Part 20: Selection [1/66]
Econometric Analysis of Panel Data
William Greene
Department of Economics
Stern School of Business
Part 20: Selection [2/66]
Econometric Analysis of Panel Data
20. Sample Selection and Attrition
Part 20: Selection [3/66]
Received Sunday, April 27, 2014
I have a paper regarding strategic alliances between firms, and their impact on firm risk. While
observing how a firm’s strategic alliance formation impacts its risk, I need to correct for two types of
selection biases. The reviews at Journal of Marketing asked us to correct for the propensity of firms to
enter into alliances, and also the propensity to select a specific partner, before we examine how the
partnership itself impacts risk.
Our approach involved conducting a probit of alliance formation propensity, take the inverse mills and
include it in the second selection equation which is also a probit of partner selection. Then, we include
inverse mills from the second selection into the main model. The review team states that this is not
correct, and we need an MLE estimation in order to correctly model the set of three equations. The
Associate Editor’s point is given below. Can you please provide any guidance on whether this is a
valid criticism of our approach. Is there a procedure in LIMDEP that can handle this set of three
equations with two selection probit models?
AE’s comment:
“Please note that the procedure of using an inverse mills ratio is only consistent when the main
equation where the ratio is being used is linear. In non-linear cases (like the second probit used by the
authors), this is not correct. Please see any standard econometric treatment like Greene or Wooldridge.
A MLE estimator is needed which will be far from trivial to specify and estimate given error
correlations between all three equations.”
Part 20: Selection [4/66]
Hello Dr. Greene,
My name is xxxxxxxxxx and I go to the University of xxxxxxxx.
I see that you have an errata page on your website of your econometrics book 7th edition.
It seems like you want to correct all mistakes so I think I have spotted a possible
proofreading error.
On page 477 (theorem 13.2) you want to show that theta is consistent and you say that
"But, at the true parameter values, qn(θ0) →0. So, if (13-7) is true, then it must follow
that qn(θˆGMM) →θ0 as well because of the identification assumption"
I think in the second line it should be qn(θˆGMM) → 0, not θ0.
Part 20: Selection [5/66]
I also have a questions about nonlinear GMM - which is more or less nonlinear IV technique
I suppose.
I am running a panel non-linear regression (non-linear in the parameters) and I have L
parameters and K exogenous variables with L>K.
In particular my model looks kind of like this: Y = b1*X^b2 + e, and so I am trying to
estimate the extra b2 that don't usually appear in a regression.
From what I am reading, to run nonlinear GMM I can use the K exogenous variables to
construct the orthogonality conditions but what should I use for the extra, b2 coefficients?
Just some more possible IVs (like lags) of the exogenous variables?
I agree that by adding more IVs you will get a more efficient estimation, but isn't it only the
case when you believe the IVs are truly uncorrelated with the error term?
So by adding more "instruments" you are more or less imposing more and more restrictive
assumptions about the model (which might not actually be true).
I am asking because I have not found sources comparing nonlinear GMM/IV to nonlinear
least squares. If there is no homoscadesticity/serial correlation what is more efficient/give
tighter estimates?
Part 20: Selection [6/66]
Part 20: Selection [7/66]
Dueling Selection Biases –
From two emails, same day.


“I am trying to find methods which can deal
with data that is non-randomised and
suffers from selection bias.”
“I explain the probability of answering
questions using, among other independent
variables, a variable which measures knowledge
breadth. Knowledge breadth can be constructed
only for those individuals that fill in a skill
description in the company intranet. This is
where the selection bias comes from.
Part 20: Selection [8/66]
The Crucial Element

Selection on the unobservables




Selection into the sample is based on both
observables and unobservables
All the observables are accounted for
Unobservables in the selection rule also appear in
the model of interest (or are correlated with
unobservables in the model of interest)
“Selection Bias”=the bias due to not accounting
for the unobservables that link the equations.
Part 20: Selection [9/66]
A Sample Selection Model





Linear model
2 step
ML – Murphy & Topel
Binary choice application
Other models
Part 20: Selection [10/66]
Canonical Sample Selection Model
Regression Equation
y*=x +
Sample Selection Mechanism
d*=z+u; d=1[d* > 0] (probit)
y = y* if d = 1; not observed otherwise
Is the sample 'nonrandomly selected?'
E[y*|x,d=1] = x +E[ | x, d  1]
= x +E[ | x,u  z]
= x   something if Cor[,u|x]  0
A left out variable problem (again)
Incidental truncation
Part 20: Selection [11/66]
Applications

Labor Supply model:








y*=wage-reservation wage
d=labor force participation
Attrition model: Clinical studies of medicines
Survival bias in financial data
Income studies – value of a college application
Treatment effects
Any survey data in which respondents self select to
report
Etc…
Part 20: Selection [12/66]
Estimation of the Selection Model

Two step least squares




Inefficient
Simple – exists in current software
Simple to understand and widely used
Full information maximum likelihood



Efficient
Simple – exists in current software
Not so simple to understand – widely misunderstood
Part 20: Selection [13/66]
Heckman’s Model
y i *=x iβ+i
di *=zi γ+ui ; di =1[di * > 0] (probit)
y i = y i * if di = 1; not observed otherwise
[i ,ui ]~Bivariate Normal[0,0,2 , ,1]
E[y i *|x i ,di =1] = x iβ+E[i | x i , di  1]
= x iβ+E[i | x i ,ui  zi γ]
 (zi γ) 
= x iβ   



(
z
γ
)

i

= x iβ  i
Least squares is biased and inconsistent again. Left out variable
Part 20: Selection [14/66]
Two Step Estimation
Step 1: Estimate the probit model
di *=zi γ+ui ; di =1[di * > 0] (probit).
 (zi γ
ˆ) 
ˆ
ˆ. Now compute i  
Estimation of γ by γ

ˆ) 
 (zi γ
Step 2: Estimate the regression model with estimated regressor
y i *=x iβ+i
y i = y i * if di = 1; not observed otherwise
E[y i *|x i ,di =1] = x iβ+E[i | x i , di  1]
The “LAMBDA”
= x iβ  i
ˆi .
Linearly regress y i on x i , 
Step2a. Fix standard errors (Murphy and Topel). Estimate 
and  using ˆ
 and e'e/n
Part 20: Selection [15/66]
FIML Estimation
logL   d0 log   zi  
 1
 i2   zi   i /   

  d1 log 
exp  2   
2

  2
1
 2  

i  y i  x iβ
Let
  1 / , =- /, =

1-2
logL   d0 log   zi  


 

2
 1
  d1 log 
exp    y i  x iδ    ( 1  2 )zi   (y i  x iδ) 
 2

 2

Note : no inverse Mills ratio appears anywhere in the model.
Part 20: Selection [16/66]
Classic Application

Mroz, T., Married women’s labor supply,
Econometrica, 1987.



A (my) specification



N =753
N1 = 428
LFP=f(age,age2,family income, education, kids)
Wage=g(experience, exp2, education, city)
Two step and FIML estimation
Part 20: Selection [17/66]
Selection Equation
+---------------------------------------------+
| Binomial Probit Model
|
| Dependent variable
LFP
|
| Number of observations
753
|
| Log likelihood function
-490.8478
|
+---------------------------------------------+
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
---------+Index function for probability
Constant|
-4.15680692
1.40208596
-2.965
.0030
AGE
|
.18539510
.06596666
2.810
.0049
42.5378486
AGESQ
|
-.00242590
.00077354
-3.136
.0017
1874.54847
FAMINC |
.458045D-05
.420642D-05
1.089
.2762
23080.5950
WE
|
.09818228
.02298412
4.272
.0000
12.2868526
KIDS
|
-.44898674
.13091150
-3.430
.0006
.69588313
Part 20: Selection [18/66]
Heckman Estimator and MLE
Part 20: Selection [19/66]
Extension – Treatment Effect
What is the value of an elite college education?
di *=zi γ+ui ; di=1[di * > 0] (probit)
y i *=x iβ+di  i observed for everyone
[i ,ui ]~Bivariate Normal[0,0,2 , ,1]
E[y i *|x i ,di=1] = x iβ+di +E[i | x i , di  1]
= x iβ++E[i | x i ,ui  zi γ]
 (zi γ) 
= x iβ+   

 (zi γ) 
= x iβ+  i
E[y i *|x i ,di=0] = x iβ+di +E[i | x i , di  0]
 (zi γ) 
= x iβ   

 (zi γ) 
Least squares is still biased and inconsistent. Left out variable
Part 20: Selection [20/66]
Sample Selection
An approach modeled on Heckman's model
Regression Equation:
Prob[y=j|x,u]=P(λ); λ=exp(x β+θu)
Selection Equation:
d=1[zδ+ε>0] (The usual probit)
[u,ε]~n[0,0,1,1,ρ] (Var[u] is absorbed in θ)
Estimation:
Nonlinear Least Squares: [Terza (1998, see cite in text).]
Φ(zδ+ρ)
E[y|x,d=1]=exp(x β+θρ2 )
Φ(zδ)
FIML using Hermite quadrature: [Greene (Stern wp, 97-02, 1997)]
Part 20: Selection [21/66]
Extensions – Binary Data
Application: In German Health care data, insurance choices
di *=zi γ+ui ; di=1[di * > 0] (probit)
y i *=x iβ+i , y i=1[y i * > 0] (probit)
y i = y i * if di = 1; not observed otherwise
[i ,ui ]~Bivariate Normal[0,0,1, ,1]
Estimation:
(1) Two step? (Wooldridge text)
(2) FIML (Stata, LIMDEP)
Part 20: Selection [22/66]
Panel Data and Selection
Selection equation with time invariant individual effect
dit  1[zit γ  i  it  0]
Observation mechanism: (y it , x it ) observed when dit  1
Primary equation of interest
Common effects linear regression model
y it | (dit  1)  x it β  i  it
" Selectivity " as usual arises as a problem when the unobservables
are correlated; Corr(it , it )  0.
The common effects, i and i make matters worse.
Part 20: Selection [23/66]
Panel Data and Sample Selection Models:
A Nonlinear Time Series
I. 1990-1992: Fixed and Random Effects
Extensions
II. 1995 and 2005: Model Identification through
Conditional Mean Assumptions
III. 1997-2005: Semiparametric Approaches based
on Differences and Kernel Weights
IV. 2007: Return to Conventional Estimators, with
Bias Corrections
Part 20: Selection [24/66]
Panel Data Sample Selection Models
Verbeek, Economics Letters, 1990.
dit  1[zit γ  w i  it  0] (Random effects probit)
y it | (dit  1)  x it β  i  it ; (Fixed effects regression)
Proposed "marginal likelihood" based on joint normality

zit γ + it  ui,1  ditui,2 
 f(ui,1 ,ui,2 )dui,1dui,2
logL i     t 1  (2dit  1)
 
2
2


 (1  dit  )


it  ( /  )dit (y it  y i )  ( x it  x i )'β 
(Integrate out the random effects; difference out the fixed effects.)


Ti
ui,1 ,ui,2 are time invariant uncorrelated standard normal variables
How to do the integration? Natural candidate for simulation.
(Not mentioned in the paper. Too early.)
[Verbeek and Nijman: Selectivity "test" based on this model, International
Economic Review, 1992.]
Part 20: Selection [25/66]
Zabel – Economics Letters


Inappropriate to have a mix of FE and RE
models
Two part solution




Treat both effects as “fixed”
Project both effects onto the group means of the
variables in the equations
Resulting model is two random effects equations
Use both random effects
Part 20: Selection [26/66]
Selection with Fixed Effects
yit *  i  xit   it , i  xi  wi , wi ~ N [0,1]
dit *  i  zit   uit , i  zi  vi , vi ~ N [0,1]
(it , uit ) ~ N 2 [(0, 0), (2 ,1, )].
Li 

 

dit  0
  zit   zi  vi  (vi )dvi
 z   z  v  ( / )  1  
i
i
it
    d 1   it
  it

-  
it
1  2

   
it  yit  xit   xi  wi



 2 (vi , wi )dvi dwi

Part 20: Selection [27/66]
Practical Complications
The bivariate normal integration is actually the
product of two univariate normals, because in the
specification above, vi and wi are assumed to be
uncorrelated. Vella notes, however, “… given the
computational demands of estimating by maximum
likelihood induced by the requirement to evaluate
multiple integrals, we consider the applicability of
available simple, or two step procedures.”
Part 20: Selection [28/66]
Simulation
The first line in the log likelihood is of the form
Ev[d=0(…)] and the second line is of the form
Ew[Ev[(…)(…)/]]. Using simulation instead, the
simulated likelihood is
LSi 
1 R

R r 1
dit  0
  zit   zi  vi ,r 
 zit   zi  vi ,r  ( / )it ,r  1  it ,r 
 
 dit 1  

2


1 




 yit  xit   xi  wi ,r
1 R


R r 1
it ,r

Part 20: Selection [29/66]
Correlated Effects
Suppose that wi and vi are bivariate standard normal with
correlation vw. We can project wi on vi and write
wi = vwvi + (1-vw2)1/2hi
where hi has a standard normal distribution. To allow
the correlation, we now simply substitute this expression
for wi in the simulated (or original) log likelihood, and
add vw to the list of parameters to be estimated. The
simulation is then over still independent normal variates,
vi and hi.
Part 20: Selection [30/66]
Conditional Means
Part 20: Selection [31/66]
A Feasible Estimator
Part 20: Selection [32/66]
Estimation
Part 20: Selection [33/66]
Kyriazidou - Semiparametrics
Assume 2 periods
Estimate selection equation by FE logit
Use first differences and weighted least squares:
1
ˆ i xi yi 
ˆ = Ni=1di1di2  i xi xi  Ni=1di1di2 

1  w iˆ 
ˆ
i  K 
kernel function.

h  h 
Use with longer panels - any pairwise differences
Extensions based on pairwise differences by RochinaBarrachina and Dustman/Rochina-Barrachina (1999)
Part 20: Selection [34/66]
Bias Corrections


Val and Vella, 2007 (Working paper)
Assume fixed effects


Bias corrected probit estimator at the first step
Use fixed probit model to set up second step
Heckman style regression treatment.
Part 20: Selection [35/66]
Postscript

What selection process is at work?




All of the work examined here (and in the literature) assumes
the selection operates anew in each period
An alternative scenario: Selection into the panel, once, at
baseline.
Why aren’t the time invariant components correlated?
(Greene, 2007, NLOGIT development)
Other models


All of the work on panel data selection assumes the main
equation is a linear model.
Any others? Discrete choice? Counts?
Part 20: Selection [36/66]
Attrition


In a panel, t=1,…,T individual I leaves the
sample at time Ki and does not return.
If the determinants of attrition (especially the
unobservables) are correlated with the variables
in the equation of interest, then the now
familiar problem of sample selection arises.
Part 20: Selection [37/66]
Application of a Two Period Model




“Hemoglobin and Quality of Life in Cancer
Patients with Anemia,”
Finkelstein (MIT), Berndt (MIT), Greene (NYU),
Cremieux (Univ. of Quebec)
1998
With Ortho Biotech – seeking to change labeling
of already approved drug ‘erythropoetin.’
r-HuEPO
Part 20: Selection [38/66]
QOL Study

Quality of life study




yit = self administered quality of life survey, scale = 0,…,100
xit = hemoglobin level, other covariates



Treatment effects model (hemoglobin level)
Background – r-HuEPO treatment to affect Hg level
Important statistical issues





i = 1,… 1200+ clinically anemic cancer patients undergoing
chemotherapy, treated with transfusions and/or r-HuEPO
t = 0 at baseline, 1 at exit. (interperiod survey by some patients was
not used)
Unobservable individual effects
The placebo effect
Attrition – sample selection
FDA mistrust of “community based” – not clinical trial based statistical
evidence
Objective – when to administer treatment for maximum marginal
benefit
Part 20: Selection [39/66]
Dealing with Attrition


The attrition issue: Appearance for the second interview
was low for people with initial low QOL (death or
depression) or with initial high QOL (don’t need the
treatment). Thus, missing data at exit were clearly
related to values of the dependent variable.
Solutions to the attrition problem

Heckman selection model (used in the study)



Prob[Present at exit|covariates] = Φ(z’θ) (Probit model)
Additional variable added to difference model i = Φ(zi’θ)/Φ(zi’θ)
The FDA solution: fill with zeros. (!)
Part 20: Selection [40/66]
An Early Attrition Model
Hausman, J. and Wise, D., "Attrition Bias in Experimental and Panel Data:
The Gary Income Maintenance Experiment," Econometrica, 1979.
A two period model:
Structural response model (Random Effects Regression)
y i1  x i1β  i1  ui
y i2  x i2β  i2  ui
Attrition model for observation in the second period (Probit)
zi2 *  y i2  x i2θ  wi2 α  v i2
zi2  1(zi2 *  0)
Endogeneity "problem"
12  Corr[i1  ui , i2  ui ]  u2 /(2  u2 )
  Corr[v i2 , i2  ui ]
 Corr[v i2  (i2  ui ), i2  ui )
Part 20: Selection [41/66]
Methods of Estimating the Attrition Model





Heckman style “selection” model
Two step maximum likelihood
Full information maximum likelihood
Two step method of moments estimators
Weighting schemes that account for the
“survivor bias”
Part 20: Selection [42/66]
Selection Model
Reduced form probit model for second period observation equation
zi2 *  x i2 (  )  wi  (i2  ui  v i )
 ri2   hi2
zi2
 1(zi2 *  0)
Conditional means for observations observed in the second period
(ri2 )
E[y i2 | x i2 , zi2  1]  x i2  (12  )
(ri2 )
First period conditional means for observations observed in the
second period
(ri2 )
E[y i1 | x i1 , zi2  1]  x i1  (12  )
(ri2 )
(1) Estimate probit equation
(2) Combine these two equations with a period dummy variable, use
OLS with a constructed regressor in the second period
THE TWO DISTURBANCES ARE CORRELATED.
TREAT THIS IS A SUR MODEL. (EQUIVALENT TO MDE)
Part 20: Selection [43/66]
Maximum Likelihood
LogL i 
(y i1  x i1)2
 log 2
 log  
2
22

[(y i2  12 y i1 )  (x12  12 x i1 ))]2 
2
log   log 1  12 

2
2
2

(1


)

12



zi2 

 r   ( /  )(y  x  ) 

i2
i2



 log   i2
2




1







 r   (  /  )(y  x  )  
12

i1
i1

 (1  zi2 ) log    i2
2



1  12 



(1) See H&W for FIML estimation
(2) Use the invariance principle to reparameterize
(3) Estimate  separately and use a two step ML with Murphy and
Topel correction of asymptotic covariance matrix.
Part 20: Selection [44/66]
Part 20: Selection [45/66]
A Model of Attrition


Nijman and Verbeek, Journal of Applied
Econometrics, 1992
Consumption survey (Holland, 1984 – 1986)


Exogenous selection for participation (rotating
panel)
Voluntary participation (missing not at random –
attrition)
Part 20: Selection [46/66]
Attrition Model
The main equation
yi,t  0  xi,t  i  i,t , Random effects consumption function
i  xi  ui ,
Mundlak device; ui uncorrelated with X i
yi,t  0  xi,t  xi  ui  i,t , Reduced form random effects model
The selection mechanism
ait  1[individual i asked to participate in period t] Purely exogenous
ait may depend on observables, but does not depend on unobservables
rit  1[individual i chooses to participate if asked] Endogenous.
rit is the endogenous participation dummy variable
ait  0  rit  0
ait  1  the selection mechanism operates
Part 20: Selection [47/66]
Selection Equation
The main equation
yi,t  0  xi,t  xi  ui  i,t , Reduced form random effects model
The selection mechanism
rit  1[individual i chooses to participate if asked] Endogenous.
rit is the endogenous participation dummy variable
ait  0  rit  0
ait  1  the selection mechanism operates
rit  1[  0  xi,t   xi  zi,t   v i  w i,t  0] all observed if ait  1
State dependence: z may include ri,t-1
Latent persistent unobserved heterogeneity: 2v  0.
"Selection" arises if Cov[i,t ,w i,t ]  0 or Cov[ui,v i ]  0
Part 20: Selection [48/66]
Estimation Using One Wave



Use any single wave as a cross section with
observed lagged values.
Advantage: Familiar sample selection model
Disadvantages


Loss of efficiency
“One can no longer distinguish between state
dependence and unobserved heterogeneity.”
Part 20: Selection [49/66]
One Wave Model
A standard sample selection model.
yit  0  xit  xi  (ui  it )
rit  1[  0  xit   xi  1ri,t 1  2ai,t 1  (v i  w it )  0]
With only one period of data and ri,t-1 exogenous,
this is the Heckman sample selection model.
If > 0, then ri,t-1 is correlated with v i and the Heckman
approach fails.
An assumption is required:
(1) Include ri,t-1 and assume no unobserved heterogeneity
(2) Exclude ri,t-1 and assume there is no state dependence.
In either case, now if Cov[(ui  it ),(v i  w it )] we can use OLS.
Otherwise, use the maximum likelihood estimator.
Part 20: Selection [50/66]
Maximum Likelihood Estimation


See Zabel’s model in slides 20 and 23.
Because numerical integration is required in one
or two dimensions for every individual in the
sample at each iteration of a high dimensional
numerical optimization problem, this is, though
feasible, not computationally attractive.


The dimensionality of the optimization is irrelevant
This is much easier in 2008 than it was in 1992
(especially with simulation) The authors did the
computations with Hermite quadrature.
Part 20: Selection [51/66]
Testing for Selection?

Maximum Likelihood Results




Covariances were highly insignificant.
LR statistic=0.46.
Two step results produced the same conclusion
based on a Hausman test
ML Estimation results looked like the two step
results.
Part 20: Selection [52/66]
A Dynamic Ordered Probit Model
Part 20: Selection [53/66]
Random Effects Dynamic
Ordered Probit Model
Random Effects Dynamic Ordered Probit Model
hit *  xit    Jj1 jhi,t 1( j)  i  i,t
hi,t  j if  j-1 < hit * <  j
hi,t ( j)  1 if hi,t = j
Pit,j  P[hit  j]  ( j  xit   Jj1 jhi,t 1( j)   i )
 ( j1  xit   Jj1 jhi,t 1( j)  i )
Parameterize Random Effects
i  0   Jj11,jhi,1( j)  xi  ui
Simulation or Quadrature Based Estimation
lnL= i=1 ln
N
i

Ti
t 1
Pit,j f(  j )d j
Part 20: Selection [54/66]
A Study of Health Status in the Presence
of Attrition
“THE DYNAMICS OF HEALTH IN THE BRITISH
HOUSEHOLD PANEL SURVEY,”
Contoyannis, P., Jones, A., N. Rice
Journal of Applied Econometrics, 19, 2004,
pp. 473-503.
 Self assessed health
 British Household Panel Survey (BHPS)


1991 – 1998 = 8 waves
About 5,000 households
Part 20: Selection [55/66]
Attrition
Part 20: Selection [56/66]
Testing for Attrition Bias
Three dummy variables added to full model with unbalanced panel
suggest presence of attrition effects.
Part 20: Selection [57/66]
Attrition Model with IP Weights
Assumes (1) Prob(attrition|all data) = Prob(attrition|selected variables) (ignorability)
(2) Attrition is an ‘absorbing state.’ No reentry.
Obviously not true for the GSOEP data above.
Can deal with point (2) by isolating a subsample of those present at wave 1 and the
monotonically shrinking subsample as the waves progress.
Part 20: Selection [58/66]
Probability Weighting Estimators



A Patch for Attrition
(1) Fit a participation probit equation for each
wave.
(2) Compute p(i,t) = predictions of participation
for each individual in each period.


Special assumptions needed to make this work
Ignore common effects and fit a weighted
pooled log likelihood: Σi Σt [dit/p(i,t)]logLPit.
Part 20: Selection [59/66]
Inverse Probability Weighting
Panel is based on those present at WAVE 1, N1 individuals
Attrition is an absorbing state. No reentry, so N1  N2  ...  N8.
Sample is restricted at each wave to individuals who were present at
the previous wave.
d it = 1[Individual is present at wave t].
d i1 = 1  i, d it  0  d i ,t 1  0.
xi1  covariates observed for all i at entry that relate to likelihood of
being present at subsequent waves.
(health problems, disability, psychological well being, self employment,
unemployment, maternity leave, student, caring for family member, ...)
Probit model for d it  1[xi1  wit ], t = 2,...,8. ˆ it  fitted probability.
t
Assuming attrition decisions are independent, Pˆit   s 1 ˆ is
ˆ  d it
Inverse probability weight W
it
Pˆit
Weighted log likelihood
logLW   i 1  t 1 log Lit (No common effects.)
N
8
Part 20: Selection [60/66]
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
Part 20: Selection [61/66]
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.
•
•
•
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)
Part 20: Selection [62/66]
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.
Part 20: Selection [63/66]
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
Part 20: Selection [64/66]
Spatial Weights
1
cij  2 ,
dij
d ij  Euclidean distance
Band of 7 neighbors is used
Row standardized.
Part 20: Selection [65/66]
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.
Part 20: Selection [66/66]