IV methods for Tobit models

Andrew Chesher
Dongwoo Kim
Adam M. Rosen

The Institute for Fiscal Studies
Department of Economics, UCL
cemmap working paper CWP26/21

IV Methods for Tobit Models
Andrew Cheshery

Dongwoo Kimz

UCL and CeMMAP

Simon Fraser University

Adam M. Rosenx
Duke University and CeMMAP
June 27, 2021

Abstract
This paper studies models of processes generating censored outcomes with endogenous explanatory variables and instrumental variable restrictions. Tobit-type left censoring at zero is the primary focus in the exposition. The models studied here areunrestrictive relative to others widely used in practice, so they are relatively robust
to misspeci…cation. The models do not specify the process determining endogenous
explanatory variables and they do not embody restrictions justifying control function
approaches. The models can be partially or point identifying. Identi…ed sets are characterized and it is shown how inference can be performed on scalar functions of partially
identi…ed parameters when exogenous variables have rich support. In an application
using data on UK household tobacco expenditures inference is conducted on the coe¢ cient of an endogenous total expenditure variable with and without a Gaussian
distributional restriction on the unobservable and compared with the results obtainedusing a point identifying complete triangular model.
We thank Xinyue Bei for providing excellent research assistance. Financial support from the UK Economic and Social Research Council through grants (RES-589-28-0001, RES-589-28-0002 and ES/P008909/1)
to the ESRC Centre for Microdata Methods and Practice (CeMMAP) is gratefully acknowledged.
yAddress: Department of Economics, University College London, Gower Street, London WC1E 6BT,
United Kingdom. Email: andrew.chesher@ucl.ac.uk.
zAddress: Department of Economics, Simon Fraser University, West Mall Centre, SFU, 8888 University
Drive Burnaby, BC Canada, V5A 1S6. Email: dongwook@sfu.ca.
xAddress: Department of Economics, Duke University, 213 Social Sciences, Box 90097 Durham, NC
27708, United States. Email: adam.rosen@duke.edu.

1Keywords: censored models, endogeneity, incomplete models, instrumental variables,
partial identi…cation, random sets.
JEL classi…cation: C10, C24, C26, C51.

1

Introduction

This paper develops and applies results on the identifying power of models for continuous
censored outcomes, focussing on cases with left censoring at a …xed known value. In econometrics the Tobit model, Tobin (1958), is a well known example of such a model.
In contrast to the classical Tobit model, the focus is on cases in which one or more of
the explanatory variables is endogenous and there are exogenous variables that satisfy exclusion restrictions that limit their degree of in‡uence on the censored outcomes. These
instrumental variables (IVs) are restricted to be distributed to some degree independently of
the unobserved variables in the structural equation for the censored outcomes. A variety of
independence restrictions are considered, including mean, quantile, and full stochastic independence restrictions, with both parametric and nonparametric distributional restrictions.
This paper is unlike most that allow for endogenous explanatory variables in situations
in which outcomes are censored because there is no speci…cation of the determination of
the endogenous explanatory variables as either a deterministic or stochastic function of
the exogenous explanatory variables and instruments. Consequently the models studied
here are incomplete for the determination of endogenous explanatory variables. The IV
approach explored here is an alternative to approaches that use complete models, which
are prey to misspeci…cation if they introduce an incorrect speci…cation of the genesis of
endogenous explanatory variables. STATA’s ivtobit command,1 despite its name, computes
estimates of parameters of Tobit models with endogenous explanatory variables using acomplete triangular model assuming Gaussian unobserved variables. That attack does notdeliver consistent estimates when endogenous variables are discrete or a¤ected by multiple
sources of heteorgeneity. By contrast, the IV approach employed here can be used whenendogenous variables are discrete or when the way in which endogenous variables’ values
are generated is unrestricted. The IV approach is a useful alternative to control function
approaches which require endogeneity to be absent once there is conditioning on identi…able
functions of observed variables. This paper shows how inferences can be drawn using censoreddata and weakly restrictive robust IV models.
1

StataCorp (2019).

2

The models studied are single equation IV models for a censored outcome variable. They
fall in the class of generalized instrumental variable models analyzed in Chesher and Rosen(2017), henceforth CR17. We use techniques developed in that paper to carry out identi…cation analysis, and we show how to implement the high level results given in CR17 inmodels for censored outcomes. The models are in general partially identifying but they canbe point identifying, and it is usually not possible to determine identi…cation status using
the data available in applications. We conduct inference on scalar functions or subvectors ofpartially identi…ed parameters following the approach of Belloni et al. (2018) based on a selfnormalized critical value as in Chernozhukov et al. (2019), which is appropriate for inference
based on a very large number of moment inequalities. The approach applies regardless ofidenti…cation status, and we propose a method for implementation when exogenous variables
have rich support. We illustrate with an application to UK household survey data recording
tobacco expenditures in which around 70% of households record zero expenditures.
The main focus in this paper is on IV Tobit models with left censoring at zero, with andwithout a Gaussian distributional restriction on the scalar unobserved variable. However it
is straightforward to extend to cases with right censored outcomes and to cases in which
the censoring value is stochastic, distributed independently of the unobserved variable in the
censored outcome equation.2 Leading examples with right censoring arise when the censoredoutcome is the time until some event occurs, so the methods developed here …nd applicationin models of durations admitting endogenous explanatory variables.3
As noted, much of the prior literature studying models with censored outcomes andendogenous explanatory variables relies on a complete model for the determination of endogenous variables. Examples include the fully parametric speci…cations studied in Heckman (1978), Nelson and Olson (1978), Amemiya (1979), Smith and Blundell (1986), Newey(1987), and Blundell and Smith (1989) that made early contributions to the study of limited
dependent variable models (including Tobit models in particular) permitting endogenous
explanatory variables and enabling consistent estimation by way of control function approaches and marginal or conditional maximum likelihood procedures.4 Control function
approaches for semiparametric triangular models for censored outcomes are provided in Das(2002), Blundell and Powell (2007), and Chernozhukov et al. (2015). These papers do not2

Cases in which the censoring variable is also endogenous are considered in ongoing research.
See for example Lancaster and Chesher (1984), Lancaster (1985), Olsen and Farkas (1989), Frandsen(2015), and Wrenn et al. (2017).
4
Comparisons of the e¢ ciency of di¤erent procedures are provided in Newey (1987) and Blundell andSmith (1989).
33require parametric distributional restrictions on unobservable heterogeneity, with Das (2002)
employing symmetry restrictions and Blundell and Powell (2007) and Chernozhukov et al.
(2015) using conditional quantile restrictions.
Our approach is in the spirit of Manski and Tamer (2002), which pioneered the useof incomplete models for censored outcomes or covariates and used partial identi…cation
analysis. That paper characterized identi…ed sets and proposed consistent set estimators fora variety of models with censored variables, in which the censoring process is not speci…ed.
In the IV Tobit models studied here, the censoring process is speci…ed, but endogenous
explanatory variables are permitted and it is the lack of speci…cation of their determinationthat renders the models incomplete. In this respect the models studied here have similarities
with the model studied in Hong and Tamer (2003). That model employs conditional quantile
restrictions with a censored outcome variable and does not impose a speci…cation of theprocess determining values of endogenous explanatory variables. The model of Hong and
Tamer (2003) is incomplete but the focus there is on settings in which the parameters of themodel are point identi…ed. Su¢ cient conditions for point identi…cation are proposed alongwith a point estimator, and the asymptotic properties of the estimator are characterized. The
support conditions shown to guarantee point identi…cation are strong and there are manycases arising in practice in which they will not be satis…ed. We characterize sharp identi…edsets for model parameters applicable when these support conditions are not guaranteed
to hold, and we illustrate the use of inference that is robust to the possibility of partial
identi…cation.
There is also research that considers the di¤erent but important problems of endogenous
censoring of explanatory variables. This includes Khan and Tamer (2009), Khan et al.
(2011), and Section 7 of Chesher and Rosen (2020b).
This paper makes several contributions to the literature on models of censoring withendogenous explanatory variables and instrumental variable restrictions.
1. As already pointed out, the vast majority of the previous papers on this topic consider
models that, unlike those considered here, require a complete speci…cation for thedetermination of endogenous explanatory variables. Our analysis thus shows what
can still be learned when the speci…cation of the genesis of endogenous explanatory
variables is dropped.
2. We consider the use of more or less demanding restrictions on the distribution ofunobservable heterogeneity conditional on instruments. This can be used to assess4

(for example) how robust empirical …ndings are to relaxation from a full stochastic
independence restriction to selected conditional quantile restrictions.
3. Our analysis is robust to the possibility of partial identi…cation, and is thus applicable when data are not compatible with conditions that are known to ensure point
identi…cation.
4. We show how to conduct inference on functions or subvectors of parameters partially
identi…ed by moment inequalities in these IV Tobit models, using recent developments
in Chernozhukov et al. (2019) and Belloni et al. (2018) allowing for a large number ofmoment inequalities relative to the sample size, as encountered in our application.
5. We show how quantile independence restrictions at multiple quantiles can be incorporated. This enables the study of the increase in identifying power as one moves frominvoking a single conditional quantile restriction to successively more quantile restrictions, approaching full stochastic independence as more such restrictions are imposed.
6. We show how to determine the identifying power of an IV model for censored outcomesunder a stochastic independence restriction with a nonparametric speci…cation of the
distribution of the unobservable variable in the structural equation for the censoredoutcome.
The paper proceeds as follows. In the following section we present the class of IV Tobit
models studied. In Section 3 we characterize the identi…ed set of structures –combinationsof structural functions and distributions of unobservable heterogeneity –that are compatible
with the censored outcome model. The identi…ed set is shown to comprise those structuresthat lie in the intersection of two sets, each de…ned by a collection of conditional moment
inequalities. We show how under some circumstances certain subsets of the inequalities
reduce to moment equalities, and we show how exclusion restrictions can be incorporated
into the characterization of the identi…ed set. In Section 4 we analyze the identifying content of various restrictions on the joint distribution of exogenous variables and unobservable
heterogeneity, such as conditional mean, conditional quantile, and stochastic independence
restrictions. In Section 5 we describe how inference is carried out using results from Chernozhukov et al. (2019) and Belloni et al. (2018), and we propose a practical approach forapplication of the inference method when an identi…ed set is characterized by conditional
moment inequalities with continuous conditioning variables. We illustrate with an application in which we focus on conducting inference on the e¤ect of total household nondurable
5expenditure on the share of expenditure spent on tobacco, previously considered using acontrol function approach in Adams et al. (2019). All proofs and …gures are provided in in
the Appendix.

2

The IV Tobit model

Scalar endogenous outcome Y1 , possibly endogenous vector Y2 , exogenous vector Z 2 RZ ,and unobserved scalar U 2 R satisfy:
Y1 = m(Y2 ; Z; U )Y1 = max(Y1 ; 0),(1)where the function m is strictly increasing in its third argument (U ) and for all y2 and z,m(y2 ; z; 1) 0. There is the inverse function m 1 (y2 ; z; y1 ) such that for all y1 , y2 and zm(y2 ; z; m 1 (y2 ; z; y1 )) = y1 :In a leading case of interest the function m is linearm(y2 ; z; u) = y2 + z + u,
as in the classical Tobit model which has U N (0; 2 ) independent of Z and no endogenous
explanatory variables. In the linear case the inverse function ism 1 (y2 ; z; y1 ) = y1y2z:

We consider cases in which the model embodies a parametric speci…cation of the distribution of U , for example U N (0; 2 ), and cases in which there is no parametric speci…cation.
We consider models restricting U and Z to be stochastically independent, less restrictive
speci…cations requiring quantile independence at speci…ed quantiles, or, alternatively, mean
independence restrictions.
We cast the problem into the Generalized Instrumental Variable framework set out inCR17 in which a structure, (m; GU jZ ) comprises two components, namely (i) a structuralfunction and (ii) a distribution of unobservable heterogeneity conditional on each possible
value of exogenous variables. The …rst of these components, the structural function, determines which combinations of (Y; Z; U ) can jointly occur. In the IV Tobit model this is

6

fully determined by the function m.5 The second component of a structure is a collection of
conditional distributions of U given Z, denoted
GU jZ
where for any set S

GU jZ ( jz) : z 2 RZ

R
GU jZ (Sjz)

P[U 2 SjZ = z]:

A model, A, comprises a list of restrictions on structures, de…ning a set of admissible
structures, MA , which satisfy the restrictions. A model’s restrictions may limit the dependence between U and Z and may require that the function m satis…es conditions additional
to those so far imposed, for example functional form and exclusion restrictions. When U
and Z are stochastically independent a collection GU jZ is a singleton fGU g where GU is themarginal distribution of U . A model can additionally impose parametric restrictions on the
distribution of U .
In summary, throughout the paper a model is referred to as an IV Tobit Model if itsatis…es the following de…nition.
De…nition 1 An IV Tobit Model A comprises a set of restrictions on the process generating
observed variables Y1 , Y2 , and Z such that (1) holds for some unobservable variable U residing
on the same probability space as (Y1 ; Y2 ; Z). The function m and conditional distributions ofU given Z are restricted to belong to some set, MA , of admissible pairs (m; GU jZ ).
A variety of IV Tobit Models are considered, imposing di¤erent sets of restrictions on the~ U jZ (tjz) GU jZ (( 1; t] jz) is used to denotepair (m; GU jZ ). When convenient, notation G
the conditional cumulative distribution function of U given Z = z associated with GU jZ ( ),and g~U jZ ( jz) is used to denote the corresponding conditional density when conditions are
such that the density exists.
5

In CR17 notation h : RY ZU ! R is used to denote a structural function de…ned on the support of(Y; Z; U ) such that h (Y; Z; U ) = 0 with probability one. In the the IV Tobit model there is a uniquemapping from values of (Y2 ; Z; U ) to values of Y and it is more natural to work with this function, m,
directly. In the notation of CR17, one would de…ne h (Y; Z; U ) Y1 max (0; m (Y2 ; Z; U )).

7

3

Identi…cation

3.1

Characterizations of identi…ed sets

CR17 gives characterizations of identi…ed sets of structures from which identi…ed sets ofstructural features are obtained by projection. The characterizations make use of residual
sets associated with a structure (m; GU jZ ). A residual set is the set of values of unobserved
U that, for a structural function m, can deliver the value y of endogenous Y when exogenous
Z is equal to z. In the IV Tobit model the residual sets are singleton sets when y1 > 0 andsemi-in…nite intervals when y1 = 0, as follows.
U(y; z; m) =(( 1; m 1 (y2 ; z; 0)] ; y1 = 0
fm 1 (y2 ; z; y1 )g; y1 > 0

Let FY jZ fFY jZ ( jz) : z 2 RZ g denote the collection of conditional distributions of Y
given Z, where for any set6 Y RY
FY jZ (Yjz)P[Y 2 YjZ = z].

Data is informative about this collection of distributions, which is assumed to be identi…ed.
The following Proposition characterizes the identi…ed set of structures given knowledge of
FY jZ .
Proposition 1 The identi…ed set of structures delivered by an IV Tobit model, A, and acollection of conditional distributions of Y given Z, FY jZ fFY jZ ( jz) : z 2 RZ g isI(FY jZ ; RZ ; A) = I1 (FY jZ ; RZ ; A) \ I2 (FY jZ ; RZ ; A),

(2)

where
I1 (FY jZ ; RZ ; A)(m; GU jZ ) 2 MA : 8t 2 R; GU jZ (( 1; t]jz)

6

C(z; t; m) a.e. z 2 RZ ,(3)

To keep the notation uncluttered the support of Y is taken to be independent of z. This can easily berelaxed.

8

I2 (FY jZ ; RZ ; A)(m; GU jZ ) 2 MA : 8[t1 ; t2 ]R;

GU jZ ([t1 ; t2 ]jz)
(z; t1 ; t2 ; m)B(z; t; m)B(z; t2 ; m)

P [0 < Y1
P[Y1 = 0 ^ 0D(z; t; m)(z; t1 ; t2 ; m) a.e. z 2 RZ , (4)B(z; t1 ; m),m(Y2 ; Z; t)jZ = z],m(Y2 ; Z; t)jZ = z],andC(z; t; m)P [Y1m(Y2 ; Z; t)jZ = z] = D(z; t; m) + B(z; t; m).

The proof of the Proposition follows from application of the inequality
GU jZ (Sjz)

P[U(Y; Z; h)

SjZ = z](5)

to sets S comprising certain intervals on the real line. The probability P[U(Y; Z; h) SjZ =z], termed a containment probability, is the conditional probability given Z = z of theoccurrence of a value of Y that can occur only when unobserved U takes a value in theinterval S.7 The expressions (z; t1 ; t2 ; m), B(z; t; m), D(z; t; m), and C(z; t; m) de…ned in
the Proposition help to provide concise representations for these containment probabilities.
Speci…cally, for intervals ( 1; t] that are unbounded below:
P[U(Y; Z; m)
For intervals [t1 ; t2 ], with t1 >( 1; t] jZ = z] = C(z; t; m):

1 but with t2

P[U(Y; Z; m)t1 unrestricted:[t1 ; t2 ]jZ = z] =(z; t1 ; t2 ; m) ,from which it also follows that for intervals [t1 ; 1) with t1 >P[U(Y; Z; m)[t1 ; 1)jZ = z] = P [Y1 > 0jz]

7

(6)

(7)

1:
B(z; t1 ; m):(8)

Other characterizations of identi…ed sets are available. One such will be employed when we consider theforce of the restriction that unobserved U is mean independent of Z. All of the characterizations follow fromthe result that a structure (m; GU jZ ) is in the identi…ed set if and only if for all z in the support of Z the
distribution GU jZ ( jz) is selectionable with respect to the distribution of the random set U(Y; Z; m) inducedby the distribution of (Y; Z) delivered by the process under study. De…nitions and details are in CR17.

9In addition, it will be useful later to have an expression for the containment probability
which applies for intervals [t1 ; t2 ] with t1 …nite or in…nite, as follows.
P[U(Y; Z; m)[t1 ; t2 ]jZ = z] = 1[t1 =1]

P[Y1 = 0 ^ 0P[Y1 > 0 ^ m(Y2 ; Z; t1 )3.2

m(Y2 ; Z; t2 )jZ = z]+
Y1

m(Y2 ; Z; t2 )jZ = z] (9)Characterizations using singly-in…nite systems of moment inequalities

The sets I1 (FY jZ ; RZ ; A) and I2 (FY jZ ; RZ ; A) are determined by systems of respectively
singly- and doubly-in…nite moment inequalities. Under additional restrictions that imply
that B(z; t; m) is everywhere di¤erentiable in t the doubly-in…nite system can be replacedwith an equivalent singly-in…nite system. This can have computational advantages.
eU jZ ( jz) denoting the conditional cumulative distribution function of U given
With GZ = z the condition
GU jZ ([t1 ; t2 ]jz)
(z; t1 ; t2 ; m)that appears in I2 (FY jZ ; RZ ) can be expressed as
eU jZ (t2 jz)
GeU jZ (t1 jz)
G(z; t1 ; t2 ; m) .

From this, with a di¤erentiability restriction, Proposition 2 provides a singly-in…nite moment
inequality characterization for the set I2 (FY jZ ; RZ ; A) originally de…ned in (4).
Proposition 2 Suppose m(y2 ; z; t) is everywhere di¤erentiable with respect to t for all valuesof (y2 ; z) and that U is continuously distributed given Z with conditional density geU jZ ( jz).
Then
I2 (FY jZ ; RZ ; A) = (m; GU jZ ) 2 MA : 8t 2 R; geU jZ (tjz)

where b(z; t; m)3.3b(z; t; m) a.e. z 2 RZ ,(10)

rt B(z; t; m) is the partial derivative of B(z; t; m) with respect to t.

Upper and lower bounds and moment equalities

The containment inequality (5) used to produce Proposition 1 provides a lower bound on the
distribution of U . It is shown in CR17 that applying the inequality in (5) to the complement10

S c of a set S delivers the following inequality satis…ed by all structures in the identi…ed set
for all z 2 RZ and all closed sets S on the support of U .8
GU jZ (Sjz)

P[U(Y; Z; m) \ S =6 ;jZ = z]

(11)

So, for all structures in the identi…ed set I(FY jZ ; RZ ; A) the inequalities
P[U(Y; Z; m) \ S =6 ?jZ = z]

GU jZ (Sjz)

P[U(Y; Z; m)

SjZ = z](12)

hold for all z 2 RZ and all intervals, S, on the real line.
There are moment equalities in the characterization of the identi…ed set of structures
when there are z and S such that the probabilities on the left and right hand sides above
are equal. In a su¢ ciently restrictive model and for particular collections of distributions
FY jZ and support RZ , there is the possibility that these moment inequalities deliver point
identi…cation.
In the IV Tobit model the upper bounding probability P[U(Y; Z; m) \ S 6= ;jZ = z] is asfollows.
P[U(Y; Z; m) \ [t1 ; t2 ] 6= ?jZ = z] = P[(Y1 = 0) ^ (0m(Y2 ; Z; t1 ))jZ = z]

+ P[(Y1 > 0) ^ (m(Y2 ; Z; t1 )Y1

m(Y2 ; Z; t2 ))jZ = z]: (13)Considering (6), (7), and (13), bounding probabilities in (12) are equal for semi-in…nite
intervals ( 1; t] when z and m are such thatP [Y1 = 0jZ = z] = P [Y1 = 0 ^ 0m(Y2 ; Z; t)jZ = z]

and for …nite intervals [t1 ; t] when z and m are such thatP [Y1 = 0 ^ 0m(Y2 ; Z; t)jZ = z] = 0:Both conditions are satis…ed when m(Y2 ; Z; t) > 0 almost surely conditional on Z = z. A
leading case in which this can occur is when the endogenous explanatory variable Y2 hasbounded support and the function m is unbounded above as t becomes large.
Conditions such that the inequalities (12) reduce to equalities for some values of z, can
8Here ; denotes the empty set. The probability on the right hand side is known as a capacity functionalor the hitting probability of the random set.

11

be the basis for establishing su¢ cient conditions for point identi…cation. For example, inmodels for censored outcomes with a conditional median restriction (Hong and Tamer, 2003,
p. 908) provide support conditions under which certain resulting moment equalities canpestablish point identi…cation and a n-consistent and asymptotically normal estimator ofmodel parameters when m is linear in parameters. Under the restriction that med (U jZ) = 0
and m(Y2 ; Z; 0) > 0 almost surely conditional on Z = z with m linear in parameters andadditive U there is in our notation
med (Y1

Y2 + Z jZ = z) = 0,

which corresponds to the moment equality delivered by the inequalities (12) applied to theset ( 1; t]. A condition requiring that the set of values of z 2 RZ such that m(Y2 ; Z; 0) > 0
almost surely conditional on Z = z has positive measure, in conjunction with a condition
requiring su¢ cient variation in included endogenous variables conditional on instruments, isthen used to establish su¢ cient conditions for point identi…cation in Lemma 2 of Hong and
Tamer (2003).

3.4

Restrictions on the in‡uence of exogenous Z

So far no restrictions on the dependence between U and Z or on the in‡uence of z on
the values taken by the function m(y2 ; z; u) have been considered. So far as the latter isconcerned consider the restriction requiring m to depend on z solely through the variation
in a function, w(z), that arises as z varies across the support of Z.
Restriction ZD: Restricted Z dependence
9w( ) s:t: 8(z; z 0 ) 2 RZ

RZ ; 8(y2 ; u);w(z) = w(z 0 ) =) m(y2 ; z; u) = m(y2 ; z 0 ; u)A case which commonly arises involves exclusion restrictions with z = (z1 ; z2 ) and w(z) = z1 .
Index restrictions in which m only varies with z through variation in some parametric linear
functions of z are also commonly employed.
De…ne the set of values that w(z) can take as z varies across its support
W(RZ )

fw(z) : z 2 RZ g

12

and for each element, w, of this set de…ne the set of values of z such that w(z) = w:
Z(w; RZ )fz 2 RZ : w(z) = wg:

When Z is excluded from the structural function de…ne w(z) = z in which case W(RZ ) =RZ and Z(w; RZ ) = RZ .
In the case in which there is a stochastic independence condition so that GU jZ = fGU geU (t) and geU (t) are
the sets I1 (FY jZ ; RZ ; A) and I2 (FY jZ ; RZ ; A) are then as follows, where G
respectively the marginal distribution and density functions of U .
I1 (FY jZ ; RZ ; A) =I2 (FY jZ ; RZ ; A) =(eU (t)(m; GU jZ ) : 8w 2 W(RZ ); t 2 R; G(

m; GU jZ : 8w 2 W(RZ ); t 2 R; geU (t)supz2Z(w;RZ )supz2Z(w;RZ ))C(z; t; m) ,)b(z; t; m) ,and the identi…ed set of structures is

I(FY jZ ; RZ ; A) = I1 (FY jZ ; RZ ; A) \ I2 (FY jZ ; RZ ; A).94

The impact of restrictions on the dependence between U and Z

In this section we consider the identifying power of a conditional mean independence restriction, a conditional quantile independence restriction focussing on median independence, astochastic independence restriction with no parametric speci…cation of the distribution GU
and stochastic independence restriction with U restricted to be Gaussian.
We present characterizations of identi…ed sets for models in which Y1 = max(0; m(Y2 ; Z; U ))and illustrate for the case in which m(Y2 ; Z; U ) = + Y2 + U . In the illustrations probabilities delivered by two speci…c structures are employed, as follows.10
9In order to deal with possibilities of zero measure sets and conditions required to hold almost everywhere,
here and througout the paper the sup and inf operators are to be understood to mean “essential supreumum”
and “essential in…mum” when applied to functions of realizations of random variables. So for instance
sup f (z) indicates the smallest value c such that f (Z) c with probability one given Z 2 Z.
z2Z
10

In order to fully determine the conditional distributions of Y jZ = z for each z 2 Rz it is necessary to
specify complete structures in which the process delivering Y2 is speci…ed. However, the equation relating
Y2 to Z and a stochastic unobservable is not a restriction used by the single equation IV Tobit model, and13

Structure 1
Y1 = max(0; b + aY2 + U1 )Y2 = d0 + d1 Z + U2
Structure 2
Y1 = max(0; b + aY2 + U1 )Y2 = g(Z; U2 )e( d0 d1 Z U2 )g(Z; U2 ) = k
In both cases, for all Z 2 RZ
U=

"

U1
U2

#

N2

"

0
0

# "
;

s11 s12s12 s22#!

and in all the examples the support of scalar Z is
RZ = f 1; 0:9; 0:8; : : : ; 0; : : : ; 0:8; 0:9; 1g;
It is important to understand that these are speci…cations of complete structures. The
incomplete models that we consider do not employ all the restrictions that are embodied in
the structures. In particular none of the models specify a structural equation for Y2 and in
only one of the models is U1 restricted to be Gaussian.
A crucial feature of Structure 2 is that Y2 is bounded above. Containment and otherprobabilities are calculated to high accuracy using numerical integration procedures.

4.1

Mean independenceFirst consider models in which U is restricted to be mean independent of the instrumental
variables. Absent censoring such a model would be point identifying under a suitable rank
condition.
RESTRICTION MI - Mean Independence: Let GU jZ comprise all collections of conditional distributions for U given Z, GU jZ , satisfying E[U jz] = 0, a.e. z 2 RZ .
is thus not brought to bear in the identi…cation analysis.

14

Table 1: Parameter values employed in numerical illustrations using Structures 1 and 2
Structure 1
Parameter Case 1 Case 2 Case 3
b
1:0
1:0
1:0
a
1:0
1:0
1:0
k
d0
0:5
0:25
0:0
d1
1:0
0:5
0:25
s11
2:0
1:5
1:0
s12
0:5
0:375
0:25
s22
1:0
1:0
1:0

Structure 2
Case 1
0:0
1:0
1:0
0:0
1:0
1:0
0:5
1:0

Manski and Tamer (2002) used mean independence restrictions conditional on included
exogenous variables in regressions with censored outcomes or covariates. Here we impose
an IV version of a conditional mean restriction, conditioning an included exogenous variable
and instruments. This is also Restriction MI in CR17 except that here, to simplify, thevalue of the conditional expectation is restricted to be zero rather than a member of a
speci…ed set of values. Modifying Theorem 5 of CR17 delivers the result that the identi…ed
set for structural function m comprises those functions m such that zero is an element of the
Aumann expectation of U(Y; Z; m) conditional on Z = z a.e. z 2 RZ .
Proposition 3 Under Restriction MI the identi…ed set for the function m is
fm : 0E[m 1 (Y2 ; Z; Y1 )jZ = z] a.e. z 2 RZ g:

In the case in which m(y2 ; z; u) is linear and equal to y2 + z + u there ism 1 (y2 ; z; y1 ) = y1y2z:De…ne W( ), the set of values that z can take as z varies across its support11W( )11f z : z 2 RZ g

The set W( ) depends upon RZ but this is not shown in the notation.

15

and for each element, w, of this set de…ne the set of values of z such that z = w:
Z(w; ) = fz 2 RZ : z = wg:
The identi…ed set for ( ; ) is as follows.
( ; ) : 8w 2 W( );0infz2Z(w; )(E[Y1 jz]

E[Y2 jz]z)This is an intersection of linear half spaces and is therefore a convex set.
In the case in which there are no included exogenous variables in the structural equation
for the censored outcome, so z is simply a scalar intercept term denoted , the identi…ed
set is as follows.
( ; ): 0
inf (E[Y1 jz]
E[Y2 jz]
) :
(14)
z2RZ

Figure 1 shows an example of this set for Structure 1 with the parameter values shown inthe column headed Case 1 in Table 1.
The identi…ed set comprises the region below all of the blue drawn straight lines. The
projection of the set onto the space ofis the entire real line. The projection of theset onto the space of is the entire real line unless there exist z and z 0 in RZ such thatE[Y2 jz] 0 E[Y2 jz 0 ] (a condition which holds with strong inequalities in the case pictured)in which case the projection is a semi-in…nite interval with …nite upper limit. The value ofand in the structure that generates the probabilities used in the calculations is the green
plotted point.
The identi…ed sets under mean independence are similar under the other cases of Structure 1 and in Structure 2 as will be shown shortly.
Clearly a conditional expectation restriction does not lead to particularly informative
identi…ed sets. We now turn to consider the identifying power of conditional quantile restrictions. These can be much more informative.

4.2

Quantile independence

We now study the power of the following quantile independence restriction.
RESTRICTION QI - Quantile Independence: Let GU jZ comprise all collections ofconditional distributions of U given Z satisfying P[Uqj jz] = j , a.e. z 2 RZ for all
j2J
f1; :::; Jg wheref 1 ; :::; J g is a collection of speci…ed known values, and some16

collection of values fq1 ; :::; qJ g 2 Q withof possible values of fq1 ; :::; qJ g.

j and qj both increasing in j and Q a speci…ed set

Restriction QI restricts the J conditional quantiles of U given Z = z speci…ed by to
be invariant with respect to z. Q is a known set of possible values for these conditional
quantiles. For example, a conditional median restriction corresponds to = f0:5g and theusual normalization that this conditional median is zero is then captured by setting Q = f0g.
In this case J = 1. However, Restriction QI allows one to restrict the conditional distributionsof U to be invariant at multiple quantiles. For instance, specifying= f0:25; 0:5; 0:75g
and Q = ffq1 ; q2 ; q3 g 2 R3 : q2 = 0g constitutes a conditional quantile restriction at J = 3
quantiles, with the 0:5 quantile set to zero a typical location normalization. In many cases
all but one (normalized) value qj will be unrestricted in which case they can either be added
to the list of unknown model parameters or treated as nuisance parameters.
Sharp characterization of the identi…ed set of structures will require consideration of all
test sets comprising intervals of the form ( 1; qj ] for all j = 1; :::; J and [qj ; qj+1 ] for all
j = 0; :::; J where q0 = 1 and qJ+1 1.
Proposition 4 Let Restriction QI hold. Then the identi…ed set of structral functions delivered by the IV Tobit Model is the set of functions m 2 M such that for some fq1 ; :::; qJ g 2 Q:
(1) 8j 2 J C (z; qj ; m)(z; qj ; qj+1 ; m)j , and (2) 8j 2 f0; 1; :::; Jgj+1j.
~Note that if G(tjz;m) and B(z; t; m) are di¤erentiable with respect to t then the …nal
condition here is equivalent tog~(tjz; m) b(z; t; m)holding for all t and z whereg~(tjz; m)~rt G(tjz;m)b(z; t; m)rt B(z; t; m):

In the application to tobacco expenditure shares in Section 5.2 multiple quantile restrictions are considered as a means of relaxing the restriction that U have a Gaussian distribution
independent of Z.
In numerical illustrations considered now Restriction QI is imposed encompassing a single
median independence restriction such that J = 1 and 1 = 0:5 with the normalization q1 = 0.
Thus Q ={0} andQ = f( 1; 0); (0; 1))g ,17and for all z 2 RZ
GU jZ (( 1; 0]jz) = 0:5,
GU jZ ([0; 1)jz) = 0:5,
leading respectively to the inequality from (6)P[Y1 = 0 ^ 0m(Y2 ; Z; 0)jZ = z] + P[Y1 > 0 ^ Y1

m(Y2 ; Z; 0)jZ = z]

0:5,

and the inequality from (8)P[Y1 = 0jz] + P[0 < Y1

m(Y2 ; Z; 0)jZ = z]

0:5

which together deliver the identi…ed set of structural functions, m, under the median independence restriction, as follows.
fm : P[Y1 = 0 ^ 0m(Y2 ; Z; 0)jZ = z] + P[Y1 > 0 ^ Y1

m(Y2 ; Z; 0)jZ = z]

0:5
P[Y1 = 0jz] + P[0 < Y1

m(Y2 ; Z; 0)jZ = z] a.e. z 2 RZ g.

In the case in which m(y2 ; z; u) is linear and equal to y2 + + u the identi…ed set of valuesof and is as follows.
f( ; ) : P[Y1 = 0 ^ 0

Y2 + jZ = z] + P[Y1 > 0 ^ Y1

Y2 + jZ = z]

0:5
P[Y1 = 0jz] + P[0 < Y1

Y2 + jZ = z] a.e. z 2 RZ g.

Identi…ed sets of values of and under a zero median independence restriction QI are
calculated and displayed for each of the lists of parameter values shown in the columns ofTable 1 relating to Structure 1.
In each case the identi…ed set obtained under a zero conditional mean independence
restriction is shown as the region below all the blue drawn lines and the value of and in
the structure generating the probabilities is the green plotted point.
Figure 2 shows the results using the parameter values in the column Case 1. These are
the parameter values that deliver the results on the mean independence restriction shown in
18

Figure 1. The identi…ed set obtained in this Case under the median independence restrictionis the extremely small pink …lled region. At the parameter values of Case 1 the median
independence restriction is very powerful.
Figure 3 shows the results using the parameter values in the column Case 2 of Table 1. In
this case the instrumental variable has lower predictive power for endogenous Y2 because d1
is closer to zero. The pink …lled region is the identi…ed set under the median independence
restriction, much larger than in Case 1 but still very informative compared with the set
obtained under the conditional mean independence restriction. In Case 3, shown in Figure
4 the identi…ed set under the median independence restriction is larger still, containing
unboundedly large positive values of and unboundedly large negative values of , but it
remains informative relative to the identi…ed set obtained under the mean independence
restriction.
Figure 5 shows identi…ed sets under mean independence and quantile independence restrictions using probabilities delivered by Structure 2 with the parameter values shown in
the …nal column of Table 1. Also shown in Figure 5 is the projection onto the space of ( ; )of the identi…ed set for ( ; ; ) obtained under a parametric Gaussian speci…cation of the
distribution of U in which the variance parameter is 2 . This is a bounded set unlike thesets obtained at these parameter values under mean or quantile independence restrictions.
The median independence restriction delivers substantially smaller identi…ed sets thanthe mean independence restriction in the cases studied and the identi…ed sets it delivers canbe very small indeed.

4.3

Stochastic independence

Now consider a restriction requiring U and Z to be independently distributed but with noparametric speci…cation of the distribution of U .
RESTRICTION NPSI - Nonparametric Stochastic Independence: The randomvariables U and Z are independently distributed and GU jZ is the singleton set fGU g.
Recall the characterization of identi…ed sets using singly in…nite collections of inequalities
given in (3) and (10), repeated here for convenience with the restriction NPSI imposed. The
identi…ed set of structures, I(FY jZ ; RZ ; A), is the intersection of these two sets.
I(FY jZ ; RZ ; A) = I1 (FY jZ ; RZ ; A) \ I2 (FY jZ ; RZ ; A)

19

(15)

I1 (FY jZ ; RZ ; A) =I2 (FY jZ ; RZ ; A) =eU (t)(h; GU jZ ) 2 MA : 8t 2 <; G(h; GU jZ ) 2 MA : 8t 2 <; geU (t)sup C(z; t; m)

(16)

z2RZ

sup b(z; t; m)

(17)

z2RZ

Here geU (t) is the probability density function of U , the …rst derivative of the distributioneU (t) GU ([ 1; t]), and b(z; t; m) = rt B(z; t; m) where
function G
B(z; t; m)P[Y1 > 0 ^ Y1

m(Y2 ; Z; t)jz]

andC(z; t; m) = P[Y1 = 0 ^ 0m(Y2 ; Z; t)jz] + P[Y1 > 0 ^ Y1

m(Y2 ; Z; t)jz]:

In the linear case used in the illustrations in which there is no exogenous variable in the
structural equation, these functions are as follows.
B(z; t; )C(z; t; )P[Y1 = 0 ^ 0P[Y1 > 0 ^ Y1
Y2 +

Y2 +

+ tjz]+ tjz] + P[Y1 > 0 ^ Y1

Y2 ++ tjz]:

Rather than m we use = ( ; ) as an argument of these functions in this linear case.
Absent additional restrictions on the distribution of U the distribution of observable variables
contains no information about the value of the constant term, , so, in determining theidenti…ed set of values of , can be set equal to an arbitrary value. It is set to zero in the
numerical illustrations.
We now develop a method for calculating an outer set for the identi…ed set of structures
onto the space of structural functions under the NPSI restriction. For this purpose partition
the support of U into N intervals: ( 1; t1 ], (t1 ; t2 ], : : : ; (tN 1 ; 1) where N is large. For
each n = 1; :::; N de…nepn GU ((tn 1 ; tn ]) ,(18)
where it is understood that (t0 ; t1 ] means ( 1; t1 ] and (tN 1 ; tN ] means (tN 1 ; 1).
It follows from Proposition 1 that for any structural function m for which there exists
a distribution GU such that (m; gU ) is in the identi…ed set, there must exist probabilities20p1 ; : : : ; pN each nonnegative and summing to one such that
8n = 1; :::; N :8n = 1; :::; N :

pn

n
X

pi

(19)

C(z; tn ; m),i=1

B(z; tn ; m)(20)

B(z; tn 1 ; m),for almost every z 2 RZ . Let IM FY jZ ; RZ ; A; T denote the set of admissible m that satisfy these inequalities applied to T ft0 ; t1 ; :::; tN g. Inequalities of the form (19) correspondto those de…ning I1 (FY jZ ; RZ ; A) in the statement of Proposition 1 with t = tn . Inequalitiesof the form (20) are those characterizing I1 (FY jZ ; RZ ; A) with tn 1 and tn in place of t1
and t2 , respectively.12 Indeed, Proposition 1 implies that the identi…ed set for m are thoseadmissible structural functions that satisfy these inequalities for all tn and tn 1 for some possible distribution GU , with GU ((tn 1 ; tn ]) replacing pn , almost surely. Indeed, it is precisely
the use of only a …nite set of intervals (tn 1 ; tn ] that makes the resulting characterization
nonsharp.
For any such sequence of N intervals the conditions determining when structural function
m is in IM FY jZ ; RZ ; A; T can be checked by solving a linear program. We set out how
this can be done using the method developed in Theorem 4 of Chesher and Rosen (2020a)
in the study of interdependent determination of discrete outcomes. De…ne the following
quantities.13
A

1 N

2

12

[1; 1; : : : ; 1]
| {z }
N times

3

B
6 . 7
. 7
B
=6
4 . 5
R(2N 1) N
B

x

N 1

[p1 ; : : : ; pN ]023c(z1 ; m)
67..
7c(m) = 6.
4
5
R(2N 1) 1c(zR ; m)b=1

R = ](RZ )

(21)

(22)

These are equivalently conditional containment probabilities applied to intervals of the form ( 1; tn ]and (tn 1 ; tn ], respectively.
13This is set out in the notation in Chesher and Rosen (2020a) which employs matrix B, here denoted B,
not to be confused with the function B(z; t; m).

21

2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4

1
0
..
.

0
1

0
0
..
.

3
0
7
0 7
.. 7
. 7
7
7
7
7
7
0 7
7
1 7
7
7
0 7
7
0 7
7
.. 7
. 7
7
7
0 5
0

2

B(z; t0 ; m)B(z; t1 ; m)B(z; t1 ; m)B(z; t2 ; m)

3

6
7
6
7
6
7
.
..
6
7
.
.
.
6
7
6
7
..
..
6
7
.
.
6
7
6
7
6 B(z; tN 2 ; m) B(z; tN 1 ; m) 7
0 0
1
6
7
B
=
c(z; m) = 60 00B(z; tN 1 ; m) B(z; tN ; m) 76
7
(2N 1) N(2N 1) 1
6
7
6
7
1 0 0
0
C(z;
t
1 ; m)
6
7
6
7
1 1 0
0
C(z; t2 ; m)
6
7
6
7
.. ..
..
.
..
6
7
. .
.
6
7
6
7
1 1 1
0
C(z; tN 2 ; m)
4
5
1 1 1
1
C(z; tN 1 ; m)(23)
Application of Theorem 4 of Chesher and Rosen (2020a) to the characterization of IM FY jZ ; RZ ; T
given by (19) and (20) then yields the following result.
Proposition 5 Let T ft0 ; t1 ; :::; tN g be an increasing sequence of scalars with tN = t0 =1. In an IV Tobit Model A restricting structural function m to the set M0A and in which
Restriction NPSI holds, the set
IM FY jZ ; RZ ; A; T = m 2 M0A : v (m)0comprises bounds on m, where v (m) is the value attained by the linear program
(24)

min v
s;t;v

subject to the constraintssA + tB
t

0;

(25)
(26)

0

and
s + t c(m)where s 2 R1 , t 2 RK , v 2 R, and K = R(2N
14

v,

(27)

1).14

The proof of this proposition given in Chesher and Rosen (2020a) makes use of a version of Farkas’
Alternative. Again we stay with the notation in Chesher and Rosen (2020a) which employs a decisionvariable t in R2, not to be confused with tn used to signify boundaries of intervals that partition the support

22

The linear program of Proposition 5 is easy to check for any given structural function mgiven a choice of points T de…ning a partition of the real line. The outer set approximation
to the identi…ed set of structural functions can be made closer to the identi…ed set by …ner
choice of T . A large number of points t1 ; :::; tN can be used because linear programs can besolved very quickly on modern computers. The outer set further has the interpretation of the
sharp identi…ed set under an additional restriction that the distribution G has a piecewise
constant density on each of the N intervals de…ned by successive points in T .
This procedure has been implemented using probabilities generated by the Gaussian
triangular structures 1 and 2 in which the structural function m is parametrically speci…edand indexed by parameter .15 The results obtained using Structures 1 and 2 are shown inrespectively Table 2 and Table 3. In all cases the value of is normalized to zero16 since
the location of GU is unrestricted. The tables show parameter values of structures used togenerate probabilities in these calculations, N , the number of intervals on the support of U ,and the projection of the identi…ed set onto the space of which is an interval [ ; ]. The
set is an outer region because we approximate the distribution of U using a …nite numberof intervals. In addition the minimum and maximum conditional censoring probabilities
p0 (z)P [Y = 0jZ = z] with respect to z 2 RZ are provided, as well as the marginal
probability p0 P [Y = 0] when Z is distributed with equal mass on each point of RZ .
These projections are, to the accuracy obtained in the calculations, population values of[ ; ], not estimates. This is so because the calculations employ probabilities delivered by
the structures, not estimates of the probabilities. When applying this method to data, notdone here, it will be necessary to account for sampling variation to perform inference. Thisis the topic of research in progress.
4.3.1

Structure 1

First consider the results shown in Table 2 using probabilities generated by Structure 1
which is a triangular Gaussian structure with Y2 a linear function of Z and U2 and sounbounded below and above. The support of Z is bounded and the model employed imposes
the restriction that U and Z are stochastically independent.
of U .
15
Linear programs are solved using R’s (R Core Team (2020)) package lpSolveAPI (Konis andSchwendinger (2020)). Projections onto the space of are calculated using R’s uniroot function. Someof the programs are large. In Case 6 (see Table 2) there are 21 (2 1000 1) + 2 = 41977 decision variables
and 1001 constraints. No calculation takes longer than a few minutes to run on an iMac with a 4.2GHz Intel
Core i7 processor and 32 GB memory.
16
This is a normalization - any value of can be used.

23

Table 2: Projections of the identi…ed set onto the space ofunder a stochastic independence restriction and a nonparametric speci…cation of the distribution of U with thenumber of intervals N = 100. Structure 1. Censoring probabilities p`0 , p0 , and pu0 denotemin P [Y = 0jZ = z], P [Y = 0], and max P [Y = 0jZ = z], respectively.
z2RZ

z2RZ

Structure 1 parameter valuesba d0 d1 s11 s12 s22
0:0 1 0 1
1
0:5 1
2:0 1 0 1
1
0:5 1
1:0 1 0 1
1
0:5 1
1:0 1 0 2
1
0:5 1
1:0 1 0 1:5 1
0:5 1
1:5 1 0 1:5 1
0:5 1

Case
4
5
6
7
8
9

Censoring Probabilities
p`0
p0
pu0
0:28 0:50
0:72
0:04 0:14
0:28
0:50 0:71
0:88
0:28 0:68
0:96
0:39 0:69
0:93
0:50 0:78
0:96

Interval for
0:93
0:99
0:84
0:97
0:91
0:86

1:01
1:00
1
1:00
1:02
1

Table 3: Projections of the identi…ed set onto the space ofunder a stochastic independence restriction and a nonparametric speci…cation of the distribution of U with thenumber of intervals N = 50. Structure 2. Censoring probabilities p`0 , p0 , and pu0 denotemin P [Y = 0jZ = z], P [Y = 0], and max P [Y = 0jZ = z], respectively.
z2RZ

Case
1
2
3
4
5

z2RZ

b
0
10
0
0
0

Structure 2 parameter values
Censoring Probabilities
a k d0 d1 s11 s12 s22
p`0
p0
pu0
1 1 0 1 10:5 1
0:34 0:54
0:76
1 1 0 1 10:5 1
0:00 0:02
0:09
1 10 0 1 1
0:5 1
0:00 0:03
0:11
1 1 0 5 10:5 1
0:16 0:57
1:00
1 1 0 1 1
0:0 0:1
0:27 0:55
0:92

24

Interval for
0:38
0:98
0:98
1:00
1:00

1:77
1:00
1:00
1:00
1:00

In Case 4 the interval for is short. Increasing N to 1000 reduces the interval but only
slightly to [0:96; 1:00]. Case 5 has a large intercept (b = 2) in the structural equation for Y1
so the marginal probability of censoring is small (0:14). For some values of Z the conditional
censoring probability p0 (z) is indeed quite small, with a minimum of 0:04. For values of Z forwhich the conditional censoring probability is very small, the distribution of Y given Z = z
will be close to that produced by a linear IV model which is point identifying under a rank
condition. Thus, while the identi…ed set for depends on the entire collection of conditional
distributions FY jZ , values of Z that produce small conditional censoring probabilities shouldbe helpful in achieving narrow intervals for . We see that in Case 5 the identi…ed interval
for is indeed extremely short. By contrast in Case 6 the low value of the intercept (b = 1)
causes the probability of censoring to be high (0:71). The high intercept similarly induceshigher conditional censoring probabilities than for Case 5, ranging from 0:50 to 0:88, and we
observe that the identi…ed interval is unbounded above.
In Case 7 the intercept is held at the Case 6 value but the e¤ect of the instrument on
endogenous Y2 is much larger (d1 = 2). In this case the conditional censoring probability
P [Y1 = 0jZ = z] ranges from 0:28 to 0:96. Even though the marginal probability of censoring
remains high (0:68), the identi…ed interval is very short. In Case 8 the coe¢ cient on the
instrument in the equation for Y2 is reduced from 2:0 to 1:5 and the identi…ed interval
lengthens slightly, but when the intercept is further reduced to 1:5 (Case 9) causing theprobability of censoring to rise to 0:78 the identi…ed interval for becomes unbounded above.
4.3.2

Structure 2

Now consider the results obtained using probabilities generated by Structure 2 which is atriangular Gaussian structure with Y2 a nonlinear function of Z and U2 , unbounded below
and bounded above at the value of the parameter k.
The parameter values in Case 1 are the values delivering the identi…ed sets of valuesof ( ; ) drawn in Figure 5 for mean and quantile independence restrictions and for stochastic independence with U restricted Gaussian. The identi…ed interval for is [0:38; 1:77],
which is bounded unlike the sets obtained under mean and median independence restrictions.
Increasing the number of intervals to N = 100 has no e¤ect on the interval.
The intercept is increased in Case 2, from b = 0 to b = 10, reducing the conditional
censoring probabilities to all lie between 0:00 and 0:09. This causes the length of the identi…ed
interval to become close to zero. The result is similar on returning the intercept to b = 0
and increasing the value of k to 10 as in Case 3. In Cases 4 and 5 it remains that b = 0
25

and we observe the e¤ects of (i) increasing the coe¢ cient on Z in the equation for Y2 to
d1 = 5, and (ii) reducing the variance of the unobserved variable in the equation for Y2 froms22 = 1 to s22 = 0:1, respectively. In these last two cases the marginal censoring probability
is relatively high 0:54, but the instrument has great predictive power for Y2 . In these cases
the minimum censoring probabilities p`0 are small and indeed the identi…ed interval is very
short, nearly a point.

4.4

Parametric restrictions - the Gaussian IV Tobit model

Finally, before coming to our empirical application, we consider models in which a stochastic
independence restriction is imposed and, additionally, the distribution of U is restricted tobelong to a parametric family of distributions. In the examples here a Gaussian distribution
is employed as in the classic Tobit model.
RESTRICTION GaussSI - Stochastic Independence - Gaussian U: The randomvariables U and Z are independently distributed and GU jZ is the singleton set fGU g where
GU is the N (0; 2 ) distribution.
eU ( )In these illustrations we use the characterization given in (3), (10) and (15) with G
and geU ( ) respectively the distribution and density function of a N (0; 2 ) random variable.
For Structures 1 and 2 at selected values of the parameters = ( ; ; ) we calculate thevalue of the derivative b(z; t; ) and the probability C(z; t; ) and examine the inequalities
over a long sequence of values of t pronouncing the value of in the identi…ed set if none of
the inequalities are violated. The sequences of values of t aref +1(") : " 2 f1=n; : : : ; (n1)=ngg

(28)

1
where
is the standard normal quantile function. In the illustrations we set = 0, = 2
and n = 500.17
For the parameter values in Case 4 of Structure 1 shown in Table 2 the stochastic independence model is not point identifying but the Gaussian model is. We determine this byrandomly sampling18 points uniformly distributed on a sphere with small radius r centeredat the probability generating value of , namely (0; 1; 1). When the radius of the sphere isas small as 0:001 we …nd no points on the sphere that lie inside the identi…ed set, which we
17

Smaller values of n are used to determine the rough location and extent of an identi…ed set and thenresults are re…ned using n = 500.
18
We use the function runif_on_sphere available in the R package uniformly, Laurant (2018).

26

take to indicate point identi…cation bearing in mind the slight inaccuracies arising because
we are conducting …nite precision arithmetic and in particular calculating some probabilities
using numerical integration routines. The situation is the same if the value of the intercept,
b, in Structure 1 is reduced to 0:5. However when bis reduced further to 1 (Case 6 inTable 2) there is no longer point identi…cation under the Gaussian restriction.
Figure 6 shows projections of the identi…ed set for ( ; ; ) onto the space of each pairof parameters in turn for Structure 2 in which Y2 is bounded above. The solid …lled regions
are convex hulls of points found lying in the projections.19 Parameter values for this caseare those in Case 1 of Table 3.20
Figure 5 shows identi…ed sets for ( ; ) under mean and median independence restrictions
at these parameter values and additionally the projection onto the space of ( ; ) of theidenti…ed set for ( ; ; ) obtained under the Gaussian stochastic independence restriction(…lled in magenta).21 The identi…ed interval for under stochastic independence absent
the Gaussian restriction, maintaining the stochastic independence condition is [0:38; 1:77] see Row 1 of Table 3. On additionally imposing the Gaussian restriction this is reduced to[0:47; 1:42] as shown in Figure 5.
Calculations with both structures show that the IV Tobit model can be e¤ectively point
identifying and that when it is not the model can still be highly informative about the valuesof structural parameters. We now turn to an application involving real data and consider
issues of estimation and inference.

5

Implementation

This section …nishes with an application to a Tobit model of tobacco expenditure using UK
survey data from the period 2000-2009. First we set out the method employed to calculate
con…dence regions on projections of the identi…ed sets and we explain how implementation
is done when, as in the application, exogenous variables have rich support.
19
There is no evidence to suggest the projections are not convex. The parameter values whose membershipof the identi…ed set was evaluated were obtained in a trail and error process by randomly sampling points inspheres of varying radii centred on points lying centrally in the identi…ed set. The function runif_in_spherein R’s uniformly package was employed.
20
Also shown in the …nal column of Table 1.
21
This is the projection shown in the ( ; ) and ( ; ) panes of Figure 6

27

5.1

Inference

We employ a procedure proposed in Belloni et al. (2018) (BBC18) to calculate 95% con…dence regions for individual parameter components of partially identi…ed parameter vectors.
We use a self-normalized critical value, shown to be asymptotically valid in Chernozhukov
et al. (2019) (CCK19). Belloni et al. (2018) additionally provide theoretical justi…cation foralternative critical values using a bootstrap procedure that can further re…ne these con…dence
sets. We employ the self-normalized critical value for its computational simplicity.
In applications there will typically be many exogenous variables, Z, and some of these
may be continuous. In this circumstance it is hard to make progress using conditional
moment inequalities in which conditioning is on Z taking singleton values.22 Instead we
conduct inference on outer regions obtained from moment inequalities in which conditioning
is on Z taking a value in a set of values, Z.
In the models that we employ the moment inequalities arising when an interval [t1 ; t2 ] isconsidered arew(t1 ; t2 ; ; z) p(t1 ; t2 ; ; z):(29)Here, employing (9),w(t1 ; t2 ; ; z)1[t1 =

1]

P[Y1 = 0 ^ 0

Y2 + Z + t2 )jZ = z]+

P[Y1 > 0 ^ Y2 + Z + t1

Y1

Y2 + Z + t2 jZ = z]andp(t1 ; t2 ; ; z)

GU jZ ([t1 ; t2 ]jz]

is the probability mass placed on the interval [t1 ; t2 ] by the distribution of U given Z = z
admitted by the model. This may depend on the value of the parameter .
We estimate using the GaussSI model for whicht2

p(t1 ; t2 ; ; z) =t1

and we also estimate using two models in which the Gaussian restriction is dropped andquantile independence is imposed at 3, 5 or 7 quantiles associated with selected probabili22

When Z has rich support it will be di¢ cult to obtain accurate estimates of conditional probabilities. Kernel or sieve estimation would lead to estimated moment functions that are not simple means of contributions
which is required when the BBC18 procedure is used.

28

ties. In these two cases, as set out at the start of Section 4.2, t1 and t2 are unknown valuesof quantiles at the selected speci…ed probabilities and p(t1 ; t2 ; ; z) is the di¤erence between
those probabilities, independent of z. In these two cases the unknown values of the quantiles are elements of the parameter vector . In all the cases considered in the applicationp(t1 ; t2 ; ; z) does not depend on z which we make explicit now by writing the probability asp(t1 ; t2 ; ).
Let the distribution function of Z be FZ (z). If for some value of , t1 and t2 the moment
conditions (29) hold for all z there is, for all sets Z RZ
Z

w(t1 ; t2 ; ; z)dFZ (z)p(t1 ; t2 ; )Z

Z

dFZ (z)Z

and thus the moment conditions imply that, withw(t~ 1 ; t2 ; ; Z)1[t1 =

1]

P[Y1 = 0 ^ 0

Y2 + Z + t2 ) ^ Z 2 Z]+P[Y1 > 0 ^ Y2 + Z + t1

Y2 + Z + t2 ^ Z 2 Z]

Y1

the inequalityw(t~ 1 ; t2 ; ; Z)p(t1 ; t2 ; )P[Z 2 Z]

holds for that and and the interval [t1 ; t2 ] and for all sets Z RZ .
Introducing the notation employed in BBC18, let the functions of moments required to
be nonpositive at a parameter value in an identi…ed set be denoted mj ( ), j 2 J .
k(j) k(j)The test set employed in constructing mj ( ) is the interval [t1 ; t2 ] where fk(j) : j 2
J g is a list of indexes identifying the endpoints of intervals and a set of values of Z, Z l(j) ,is employed where fl(j) : j 2 J :g is a list of indexes identifying sets Z l(j) RZ .23
A moment function mj ( ) therefore has the following form.
hk(j)mj ( ) = E 1[t1 =

1]

k(j)

1[Y1 = 0 ^ 0k(j)1[Y1 > 0 ^ Y2 + Z + t1

Y2 + Z + t2 ) ^ Z 2 Zl(j) ]+
Y1

k(j)

Y 2 + Z + t2

^ Z 2 Zl(j) ]p(t1 ; t2 ; )1[Z 2 Zl(j) ]23

When selected quantile independence is imposed the unique unknown elements in ftk(j) gJj=1 are parameters, elements of .

29

With data f(Y1i ; Y2i ; Zi ) : i 2 f1; : : : ; N gg de…ne the estimatorm^ j( )N

1

N
X

mji ( )

i=1

where the contributions, mji ( ), are obtained by replacing Y1 , Y2 , and Z in the expression
whose expectation appears in mj ( ) by realized values, respectively Y1i , Y2i , and Zi De…nean estimator of the variance of N 1=2 m^ j( )^ 2j (

)

1

N

N
X

(mij ( )m^ j ( ))2 :i=1

Using the self-normalization-based critical value given in BBC18 and CCK19 the 100(1)% con…dence region for the projection of the identi…ed set onto the space of an elementisk ofCI( k ; ) fr : Tn;k (r) cN (J; )g
(30)
where
Tn;k (r)infmax

f : k =rg j2f1;:::Jg

N 1=2

m
^ j( )^j ( ),(31)

and where cN (J; ) is the critical value:1cN (J; )

p
1

(1
1 (1

=J)
=J)2 =N:with denoting the standard Gaussian cumulative distribution function.
In implementation we obtain sets Z by specifying intervals for each exogenous variable
and the connected unions of those intervals. The sets Z that we employ are the acrossvariable intersections of these connected unions. An example is given in the next section.
In the models that restrict quantile independence at speci…ed quantile probabilities thetest sets comprise all intervals with endpoints selected from the set of (unknown) values of
the quantiles augmented with plus and minus in…nity.
In models that impose the GaussSI restriction a collection of test sets (intervals) is obtained by choosing a large positive integer n and de…ning a sequence of n values:
W(n; ; )f +1(") : " 2 f0; 1= (n301) ; : : : ; (n2)= (n1) ; 1gg

1
where
is the standard normal quantile function. The values of (n; ; ) are chosen todeliver good coverage of the main part of the support of U .24 The collection of test sets isf[t1 ; t2 ] : t1 < t2 ^ t1 2 W(n; ; ) ^ t2 2 W(n; ; )g:

5.2

(32)

Application: tobacco expenditure share

In this section, inspired by Adams et al. (2019) (ABBC), we present con…dence regions andestimates of parameters of models for the share of household nondurable expenditure spent
on tobacco.
ABBC estimate a Tobit model for tobacco expenditures as a share of total nondurable
expenditure with explanatory variables log total expenditure on nondurables (potentiallyendogenous), an OECD equivalence scale25 , with log household disposable income as an
excluded instrumental variable.
The data used in ABBC come from the UK Family Expenditure Survey (FES) 1980-2000
and are a sample of households with head of household aged 25-35 years in 1980. We do
not aim to reproduce their analysis, and while we use the same explanatory variables anddata source, aiming for reasonably large samples we focus on households in the FES and its
successor surveys26 in 2000-09 with head of household aged 25-60 at the time of observation.27
We conduct separate analysis of the periods 2000-04 and 2005-09 in which respectively 68%
and 74% of households record zero tobacco expenditures.
ABBC take what is described as a quantile control function approach which employs amodel in which nondurable expenditure is exogenous when there is conditioning on a control
function which depends on nondurable expenditures, log household disposable income andthe equivalence scale.28 This control function restriction can arise in a complete triangular model in which there is a structural equation for log nondurable expenditure with, as
explanatory variables, log disposable household income and the equivalence scale, and an
unobserved variable which along with the unobserved variable in the structural equation for
the tobacco expenditure share is jointly independent of the exogenous variables.
24

In the application U is speci…ed N (0; 1) so this is easily done.
The OECD equivalence scale is 1+0.7 the number of adults in excess of one + 0.5* the number ofchildren.
26
These are, from 2001, the Expenditure and Food Survey and from 2008, the Living Costs and Food
Survey.
27
We exclude households comprising more than one tax unit, all of which have one or two adult members,
and households with disposable weekly income recorded as £ 20 or less.
28
Expenditures and income are recorded in UK pounds per week.
25

31

Here we employ the incomplete, single equation, IV model introduced in this paper. The
model has the structural equation, as in ABBC,
Y1 = max(0;

0+

Y2 +

1 Z1 +

U1 )where Y1 denotes tobacco expenditure share, Y2 denotes log nondurable expenditure andthere are exogenous variables Z1 (the OECD equivalence scale), and Z2 (log household disposable income), the latter excluded from the structural equation for the tobacco expenditure
share. The model places no restriction on the process delivering Y2 . The partial correlation
between log nondurable expenditure and log disposable household income given the OECDequivalence scale, rY2 Z2 :Z1 , is 0:64 in 2000-04 and 0:62 in 2005-09.
We calculate con…dence regions for the value of using the method set out in Section
5.1.29 We …rst consider a model requiring U1 to have a Gaussian distribution independent of
Z = (Z1 ; Z2 ) and then drop the Gaussian restriction and turn to models imposing conditional
quantile independence restrictions.
We compare with estimates of obtained using a classical Tobit model making no allowance for endogeneity and with estimates of obtained using a complete, point identifying,
triangular model in which there is the additional structural equation
Y2 =

0+

1 Z1 +

2 Z2 + U2

and the restriction that (U1 ; U2 ) have a Gaussian distribution independent of Z1 and Z2 .
In the calculations using the IV model, sets of values of the exogenous variables are
obtained as follows. Lists of intervals of values of exogenous variables are constructed: for
Z1 , the OECD equivalence scale,2
3
1:0 1:0
6
7
4 1:5 2:7 5
3:0 5:7
29

Notice that in the equation for Y1 the parameter multiplies U1 which is restricted to be N (0; 1) when
the GaussSI restriction is imposed. By this device we are able to de…ne intervals [t1 ; t2 ] that span the e¤ective
range of U1 at all values of the parameters.

32and for Z2 , log disposable household income,
2

3

Q(0)
Q(0:125)
67
6 Q(0:125) Q(0:250) 7
6
7
6
7
..
..
6
7
.
.
6
7
6 Q(0:750) Q(0:875) 7
4
5
Q(0:875)Q(1)where Q(p) is the sample p-quantile in the data under consideration. For each of Z1 and Z2
in turn a list of sets (denoted respectively Z1 and Z2 ) containing all connected unions of theintervals is constructed. The sets employed in the calculations are the sets in the collection
ffZ : Z1 2 Z1 ^ Z2 2 Z2 g : Z1 2 Z1 ; Z2 2 Z2 g:
In the calculations reported here there are 216 such sets.
In analysis using the Gaussian model the test sets (intervals) employed comprise allintervals with endpoints chosen from the list: ( 1; 1:35; 0; 1:35; 1), excluding the interval( 1; 1). There are 9 such intervals and combining test sets and Z sets leads to 1; 944
moment inequalities.
For each of the samples from the periods 2000-04 and 2005-09, respectively, there is novalue of found that satis…es all 1944 sample moment inequalities. Equivalently, there isno value of such that Tn ( ) = 0.30 Given statistical uncertainty due to sampling variationand the large number of moment inequalities, this is unsurprising. We thus report both halfmedian-unbiased set estimates and 95% con…dence intervals for using the inference method
described in Section 5.1. The half-median-unbiased set estimates employ a conservative
median-bias correction such that the upper (lower) endpoint of the interval is lower than(greater than) the upper (lower) bound of the population bounds for with probability nogreater than 1=2 asymptotically. These correspond to 50% con…dence intervals, as advocatedby Chernozhukov et al. (2013).
Results using the GaussSI restriction are shown in Table 4. For comparison results using
two point identifying models are also provided. These are (1) maximum likelihood (ML)estimates of and estimated standard errors obtained using a simple Gaussian Tobit model,
30

Relative to equation (31) the second subscript k indexing the parameter component in Tn;k ( ) has been
dropped in a slight abuse of notation. The parameter component under consideration is to be understood
to be throughout.

33

Table 4: Estimates and con…dence intervals for the coe¢ cient on log total expenditure in
determination of tobacco expenditure share. IV estimates are half-median-unbiased interval
estimates. Triangular and Tobit results obtained by maximum likelihood.
Years

N

2000-04 18473

2005-09 15885

%0
68

74

estimate
std err
95% CI
estimate
std err
95% CIIV model GaussSI[ 0:199; 0:117]

Triangular model
0:130(0.0043)(0.0027)[ 0:211; 0:114][ 0:299; 0:103][ 0:139; 0:122]
0:121[ 0:053; 0:043]
0:043(0.0046)(0.0028)[ 0:130; 0:112][ 0:048; 0:038][ 0:339; 0:096]

Tobit model
0:048

which makes no allowance for endogeneity of nondurable expenditure, and (2) ML estimatesof obtained using the complete two equation triangular model with estimated standard
errors and 95% con…dence intervals.31
The Tobit estimates of obtained without accounting for endogeneity are slightly lessthan one third of the magnitude of the estimates obtained using the complete triangularmodel. The hypothesis of exogeneity of log nondurable expenditure is soundly rejected withtests delivering p-values less than 0:0001.
The incomplete IV model delivers somewhat wider con…dence regions than the complete
triangular model. This is to be expected as the IV model uses only a subset of the restrictions
used in the triangular model. The IV model estimates are robust to failure of the additional
restrictions embodied in the complete triangular model and the IV model delivers encouragingly informative con…dence regions. In both periods the 95% con…dence regions deliveredby the complete triangular model are subsets of the 95% con…dence regions produced using
the IV model.
Dropping the Gaussian restriction we imposed conditional quantile independence restrictions, in three distinct cases restricting quantiles of U1 at three quantile probabilities(0:25; 0:5; 0:75), …ve quantile probabilities (0:167; 0:333; 0:5; 0:666; 0:833), and at seven quantile probabilities (0:125; 0:25; 0:375; 0:5; 0:635; 0:75; 0:875), to be independent of the values ofthe exogenous variables, Z, in all cases normalizing the median by setting it equal to zero.
As explained in Section 4.2 this is done by including as unknown parameters the unknown
values of the quantiles at the nominated probabilities.
Restricting quantile independence at J quantile probabilities, p1 ; : : : ; pJ , denote the values
31

These estimates are calculated using the Stata 16 commands tobit and ivtobit, StataCorp (2019). The
triangular two equation model estimated using the ivtobit command restricts the unobservable variablesin the two structural equations to be jointly Gaussian, independent of Z1 and Z2 .

34

Table 5: Half-median-unbiased interval estimates and 95% con…dence intervals for the coef…cient on log total expenditure in determination of tobacco expenditure share under theGaussSI restriction and with quantile independence restrictions at 3, 5 and 7 quantiles.
Years
2000-04
2005-09

estimate
95% CI
estimate
95% CI

GaussSI[[[[

0:199;
0:211;
0:299;
0:339;

0:177]
0:114]
0:103]
0:096]

Quantile independence restricting:
3 quantiles
5 quantiles
7 quantiles[ 0:448; 0:078][ 0:227; 0:093][ 0:198; 0:108][ 0:641; 0:070][ 0:258; 0:091][ 0:229; 0:103]( 1; 0:052]( 1; 0:075][ 0:492; 0:088]( 1; 0:045]( 1; 0:072]( 1; 0:085]

of the quantiles at these probabilities by q1 ; : : : ; qJ . As set out in Proposition 4, the core
determining collection of test sets comprises the collection of intervals:f( 1; qj ];j 2 f1; : : : ; Jgg

together with the collection of intervals
f[qj ; qj+1 ] : j 2 f1; : : : ; J

1gg :In the estimations reported here all connected unions of these intervals were employed as
test sets. We include additional test sets because it is possible that the functions of moments
appearing in the inequalities delivered by some of the unions of core determining sets are
more accurately estimated than the moment functions arising if only core determining test
sets are employed. There is no great computational cost unless J is large and there is likely
improved …nite sample performance.
Placing conditional independence restrictions on 3, 5 and 7 quantiles leads to identi…ed sets de…ned by respectively 1944, 4320 and 7560 moment inequalities. The parameters in the model employing quantile independence restrictions at J quantile probabilities are 0 ; 1 ; ; q1 ; : : : ; q(J+1)=2 1 ; q(J+1)=2+1 ; : : : ; qJ the value of q(J+1)=2 associated withp(J+1)=2+1 = 0:5 being normalized, set equal to zero.
The half-median-unbiased interval estimates and 95% con…dence intervals obtained under
the conditional quantile independence restrictions are shown in Table 5 which, for comparison, also includes the con…dence regions obtained under the Gaussian stochastic independence restriction.
For the period 2000-04 the 95% con…dence regions and the median-bias-corrected set
estimates under conditional independence restrictions on 7 quantiles are only slightly longer
35

then the regions obtained under the GaussSI restriction. Reducing the strength of theconditional quantile independence restriction, restricting just 5 quantiles leads to only amoderate increase in the length of the con…dence regions, and even when independence
restrictions are placed on just 3 quantiles the con…dence regions remain informative. It
seems that, in the 2000-04 period, restricting the distribution of U1 to be Gaussian does not
buy much over and above independence restrictions.
For the period 2005-09 the Gaussian restriction contributes much more. Absent theGaussian restriction all but one of the con…dence regions is unbounded below, but the upperbounds are all negative, so the value of is e¤ectively signed. In this period there are morezero tobacco expenditures and the GaussSI con…dence regions are nearly twice as long as in
2000-04. There is more censoring, and there seems to be more noise in the process in 2005-09
and possibly some drift in the values of parameters.

6

Concluding remarks

When working with censored data and endogenous explanatory variables the easy way toobtain estimates of structural parameters is to employ a complete triangular model like theGaussian model underlying STATA’s ivtobit command32 or to assume directly that a valid
identi…able control function exists. When there is no economic rationale for such restrictions the IV model developed here provides a route to robust estimation. Even when more
restrictive models are thought to be appropriate the IV model can deliver useful information regarding the force of additional restrictions. The IV model can signal misspeci…cationof more restrictive models. It can deliver results when the complete model attack is notavailable, for example when endogenous variables are discrete or are determined by multiple
sources of heterogeneity.
The IV model can be partially or point identifying and it may not be possible to determine identi…cation status in particular applications. So it is important to use methods for
estimation and inference that deliver results regardless of whether there is point or partial
identi…cation, as has been done in the application presented here.
There is rarely a good economic argument for particular parametric restrictions on the
distribution of the unobserved variable U in the structural equation for a censored outcome.
Nevertheless such restrictions are frequently imposed. We have shown how estimation andinference can be done using an IV Tobit model, dispensing with the commonly used Gaussian
32

StataCorp (2019).

36

restriction, using multiple quantile independence restrictions, requiring the p-quantiles of U
given instrumental variables, Z, to be independent of Z at selected quantile probabilities.
The values of quantiles at the selected probabilities become parameters with unknown values
and we calculate con…dence regions on projections of the identi…ed set of values of theaugmented parameter vector onto the spaces of particular parameters of interest. In theapplication the procedure reveals the varying power of the Gaussian restriction.
We have shown how to calculate outer regions for identi…ed sets of structural parameter values and their projections onto the space of individual parameters under a restriction
requiring U and Z to be stochastically independent with no further restriction on the distribution of U . A parameter value lies in the identi…ed set if and only if the solution to alinear program is nonnegative. Although the program can involve a very large number of
inequalities the solution is quick to calculate. Conducting inference using this method isa research challenge not addressed here, and there remain other challenges. For example,
we have proposed and applied a procedure for conducting inference on a partially identi…ed
parameter capturing the marginal e¤ect of an endogenous variable on an outcome of interest when instrumental variables are continuously distributed, in which one calculates joint
probabilities of events de…ned by sets of values of endogenous variables and sets of valuesof instrumental variables. Finite sample performance will of course depend on the sets thatare chosen and future research to help guide these choices is warranted.

References
Adams, A., R. Blundell, M. Browning, and I. Crawford (2019): “Prices versus
Preferences: Taste Change and Revealed Preference,”IFS Working Paper.
Amemiya, T. (1979): “The Estimation of a Simultaneous Equation Tobit Model,”International Economic Review, 20, 169–181.
Belloni, A., F. Bugni, and V. Chernozhukov (2018): “Subvector Inference in PI
Models with with Many Moment Inequalities,”ArXiv:1806.11466.
Blundell, R. and J. L. Powell (2007): “Censored Regression Quantiles with Endogenous Regressors,”Journal of Econometrics, 141, 65–83.
Blundell, R. and R. J. Smith (1989): “Estimation in a Class of Simultaneous Equation
Limited Dependent Variable Models,”Review of Economic Studies, 56, 37–58.
37

Border, K. C. (2020):
2020.10.15::09.50.

“Alternative Linear Inequalities,” CalTech,versionChernozhukov, V., D. Chetverikov, and K. Kato (2019): “Inference on Causal and
Structural Parameters Using Many Moment Inequalities,” Review of Economic Studies,86, 1867–1900.
Chernozhukov, V., I. Fernandez-Val, and A. Kowalski (2015): “Quantile Regression with Censoring and Endogeneity,”Journal of Econometrics, 186, 201–221.
Chernozhukov, V., S. Lee, and A. M. Rosen (2013): “Intersection Bounds: Estimation
and Inference,”Econometrica, 81, 667–737.
Chesher, A. and A. Rosen (2017): “Generalized Instrumental Variable Models,”Econometrica, 85, 959–989.
Chesher, A. and A. M. Rosen (2020a): “Econometric Modeling of Interdependent Discrete Choice,”CeMMAP working paper CWP25/20.
— — — (2020b): “Generalized Instrumental Variable Models, Methods, and Applications,”
in The Handbook of Econometrics, ed. by S. Durlauf, L. P. Hansen, J. J. Heckman, and
R. Matzkin, Elsevier, vol. 7a.
Das, M. (2002): “Estimation and Inference in a Censored Regression Model with Endogenous Covariates,”Columbia Department of Economics Discussion Paper 0102-61.
Frandsen, B. R. (2015): “Treatment E¤ects with Censoring and Endogeneity,” Journalof the American Statistical Association, 110, 1745–1752.
Heckman, J. J. (1978): “Dummy Endogenous Variables in a Simultaneous Equation System,”Econometrica, 46, 931–959.
Hong, H. and E. Tamer (2003): “Inference in Censored Models with Endogenous Regressors,”Econometrica, 71, 905–932.
Khan, S., M. Ponomareva, and E. Tamer (2011): “Sharpness in Randomly CensoredLinear Models,”Economics Letters, 113, 23–25.
Khan, S. and E. Tamer (2009): “Inference on Endogenously Censored Regression Models
Using Conditional Moment Inequalities,”Journal of Econometrics, 152, 104–119.
38

Konis, K. and F. Schwendinger (2020): “lpSolveAPI: R Interface to ’lp_solve’ Version 5.5.2.0. R package version 5.5.2.0-17.7.” https://CRAN.R-project.org/package=
lpSolveAPI.
Lancaster, T. (1985): “Simultaneous Equations Models in Applied Search Theory,”Journal of Econometrics, 28, 113–126.
Lancaster, T. and A. Chesher (1984): “Simultaneous Equations with Endogenous
Hazards,”in Studies in Labor Market Dynamics, Studies in Contemporary Economics, vol11., ed. by G. Neumann and N. Westergå rd Nielsen, Springer, Berlin, Heidelberg.
Manski, C. F. and E. Tamer (2002): “Inference on Regressions with Interval Data on a
Regressor or Outcome,”Econometrica, 70, 519–546.
Molchanov, I. and F. Molinari (2018): Random Sets in Econometrics, Cambridge,
U.K.: Cambridge University Press.
Nelson, F. and L. Olson (1978): “Speci…cation and Estimation of a Simultaneous Equation Model with Limited Dependent Variables,”International Economic Review, 19, 695–
709.
Newey, W. K. (1987): “E¢ cient Estimation of Limited Dependent Variable Models with
Endogenous Explanatory Variables,”Journal of Econometrics, 36, 231–250.
Olsen, R. J. and G. Farkas (1989): “Endogenous Covariates in Duration Models and theE¤ect of Adolescent Childbirth on Schooling,”Journal of Human Resources, 24, 39–53.
R Core Team (2020): R: A Language and Environment for Statistical Computing, RFoundation for Statistical Computing, Vienna, Austria, ISBN 3-900051-07-0.
Smith, R. J. and R. Blundell (1986): “An Exogeneity Test for a Simultaneous Equation
Tobit Model with an Application to Labor Supply,”Econometrica, 54, 679–685.
StataCorp (2019): “Stata Statistical Software: Release 16,” College Station, TX: StataCorp LLC.
Tobin, J. (1958): “Estimation of Relationships for Limited Dependent Variables,” Econometrica, 26, 24–36.

39

Wrenn, D. H., H. A. Klaiber, and D. A. Newburn (2017): “Confronting Price Endogeneity in a Duration Model of Residential Subdivision Development,”Journal of Applied
Econometrics, 32, 661–682.

Appendix: Proofs and Figures
Proof of Proposition 1. Applying Theorem 1, Corollary 1 of CR17 gives the following
characterization of the identi…ed set of structures delivered by a model, A, a collection of
conditional distributions of Y given Z and the support of Z:
I(FY jZ ; RZ ; A) = (m; GU jZ ) 2 MA :8S 2 F (RU ) ; GU jZ (Sjz)

P[U(Y; Z; m)

SjZ = z] a.e. z 2 RZ , (33)where F (RU ) denotes the collection of closed subsets of RU . By Lemma 1 of CR17 it follows that the requirement that the inequality holds for all closed sets S can be replaced by
the requirement that it holds for all S that are unions of sets on the support of U(Y; Z; m)conditional on Z = z. Each such set can be be written as a union of sets of the form ( 1; t]and [t1 ; t2 ], where if t1 = t2 = t, the set [t1 ; t2 ] is simply the point ftg. All such unions arethemselves either of the form S = ( 1; t] or S = [t1 ; t2 ]. The collections I1 (FY jZ ; RZ ; A)
and I2 (FY jZ ; RZ ; A) comprise those structures satisfying GU jZ (Sjz) P[U(Y; Z; m) for eachof these two types of sets S, respectively, completing the proof.
Proof of Proposition 2. Existence of b(z; t; m) rt B(z; t; m) follows from di¤erentiabilityof m(y2 ; z; t) with respect to t and the existence of the density geU jZ ( jz). The inequality
eU jZ (t2 jz)
G

can then be expressed as

Z t2
t1

eU jZ (t1 jz)
G

B(z; t2 ; m)B(z; t1 ; m).

(egU jZ (tjz)b(z; t; m))dt0,which for any z holds for all [t1 ; t2 ]R if and only if for all t, geU jZ (tjz)40b(z; t; m).

Proof of Proposition 3. Theorem 5 of CR17 implies that identi…ed set for structural function m comprises those functions m such that zero is an element of the Aumann expectationof U(Y; Z; m) conditional on Z = z a.e. z 2 RZ . Recall that the residual set in the model
under study is(( 1; m 1 (y2 ; z; 0)] ; y1 = 0
U(y; z; m) =fm 1 (y2 ; z; y1 )g; y1 > 0and let E[Ajz] denote the Aumann expectation of random set A conditional on Z = z.33
There isE[U(Y; Z; m)jz] = E[U(Y; Z; m)jz; Y1 = 0]P [Y1 = 0jz] + E[U(Y; Z; m)jz; Y1 > 0]P [Y1 > 0jz]
where the sum is a Minkowski sum.34 Considering each term in turn there is35
E[U(Y; Z; m)jz; Y1 = 0] = ( 1; E[m 1 (Y2 ; Z; 0)jz; Y1 = 0]]
which is a semi-in…nite interval and there is36
E[U(Y; Z; m)jz; Y1 > 0] = fE[m 1 (Y2 ; Z; Y1 )jz; Y1 > 0]g
which is a singleton. The Minkowski sum of a semi-in…nite interval and a singleton set is asemi-in…nite interval. The result is that the conditional (on Z) Aumann expectation of theresidual set is the semi-in…nite interval
E[U(Y; Z; m)jz] = ( 1; E[m 1 (Y2 ; Z; Y1 )jz]]
which leads directly to the result of the Proposition.
33

The Aumann expectation of a random set A is the set of expected values of all random variables A with…nite expected values having the property that with probability one A 2 A.
34
The Minkowski sum of sets A and B is the set of values obtained by adding each element of A to eachelement of B.
A + B = fa + b : a 2 A; b 2 Bg
35
36

See Example 3.14 in Molchanov and Molinari (2018).
See Example 3.12 in Molchanov and Molinari (2018).

41

Proof of Proposition 4. From Proposition 1 the identi…ed set for m; GU jZ isI(FY jZ ; RZ ; A) = I1 (FY jZ ; RZ ; A) \ I2 (FY jZ ; RZ ; A).
It will be shown that under Restriction QI the identi…ed set for structural function m, whichis the projection of I(FY jZ ; RZ ; A) onto M, here denoted M , is equivalent to the set offunctions m 2 M that satisfy conditions (1) and (2) in the statement of the Proposition.
Suppose …rst that m 2 M such that for some GU jZ satisfying Restriction QI m; GU jZ 2
I(FY jZ ; RZ ; A). Conditions (1) and (2) then hold because these are implied by the inequalities that de…ne the sets I1 (FY jZ ; RZ ; A) and I2 (FY jZ ; RZ ; A), respectively.
Now suppose that m satisi…es conditions (1) and (2) for some fq1 ; :::; qJ g 2 Q. To showthat m 2 M it will be shown by construction that there exists a collection of conditional~ jz; m) = G~ U jZ ( jz; m)distributions GU jZ ( jz; m) with cumulative distribution functions G(
for each z 2 RZ satisfying Restriction QI such that m; GU jZ ( jz; m) 2 I(FY jZ ; RZ ; A).
The inclusion of m in the notation GU jZ ( jz; m) signi…es that the associated collection of
conditional distributions GU jZ ( jz) = GU jZ ( jz; m) in the construction will in general varywith m.
Speci…cally, it needs to be shown that for almost every z 2 RZ there exists a cumulative~ jz; m) such that the following three conditions (34)–(36) hold. For
distribution function G(
all j 2 J :~ j jz; m) = j .
G(q(34)
For all t 2 R:
For all s~G(tjz;m)

C(z; t; m).

(35)t, each in R:~
G(tjz;m)~
G(sjz;
m)(z; s; t; m).

(36)

Condition (34) ensures that Restriction QI holds and conditions (35) and (36) are the conditions de…ning the identi…ed set for (m; GU jZ ), as shown in Proposition 1. Note that for (36)
it is equivalent to show that~G(tjz;m) B(z; t; m)(37)
is weakly increasing in t.
~
Construction of G(tjz;m) : R ! [0; 1] for each z; m is as follows, divided into separate
42

cases according to where argument t lies relative to q1 ; :::; qJ .
1. Case 1: t 2 ( 1; q1 ]. De…ne~G(tjz;m)

C(z; t; m) + ( 1C(z; q1 ; m)) exp ( (tq1 )) ,

> 0 is arbitrary.37 Since lim C(z; t; m) = 0 and lim exp ( (t

where

t! 1

t! 1

q1 )) =~
~ 1 jz; m) = 1 . G(tjz;~
0 it follows that lim G(tjz;m) = 0. There is also G(q
m) is
t! 1
an increasing function of t because it is the sum of two increasing functions of t,~G(tjz;m) C(z; t; m) by de…nition and~G(tjz;
m)

B(z; t; m) = D(z; t; m) + ( 2C(z; q2 ; m)) exp ( (tq2 ))is an increasing function of t because it is the sum of two increasing functions of t.
2. Case 2: t 2 [qj ; qj+1 ], each j = 1; :::; JLj (z; t; m)

1. De…neB(z; t; m) +j

B(z; qj ; m)which is an increasing function of t with Lj (z; t; m) B(z; t; m) constant and Lj (z; qj ; m) =j . Condition (2) ensures
j+1

B(z; qj+1 ; m)from which it follows that Lj (z; qj+1 ; m)Mj (z; t; m)C(z; t; m) + ( j+1jB(z; qj ; m),

j+1 . De…neC(z; qj+1 ; m))This is an increasing function of t with Mj (z; t; m)C(z; qj ; m), and Mj (z; qj+1 ; m) = j+1 . De…ne(t qj ):
(qj+1 qj )C(z; t; m), Mj (z; qj ; m) =~G(tjz;m) = max(Lj (z; t; m); Mj (z; t; m)).
There is~ j jz; m) = max( j ; C(z; qj ; m)) =
G(q
37

j,

~
Construction of G(tjz;m) employing functions other than the exponential function could also be used.

43

and
~ j+1 jz; m) = max(Lj (z; qj+1 ; m);G(qj+1 ) =

j+1 ,because as just shown, Lj (z; qj+1 ; m)
j+1 . This is an increasing function of t in
the interval [qj ; qj+1 ] because it is the maximum of two increasing functions of t, and~~G(tjz;m) C(z; t; m) because G(tjz;m) is the maximum of two functions one of which
is at least equal to C(z; t; m) in the interval under consideration. Moreover,~G(tjz;
m)

B(z; t; m) =maxj

B(z; qj ; m); D(z; t; m) + ( j+1

C(z; qj+1 ; m))(t qj )
(qj+1 qj )which is increasing in t because it is the maximum of two increasing functions of t,
verifying condition (37).
Case 3: t 2 [qJ ; 1). De…ne~G(tjz;m)max (C(z; t; m); B(z; t; m)B(z; qJ ; m) +J) :There is~ J jz; m) = max(C(z; qJ ; m);
G(q

J) =

J,

and then~lim G(tjz;m) = max(1; B(z; 1; m)t!1

B(z; qJ ; m) +J) = 1

because from condition (2)J

B(z; qJ ; m)J+1B(z; qJ+1 ; m)which implies
B(z; qJ+1 ; m)B(z; qJ ; m) +J

J+1 = 1:~
G(tjz;m) is an increasing function of t because it is the maximum of two increasing
functions of t. Finally~G(tjz;
m)

B(z; t; m) = max (D(z; t; m); B(z; qJ ; m) +44

J)

which is an increasing function of t.
We have shown that for any z 2 RZ and any function m satisfying the conditions of the~
Proposition the piecewise function G(tjz;m) de…ned above can be constructed and we haveshown that it satis…es conditions (34), (35) and (36) Therefore any function m satisfying
the conditions of the Proposition is contained in the identi…ed set of structural functionsdelivered by the model.
Proof of Proposition 5. Suppose that m is in the projection of the identi…ed set ofstructures m; GU jZ deliverd by the IV Tobit Model A under consideration. Then under
Restriction NPSI there exists a distribution G such that (m; fGg) 2 I(FY jZ ; RZ ; A) and as
explained in the text it follows from Proposition 1 that (19) and (20) hold with p1 ; :::; pN
as de…ned in (18) as a function of that distribution G. The existence of a vector of proper
probabilities p = (p1 ; :::; pN ) such that (19) and (20) hold almost surely is equivalent to theexistence of p 2 RN satisfying
Ap = b;Bpc (m) ;

p0,

a linear program in p. Then, applying the same steps as in Chesher and Rosen (2020a),and in particular using the version of Farkas’s Alternative provided in Border (2020) –seeparagraph 12, Section 1.4 – such probabilities exist if and only if there is no solution for(s; t), s 2 R, t 2 RK , v 2 R, to the system
sA + tBt

0,0,s + t c(m) < 0,which is equivalent there being a nonnegative value of the linear program (24) subject to(25), (26), and (27).

45

1.0
0.8
0.6

β

1.2

1.4

Figure 1: Structure 1, Y2 linear in Z and U2 . The identi…ed set of values of and ina linear IV Tobit model under a zero conditional expectation restriction, E[U jz] = 0 for
all z: The identi…ed set comprises the region below all the drawn lines whose formula is= E[Y1 jz]
E[Y2 jz] for z 2 f 1; 0:9; : : : ; 0:9; 1g. The value of a and b generating theprobabilities used to calculate the set is a = 1, b = 1, plotted in green. The parameter value
used to calculate the conditional expectations is Case 1 in Table 1.

0.6

0.8

1.0

1.2
α

46

1.4

1.6

1.0
0.8
0.6

β

1.2

1.4

Figure 2: Structure 1, Y2 linear in Z and U2 . The pink …lled region is the identi…ed set ofvalues of and in a linear IV Tobit model under a zero conditional median restriction.
The region below the blue drawn lines is the identi…ed set under a zero conditional mean
restriction. The parameter values in the triangular Gaussian structure generating the probabilities used in this calculation are shown in the column Case 1 in Table 1. The value of aand b generating the probabilities used to calculate the set is a = 1, b = 1, plotted in green.

0.6

0.8

1.0

1.2
α

47

1.4

1.6

1.0
0.8
0.6

β

1.2

1.4

Figure 3: Structure 1, Y2 linear in Z and U2 . The pink …lled region is the identi…ed set ofvalues of and in a linear IV Tobit model under a zero conditional median restriction.
The region below the blue drawn lines is the identi…ed set under a zero conditional mean
restriction. The parameter values in the triangular Gaussian structure generating the probabilities used in this calculation are shown in the column Case 2 in Table 1. The value of aand b generating the probabilities used to calculate the set is a = 1, b = 1, plotted in green.

0.6

0.8

1.0

1.2
α

48

1.4

1.6

1.0
0.8
0.6

β

1.2

1.4

Figure 4: Structure 1, Y2 linear in Z and U2 . The pink …lled region is the identi…ed set ofvalues of and in a linear IV Tobit model under a zero conditional median restriction.
The region below the blue drawn lines is the identi…ed set under a zero conditional mean
restriction. The parameter values in the triangular Gaussian structure generating the probabilities used in this calculation are shown in the column Case 3 in Table 1. The value of aand b generating the probabilities used to calculate the set is a = 1, b = 1, plotted in green.

0.6

0.8

1.0

1.2
α

49

1.4

1.6

0.0
-0.5

Mean independence
Median independence
Stochastic independence & Gaussian U
-1.0

β

0.5

1.0

Figure 5: Structure 2, Y2 bounded above. The pink …lled region is the identi…ed set of valuesof and in a linear IV Tobit model under a zero conditional median restriction. The
magenta …lled region is the projection of the identi…ed set onto the space of ( ; ) when U is
restricted to be Gaussian, independent of Z with mean 0 and unknown variance. The region
below the blue drawn lines is the identi…ed set under a zero conditional mean restriction.
The parameter values in the triangular Gaussian structure generating the probabilities usedin this calculation are shown in the column Structure 2 Case 1 in Table 1. The value of aand b generating the probabilities used to calculate the set is a = 1, b = 0, plotted in green.

0.0

0.5

1.0
α

50

1.5

β

0.0 0.1 0.2 0.3 0.4 0.5
-0.2

-0.2

β

0.0 0.1 0.2 0.3 0.4 0.5

Figure 6: IV tobit model with U restricted Gaussian independent of Z, Structure 2 Case 1:
projections of the identi…ed set onto the space of each pair of parameters in turn. Values ofthe parameters generating probabilities are plotted in green. One dimensional projections
are drawn in pale blue. Red …lled regions are convex hulls of points found to lie in the
projections.

0.4

0.6

0.8

1.0

1.2

1.4

0.90

1.00

1.20

1.10

1.20

1.2
1.0
α
0.8
0.6
0.4

0.4

0.6

0.8

α

1.0

1.2

1.4

σ

1.4

α

1.10

-0.2

0.0 0.1 0.2 0.3 0.4 0.5

0.90

β

1.20
0.90

1.00

σ

1.10

1.20

σ

1.10
0.90

1.00

σ

-0.2

0.0 0.1 0.2 0.3 0.4 0.5
β

1.00

0.4

0.6

0.8

1.0
α

51

1.2

1.4