1. Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business

2. Econometric Analysis of Panel Data 17. Spatial Autoregression and Autocorelation

3. Nonlinear Models with Spatial Data William Greene Stern School of Business, New York University Washington D.C. July 12, 2013

4. Applications  School District Open Enrollment: A Spatial Multinomial Logit Approach; David Brasington, University of Cincinnati, USA, Alfonso Flores-Lagunes, State University of New York at Binghamton, USA, Ledia Guci, U.S. Bureau of Economic Analysis, USA  Smoothed Spatial Maximum Score Estimation of Spatial Autoregressive Binary Choice Panel Models; Jinghua Lei, Tilburg University, The Netherlands  Application of Eigenvector-based Spatial Filtering Approach to a Multinomial Logit Model for Land Use Data; Takahiro Yoshida & Morito Tsutsumi, University of Tsukuba, Japan  Estimation of Urban Accessibility Indifference Curves by Generalized Ordered Models and Kriging; Abel Brasil, Office of Statistical and Criminal Analysis, Brazil, & Jose Raimundo Carvalho, Universidade Federal do Cear´a, Brazil  Choice Set Formation: A Comparative Analysis, Mehran Fasihozaman Langerudi, Mahmoud Javanmardi, Kouros Mohammadian, P.S Sriraj, University of Illinois at Chicago, USA, & Behnam Amini, Imam Khomeini International University, Iran  Not including semiparametric and quantile based linear specifications

5. Also On Our Program  Ecological fiscal incentives and spatial strategic interactions: the José Gustavo Féres Institute of Applied Economic Research (IPEA) Sébastien Marchand_ CERDI, University of Auvergne Alexandre Sauquet_ CERDI, University of Auvergne (Tobit)  The Impact of Spatial Planning on Crime Incidence : Evidence from Koreai Hyun Joong Kimii Ph.D. Candidate, Program in Regional Information, Seoul National University & Hyung Baek Lim Professor, Dept. of Community Development SungKyul University (Spatially Autoregressive Probit)  Spatial interactions in location decisions: Empirical evidence from a Bayesian Spatial Probit model Adriana Nikolic, Christoph Weiss, Department of Economics, Vienna University of Economics and Business  A geographically weighted approach to measuring efficiency in panel data: The case of US saving banks Benjamin Tabak, Banco Central do Brasil, Brazil, Rogerio B. Miranda, Universidade Catolica de Brasılia, Brazil, & Dimas M Fazio, Universidade de Sao Paulo (Stochastic Frontier) 

6.  Spatial Autoregression in a Linear Model 

7.  Spatial Autocorrelation in Regression 

8. Panel Data Applications 

9. 

10. Analytical Environment  Generalized linear regression  Complicated disturbance covariance matrix  Estimation platform: Generalized least squares, GMM or maximum likelihood.  Central problem, estimation of 

11. Practical Obstacles  Numerical problem: Maximize logL involving sparse (I- W)  Inaccuracies in determinant and inverse  Appropriate asymptotic covariance matrices for estimators  Estimation of. There is no natural residual based estimator  Potentially very large N – GIS data on agriculture plots  Complicated covariance structures – no simple transformations to Gauss-Markov form 

12. Klier and McMillen: Clustering of Auto Supplier Plants in the United States. JBES, 2008  Binary Outcome: Y=1[New Plant Located in County]

13. Outcomes in Nonlinear Settings  Land use intensity in Austin, Texas – Discrete Ordered Intensity = ‘1’ < ‘2’ < ‘3’ < ‘4’  Land Usage Types, 1,2,3 … – Discrete Unordered  Oak Tree Regeneration in Pennsylvania – Count Number = 0,1,2,… (Excess (vs. Poisson) zeros)  Teenagers in the Bay area: physically active = 1 or physically inactive = 0 – Binary  Pedestrian Injury Counts in Manhattan – Count  Efficiency of Farms in West-Central Brazil – Stochastic Frontier  Catch by Alaska trawlers - Nonrandom Sample 

14. Nonlinear Outcomes Models  Discrete revelation of choice indicates latent underlying preferences  Binary choice between two alternatives  Unordered choice among multiple choices  Ordered choice revealing underlying strength of preferences  Counts of events  Stochastic frontier and efficiency  Nonrandom sample selection 

15. Modeling Discrete Outcomes  “Dependent Variable” typically labels an outcome  No quantitative meaning  Conditional relationship to covariates  No “regression” relationship in most cases.  Models are often not conditional means.  The “model” is usually a probability  Nonlinear models – usually not estimated by any type of linear least squares  Objective of estimation is usually partial effects, not coefficients. 

16. Nonlinear Spatial Modeling  Discrete outcome y it = 0, 1, …, J for some finite or infinite (count case) J.  i = 1,…,n  t = 1,…,T  Covariates x it  Conditional Probability (y it = j) = a function of x it. 

17. 

18. Issues in Spatial Discete Choice  A series of Issues  Spatial dependence between alternatives: Nested logit  Spatial dependence in the LPM: Solves some practical problems. A bad model  Spatial probit and logit: Probit is generally more amenable to modeling  Statistical mechanics: Social interactions – not practical  Autologistic model: Spatial dependency between outcomes vs. utilities.  Variants of autologistic: The model based on observed outcomes is incoherent (“self contradictory”)  Endogenous spatial weights  Spatial heterogeneity: Fixed and random effects. Not practical?  The models discussed below 

19. Two Platforms  Random Utility for Preference Models Outcome reveals underlying utility  Binary: u* =  ’x y = 1 if u* > 0  Ordered: u* =  ’x y = j if  j-1 < u* <  j  Unordered: u*(j) =  ’x j, y = j if u*(j) > u*(k)  Nonlinear Regression for Count Models Outcome is governed by a nonlinear regression  E[y|x] = g( ,x) 

20. Maximum Likelihood Estimation Cross Section Case: Binary Outcome 

21. Cross Section Case: n Observations 

22. Spatially Correlated Observations Correlation Based on Unobservables 

23. Spatially Correlated Observations Based on Correlated Utilities 

24. LogL for an Unrestricted BC Model 

25. Spatial Autoregression Based on Observed Outcomes 

26. See, also, Maddala (1983) From Klier and McMillen (2012) 

27. Solution Approaches for Binary Choice  Approximate the marginal density and use GMM (possibly with the EM algorithm)  Distinguish between private and social shocks and use pseudo-ML  Parameterize the spatial correlation and use copula methods  Define neighborhoods – make W a sparse matrix and use pseudo-ML  Others … 

28. 

29. 

30. GMM in the Base Case with  = 0 Pinske, J. and Slade, M., (1998) “Contracting in Space: An Application of Spatial Statistics to Discrete Choice Models,” Journal of Econometrics, 85, 1, 125-154. Pinkse, J., Slade, M. and Shen, L (2006) “Dynamic Spatial Discrete Choice Using One Step GMM: An Application to Mine Operating Decisions”, Spatial Economic Analysis, 1: 1, 53 — 99.  See, also, Bertschuk, I., and M. Lechner, 1998. “Convenient Estimators for the Panel Probit Model.” Journal of Econometrics, 87, 2, pp. 329–372

31. GMM in the Spatial Autocorrelation Model 

32. Using the GMM Approach  Spatial autocorrelation induces heteroscedasticity that is a function of   Moment equations include the heteroscedasticity and an additional instrumental variable for identifying .  LM test of  = 0 is carried out under the null hypothesis that  = 0.  Application: Contract type in pricing for 118 Vancouver service stations. 

33. 

34. 

35. 

36. An LM Type of Test?  If  = 0, g  = 0 because A ii = 0  At the initial logit values, g  = 0  If  = 0, g = 0. Under the null hypothesis the entire score vector is identically zero.  How to test  = 0 using an LM test?  Same problem shows up in RE models  But, here,  is in the interior of the parameter space! 

37. Pseudo Maximum Likelihood  Maximize a likelihood function that approximates the true one  Produces consistent estimators of parameters  How to obtain standard errors?  Asymptotic normality? Conditions for CLT are more difficult to establish. 

38. Pseudo MLE 

40. Covariance Matrix for Pseudo-MLE 

41. ‘Pseudo’ Maximum Likelihood Smirnov, A., “Modeling Spatial Discrete Choice,” Regional Science and Urban Economics, 40, 2010. 

42. Pseudo Maximum Likelihood  Bases correlation on underlying utilities  Assumes away the correlation in the reduced form  Makes a behavioral assumption  Still requires inversion of (I-  W)  Computation of (I-  W) is part of the optimization process -  is estimated with .  Does not require multidimensional integration (for a logit model, requires no integration) 

43. Other Approaches  Beron and Vijverberg (2003): Brute force integration using GHK simulator in a probit model. Impractical.  Case (1992): Define “regions” or neighborhoods. No correlation across regions. Produces essentially a panel data probit model. (Also Wang et al. (2013))  LeSage: Bayesian - MCMC  Copula method. Closed form. See Bhat and Sener, 2009. Case A (1992) Neighborhood influence and technological change. Economics 22:491–508 Beron KJ, Vijverberg WPM (2004) Probit in a spatial context: a monte carlo analysis. In: Anselin L, Florax RJGM, Rey SJ (eds) Advances in spatial econometrics: methodology, tools and applications. Springer, Berlin 

44. See also Arbia, G., “Pairwise Likelihood Inference for Spatial Regressions Estimated on Very Large Data Sets” Manuscript, Catholic University del Sacro Cuore, Rome, 2012. 

45. Partial MLE (Looks Like Case, 1992) 

46. Bivariate Probit  Pseudo MLE  Consistent  Asymptotically normal?  Resembles time series case  Correlation need not fade with ‘distance’  Better than Pinske/Slade Univariate Probit?  How to choose the pairings? 

47. 

48. 

49. LeSage Methods - MCMC Bayesian MCMC for all unknown parameters Data augmentation for unobserved y* Quirks about sampler for rho. 

50. An Ordered Choice Model (OCM) 

51. OCM for Land Use Intensity 

52. A Dynamic Spatial Ordered Choice Model Wang, C. and Kockelman, K., (2009) Bayesian Inference for Ordered Response Data with a Dynamic Spatial Ordered Probit Model, Working Paper, Department of Civil and Environmental Engineering, Bucknell University. 

53. Data Augmentation 

54. Unordered Multinomial Choice 

55. Spatial Multinomial Probit Chakir, R. and Parent, O. (2009) “Determinants of land use changes: A spatial multinomial probit approach, Papers in Regional Science, 88, 2, 328-346. 

56. 

57.  Mixed Logit Models for Type of Residential Unit

58. First Law of Geography: [Tobler (1970)] ‘‘Everything is related to everything else, but near things are more related than distant things”. 

59. * http://www.openloc.eu/cms/storage/openloc/workshops/UNITN/20110324-26/Giuliani/Giuliani_slides.pdf * Arbia, G., R. Benedetti, and G. Espa. 1996. Effects of the MAUP on image classification. Geographical Systems 3:123–41. * http://urizen-geography.nsm.du.edu/~psutton/AAA_Sutton_WebPage/Sutton/Courses/ Geog_4020_Geographic_Research_Methodology/SeminalGeographyPapers/TOBLER.pdf Is there a second law of geography? Heisenberg?

60. Location Choice Model  Common omitted geographic features embedded in the random utility functions  Cross – nested multinomial logit model 

61. Does the model extension matter?

62.  Does the model extension matter?

63. Canonical Model for Counts Rathbun, S and Fei, L (2006) “A Spatial Zero-Inflated Poisson Regression Model for Oak Regeneration,” Environmental Ecology Statistics, 13, 2006, 409-426 

64. Zero Inflation  There are two states  Always zero  Zero is one possible value, or 1,2,…  Prob(0) = Prob(state 1) + Prob(state 2) P(0|state 2)  Used here as a functional form issue – too many zeros. 

65. Numbers of firms locating in Texas counties: Count data (Poisson) Bicycle and pedestrian injuries in census tracts in Manhattan. (Count data and ordered outcomes)  A Blend of Ordered Choice and Count Data Models

66. Kriging 

67. 

68. Spatial Autocorrelation in a Sample Selection Model  Alaska Department of Fish and Game.  Pacific cod fishing eastern Bering Sea – grid of locations  Observation = ‘catch per unit effort’ in grid square  Data reported only if 4+ similar vessels fish in the region  1997 sample = 320 observations with 207 reported full data Flores-Lagunes, A. and Schnier, K., “Sample Selection and Spatial Dependence,” Journal of Applied Econometrics, 27, 2, 2012, pp. 173-204. 

69. Spatial Autocorrelation in a Sample Selection Model  LHS is catch per unit effort = CPUE  Site characteristics: MaxDepth, MinDepth, Biomass  Fleet characteristics:  Catcher vessel (CV = 0/1)  Hook and line (HAL = 0/1)  Nonpelagic trawl gear (NPT = 0/1)  Large (at least 125 feet) (Large = 0/1) 

70. Spatial Autocorrelation in a Sample Selection Model 

71. Spatial Autocorrelation in a Sample Selection Model 

72. Spatial Weights 

73. Two Step Estimation   

74. Spatial Stochastic Frontier 

75. 247 Spanish Dairy Farms, 6 Years 

76. A True Random Effects Model* Spatial Stochastic Frontier Models; Accounting for Unobserved Local Determinants of Inefficiency. Schmidt, Moriera, Helfand, Fonseca; Journal of Productivity Analysis, 2009.  * Greene, W., (2005) "Reconsidering Heterogeneity in Panel Data Estimators of the Stochastic Frontier Model", Journal of Econometrics, 126(2), 269-303

77.  Spatial Frontier Models

78. Estimation by Maximum Likelihood  Cost Model Production Model

79. LeSage (2000) on Timing “The Bayesian probit and tobit spatial autoregressive models described here have been applied to samples of 506 and 3,107 observations. The time required to produce estimates was around 350 seconds for the 506 observations sample and 900 seconds for the case involving 3,107 observations. … (inexpensive Apple G3 computer running at 266 Mhz.)” 

80. Time and Space (In Your Computer) 

81. 