1 Introduction 1
1.1 Spatial dependence . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The spatial autoregressive process . . . . . . . . . . . . . . . 8
1.2.1 Spatial autoregressive data generating process . . . . . 12
1.3 An illustration of spatial spillovers . . . . . . . . . . . . . . . 16
1.4 The role of spatial econometric models . . . . . . . . . . . . 20
1.5 The plan of the text . . . . . . . . . . . . . . . . . . . . . . . 22
2 Motivating and Interpreting Spatial Econometric Models 25
2.1 A time-dependence motivation . . . . . . . . . . . . . . . . . 25
2.2 An omitted variablesmotivation . . . . . . . . . . . . . . . . 27
2.3 A spatial heterogeneity motivation . . . . . . . . . . . . . . . 29
2.4 An externalities-based motivation . . . . . . . . . . . . . . . 30
2.5 Amodel uncertaintymotivation . . . . . . . . . . . . . . . . 30
2.6 Spatial autoregressive regression models . . . . . . . . . . . . 32
2.7 Interpreting parameter estimates . . . . . . . . . . . . . . . . 33
2.7.1 Direct and indirect impacts in theory . . . . . . . . . 34
2.7.2 Calculating summary measures of impacts . . . . . . . 39
2.7.3 Measures of dispersion for the impact estimates . . . . 39
2.7.4 Partitioning the impacts by order of neighbors . . . . 40
2.7.5 Simplified alternatives to the impact calculations . . . 41
2.8 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Maximum Likelihood Estimation 45
3.1 Model estimation . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Estimates of dispersion for the parameters . . . . . . . . . . 54
3.3 Omitted variables with spatial dependence . . . . . . . . . . 60
3.4 An applied example . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.1 Coefficient estimates . . . . . . . . . . . . . . . . . . . 69
3.4.2 Cumulative effects estimates . . . . . . . . . . . . . . 70
3.4.3 Spatial partitioning of the impact estimates . . . . . . 72
3.4.4 A comparison of impacts from different models . . . . 73
3.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 75
4 Log-determinants and Spatial Weights 77
4.1 Determinants and transformations . . . . . . . . . . . . . . . 77
4.2 Basic determinant computation . . . . . . . . . . . . . . . . 81
4.3 Determinants of spatial systems . . . . . . . . . . . . . . . . 84
4.3.1 Scalings and similarity transformations . . . . . . . . 87
4.3.2 Determinant domain . . . . . . . . . . . . . . . . . . . 88
4.3.3 Special cases . . . . . . . . . . . . . . . . . . . . . . . 89
4.4 Monte Carlo approximation of the log-determinant . . . . . . 96
4.4.1 Sensitivity of ρ estimates to approximation . . . . . . 100
4.5 Chebyshev approximation . . . . . . . . . . . . . . . . . . . . 105
4.6 Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.7 Determinant bounds . . . . . . . . . . . . . . . . . . . . . . . 108
4.8 Inverses and other functions . . . . . . . . . . . . . . . . . . 110
4.9 Expressions for interpretation of spatial models . . . . . . . . 114
4.10 Closed-form solutions for single parameter spatial models . . 116
4.11 Forming spatial weights . . . . . . . . . . . . . . . . . . . . . 118
4.12 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 120
5 Bayesian Spatial Econometric Models 123
5.1 Bayesianmethodology . . . . . . . . . . . . . . . . . . . . . . 124
5.2 Conventional Bayesian treatment of the SAR model . . . . . 127
5.2.1 Analytical approaches to the Bayesian method . . . . 127
5.2.2 Analytical solution of the Bayesian spatial model . . . 130
5.3 MCMC estimation of Bayesian spatial models . . . . . . . . 133
5.3.1 Sampling conditional distributions . . . . . . . . . . . 133
5.3.2 Sampling for the parameter ρ . . . . . . . . . . . . . . 136
5.4 TheMCMC algorithm . . . . . . . . . . . . . . . . . . . . . 139
5.5 An applied illustration . . . . . . . . . . . . . . . . . . . . . 142
5.6 Uses for Bayesian spatial models . . . . . . . . . . . . . . . . 145
5.6.1 Robust heteroscedastic spatial regression . . . . . . . 146
5.6.2 Spatial effects estimates . . . . . . . . . . . . . . . . . 149
5.6.3 Models with multiple weight matrices . . . . . . . . . 150
5.7 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 152
6 Model Comparison 155
6.1 Comparison of spatial and non-spatial models . . . . . . . . 155
6.2 An applied example ofmodel comparison . . . . . . . . . . . 159
6.2.1 The data sample used . . . . . . . . . . . . . . . . . . 161
6.2.2 Comparing models with different weight matrices . . . 161
6.2.3 A test for dependence in technical knowledge . . . . . 163
6.2.4 A test of the common factor restriction . . . . . . . . 164
6.2.5 Spatial effects estimates . . . . . . . . . . . . . . . . . 165
6.3 Bayesian model comparison . . . . . . . . . . . . . . . . . . . 168
6.3.1 Comparing models based on different weights . . . . . 169
6.3.2 Comparing models based on different variables . . . . 173
6.3.3 An applied illustration of model comparison . . . . . . 175
6.3.4 An illustration of MC3 and model averaging . . . . . 178
6.4 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 184
6.5 Chapter appendix . . . . . . . . . . . . . . . . . . . . . . . . 185
7 Spatiotemporal and Spatial Models 189
7.1 Spatiotemporal partial adjustment model . . . . . . . . . . . 190
7.2 Relation between spatiotemporal and SAR models . . . . . . 191
7.3 Relation between spatiotemporal and SEM models . . . . . . 196
7.4 Covariancematrices . . . . . . . . . . . . . . . . . . . . . . . 197
7.4.1 Monte Carlo experiment . . . . . . . . . . . . . . . . . 200
7.5 Spatial econometric and statistical models . . . . . . . . . . 201
7.6 Patterns of temporal and spatial dependence . . . . . . . . . 203
7.7 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 207
8 Spatial Econometric Interaction Models 211
8.1 Interregional flows in a spatial regression context . . . . . . . 212
8.2 Maximum likelihood and Bayesian estimation . . . . . . . . 218
8.3 Application of the spatial econometric interaction model . . 223
8.4 Extending the spatial econometric interaction model . . . . . 228
8.4.1 Adjusting spatial weights using prior knowledge . . . . 229
8.4.2 Adjustments to address the zero flow problem . . . . . 230
8.4.3 Spatially structured multilateral resistance effects . . . 232
8.4.4 Flows as a rare event . . . . . . . . . . . . . . . . . . . 234
8.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 236
9 Matrix Exponential Spatial Models 237
9.1 TheMESS model . . . . . . . . . . . . . . . . . . . . . . . . 237
9.1.1 The matrix exponential . . . . . . . . . . . . . . . . . 238
9.1.2 Maximum likelihood estimation . . . . . . . . . . . . . 239
9.1.3 A closed form solution for the parameters . . . . . . . 240
9.1.4 An applied illustration . . . . . . . . . . . . . . . . . . 241
9.2 Spatial error models using MESS . . . . . . . . . . . . . . . 243
9.2.1 Spatial model Monte Carlo experiments . . . . . . . . 246
9.2.2 An applied illustration . . . . . . . . . . . . . . . . . . 247
9.3 A Bayesian version of the model . . . . . . . . . . . . . . . . 250
9.3.1 The posterior for α . . . . . . . . . . . . . . . . . . . . 250
9.3.2 The posterior for β . . . . . . . . . . . . . . . . . . . . 252
9.3.3 Applied illustrations . . . . . . . . . . . . . . . . . . . 253
9.4 Extensions of the model . . . . . . . . . . . . . . . . . . . . . 255
9.4.1 More flexible weights . . . . . . . . . . . . . . . . . . . 255
9.4.2 MCMC estimation . . . . . . . . . . . . . . . . . . . . 256
9.4.3 MCMC estimation of the model . . . . . . . . . . . . 257
9.4.4 The conditional distributions for β, σ and V . . . . . . 258
9.4.5 Computational considerations . . . . . . . . . . . . . . 259
9.4.6 An illustration of the extended model . . . . . . . . . 260
9.5 Fractional differencing . . . . . . . . . . . . . . . . . . . . . . 265
9.5.1 Empirical illustrations . . . . . . . . . . . . . . . . . . 270
9.5.2 Computational considerations . . . . . . . . . . . . . . 275
9.6 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 277
10 Limited Dependent Variable Spatial Models 279
10.1 Bayesian latent variable treatment . . . . . . . . . . . . . . . 281
10.1.1 The SAR probit model . . . . . . . . . . . . . . . . . . 283
10.1.2 An MCMC sampler for the SAR probit model . . . . 284
10.1.3 Gibbs sampling the conditional distribution for y∗ . . 285
10.1.4 Some observations regarding implementation . . . . . 287
10.1.5 Applied illustrations of the spatial probit model . . . . 289
10.1.6 Marginal effects for the spatial probit model . . . . . . 293
10.2 The ordered spatial probit model . . . . . . . . . . . . . . . 297
10.3 Spatial Tobitmodels . . . . . . . . . . . . . . . . . . . . . . 299
10.5 An applied illustration of spatial MNP . . . . . . . . . . . . 312
10.5.1 Effects estimates for the spatial MNP model . . . . . 314
10.6 Spatially structured effects probitmodels . . . . . . . . . . . 316
10.7 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 320