Spatial Discrete Choice Models

Professor William GreeneStern School of Business, New York University

Spatial Correlation

Per Capita Income in Monroe County, New York, USA

Spatially Autocorrelated Data

The Hypothesis of Spatial Autocorrelation

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

Linear Spatial Autocorrelation

ii

( ) ( ) , N observations on a spatially

arranged variable

'contiguity matrix;' 0

spatial a

x i W x i ε

W W

W must be specified in advance. It is not estimated.

2

1

2 -1

utocorrelation parameter, -1 < < 1.

E[ ]= Var[ ]=

( ) [ ]

E[ ]= Var[ ]= [( ) ( )]

ε 0, ε I

x i I W ε = Spatial "moving average" form

x i, x I W I W

Testing for Spatial Autocorrelation

Spatial Autocorrelation

2

2 2

.

E[ ]= Var[ ]=

E[ ]=

Var[ ]

y Xβ Wε

ε| X 0, ε| X I

y| X Xβ

y| X = WW

A Generalized Regression Model

Spatial Autoregression in a Linear Model

2

1

1 1

1

2 -1

+ .

E[ ]= Var[ ]=

[ ] ( )

[ ] [ ]

E[ ]=[ ]

Var[ ] [( ) ( )]

y Wy Xβ ε

ε| X 0, ε| X I

y I W Xβ ε

I W Xβ I W ε

y| X I W Xβ

y| X = I W I W

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

Panel Data Application

it i it

E.g., N countries, T periods (e.g., gasoline data)

y c

= N observations at time t.

Similar assumptions

Candidate for SUR or Spatial Autocorrelation model.

it

t t t

x β

ε Wε v

Spatial Autocorrelation in a Panel

Alternative Panel Formulations

i,t t 1 i it

i

Pure space-recursive - dependence pertains to neighbors in period t-1

y [ ] regression +

Time-space recursive - dependence is pure autoregressive and on neighbors

in period t-1

y

Wy

,t i,t-1 t 1 i it

i,t i,t-1 t i it

y + [ ] regression +

Time-space simultaneous - dependence is autoregressive and on neighbors

in the current period

y y + [ ] regression +

Time-space dynamic -

Wy

Wy

i,t i,t-1 t i t 1 i it

dependence is autoregressive and on neighbors

in both current and last period

y y + [ ] + [ ] regression + Wy Wy

Analytical Environment

Generalized linear regression Complicated disturbance covariance

matrix Estimation platform

· Generalized least squares· Maximum likelihood estimation when

normally distributed disturbances (still GLS)

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.

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

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

Simple Binary Choice: Insurance

Redefined Multinomial Choice

Fly Ground

Multinomial Unordered Choice - Transport Mode

Health Satisfaction (HSAT)

Self administered survey: Health Care Satisfaction? (0 – 10)

Continuous Preference Scale

Ordered Preferences at IMDB.com

Counts of Events

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

Simple Binary Choice: Insurance

Decision: Yes or No = 1 or 0Depends on Income, Health, Marital Status, Gender

Multinomial Unordered Choice - Transport Mode

Decision: Which Type, A, T, B, C. Depends on Income, Price, Travel Time

Health Satisfaction (HSAT)

Self administered survey: Health Care Satisfaction? (0 – 10)

Outcome: Preference = 0,1,2,…,10Depends on Income, Marital Status, Children, Age, Gender

Counts of Events

Outcome: How many events at each location = 0,1,…,10Depends on Season, Population, Economic Activity

Nonlinear Spatial Modeling

Discrete outcome yit = 0, 1, …, J for some finite or infinite (count case) J.· i = 1,…,n· t = 1,…,T

Covariates xit .

Conditional Probability (yit = j) = a function of xit.

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) = ’xj , y = j if u*(j) > u*(k)

Nonlinear Regression for Count Models Outcome is governed by a nonlinear regression· E[y|x] = g(,x)

Probit and Logit Models

Prob(y 1 or 0| ) = F( ) or [1- F( )]x x xi i i iθ θ

Implied Regression Function

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 -0.42085 -2.662 -0.25179 -2.600 0.00960 0.078

X1 0.02365 7.205 0.01445 7.257 0.01878 7.129

X2 -0.44198 -2.610 -0.27128 -2.635 -0.32343 -2.536

X3 0.63825 8.453 0.38685 8.472 0.52280 8.407

Log-L -2097.48 -2097.35 -2098.17

Log-L(0) -2169.27 -2169.27 -2169.27

+ 1 (X1+1) + 2 (X2) + 3 X3 (1 is positive)

Effect on Predicted Probability of an Increase in X1

Estimated Partial Effects vs. Coefficients

Applications: Health Care UsageGerman Health Care Usage Data, 7,293 Individuals, Varying Numbers of PeriodsVariables in the file areData 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 / 10000. (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

An Estimated Binary Choice Model

An Estimated Ordered Choice Model

An Estimated Count Data Model

210 Observations on Travel Mode Choice

CHOICE ATTRIBUTES CHARACTERISTICMODE TRAVEL INVC INVT TTME GC HINCAIR .00000 59.000 100.00 69.000 70.000 35.000TRAIN .00000 31.000 372.00 34.000 71.000 35.000BUS .00000 25.000 417.00 35.000 70.000 35.000CAR 1.0000 10.000 180.00 .00000 30.000 35.000AIR .00000 58.000 68.000 64.000 68.000 30.000TRAIN .00000 31.000 354.00 44.000 84.000 30.000BUS .00000 25.000 399.00 53.000 85.000 30.000CAR 1.0000 11.000 255.00 .00000 50.000 30.000AIR .00000 127.00 193.00 69.000 148.00 60.000TRAIN .00000 109.00 888.00 34.000 205.00 60.000BUS 1.0000 52.000 1025.0 60.000 163.00 60.000CAR .00000 50.000 892.00 .00000 147.00 60.000AIR .00000 44.000 100.00 64.000 59.000 70.000TRAIN .00000 25.000 351.00 44.000 78.000 70.000BUS .00000 20.000 361.00 53.000 75.000 70.000CAR 1.0000 5.0000 180.00 .00000 32.000 70.000

An Estimated Unordered Choice Model

Maximum Likelihood EstimationCross Section Case

Binary Outcome

Random Utility: y* = +

Observed Outcome: y = 1 if y* > 0,

0 if y* 0.

Probabilities: P(y=1|x) = Prob(y* > 0| )

x

x

= Prob( > - )

P(y=0|x) = Prob(y* 0| )

= Prob( - )

Likelihood for the sample = joint probability

x

x

x

i i1

i i1

= Prob(y=y| )

Log Likelihood = logProb(y=y| )

x

x

n

i

n

i

Cross Section Case

1 1 1 1

2 2 2 2

1 1

2 2

| or >

| or > Prob Prob

... ...

| or >

Prob( or > )

Prob( or > ) =

...

Prob( or >

x x

x x

x x

x

x

x

n n n n

n

y j

y j

y j

)

We operate on the marginal probabilities of n observations

n

Log Likelihoods for Binary Choice Models

1

2

Logl( | )= logF 2 1

Probit

1 F(t) = (t) exp( t / 2)dt

2

(t)dt

Logit

exp(t) F(t) = (t) =

1 exp(t)

X,y xn

i ii

t

t

y

Spatially Correlated ObservationsCorrelation Based on Unobservables

1 1 1 1 1

2 2 2 2 2 2

u u 0

u u 0 ~ f ,

... ... ... ...

u u 0

In the cross section case, = .

= the usual spatial weight matrix .

x

x

x

W WW

WW I

n n n n n

y

y

y

Now, it is a full

matrix. The joint probably is a single n fold integral.

Spatially Correlated ObservationsCorrelated Utilities

* *1 1 1 11 1

* *12 2 2 22 2

* *

... ...... ...

In the cross section case

= the usual spatial weight matrix .

x x

x x

x x

W I W

Wn n n nn n

y y

y y

y y

, = . Now, it is a full

matrix. The joint probably is a single n fold integral.

W I

Log Likelihood

In the unrestricted spatial case, the log likelihood is one term,

LogL = log Prob(y1|x1, y2|x2, … ,yn|xn)

In the discrete choice case, the probability will be an n fold integral, usually for a normal distribution.

LogL for an Unrestricted BC Model

1

1 1 1 2 12 1 1 1

2 2 1 2 21 2 2 2

1 1 2 2

1 ...

1 ...LogL( | )=log ...

... ... ... ... ... ...

... 1

1 if y = 0 and

x xX,y

n

n n

n nn

n n n n n n n

i i

q q q w q q w

q q q w q q wd

q q qw q q w

q

+1 if y = 1.

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.

i

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 …

Pseudo Maximum LikelihoodSmirnov, A., “Modeling Spatial Discrete Choice,” Regional Science and Urban Economics, 40, 2010.

1 1

1

0

Spatial Autoregression in Utilities

* * , 1( * ) for all n individuals

* ( ) ( )

( ) ( ) assumed convergent

=

= + where

t

t

y Wy X y y 0

y I W X I W

I W W

A

D A-D

1

= diagonal elements

*

Private Social

Then

aProb[y 1 or 0| ] F (2 1) , p

nj ij j

i ii

yd

D

y AX D A-D

Suppose individuals ignore the social "shocks."

xX

robit or logit.

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)

GMMPinske, 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.

1

*= + , = +

= [ - ]

= u

Cross section case: =0

Probit Model: FOC for estimation of is based on the

ˆ generalized residuals ui

y W u

I W u

A

Xθ ε

1

= y [ | ]

( ( )) ( ) =

( )[1 ( )]

Spatially autocorrelated case: Moment equations are still

valid. Complication is computing the variance of the

i i

n i i iii

i i

E y

y

x xx 0

x x

moment

equations, which requires some approximations.

GMM

1

*= + , = +

= [ - ]

= u

Autocorrelated Case: 0

Probit Model: FOC for estimation of is based on the

ˆ generalized residuals u

y W u

I W u

A

Xθ ε

1

= y [ | ]

( ) ( ) =

1( ) ( )

i i i

i ii

n ii iiii

i i

ii ii

E y

ya a

a a

x x

z 0x x

Requires at least K+1 instrumental variables.

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.

Copula Method and ParameterizationBhat, C. and Sener, I., (2009) “A copula-based closed-form binary logit choice modelfor accommodating spatial correlation across observational units,” Journal of Geographical Systems, 11, 243–272

* *

1 2

Basic Logit Model

y , y 1[y 0] (as usual)

Rather than specify a spatial weight matrix, we assume

[ , ,..., ] have an n-variate distribution.

Sklar's Theorem represents the joint distribut

i i i i i

n

x

1 1 2 2

ion in terms

of the continuous marginal distributions, ( ) and a copula

function C[u = ( ) ,u ( ) ,...,u ( ) | ]i

n n

Copula Representation

Model

Likelihood

Parameterization

Other Approaches

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).

Case A (1992) Neighborhood influence and technological change. Economics 22:491–508Beron 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

Ordered Probability Model

1

1 2

2 3

J-1 J

j-1

y* , we assume 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*

In general: y = j if < y*

βx x

j

-1 o J j-1 j,

, j = 0,1,...,J

, 0, , j = 1,...,J

Outcomes for Health Satisfaction

A Spatial Ordered Choice ModelWang, 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.

* *1

* *1

Core Model: Cross Section

y , y = j if y , Var[ ] 1

Spatial Formulation: There are R regions. Within a region

y u , y = j if y

Spatial he

i i i i j i j i

ir ir i ir ir j ir j

βx

βx2

2

1 2 1

teroscedasticity: Var[ ]

Spatial Autocorrelation Across Regions

= + , ~ N[ , ]

= ( - ) ~ N[ , {( - ) ( - )} ]

The error distribution depends on 2 para

ir r

v

v

u Wu v v 0 I

u I W v 0 I W I W2meters, and

Estimation Approach: Gibbs Sampling; Markov Chain Monte Carlo

Dynamics in latent utilities added as a final step: y*(t)=f[y*(t-1)].

v

OCM for Land Use Intensity

OCM for Land Use Intensity

Estimated Dynamic OCM

Unordered Multinomial Choice

Underlying Random Utility for Each Alternative

U(i,j) = , i = individual, j = alternative

Preference Revelation

Y(i) = j if and only if U(i,j) > U(i,k

j ij ij

Core Random Utility Model

x

1

1

) for all k j

Model Frameworks

Multinomial Probit: [ ,..., ] ~N[0, ]

Multinomial Logit: [ ,..., ] ~ type I extreme valueJ

J iid

Multinomial Unordered Choice - Transport Mode

Decision: Which Type, A, T, B, C. Depends on Income, Price, Travel Time

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

1

Utility Functions, land parcel i, usage type j, date t

U(i,j,t)=

Spatial Correlation at Time t

Modeling Framework: Normal / Multinomial Probit

Estimation: MCMC

jt ijt ik ijt

n

ij il lkl

x

w

- Gibbs Sampling

Modeling Counts

Canonical Model

Poisson Regression

y = 0,1,...

exp( ) Prob[y = j|x] =

!

Conditional Mean = exp( x)

Signature Feature: Equidispersion

Usual Alternative: Various forms of Negative Binomial

Spatial E

j

j

1

ffect: Filtered through the mean

= exp( x + )

=

i i i

n

i im m imw

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