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Transcript of Spatial Discrete Choice Models - NYUpages.stern.nyu.edu/~wgreene/SpatialDiscreteChoiceModels.pdf ·...
Spatial Discrete Choice Models
Professor William Greene
Stern School of Business, New York University
SPATIAL ECONOMETRICS ADVANCED INSTITUTE
University of Rome
May 23, 2011
Spatial Correlation
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Per Capita Income in Monroe County, New York, USA
Spatially Autocorrelated Data
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
The Hypothesis of Spatial Autocorrelation
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Testing for Spatial Autocorrelation
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Spatial Autocorrelation in a Panel
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Analytical Environment
Generalized linear regression
Complicated disturbance covariance matrix
Estimation platform
Generalized least squares
Maximum likelihood estimation when normally distributed disturbances (still GLS)
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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.
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Simple Binary Choice: Insurance
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Redefined Multinomial Choice
Fly Ground
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Multinomial Unordered Choice - Transport Mode
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Health Satisfaction (HSAT)
Self administered survey: Health Care Satisfaction? (0 – 10)
Continuous Preference Scale
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Ordered Preferences at IMDB.com
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Counts of Events
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Simple Binary Choice: Insurance
Decision: Yes or No = 1 or 0 Depends on Income, Health, Marital Status, Gender
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Multinomial Unordered Choice - Transport Mode
Decision: Which Type, A, T, B, C. Depends on Income, Price, Travel Time
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Counts of Events
Outcome: How many events at each location = 0,1,…,10 Depends on Season, Population, Economic Activity
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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.
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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)
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Probit and Logit Models
Prob(y 1 or 0| ) = F( ) or [1- F( )]x x xi i i iθ θ
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Implied Regression Function
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
+ 1 (X1+1) + 2 (X2) + 3 X3 (1 is positive)
Effect on Predicted Probability of an Increase in X1
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Estimated Partial Effects vs. Coefficients
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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 / 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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
An Estimated Binary Choice Model
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
An Estimated Ordered Choice Model
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
An Estimated Count Data Model
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
210 Observations on Travel Mode Choice
CHOICE ATTRIBUTES CHARACTERISTIC
MODE TRAVEL INVC INVT TTME GC HINC
AIR .00000 59.000 100.00 69.000 70.000 35.000
TRAIN .00000 31.000 372.00 34.000 71.000 35.000
BUS .00000 25.000 417.00 35.000 70.000 35.000
CAR 1.0000 10.000 180.00 .00000 30.000 35.000
AIR .00000 58.000 68.000 64.000 68.000 30.000
TRAIN .00000 31.000 354.00 44.000 84.000 30.000
BUS .00000 25.000 399.00 53.000 85.000 30.000
CAR 1.0000 11.000 255.00 .00000 50.000 30.000
AIR .00000 127.00 193.00 69.000 148.00 60.000
TRAIN .00000 109.00 888.00 34.000 205.00 60.000
BUS 1.0000 52.000 1025.0 60.000 163.00 60.000
CAR .00000 50.000 892.00 .00000 147.00 60.000
AIR .00000 44.000 100.00 64.000 59.000 70.000
TRAIN .00000 25.000 351.00 44.000 78.000 70.000
BUS .00000 20.000 361.00 53.000 75.000 70.000
CAR 1.0000 5.0000 180.00 .00000 32.000 70.000
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
An Estimated Unordered Choice Model
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Maximum Likelihood Estimation Cross 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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Spatially Correlated Observations Correlation 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
W
W I
n n n n n
y
y
y
Now, it is a full
matrix. The joint probably is a single n fold integral.
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Spatially Correlated Observations Correlated 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
W
n 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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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.
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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 x
X, yn
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 q w 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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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 …
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Pseudo Maximum Likelihood
Smirnov, 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 i
i
yd
D
y AX D A -D
Suppose individuals ignore the social "shocks."
xX
robit or logit.
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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)
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
GMM 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.
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.
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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 ii
ii
i i
ii ii
E y
ya a
a a
x x
z 0x x
Requires at least K +1 instrumental variables.
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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.
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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, 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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Copula Representation
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Model
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Likelihood
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Parameterization
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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 22:491–508 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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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*
β x x
j
-1 o J j-1 j,
, j = 0,1,...,J
, 0, , j = 1,...,J
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Outcomes for Health Satisfaction
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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.
* *
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
β x
β x
2
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 W
2meters, 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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
OCM for Land Use Intensity
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
OCM for Land Use Intensity
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Estimated Dynamic OCM
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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 value
J
J iid
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Multinomial Unordered Choice - Transport Mode
Decision: Which Type, A, T, B, C. Depends on Income, Price, Travel Time
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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.
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Modeling Counts
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
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
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Introduction Linear Spatial Modeling Discrete Choices Nonlinear Models Spatial Binary Choice Ordered Choice Multinomial Choice Count Data
Grazie!