Nonlinear Models with Spatial Data
William Greene
Stern School of Business, New York University
Washington D.C.
July 12, 2013
Binary Outcome: Y=1[New
Plant Located in County]
Klier and McMillen:
Clustering of Auto
Supplier Plants in the
United States. JBES, 2008

On Our Program

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

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)

 Spatial Autoregression in a Linear Model
y
= Wy + Xβ  ε.
E[ε | X ]
= 0, Var[ε | X ]=2 I
y
= [I  W]1 (Xβ  ε )
= [I  W]1 Xβ  [I  W]1 ε
E[y | X ]
= [I  W]1 Xβ
Var[y | X ] = 2 [(I  W)(I  W)]-1
Estimators: Various forms of generalized least squares.
Maximum likelihood | ε ~ Normal[0,]

 Spatial Autocorrelation in Regression
y  Xβ  (I - W)ε.
E[ε | X ]=0,
Var[ε | X]=2 I
E[y | X ]=Xβ
Var[y | X] = 2 (I - W)(I - W)
A Generalized Regression Model

ˆ  X   (I - W )(I - W ) X
β
2

ˆ 
1


1
X   (I - W )(I - W ) y

1

1
ˆ   (I - W)(I - W )1 y - Xβ
ˆ
y - Xβ
N

ˆ  The subject of much research


Panel Data Applications
(many at this meeting)
E.g., N countries, T periods
y it  xitβ  c i  it
ε t  Wε t  v t
= N observations at time t.
Similar assumptions
Candidate for SUR or Spatial Autocorrelation model.


Analytical Environment
 Generalized linear regression
 Complicated disturbance covariance matrix
 Estimation platform: Generalized least
squares, GMM or maximum likelihood.
 Central problem, estimation of 

Practical Obstacles
 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

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 in a nonrandom sample

Nonlinear Outcomes
 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

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.

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.


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

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)

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) = 1 - P(y=1|x)
 Likelihood for the sample = joint probability
=
 Log Likelihood
=


n
i=1
n
i=1
Prob(y=y i|x i )
logProb(y=y i|x i )

Cross Section Case: n Observations
 y1=j | x1 
 1  or > x1   Prob(1  or > x1 ) 



 



y
=j
|
x


or
>

x
Prob(


or
>

x
)
2
2
2
2 
Prob  2
 Prob  2
= 


 

... 
...
...



 



y
=j
|
x


or
>

x
Prob(


or
>

x
)
n
n
n
n 
 n
 n

Operate on the marginal probabilities of n observations
LogL(|X, y )=
 Probit
 Logit

n
i=1
logF  2y i  1 x i 
F(t) = (t) 

t
1
exp(  t / 2)dt 
2
exp(t)
F(t) =  (t) =
1  exp(t)

2

t

(t)dt

Spatially Correlated Observations
Correlation Based on Unobservables
y1  x1  u1
y 2  x 2  u2
...
y n  x n  un
 0 

 u1 
 1 
 

 
 
u

0
 2    I  W   2  ~ f   ,  I  W  I  W  
 ... 

 ... 
 ... 
 

 
 
u

0
 n
 n
 

W = the usual spatial weight matrix.
In the cross section case, W= 0. Now, it is a full matrix.
The joint probably is a single n fold integral.

Spatially Correlated Observations
Based on Correlated Utilities
 y1* 
 y1*   x1  1 
 x1  1 
 *
 * 



 y 2   W  y 2    x 2  2   I  W 1  x 2  2 


 ... 
 ...  



...
...
 * 
 *  



 x n  n 
 yn 
 y n   x n  n 
W = the usual spatial weight matrix.
In the cross section case, W = 0. Now, it is a full
matrix. The joint probably is a single n fold integral.

LogL for an Unrestricted BC Model
LogL(|X, y )=
 q11   1


q

2 2   q1q2 w 21


n
 ...   ...


q

 n n   qnq1wn1
q1q2 w12 ... q1qn w1n    1 
  



 xn
 x1
1
... q2qn w 2n    2 
log ...
d




 ... 
...
...
...
  
qnq2 wn2 ...
1    n 
qi  1 if y i = 0 and +1 if y i = 1 = 2y i  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.

Spatial Autoregression Based on
Observed Outcomes
 y1* 
 y1   x1  1 
 *
  



x


y
y
2
2 
 2   W  2   
 ... 
 ...  

...
 * 
  



x


y
y
n
n 
 n
 n 
y i  1[y i*  0]
y1  1[(w12 y 2  w13 y 3  ...)  x1  1  0]
y 2  1[(w 21 y1  w 23 y 3  ...)  x 2  2  0] etc.
The model based on observables is more reasonable.
There is no reduced form unless is W lower triangular.
This model is not identified. (It is "incoherent.")

See, also,
Maddala (1983)
From Klier and
McMillen (2012)

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 …


Spatial autocorrelation in the heterogeneity
yi*  x i  i   ji w ij j


 x i
*
yi  1 [yi  0], Prob  y i  1   
 Var i   ji w ij j

or







yi*  x i  ui
  x

  x i 
*
i
  
yi  1 [yi  0]Prob  y i  1   


 Var  ui  
 i 


i2  1  2  ji w ij2

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.
y*=X+,  = Wε+u
= [I-W]1u
= Au
Cross section case: =0
Probit Model: FOC for estimation of  is based on the
ˆi = y i  E[i | y i ]
generalized residuals u

n
i=1
 (y  (x i ))(x i ) 
xi  i
 =0



(

x
)[1


(

x
)]
i
i


See, also, Bertschuk, I., and M. Lechner, 1998. “Convenient Estimators for the Panel
Probit Model.” Journal of Econometrics, 87, 2, pp. 329–372

GMM in the Spatial Autocorrelation Model
y*=X+,  = Wε+u
= [I-W]1u
= Au
Autocorrelated Case:   0
Moment equations are still valid. Complication is computing
the variance of the moment equations for the weighting
matrix, which requires some approximations.
Probit Model: FOC for estimation of  is based on the
ˆi = y i  E[ | y i ]
generalized residuals u

n
i=1

 x i    x i  
y


 i
 a ()     a ()  

 ii    ii  
zi  
=0

  
  
    xi  1     xi   
  aii () 
 aii ()   


Requires at least K +1 instrumental variables.

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.


A Spatial Logit Model
y* = X+ e, e = We +  = (I-W )1
2

d = 1[y  0], Var[e] = (I-W) (I-W)  , ii  i
*
1
 xi 
*

Prob(yi  1)   
    x i    i
 i 

Algorithm
Generalized residual ui  di   i, Instruments Z
  i (1   i )xi* 
 ui /   

1
1
*
gi  

,
A
=(
I

W
)
W
(
I

W
)


x



i
 ui /     i (1   i ) 2 Aii 
i


G  [g1, g2 ,...gn ]
Iterated 2SLS (GMM) q = u(,)Z(ZZ)1Zu(,)
1. Logit estimation of β | =0, G0
ˆ  Z( ZZ)1ZG
2. u = (d - ˆ ), G
k
k

ˆ G
ˆ
3.  k  G
k k
k

1
k
ˆ u
G
k k
 ˆ 
 ˆ 
4.        k until  k is sufficiently small.
 ˆ k 1  ˆ k

An LM Type of Test?
 If  = 0, g = 0 because Aii = 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!

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.

Pseudo MLE
y*=Xθ+,  = Wε+u
= [I-W]1u
= Au
Autocorrelated Case:   0
y i*  x i  i   ji Wij j
y i  1[y i*  0]. Var[y i* ]  1  2  ji Wij2  aii ()
Implies a heteroscedastic probit.
Pseudo MLE is based on the marginal densities.
How to obtain the asymptotic covariance matrix?
[See Wang, Iglesias, Wooldridge (2013)]

Heteroscedastic Probit Approac h
Estimation and Inference
MLE: logL =

n
i1
 (2yi -1)x i 
log  



i

n  ˆ MLE     H1(  )S(  ) S=Score vector
implies the algorithm, Newton's Method.
EM algorithm essentially replaces H with XX during iterations.
(Slightly more involved for the heteroscedasticity. LHS variable
in the EM iterations is the score vector.)
To compute the asymptotic covariance, we need Var[S(  )]
Observations are (spatially) correlated! How to compute it?

Covariance Matrix for Pseudo-MLE
V  A(data, ˆ ) B(data, ˆ ) A(data, ˆ )
A(data, ˆ )  Negative inverse of Hessian
B(data, ˆ )  Covariance matrix of scores.
How to compute B(data, ˆ )
Terms are not independent in a spatial setting.

‘Pseudo’ Maximum Likelihood
Smirnov, A., “Modeling Spatial Discrete Choice,” Regional Science and Urban Economics, 40, 2010.
Spatial Autoregression in Utilities
y *  Wy *  X  , y  1( y*  0) for all n individuals
y *  (I  W )1 X  (I  W ) 1 
(I  W) 1 


t 0
(W) t assumed convergent
= A
= D + A -D
where D = diagonal elements
y *  AX  D
 A - D 
Private
Social
Suppose individuals ignore the social "shocks." Then

nj1aij   x j 
Prob[y i  1 or 0 | X ]  F (2y i  1)
 , probit or logit.
di



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)






Other Approaches
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

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.

See also Arbia, G., “Pairwise Likelihood Inference for Spatial Regressions
Estimated on Very Large Data Sets” Manuscript, Catholic University del Sacro
Cuore, Rome, 2012.

Partial MLE
(Looks Like Case, 1992)
Observation 1
 y *  x       W 

1
1
1
1j
j
j1


2
 y  1[y *  0] Var[y * ]  1  2

W

a
(

)

1
1
1
1j
11
j1


Observation 2
 y *  x      W 

2
2
2
2j
j
j2


2
 y  1[y *  0] Var[y * ]  1  2

W

a
(

)

2
2
2
2j
22
j2


Covariance of y1* and y *2 = a12 ()

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?


Core Model
y* = ρWSAR y * +Xβ + u, Spatial autoregression or
u = ρWSEMu + ε,
Spatial error model (only one at a time)
ε ~ N[0,σ 2I]
Censoring : Probit (0,1), Tobit (0,+)
Likelihood : L(ρ,β,σ ) 
2
ε  (I - ρW)y * - Xβ

 εε 
exp   2 
n /2
 2σ 
2πσ2
| I - ρW |

for SAR
ε  (I - ρW)( y * - Xβ) for SEM

LeSage Methods - MCMC
• Bayesian MCMC for all unknown
parameters
• Data augmentation for unobserved y*
• Quirks about sampler for rho.

An Ordered Choice Model (OCM)
y*  βx  , we assume x contains a constant term
y  0 if y*  0
y = 1 if 0
< y*  1
y = 2 if 1
< y*  2
y = 3 if 2
< y*  3
...
y = J if  J-1
< y*   J
In general : y = j if  j-1
< y*   j , j = 0,1,...,J
-1  ,  o  0,  J  ,  j-1   j, j = 1,...,J

OCM for Land Use Intensity

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.
Core Model: Cross Section
y i*  βx i  i ,
y i = j if  j-1  y i*   j , Var[i ]  1
Spatial Formulation: There are R regions. Within a region
y ir*  βx ir  ui  ir ,
y ir = j if  j-1  y ir*   j
Spatial heteroscedasticity: Var[ir ]  r2
Spatial Autocorrelation Across Regions
u = Wu + v , v ~ N[0,2v I]
u = (I-W)1 v ~ N[0,2v {(I-W )(I-W )} 1 ]
The error distribution depends on 2 parameters, 2v and 
Estimation Approach: Gibbs Sampling; Markov Chain Monte Carlo
Dynamics in latent utilities added as a final step: y*(t)=f[y*(t-1)].

Data Augmentation

Unordered Multinomial Choice
Core Random Utility Model
 Underlying Random Utility for Each Alternative
U(i,j) = j x 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: [1 ,...,  J ] ~ N[0, ]
Multinomial Logit: [1 ,...,  J ] ~ iid type 1 extreme value

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.
Utility Functions, land parcel i, usage type j, date t
U(i,j,t)=jt x ijt  ik  ijt
(In France) Spatial Correlation at Time t
ij   l1 willk
n
Modeling Framework: Normal / Multinomial Probit
Estimation: MCMC - Gibbs Sampling


Mixed Logit Models for Type of Residential Unit

First Law of Geography: [Tobler (1970)]
‘‘Everything is related to everything else, but near
things are more related than distant things”.

Is there a second law of geography?
Heisenberg?
* 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
Location Choice Model
 Common omitted geographic features
embedded in the random utility functions
 Cross – nested multinomial logit model

Does the model extension matter?

Does the model extension matter?

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
Poisson Regression
y = 0,1,...
exp() j
Prob[y = j|x] =
j!
Conditional Mean  = exp(x)
Signature Feature: Equidispersion
Usual Alternative: Negative Binomial
Spatial Effect: Filtered through the mean
i = exp(x i +i )
i =  m1 w im m  i
n

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.

A Blend of Ordered
Choice and Count Data
Models
Bicycle and pedestrian
injuries in census tracts in
Manhattan. (Count data and
ordered outcomes)
Numbers of firms locating in Texas
counties: Count data (Poisson)

Kriging


Spatial Autocorrelation in a
Sample Selection Model
Flores-Lagunes, A. and Schnier, K., “Sample Selection and Spatial
Dependence,” Journal of Applied Econometrics, 27, 2, 2012, pp. 173-204.
 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

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)

Spatial Autocorrelation in a
Sample Selection Model
yi*1   0  xi1  ui1
ui1   j  i cij u j1  i1
yi*2  0  xi 2  ui 2
ui 2    j  i cij u j 2  i 2
 0   12 12  
  i1 
, (?? 1  1??)
  ~ N   , 
2 
 0   12  2  
 i 2 
Observation Mechanism
yi1  1  yi*1 > 0  Probit Model
yi 2  yi*2 if yi1 = 1, unobserved otherwise.

Spatial Autocorrelation in a
Sample Selection Model
u1  Cu1  1
C = Spatial weight matrix, Cii  0.
u1  [I  C]1 1 =  (1) 1 , likewise for u 2
  () 
 , Var[u ]     (  ) 
Cov[u , u ]    () (  )
y   0  xi1   j 1 ()  i1 , Var[ui1 ]  
*
i1
N
y  0  xi 2   j 1 (  )
*
i2
N
(1)
ij
(2)
ij
N
2
1
i2
i2
j 1
2
1
N
j 1
N
i1
i2
12
j 1
(1) 2
ij
( 2) 2
ij
(1)
ij
(2)
ij

Spatial Weights
1
cij  2 ,
dij
d ij  Euclidean distance
Band of 7 neighbors is used
Row standardized.

Two Step Estimation

Probit estimated by Pinske/Slade GMM


 0  xi1




N
N
2
(1) 2
(1)
(2)
 1  j 1  ()ij  
()ij (  )ij

j 1


i 
N


(1) 2

(

)


 j 1
ij
 0  xi1




N
2
(1) 2
 1  j 1  ()ij  


 Spatial regression with included IMR in second step
(*) GMM procedure combines the two steps in one large estimation.

Spatial Stochastic Frontier
Production function model
y = βx + ε
y = βx + v - u
v  unexplained noise = N[0,1]
u = inefficiency > 0; efficiency = exp(-u)
Object of estimation is u, not 
Not a linear regression. Fit by MLE or MCMC.

247 Spanish Dairy Farms, 6 Years

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.
yij  Output of farm j in municipality i in Center-West Brazil
yij  αi +βxij +vij - u ij
(α1 ,...,α n )  conditionally autoregressive based on neighbors
αi -α k is smaller when municipalities i and k are closer together
* Greene, W., (2005) "Reconsidering Heterogeneity in Panel Data Estimators of the Stochastic
Frontier Model", Journal of Econometrics, 126(2), 269-303

Spatial Frontier Models

Cost Model
Production Model
Estimation by Maximum Likelihood

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.)”

Time and Space (In Your Computer)


Thank you!
…………………………………………….
 