementing GWR . . . . . . . . 6.3.2 Applied examples . . . . . . . . . . 6.4 Applied exercises . . . . . . . . . . . . . . 6.5 Limited dependent variable GWR models 6.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 181 183 188 193 204 206 212 214 223 228 7 Bayesian Locally linear spatial models 7.1 Bayesian spatial expansion . . . . . . . . . . . . . 7.1.1 Implementing Bayesian spatial expansion 7.1.2 Applied examples . . . . . . . . . . . . . . 7.2 Producing robust GWR estimates . . . . . . . . 7.2.1 Gibbs sampling BGWRV estimates . . . . 7.2.2 Applied examples . . . . . . . . . . . . . . 7.2.3 A Bayesian probit GWR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 230 232 234 240 244 248 256 . . . . . . . . . . CONTENTS 7.3 7.4 7.5 Extending the BGWR model . . . . . . 7.3.1 Estimation of the BGWR model 7.3.2 Informative priors . . . . . . . . 7.3.3 Implementation details . . . . . . 7.3.4 Applied Examples . . . . . . . . An applied exercise . . . . . . . . . . . . Chapter Summary . . . . . . . . . . . . vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 260 263 264 267 273 276 References 279 Econometrics Toolbox functions 285 List of Examples 1.1 2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 5.1 5.2 5.3 5.4 5.5 Demonstrate regression using the ols() function . . . . . Using sparse matrix functions . . . . . . . . . . . . . . . Solving a sparse matrix system . . . . . . . . . . . . . . Symmetric minimum degree ordering operations . . . . . Using the far() function . . . . . . . . . . . . . . . . . . Using sparse matrix functions and Pace-Barry approach Solving for rho using the far() function . . . . . . . . . . Using the sar() function with a large data set . . . . . . Using the xy2cont() function . . . . . . . . . . . . . . . Least-squares bias . . . . . . . . . . . . . . . . . . . . . Testing for spatial correlation . . . . . . . . . . . . . . . Using the sem() function with a large data set . . . . . . Using the sdm() function . . . . . . . . . . . . . . . . . Using sdm() with a large sample . . . . . . . . . . . . . Using the sac() function . . . . . . . . . . . . . . . . . . Using sac() on a large data set . . . . . . . . . . . . . . Heteroscedastic Gibbs sampler . . . . . . . . . . . . . . Metropolis within Gibbs sampling . . . . . . . . . . . . Using the far g() function . . . . . . . . . . . . . . . . . Using the far g() function . . . . . . . . . . . . . . . . . An informative prior for r . . . . . . . . . . . . . . . . . Using the coda() function . . . . . . . . . . . . . . . . . Using the raftery() function . . . . . . . . . . . . . . . . Geweke’s convergence diagnostics . . . . . . . . . . . . . Using the momentg() function . . . . . . . . . . . . . . . Testing convergence . . . . . . . . . . . . . . . . . . . . Using sem g() in a Monte Carlo setting . . . . . . . . . Using sar g() with a large data set . . . . . . . . . . . . Model specification . . . . . . . . . . . . . . . . . . . . . Gibbs sampling probit models . . . . . . . . . . . . . . . Using the sart g function . . . . . . . . . . . . . . . . . . Least-squares on the Boston dataset . . . . . . . . . . . Testing for spatial correlation . . . . . . . . . . . . . . . Spatial model estimation for the Boston data . . . . . . viii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 36 37 40 57 60 61 66 68 68 79 80 85 86 93 95 104 110 118 120 122 125 128 129 131 132 138 140 143 160 166 169 171 172 LIST OF EXAMPLES 5.6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 7.1 7.2 7.3 7.4 7.5 Right-censored Tobit Boston data . . . . Using the casetti() function . . . . . . . Using the darp() function . . . . . . . . Using darp() over space . . . . . . . . . Using the gwr() function . . . . . . . . . GWR estimates for a large data set . . . GWR estimates for the Boston data set GWR logit and probit estimates . . . . Using the bcasetti() function . . . . . . Boston data spatial expansion . . . . . . Using the bgwrv() function . . . . . . . City of Boston bgwr() example . . . . . Using the bgwr() function . . . . . . . . ix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 188 201 203 212 214 218 226 235 236 248 252 267 List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Gypsy moth counts in lower Michigan, 1991 . . . . . . . . . . . . Gypsy moth counts in lower Michigan, 1992 . . . . . . . . . . . . Gypsy moth counts in lower Michigan, 1993 . . . . . . . . . . . . Distribution of low, medium and high priced homes versus distance Distribution of low, medium and high priced homes versus living area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An illustration of contiguity . . . . . . . . . . . . . . . . . . . . . First-order spatial contiguity for 49 neighborhoods . . . . . . . . A second-order spatial lag matrix . . . . . . . . . . . . . . . . . . A contiguity matrix raised to a power 2 . . . . . . . . . . . . . . 2.1 2.2 2.3 Sparsity structure of W from Pace and Barry . . . . . . . . . . . 37 An illustration of fill-in from matrix multiplication . . . . . . . . 39 Minimum degree ordering versus unordered Pace and Barry matrix 41 3.1 3.2 Spatial autoregressive fit and residuals . . . . . . . . . . . . . . . Generated contiguity structure results . . . . . . . . . . . . . . . 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Vi estimates from the Gibbs sampler . . Conditional distribution of ρ . . . . . . First 100 Gibbs draws for ρ and σ . . . Posterior means for vi estimates . . . . . Posterior vi estimates based on r = 4 . . Graphical output for far g . . . . . . . . Posterior densities for ρ . . . . . . . . . Vi estimates for Pace and Barry dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 109 112 120 122 124 133 142 5.1 5.2 5.3 5.4 Results of plt() function for SAR logit Actual vs. simulated censored y-values Actual vs. Predicted housing values . Vi estimates for the Boston data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 167 171 178 6.1 6.2 6.3 Spatial x-y expansion estimates . . . . . . . . . . . . . . . . . . . 192 Spatial x-y total impact estimates . . . . . . . . . . . . . . . . . 193 Distance expansion estimates . . . . . . . . . . . . . . . . . . . . 194 x . . . . 4 5 6 8 9 12 18 19 20 59 69 LIST OF FIGURES xi 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 Actual versus Predicted and residuals . . . . . . . . . . GWR estimates . . . . . . . . . . . . . . . . . . . . . . . GWR estimates based on bandwidth=0.3511 . . . . . . GWR estimates based on bandwidth=0.37 . . . . . . . . GWR estimates based on tri-cube weighting . . . . . . . Boston GWR estimates - exponential weighting . . . . . Boston GWR estimates - Gaussian weighting . . . . . . Boston GWR estimates - tri-cube weighting . . . . . . . Boston city GWR estimates - Gaussian weighting . . . . Boston city GWR estimates - tri-cube weighting . . . . GWR logit and probit estimates for the Columbus data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 213 216 217 218 219 220 221 222 223 227 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 Spatial expansion versus robust estimates . . . . . . . . . . Mean of the vi draws for r = 4 . . . . . . . . . . . . . . . . Expansion vs. Bayesian expansion for Boston . . . . . . . . Expansion vs. Bayesian expansion for Boston (continued) . vi estimates for Boston . . . . . . . . . . . . . . . . . . . . . Distance-based weights adjusted by Vi . . . . . . . . . . . . Observations versus time for 550 Gibbs draws . . . . . . . . GWR versus BGWRV estimates for Columbus data set . . GWR versus BGWRV confidence intervals . . . . . . . . . . GWR versus BGWRV estimates . . . . . . . . . . . . . . . βi estimates for GWR and BGWRV with an outlier . . . . σi and vi estimates for GWR and BGWRV with an outlier t−statistics for the GWR and BGWRV with an outlier . . . Posterior probabilities for δ = 1, three models . . . . . . . . GWR and βi estimates for the Bayesian models . . . . . . . vi estimates for the three models . . . . . . . . . . . . . . . Ohio GWR versus BGWR estimates . . . . . . . . . . . . . Posterior probabilities and vi estimates . . . . . . . . . . . . Posterior probabilities for a tight prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 237 239 240 242 244 247 250 251 252 254 255 256 270 271 272 274 276 277 List of Tables 4.1 4.2 4.3 4.4 4.5 SEM model comparative estimates . SAR model comparisons . . . . . . . SEM model comparisons . . . . . . . SAC model comparisons . . . . . . . Alternative SAC model comparisons 5.1 5.2 5.3 5.4 5.5 5.6 5.7 EM versus Gibbs estimates . . . . . . . . . . . . . Variables in the Boston data set . . . . . . . . . . SAR,SEM,SAC model comparisons . . . . . . . . . Information matrix vs. numerical hessian measures SAR and SAR tobit model comparisons . . . . . . SEM and SEM tobit model comparisons . . . . . . SAC and SAC tobit model comparisons . . . . . . 6.1 DARP model results for all observations 7.1 7.2 Bayesian and ordinary spatial expansion estimates . . . . . . . . 238 Casetti versus Bayesian expansion estimates . . . . . . . . . . . . 241 xii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 144 145 146 146 . . . . . . . . . . . . . . . . . . . . . . . . of dispersion . . . . . . . . . . . . . . . . . . . . . . . . 164 168 174 175 177 179 179 . . . . . . . . . . . . . 204 Chapter 1 Introduction This chapter provides an overview of the nature of spatial econometrics. An applied approach is taken where the central problems that necessitate special models and econometric methods for dealing with spatial economic phenomena are introduced using spatial data sets. Chapter 2 describes software design issues related to a spatial econometric function library based on MATLAB software from the MathWorks Inc. Details regarding the construction and use of functions that implement spatial econometric estimation methods are provided throughout the text. These functions provide a consistent user-interface in terms of documentation and related functions that provide printed as well as graphical presentation of the estimation results. Chapter 2 describes the function library using simple regression examples to illustrate the design philosophy and programming methods that were used to construct the spatial econometric functions. The remaining chapters of the text are organized along the lines of alternative spatial econometric estimation procedures. Each chapter discusses the theory and application of a different class of spatial econometric model, the associated estimation methodology and references to the literature regarding these methods. Section 1.1 discusses the nature of spatial econometrics and how this text compares to other works in the area of spatial econometrics and statistics. We will see that spatial econometrics is characterized by: 1) spatial dependence between sample data observations at various points in space, and 2) spatial heterogeneity that arises from relationships or model parameters that vary with our sample data as we move through space. The nature of spatially dependent or spatially correlated data is taken up in Section 1.2 and spatial heterogeneity is discussed in Section 1.3. Section 1.4 takes up the subject of how we formally incorporate the locational information from spatial data in econometric models, providing illustrations based on a host of different spatial data sets that will be used throughout the text. 1 CHAPTER 1. INTRODUCTION 1.1 2 Spatial econometrics Applied work in regional science relies heavily on sample data that is collected with reference to location measured as points in space. The subject of how we incorporate the locational aspect of sample data is deferred until Section 1.4. What distinguishes spatial econometrics from traditional econometrics? Two problems arise when sample data has a locational component: 1) spatial dependence between the observations and 2) spatial heterogeneity in the relationships we are modeling. Traditional econometrics has largely ignored these two issues, perhaps because they violate the Gauss-Markov assumptions used in regression modeling. With regard to spatial dependence between the observations, recall that GaussMarkov assumes the explanatory variables are fixed in repeated sampling. Spatial dependence violates this assumption, a point that will be made clear in the Section 1.2. This gives rise to the need for alternative estimation approaches. Similarly, spatial heterogeneity violates the Gauss-Markov assumption that a single linear relationship with constant variance exists across the sample data observations. If the relationship varies as we move across the spatial data sample, or the variance changes, alternative estimation procedures are needed to successfully model this variation and draw appropriate inferences. The subject of this text is alternative estimation approaches that can be used when dealing with spatial data samples. This subject is seldom discussed in traditional econometrics textbooks. For example, no discussion of issues and models related to spatial data samples can be found in Amemiya (1985), Chow (1983), Dhrymes (1978), Fomby et al. (1984), Green (1997), Intrilligator (1978), Kelejian and Oates (1989), Kmenta (1986), Maddala (1977), Pindyck and Rubinfeld (1981), Schmidt (1976), and Vinod and Ullah (1981). Anselin (1988) provides a complete treatment of many facets of spatial econometrics which this text draws upon. In addition to discussion of ideas set forth in Anselin (1988), this text includes Bayesian approaches as well as conventional maximum likelihood methods for all of the spatial econometric methods discussed in the text. Bayesian methods hold a great deal of appeal in spatial econometrics because many of the ideas used in regional science modeling involve: 1. a decay of sample data influence with distance 2. similarity of observations to neighboring observations 3. a hierarchy of place or regions 4. systematic change in parameters with movement through space Traditional spatial econometric methods have tended to rely almost exclusively on sample data to incorporate these ideas in spatial models. Bayesian approaches can incorporate these ideas as subjective prior information that augments the sample data information. CHAPTER 1. INTRODUCTION 3 It may be the case that the quantity or quality of sample data is not adequate to produce precise estimates of decay with distance or systematic parameter change over space. In these circumstances, Bayesian methods can incorporate these ideas in our models, so we need not rely exclusively on the sample data. In terms of focus, the materials presented here are more applied than Anselin (1988), providing details on the program code needed to implement the methods and multiple applied examples of all estimation methods described. Readers should be fully capable of extending the spatial econometrics function library described in this text, and examples are provided showing how to add new functions to the library. In its present form the spatial econometrics library could serve as the basis for a graduate level course in spatial econometrics. Students as well as researchers can use these programs with absolutely no programming to implement some of the latest estimation procedures on spatial data sets. Another departure from Anselin (1988) is in the use of sparse matrix algorithms available in the MATLAB software to implement spatial econometric estimation procedures. The implementation details for Bayesian methods as well as the use of sparse matrix algorithms represent previously unpublished material. All of the MATLAB functions described in this text are freely available on the Internet at http://www.econ.utoledo.edu. The spatial econometrics library functions can be used to solve large-scale spatial econometric problems involving thousands of observations in a few minutes on a modest desktop computer. 1.2 Spatial dependence Spatial dependence in a collection of sample data means that observations at location i depend on other observations at locations j = i. Formally, we might state: yi = f (yj ), i = 1, . . . , n j = i (1.1) Note that we allow the dependence to be among several observations, as the index i can take on any value from i = 1, . . . , n. Why would we expect sample data observed at one point in space to be dependent on values observed at other locations? There are two reasons commonly given. First, data collection of observations associated with spatial units such as zip-codes, counties, states, census tracts and so on, might reflect measurement error. This would occur if the administrative boundaries for collecting information do not accurately reflect the nature of the underlying process generating the sample data. As an example, consider the case of unemployment rates and labor force measures. Because laborers are mobile and can cross county or state lines to find employment in neighboring areas, labor force or unemployment rates measured on the basis of where people live could exhibit spatial dependence. A second and perhaps more important reason we would expect spatial dependence is that the spatial dimension of socio-demographic, economic or regional activity may truly be an important aspect of a modeling problem. Regional science is based on the premise that location and distance are important forces CHAPTER 1. INTRODUCTION 4 at work in human geography and market activity. All of these notions have been formalized in regional science theory that relies on notions of spatial interaction and diffusion effects, hierarchies of place and spatial spillovers. As a concrete example of this type of spatial dependence, we use a spatial data set on annual county-level counts of Gypsy moths established by the Michigan Department of Natural Resources (DNR) for the 68 counties in lower Michigan. The North American gypsy moth infestation in the United States provides a classic example of a natural phenomena that is spatial in character. During 1981, the moths ate through 12 million acres of forest in 17 Northeastern states and Washington, DC. More recently, the moths have been spreading into the northern and eastern Midwest and to the Pacific Northwest. For example, in 1992 the Michigan Department of Agriculture estimated that more than 700,000 acres of forest land had experienced at least a 50% defoliation rate. 4 x 10 46 4.5 45.5 4 45 3.5 latitude 44.5 3 44 2.5 43.5 2 1.5 43 1 42.5 0.5 42 41.5 -86.5 0 -86 -85.5 -85 -84.5 -84 longitude -83.5 -83 -82.5 -82 Figure 1.1: Gypsy moth counts in lower Michigan, 1991 Figure 1.1 shows a contour of the moth counts for 1991 overlayed on a map outline of lower Michigan. We see the highest level of moth counts near Midland county Michigan in the center. As we move outward from the center, lower levels of moth counts occur taking the form of concentric rings. A set of k data points yi , i = 1, . . . , k taken from the same ring would exhibit a high correlation with CHAPTER 1. INTRODUCTION 5 each other. In terms of (1.1), yi and yj where both observations i and j come from the same ring should be highly correlated. The correlation of k1 points taken from one ring and k2 points from a neighboring ring should also exhibit a high correlation, but not as high as points sampled from the same ring. As we examine the correlation between points taken from more distant rings, we would expect the correlation to diminish. Over time the Gypsy moths spread to neighboring areas. They cannot fly, so the diffusion should be relatively slow. Figure 1.2 shows a similarly constructed contour map of moth counts for the next year, 1992. We see some evidence of diffusion to neighboring areas between 1991 and 1992. The circular pattern of higher levels in the center and lower levels radiating out from the center is still quite evident. 4 x 10 6 46 45.5 5 45 latitude 44.5 4 44 3 43.5 2 43 42.5 1 42 41.5 -86.5 0 -86 -85.5 -85 -84.5 -84 longitude -83.5 -83 -82.5 -82 Figure 1.2: Gypsy moth counts in lower Michigan, 1992 Finally, Figure 1.3 shows a contour map of the moth count levels for 1993, where the diffusion has become more heterogeneous, departing from the circular shape in the earlier years. Despite the increasing heterogeneous nature of the moth count levels, neighboring points still exhibit high correlations. An adequate model to describe and predict Gypsy moth levels would require that the function f () in (1.1) incorporate the notion of neighboring counties versus counties that are more distant. CHAPTER 1. INTRODUCTION 6 4 x 10 46 45.5 5 45 4 latitude 44.5 44 3 43.5 2 43 42.5 1 42 0 41.5 -86.5 -86 -85.5 -85 -84.5 -84 longitude -83.5 -83 -82.5 -82 Figure 1.3: Gypsy moth counts in lower Michigan, 1993 How does this situation differ from the traditional view of the process at work to generate economic data samples? The Gauss-Markov view of a regression data sample is that the generating process takes the form of (1.2), where y represent a vector of n observations, X denotes an nxk matrix of explanatory variables, β is a vector of k parameters and ε is a vector of n stochastic disturbance terms. y = Xβ + ε (1.2) The generating process is such that the X matrix and true parameters β are fixed while repeated disturbance vectors ε work to generate the samples y that we observe. Given that the matrix X and parameters β are fixed, the distribution of sample y vectors will have the same variance-covariance structure as ε. Additional assumptions regarding the nature of the variance-covariance structure of ε were invoked by Gauss-Markov to ensure that the distribution of individual observations in y exhibit a constant variance as we move across observations, and zero covariance between the observations. It should be clear that observations from our sample of moth level counts do not obey this structure. As illustrated in Figures 1.1 to 1.3, observations from counties in concentric rings are highly correlated, with a decay of correlation as CHAPTER 1. INTRODUCTION 7 we move to observations from more distant rings. Spatial dependence arising from underlying regional interactions in regional science data samples suggests the need to quantify and model the nature of the unspecified functional spatial dependence function f (), set forth in (1.1). Before turning attention to this task, the next section discusses the other underlying condition leading to a need for spatial econometrics — spatial heterogeneity. 1.3 Spatial heterogeneity The term spatial heterogeneity refers to variation in relationships over space. In the most general case we might expect a different relationship to hold for every point in space. Formally, we write a linear relationship depicting this as: yi = Xi βi + εi (1.3) Where i indexes observations collected at i = 1, . . . , n points in space, Xi represents a (1 x k) vector of explanatory variables with an associated set of parameters βi , yi is the dependent variable at observation (or location) i and εi denotes a stochastic disturbance in the linear relationship. A slightly more complicated way of expressing this notion is to allow the function f () from (1.1) to vary with the observation index i, that is: yi = fi (Xi βi + εi ) (1.4) Restricting attention to the simpler formation in (1.3), we could not hope to estimate a set of n parameter vectors βi given a sample of n data observations. We simply do not have enough sample data information with which to produce estimates for every point in space, a phenomena referred to as a “degrees of freedom” problem. To proceed with the analysis we need to provide a specification for variation over space. This specification must be parsimonious, that is, only a handful of parameters can be used in the specification. A large amount of spatial econometric research centers on alternative parsimonious specifications for modeling variation over space. Questions arise regarding: 1) how sensitive the inferences are to a particular specification regarding spatial variation?, 2) is the specification consistent with the sample data information?, 3) how do competing specifications perform and what inferences do they provide?, and a host of other issues that will be explored in this text. One can also view the specification task as one of placing restrictions on the nature of variation in the relationship over space. For example, suppose we classified our spatial observations into urban and rural regions. We could then restrict our analysis to two relationships, one homogeneous across all urban observational units and another for the rural units. This raises a number of questions: 1) are two relations consistent with the data, or is there evidence to suggest more than two?, 2) is there a trade-off between efficiency in the estimates and the number of restrictions we use?, 3) are the estimates biased if