Spatial Discrete Choice Models

Transcription

1 Spatial Discrete Choice Models Professor William Greene Stern School of Business, New York University SPATIAL ECONOMETRICS ADVANCED INSTITUTE University of Rome May 23, 2011

2 Spatial Correlation

3 Spatially Autocorrelated Data Per Capita Income in Monroe County, New York, USA

4 The Hypothesis of Spatial Autocorrelation

5 Spatial Discrete Choice Modeling: Agenda Linear Models with Spatial Correlation Discrete Choice Models Spatial Correlation in Nonlinear Models Basics of Discrete Choice Models Maximum Likelihood Estimation Spatial Correlation in Discrete Choice Binary Choice Ordered Choice Unordered Multinomial Choice Models for Counts

6 Linear Spatial Autocorrelation ( x i) W( x i) ε, N observations on a spatially arranged variable W ' contiguity matrix;' W 0 ii W must be specified in advance. It is not estimated. spatial autocorrelation parameter, -1 < < 1. E[ ε]= 0, Var[ ε]= I ( ) [ ] 2 1 x i I W ε = Spatial "moving average" form E[ x]= i, Var[ x]= [( I W) ( I W)] 2-1

7 Testing for Spatial Autocorrelation

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9 Spatial Autocorrelation y Xβ Wε. E[ ε X]= 0, Var[ ε X]= I E[ y X]= Xβ Var[ y X ] = 2 2 WW A Generalized Regression Model 2

10 Spatial Autoregression in a Linear Model y Wy + Xβ ε. E[ ε X]= 0, Var[ ε X]= I y I W Xβ ε 1 [ ] ( ) [ ] [ ] 1 1 I W Xβ I W ε E[ ]=[ ] 1 y X I W Xβ Var[ y X ] = [( I W) ( I W)] 2-1 2

11 Complications of the Generalized Regression Model Potentially very large N GPS data on agriculture plots Estimation of. There is no natural residual based estimator Complicated covariance structure no simple transformations

12 Panel Data Application E.g., N countries, T periods (e.g., gasoline data) y x β c it it i it ε Wε v t t t = N observations at time t. Similar assumptions Candidate for SUR or Spatial Autocorrelation model.

13 Spatial Autocorrelation in a Panel

14 Alternative Panel Formulations Pure space-recursive - dependence pertains to neighbors in period t-1 y [ Wy ] regression + i,t t1 i it Time-space recursive - dependence is pure autoregressive and on neighbors in period t-1 y y + [ Wy ] regression + i,t i,t-1 t1 i it Time-space simultaneous - dependence is autoregressive and on neighbors in the current period y y + [ Wy ] regression + i,t i,t-1 t i it Time-space dynamic - dependence is autoregressive and on neighbors in both current and last period y y + [ Wy ] + [ Wy ] regression + i,t i,t-1 t i t1 i it

15 Analytical Environment Generalized linear regression Complicated disturbance covariance matrix Estimation platform Generalized least squares Maximum likelihood estimation when normally distributed disturbances (still GLS)

16 Discrete Choices Land use intensity in Austin, Texas Intensity = 1,2,3,4 Land Usage Types in France, 1,2,3 Oak Tree Regeneration in Pennsylvania Number = 0,1,2, (Many zeros) Teenagers physically active = 1 or physically inactive = 0, in Bay Area, CA.

17 Discrete Choice Modeling Discrete outcome reveals a specific choice Underlying preferences are modeled Models for observed data are usually not conditional means Generally, probabilities of outcomes Nonlinear models cannot be estimated by any type of linear least squares

18 Discrete Outcomes Discrete Revelation of Underlying Preferences Binary choice between two alternatives Unordered choice among multiple alternatives Ordered choice revealing underlying strength of preferences Counts of Events

19 Simple Binary Choice: Insurance

20 Redefined Multinomial Choice Fly Ground

21 Multinomial Unordered Choice - Transport Mode

22 Health Satisfaction (HSAT) Self administered survey: Health Care Satisfaction? (0 10) Continuous Preference Scale

23 Ordered Preferences at IMDB.com

24 Counts of Events

25 Modeling Discrete Outcomes Dependent Variable typically labels an outcome No quantitative meaning Conditional relationship to covariates No regression relationship in most cases The model is usually a probability

26 Simple Binary Choice: Insurance Decision: Yes or No = 1 or 0 Depends on Income, Health, Marital Status, Gender

27 Multinomial Unordered Choice - Transport Mode Decision: Which Type, A, T, B, C. Depends on Income, Price, Travel Time

28 Health Satisfaction (HSAT) Self administered survey: Health Care Satisfaction? (0 10) Outcome: Preference = 0,1,2,,10 Depends on Income, Marital Status, Children, Age, Gender

29 Counts of Events Outcome: How many events at each location = 0,1,,10 Depends on Season, Population, Economic Activity

30 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.

31 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)

32 Probit and Logit Models Prob(y 1 or 0 x ) = F( θx ) or [1- F( θx )] i i i i

33 Implied Regression Function

34 Estimated Binary Choice Models: The Results Depend on F(ε) LOGIT PROBIT EXTREME VALUE Variable Estimate t-ratio Estimate t-ratio Estimate t-ratio Constant X X X Log-L Log-L(0)

35 Effect on Predicted Probability of an Increase in X1 + 1 (X1+1) + 2 (X2) + 3 X3 ( 1 is positive)

36 Estimated Partial Effects vs. Coefficients

37 Applications: Health Care Usage German Health Care Usage Data, 7,293 Individuals, Varying Numbers of Periods Variables in the file are Data downloaded from Journal of Applied Econometrics Archive. This is an unbalanced panel with 7,293 individuals. They can be used for regression, count models, binary choice, ordered choice, and bivariate binary choice. This is a large data set. There are altogether 27,326 observations. The number of observations ranges from 1 to 7. (Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987). (Downloaded from the JAE Archive) DOCTOR = 1(Number of doctor visits > 0) HOSPITAL = 1(Number of hospital visits > 0) HSAT = health satisfaction, coded 0 (low) - 10 (high) DOCVIS = number of doctor visits in last three months HOSPVIS = number of hospital visits in last calendar year PUBLIC = insured in public health insurance = 1; otherwise = 0 ADDON = insured by add-on insurance = 1; otherswise = 0 HHNINC = household nominal monthly net income in German marks / (4 observations with income=0 were dropped) HHKIDS = children under age 16 in the household = 1; otherwise = 0 EDUC = years of schooling AGE = age in years FEMALE = 1 for female headed household, 0 for male EDUC = years of education

38 An Estimated Binary Choice Model

39 An Estimated Ordered Choice Model

40 An Estimated Count Data Model

41 210 Observations on Travel Mode Choice CHOICE ATTRIBUTES CHARACTERISTIC MODE TRAVEL INVC INVT TTME GC HINC AIR TRAIN BUS CAR AIR TRAIN BUS CAR AIR TRAIN BUS CAR AIR TRAIN BUS CAR

42 An Estimated Unordered Choice Model

43 Maximum Likelihood Estimation Cross Section Case Binary Outcome Random Utility: y* = x + Observed Outcome: y = 1 if y* > 0, 0 if y* 0. Probabilities: P(y=1 x) = Prob(y* > 0 x) = Prob( > - x) P(y=0 x) = Prob(y* 0 x) = Prob( - x) Likelihood for the sample = joint probability = Prob(y=y x ) i 1 Log Likelihood = logprob(y=y x ) n n i 1 i i i i

44 Cross Section Case y1 j x1 1 or > x1 y 2 j x2 2 or > x2 Prob Prob yn j xn n or > xn Prob( 1 or > x1) Prob( 2 or > x 2) =... Prob( n or > x n ) We operate on the marginal probabilities of n observations

45 Log Likelihoods for Binary Choice Models Logl( X, y)= logf 1 2y 1 x i i Probit t 1 2 F(t) = (t) exp( t / 2)dt 2 Logit (t)dt exp(t) F(t) = (t) = 1 exp(t) n t i

46 Spatially Correlated Observations Correlation Based on Unobservables y y y W x u u 0 x u u 0 W ~ f, WW x u u n n n n n = the usual spatial weight matrix. W In the cross section case, =. I Now, it is a full matrix. The joint probably is a single n fold integral.

47 Spatially Correlated Observations Correlated Utilities * * y 1 y x1 1 x * * y 1 2 y x2 2 x W I W * * y xn n x n yn n n W = the usual spatial weight matrix. W In the cross section case, =. Now, it is a full matrix. The joint probably is a single n fold integral. I

48 Log Likelihood In the unrestricted spatial case, the log likelihood is one term, LogL = log Prob(y 1 x 1, y 2 x 2,,y n x n ) In the discrete choice case, the probability will be an n fold integral, usually for a normal distribution.

49 LogL for an Unrestricted BC Model q q1q 2w q1q nw 1n 1 q q q w 1... q q w qn n qnq1w n1 qnq 2w n n xn x n 2n 2 X, y n d LogL( )=log... q i 1 if y = 0 and i +1 if y = 1. i One huge observation - n dimensional normal integral. Not feasible for any reasonable sample size. Even if computable, provides no device for estimating sampling standard errors.

50 Solution Approaches for Binary Choice Distinguish between private and social shocks and use pseudo-ml Approximate the joint density and use GMM with the EM algorithm Parameterize the spatial correlation and use copula methods Define neighborhoods make W a sparse matrix and use pseudo-ml Others

51 Pseudo Maximum Likelihood Smirnov, A., Modeling Spatial Discrete Choice, Regional Science and Urban Economics, 40, Spatial Autoregression in Utilities y* Wy * X, y 1( y* 0) for all n individuals y* ( I W) X ( I W) t ( I W) ( W) assumed convergent t 0 = A = D + A - D where D = diagonal elements y* AX D A - D Private Social Suppose individuals ignore the social "shocks." Then n j 1 ij j Prob[yi 1 or 0 X] F (2y i 1), p d i a x robit or logit.

52 Pseudo Maximum Likelihood Assumes away the correlation in the reduced form Makes a behavioral assumption 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)

53 GMM Pinske, J. and Slade, M., (1998) Contracting in Space: An Application of Spatial Statistics to Discrete Choice Models, Journal of Econometrics, 85, 1, 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, y*= Xθ+, = Wε+ u = [ - ] = Au 1 I W u Cross section case: =0 Probit Model: FOC for estimation of ˆ generalized residuals u i is based on the = y E[ y ] n ( y i ( xi )) ( x ) i x = i 1 i 0 ( xi)[1 ( xi)] Spatially autocorrelated case: Moment equations are still valid. Complication is computing the variance of the equations, which requires some approximations. i i moment

54 GMM y*= Xθ+, = Wε+ u = [ - ] 1 I W u = Au Autocorrelated Case: 0 Probit Model: FOC for estimation of is based on the generalized residuals uˆ = y E[ y ] i i i x i x i y i n aii ( ) aii ( ) z = i 1 i 0 x i x i 1 aii ( ) aii ( ) Requires at least K +1 instrumental variables.

55 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.

56 Copula Method and Parameterization Bhat, C. and Sener, I., (2009) A copula-based closed-form binary logit choice model for accommodating spatial correlation across observational units, Journal of Geographical Systems, 11, Basic Logit Model * * y i xi i, yi 1[y i 0] (as usual) Rather than specify a spatial weight matrix, we assume [,,..., ] have an n-variate distribution. 1 2 n Sklar's Theorem represents the joint distribution in terms of the continuous marginal distributions, ( function C[u = ( ),u ( ),...,u ( ) ] n i n ) and a copula

57 Copula Representation

58 Model

59 Likelihood

60 Parameterization

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63 Other Approaches Case A (1992) Neighborhood influence and technological change. Economics 22: 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 Case (1992): Define regions or neighborhoods. No correlation across regions. Produces essentially a panel data probit model. Beron and Vijverberg (2003): Brute force integration using GHK simulator in a probit model. Others. See Bhat and Sener (2009).

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65 Ordered Probability Model y* βx, we assume x contains a constant term y 0 if y* 0 y = 1 if 0 < y* y = 2 if < y* y = 3 if < y*... y = J if < y* J-1 J In general : y = j if < y* j-1 1, 0,, j = 1,...,J -1 o J j-1 j, j, j = 0,1,...,J

66 Outcomes for Health Satisfaction

67 A 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. Core Model: Cross Section y βx, y = j if y, Var[ ] 1 * * i i i i j 1 i j i Spatial Formulation: There are R regions. Within a region y βx u, y = j if y * * ir ir i ir ir j 1 ir j Spatial heteroscedasticity: Var[ ] Spatial Autocorrelation Across Regions u Wu v v 0 I 2 = +, ~ N[, v ] u I W v 0 I W I W = ( - ) ~ N[, v {( - ) ( - )} ] The error distribution depends on 2 parameters, ir 2 r 2 v and Estimation Approach: Gibbs Sampling; Markov Chain Monte Carlo Dynamics in latent utilities added as a final step: y*(t)=f[y*(t-1)].

68 OCM for Land Use Intensity

69 OCM for Land Use Intensity

70 Estimated Dynamic OCM

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72 Unordered Multinomial Choice Core Random Utility Model Underlying Random Utility for Each Alternative U(i,j) = x j ij ij, i = individual, j = alternative Preference Revelation Y(i) = j if and only if U(i,j) > U(i,k) for all k j Model Frameworks Multinomial Probit: [,..., ] ~ N[0, ] 1 Multinomial Logit: [,..., ] ~ iid type I extreme value 1 J J

73 Multinomial Unordered Choice - Transport Mode Decision: Which Type, A, T, B, C. Depends on Income, Price, Travel Time

74 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, Utility Functions, land parcel i, usage type j, date t U(i,j,t)= x jt ijt ik ijt Spatial Correlation at Time t n w ij l 1 il lk Modeling Framework: Normal / Multinomial Probit Estimation: MCMC - Gibbs Sampling

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78 Modeling Counts

79 Canonical Model Rathbun, S and Fei, L (2006) A Spatial Zero-Inflated Poisson Regression Model for Oak Regeneration, Environmental Ecology Statistics, 13, 2006, Poisson Regression y = 0,1,... Prob[y = j x] = exp( ) j! Conditional Mean = exp( x) Signature Feature: Equidispersion Usual Alternative: Various forms of Negative Binomial Spatial Effect: Filtered through the mean = exp( x + ) i i i n = w i m 1 im m i j

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81 Grazie!