In this presentation we define two qualitatitive response models: 1) Selection Endogenous Dummy Ordered Probit model (SED-OP); 2) a Selection Endogenous Dummy Dynamic Selection Ordered Probit model (SED- DOP). The SED-OP model is a three-equation model constituted of an endogenous dummy equation, a selection equation, and a main equation which has an ordinal response form. The main feature of the model is that the endogenous dummy enters both the selection equation and the main equation. The dynamic SED-DOP model allows both the selection equation and the ordered equation to be dynamic by including lagged individual behaviour. Initial conditions are properly accounted for and free correlation among unobservables entering each of the three equations is allowed. We show how these models can be estimated in Stata using Maximum Simulated Likelihood.

Alfonso Miranda

Massimiliano Bratti

English

Research Papers in Economics

Motivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Selection endogenous dummy ordered probit,
and selection endogenous dummy dynamic
ordered probit models
Massimiliano Bratti & Alfonso Miranda
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Motivation
In many elds of applied work researchers need to model an
ordinal dependent variable that is only observed for a proportion
of the sample and that is a function of an endogenous variable
I Smoking and Education, but not everybody smokes
I Drinking and Education, but not everybody drinks
I Job type (unskilled/skilled/proesional) and Education, but
not everybody works
Selection can have dierent sources:
I Entry into an activity (smoking / drinking)
I Survey and/or item non-response
Fundamentally, the DGP of selection into missingness and the
DGP of the ordinal variable are essentially two dierent although
related processes (e.g., extensive vs. intensive margin decisions)
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Previous work
I Terza, Kenkel, Tsui-Fang and Shinichi (2008) suggest a two-
step method for estimating a selection endogenous dummy
model for an interval coded dependent variable (grouped). This
is an extension of Mullahy (1998) Modied Two Part Model
and Mullahy (1986) Hurdle Model. Despite being a two-step
approach, this is not a LIML estimator but relies on joint mul-
tivariate normality.
I Miranda and Rabe-Hesketh (2006) consider a model for an or-
dinal dependent variable with either an endogenous dummy or
sample selection. Cannot deal with the two problems at the
same time.
I Harris and Zhao (2007) suggests a zero-inated ordered probit
model, which is quite similar to Mullahy's Hurdle model and is
related to Lambert (1992) zero-inated count data models.
I Bratti and Miranda (2009) use the BCS70 and the methods
described here to analyse how higher education aects smoking
intensity in the UK.
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Selection endogenous dummy ordered probit
Let yi be the ordinal variable of interest. Variable yi is generated
according to a continuous latent variable model
y
i = x0
i + Gi + vi; (1)
where the observed response yi is determined by a threshold
model
yi =
8
> > > > > <
> > > > > :
missing if Si = 0
1 if y
it  k1 & Si = 1
2 if k1 < y
it  k2 & Si = 1
: : :
: : :
H if kH 1 < y
it & Si = 1;
Gi represents a potentially endogenous dummy and the main
response yi is only observed if a selection rule Si = 1 is met.
Both Gi and Si are always observed and fk1; ;kH 1g 2 RH 1
are constants to be estimated along other parameters.
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Selection endogenous dummy ordered probit II
The endogenous dummy and the selection dummy are also gen-
erated according to a continuous latent variable model
S
i = r0
i + 'Gi + qi
G
i = z0
i + wi;
(2)
with Si = 1(S
i > 0) and Gi = 1(G
i > 0). Although the model
is identied by functional form it is valuable to specify a set of
exclusion restrictions to avoid problems of tenuous identication.
Hence, when possible, some elements of the zi should not enter
xi or ri, and some elements of ri should not enter xi or zi.
Correlation among the three outcomes is allowed by imposing
some structure to the error terms,
wi = ui + i;
qi = 1ui + i
vi = 2ui + i;
(3)
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Selection endogenous dummy ordered probit III
To close the model we require the covariates to be all strictly
exogenous, and the idiosyncratic errors to be orthogonal given
the individual heterogeneity term u,
D(ujx;z;r) = D(u) (4)
D(jx;z;r;u) = D(ju) (5)
D(jx;z;r;u) = D(ju) (6)
D(jx;z;r;u) = D(ju) (7)
ju ? ju ? ju: (8)
To ease estimation we suppose that u, , , and  are all
independent standard normal, in which case D(ju) = D(),
D(ju) = D(), and D(ju) = D().
The factors loadings f1 2g 2 R2 are free parameters and allow
any type of correlation (positive, negative or null) between y
i ,
G
i and S
i .
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Selection endogenous dummy ordered probit IV
Correlation between unobservables entering G
i and S
i is given
by:
gs =
1 q
2
 
1 + 2
1
:
Similarly, correlation between unobservables entering G and
y
i is given by:
gy =
2 q
2
 
1 + 2
2
:
Finally, correlation between unobservables entering main re-
sponse y and selection S
i is given by:
sy =
12 q 
1 + 2
1
 
1 + 2
2
:
In this model G is exogenous wrt S if gs = 0, G is exogenous
wrt y if gy = 0, and y is observed at random if sy = 0.
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Selection endogenous dummy ordered probit V
I Estimate the system by Maximum Simulated Likelihood
I Analytical rst derivatives and numerical second derivatives
I Can also do OPG approx. of the Hessian (much faster!)
I Halton sequences cover the (0,1) interval better and require
fewer draws to achieve high precision than random samples
from uniform distribution
I Program written in Stata/Mata
I Really fast!
I Stata 10/SE + 400 Halton draws + 2,792 indv / 8,043
pers-obs + numerical 2nd derivatiives = 1.6hrs
I Stata 10/SE + 400 Halton draws + 2,792 indv / 8,043
pers-obs + OPG Hessian = less than 5min
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Selection Endogenous dummy dynamic ordered probit model
Suppose that yi is observed for two periods t = f1;2g. We now
extend the model to accommodate the fact that the outcome of
the ordinal response in period 2 can be a function of the value
that the variable took in period 1. In other words, we consider
the possibility of having autoregressive dynamics in the ordered
variable yi2 such that
y
i1 = m0
i + 1Gi + vi (9)
y
i2 = x0
i + 2Gi +
H X
j=1
j1(yi1 = j) + i; (10)
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Selection endogenous dummy dynamic ordered probit model II
As usual we suppose the model is complemented by a threshold
rule,
yit =
8
> > > > > <
> > > > > :
missing if Sit = 0
1 if y
it  k1 & Sit = 1
2 if k1 < y
it  k2 & Sit = 1
: : :
: : :
H if kH 1 < y
it & Sit = 1;
To ease presentation we suppose that G and S do not have
dynamics themselves. Further, we suppose that y is always ob-
served in the rst period and that G is a time invariant variable.
Hence, The selection and endogenous dummies are generated by
the following latent variable models,
G
it = G
i = z0
i + wi (11)
S
it = S
i = r0
i + '1Gi + qi (12)
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Selection endogenous dummy dynamic ordered probit model III
Like in the SED-OP model, we impose some structure to the
error terms,
wi = ui + i;
qi = 1ui + i
vi = 2ui + i
i = 3ui + "i;
(13)
where ui, i, i, i, and "i are all supposed to be independent
standard normal, and 1;:::;3 are free factor loadings.
Notice that the model fully recognises the fact that the initial
state dummies1(yi1 = 1);:::;1(yi1 = H) are potentially endoge-
nous in equation (10) by modelling together the initial conditions
and the dynamic equation, and allowing any type of correlation
among unobservables entering both equations. There is true
state dependance if the coecients on the initial state dummies
in (10) are dierent from zero.
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Selection endogenous dummy dynamic ordered probit model IV
Unobservables entering the system are now correlated in six
dierent ways:
g;s = 1 q
2(1+2
1)
g;y1 = 2 q
2(1+2
2)
g;y2 = 3 q
2(1+2
3)
s;y1 = 12 q
(1+2
1)(1+2
2)
s;y2 = 13 q
(1+2
1)(1+2
3)
y1;y2 = 23 q
(1+2
2)(1+2
3)
:
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Selection endogenous dummy dynamic ordered probit model V
I The initial state dummies can be included into the S with-
out further complications
I The model can be extended to allow dynamics in S and
G
I The model can deal with more than two periods with rela-
tive minor modications
I As before we use MSL for estimation
I Program written in Stata/Mata
I Really fast!
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Example I - Variables denition
We apply the SED-OP model to study the eect of higher ed-
ucation (HE) on drinking frequency using the British Cohort
Study 1970 (BCS70), 29-year follow-up survey.
Variables denition:
I yi (drinking frequency) = 1 (2 to 3 times a month); 2
(once a week); 3 (2 to 3 days a week); 4 (on most days)
I Gi (higher education) = 1 (HE); 0 (lower than HE)
I Si (usual drinker) = 1 (drinks more than 2 to 3 times a
month); 0 (drinks less than 2 to 3 times a month, i.e. less
often or in special occasions, not nowadays, never drunk)
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Example II - Descriptive statistics
Descriptive statistics, BCS70, 29-year follow-up survey
% of usual drinkers
Women Men
No HE 69.9 85.78
HE 83.56 92.02
Drinking frequency per month - usual drinkers
2-3 times once a 2-3 times
per month week a week most days Total
Women
no HE 25.04 34.4 31.63 8.92 100
HE 16.01 24.53 43.53 15.93 100
Total 21.7 30.75 36.04 11.51 100
Men
no HE 14.14 24.95 43.27 17.63 100
HE 8.95 19.13 48.35 23.57 100
Total 12.35 22.94 45.03 19.69 100
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Example III - SED-OP specication
I HE: parents' absence, parents' education, highest social class,
school type, state, BAS score, ethnicity, religion, height, teacher's
assessment of child's knowledge and parental interest in child's
education, teacher's homework style, all at age 10.
I Usual drinker: HE, parents' absence, parents' education, highest
social class, school type, state, ethnicity, religion, height, all
at age 10; height at age 30, mother's drinking during pregnancy,
homework on parents' demand, month of 29-year follow-up interview
I Drinking frequency: same controls as the selection equation (usual
drinker)
Since the selection and the main outcome variables refer to the same process
(drinking) we preferred not to impose exclusion restrictions between the two
equations. In other cases, for instance item non-reponse and panel attrition,
it can be easier to nd valid exclusion restrictions.
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Example III - SED-OP Results
Be a Drinking frequency -y- (ME)
usual drinker -s- 2-3 times once a 2-3 times most
(ME) a month week a week days
Men
HE -g- 0.156??? -0.129??? -0.109??? 0.054??? 0.183???
[0.015] [0.032] [0.021] [0.009] [0.046]
gs -0.484???[0.056]
gy -0.320???[0.091]
sy 0.310???[0.063]
No. obs. 3300
Women
HE -g- 0.259??? -0.302??? -0.087??? 0.181??? 0.208???
[0.024] [0.046] [0.010] [0.015] [0.039]
gs -0.418???[0.061]
gy -0.479???[0.087]
sy 0.401???[0.058]
No. obs. 3525
???(??) Signicant at 1% (5%). Eicker-Huber-White robust standard errors reported in square brackets.
Marginal eects (ME) are computed at the sample mean. For dummy variables, they show the change in
the relevant probability when the variable changes from 0 to 1.
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
Concluding remarks
Hence, our empirical application shows that:
I HE is endogenous wrt both drinking participation (i.e. be
an usual drinker) and drinking frequency
I Unobservables aecting drinking participation and drink-
ing frequency are positively correlated (positive selection)
I HE has a positive causal eect on both drinking par-
ticipation and drinking frequency (i.e., educated people
drink more)
I The eect is much larger for females than for males
ADMIN node  Institute of Education  University of London c Bratti&MirandaMotivation
Previous Work
SED-OP
SED-DOP
Example
Example
Concluding
remarks
References
I Bratti, M and Miranda, A (2009) Non-pecuniary returns to higher ed-
ucation: The eect on smoking intensity in the UK. Health Economics
Published Online: Jul 13 2009 12:08PM (Early View) http://dx.doi.
org/10.1002/hec.1529.
I Harris, MN and Zhao, X (2007) A zero-inated ordered probit model, with
an application to modelling tobacco consumption. Journal of Econometrics
141:1073{1099.
I Heckman, JJ (1979) Sample selection bias as a specication error. Econo-
metrica 47: 153{61.
I Lambert,D (1992) Zero-Inated Poisson Regression with an Application to
Defects in Manufacturing. Technometrics 34:1-14.
I Miranda, A. and Rabe-Hesketh, S. (2006) Maximum likelihood estimation
of endogenous switching and sample selection models for binary, ordinal,
and count variables. Stata Journal 6 (3): 285{308.
I Terza, JV and Kenkel, DS and Lin, TF and Sakata, S (2008) Care-giver
advice as a preventive measure for drinking during pregnancy: zeros, cate-
gorical outcome responses, and endogeneity. Health Economics 17: 41-54.
I Train, KE (2003). Discrete choice methods with simulation. Cambridge
university press.
ADMIN node  Institute of Education  University of London c Bratti&Miranda

A zero-in ated ordered probit model, with an application to modelling tobacco consumption.

Care-giver advice as a preventive measure for drinking during pregnancy: zeros, categorical outcome responses, and endogeneity.

Discrete choice methods with simulation. Cambridge university press.

Maximum likelihood estimation of endogenous switching and sample selection models for binary, ordinal, and count variables.

Non-pecuniary returns to higher education: The eect on smoking intensity in the UK.

Sample selection bias as a specication error.

Zero-In ated Poisson Regression with an Application to Defects in Manufacturing.

Selection-endogenous ordered probit and dynamic ordered probit models

Selection-endogenous ordered probit and dynamic ordered probit models

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