This note demonstrates identification of Unconditional Partial Effects introduced by Firpo, Fortin, and Lemieux (2009) in nonseparable triangular models with endogenous regressors via a control variable approach, as employed by Imbens and Newey (2009).

Rothe, Christoph

English

Research Papers in Economics

   09-054 
Research Group: Econometrics and Statistics  June 30, 2009 
Identification of Unconditional Partial 
Effects in Non Separable Models 
 
CHRISTOPH ROTHE 
 Identification of Unconditional Partial Effects
in Nonseparable Models
Christoph Rothe
Toulouse School of Economics
This Version: June 30, 2009
Abstract
This note demonstrates identication of Unconditional Partial Eects introduced by
Firpo, Fortin, and Lemieux (2009) in nonseparable triangular models with endogenous re-
gressors via a control variable approach, as employed by Imbens and Newey (2009).
JEL Classication: C14
Keywords: Nonseparable Models, Control Variable
Address: Toulouse School of Economics, 21 All ee de Brienne, F-31000 Toulouse, France. E-mail:
rothe@cict.fr. Website: http://www.christophrothe.net.
11 Introduction
An important feature of many interesting economic models is that they do not imply an econo-
metric specication with additively separable disturbance terms when they are taken to data.
The properties of such nonseparable models have therefore received considerable interest in
the recent literature, being investigated by Chesher (2003), Matzkin (2003), Chesher (2005),
Chernozhukov and Hansen (2005), Hoderlein and Mammen (2007), Chernozhukov, Imbens, and
Newey (2007) and Imbens and Newey (2009), amongst others. One of the most important issues
in this context is how to accommodate the presence of endogenous regressors, which are fre-
quently encountered in microeconometric applications. A possible approach is the use of control
variable techniques, which are discussed in detail by Imbens and Newey (2009). They establish
identication of various quantities of interest in triangular simultaneous equation models under
relatively general conditions. These quantities include the Average Structural Function, the
Quantile Structural Function, Average Derivatives and Policy Eects.
In this paper, we show that a further interesting class of parameters can be identied under
general conditions in their framework: the Unconditional Partial Eects. These parameters have
recently been introduced to the literature by Firpo, Fortin, and Lemieux (2009) in the exogenous
case, and correspond to the following thought experiment: suppose that every member of the
population would experience the same exogenous marginal increase in one of its observable
characteristics. How would this aect the unconditional distribution of the outcome variable?
To give a concrete example, a researcher might be interested in the eect of a marginal increase
in everybody's income on some feature of the distribution of consumption, such as its moments,
quantiles, Gini coecient or other measures of inequality. As pointed out by Firpo, Fortin, and
Lemieux (2009), such summary measures are of interest for policy analysis, where the focus is
on aggregate as opposed to individual eects of a variable.
Firpo, Fortin, and Lemieux (2009) establish that in a setting without endogenous variables,
Unconditional Partial Eects are identied under weak conditions, showing that they can be
represented by the average derivative of a projection of the recentered inuence function of the
statistic of interest on the regressors. We demonstrate that this result can be generalized to
the triangular nonseparable models discussed in Imbens and Newey (2009) using their control
variable approach (other papers that use control variable techniques in non- or semiparametric
setting include Blundell and Powell (2003), Blundell and Powell (2004), Blundell and Powell
(2007) and Florens, Heckman, Meghir, and Vytlacil (2008)). As a further contribution, this
2paper also provides a slightly dierent representation of Unconditional Partial Eects compared
to the one given in Firpo, Fortin, and Lemieux (2009). We show that these parameters can
be written as functionals of the average derivate of the conditional cumulative distribution
function (CDF) of the outcome variable given the regressors and the control variable (where
the derivate is taken with respect to the regressors). This representation is useful to give an
explicit expression for Unconditional Partial Eects when further parametric or semiparametric
restrictions are imposed on the model. This representation is by no means specic to the setting
with endogenous variables but holds under full exogeneity as well, with obvious simplications.
We illustrate this point by considering the linear quantile regression model as an example.
The remainder of this paper is organized as follows. In the next section, we describe the
model and give a precise denition of Unconditional Partial Eects. Identication is discussed
in Section 3. The nal section concludes.
2 Model and Parameters of Interest
The model we consider in this paper is essentially the same as in Imbens and Newey (2009). We
observe a scalar outcome variable of interest denoted by Y , which is linked to a random vector
X = (X1;Z1) of observable determinants and an unobserved disturbance term " through the
structural equation
Y = g(X;"): (2.1)
The subvector X1 of X is potentially endogenous and assumed to be determined through a
reduced form equation,
X1 = h(Z;) (2.2)
where  is another unobserved disturbance and Z = (Z1;Z2) is a vector of instruments that
excert inuence on X1 in a sense to be made precise below, but are independent of the error
terms. As in Imbens and Newey (2009), no restrictions on the dimesionality of " are imposed,
allowing for general forms of unobserved heterogeneity. However, for identication purposes it
will be necessary to impose such a restriction on the disturbance in (2.2), as discussed below.
To simplify the notation, we will focus in the following on the case with X = X1 consisting of
a single endogenous regressor only, but all arguments can easily be generalized to allow for the
presence of multiple endogenous regressors or additional exogenous ones.
3The parameters we are interested in correspond to the eect of a marginal increase in X on
some feature  (FY ) of the unconditional distribution of Y . That is, for some constant  6= 0,
dene the counterfactual random variable Y as
Y = g(X + ;"):
Denote the CDF of Y and Y by FY and FY;, respectively, and let  () be a functional of
interest. For example   could be the functional that maps a CDF into one of it moments, or
into its quantile function. With this notation, we can now formally dene an Unconditional
Partial Eect.
Denition 1 (Unconditional Partial Eect). For any functional   : D( 1;1) ! S, where S
is some normed space, the quantity
  = lim
!0
 (FY;)    (FY )

(2.3)
is called the Unconditional Partial Eect of X on  (FY ), provided that the limit in (2.3) exists.
3 Identication
In order to identify the Unconditional Partial Eects in models with endogeneity, we can use
control variable techniques developed in Imbens and Newey (2009). Generally speaking, a
control variable is an identied random vector that is able to absorb the dependence between
the regressors and the unobserved disturbance term in the outcome equation (2.1), in the sense
that X and " will be independent conditional on the control variable. Imbens and Newey (2009)
show that in the triangular model such a control variable is available under certain restrictions
on the second equation. We repeat their result here for completeness.
Lemma 1 (Imbens and Newey, 2009). Suppose that h(z;) is strictly increasing for all values
of z, that  is continuously distributed with strictly increasing CDF, and that Z?(";). Then
"?XjV , where V = FXjZ(X;Z).
The reason V = FXjZ(X;Z) has the properties of a control variable in our model is that the
exclusive source of dependence between X and " is their joint dependence on the disturbance
term  from equation (2.2). However, under the conditions of Lemma 1, V is simply a one-to-one
transformation of , which in turn implies the result.
4The conditional independence property can be used to derive an explicit representation for
FY;. Using the structure of the model and the law of iterated expectations, we obtain that
FY;(y) =
Z
Pr(m(X + ;")  yjX = x;V = v)dFX;V (x;v)
=
Z
Pr(m(X;")  yjX = x + ;V = v)dFX;V (x;v)
=
Z
FY jX;V (y;x + ;v)dFX;V (x;v)
= E(FY jX;V (y;X + ;V )):
This implies that the function FY; is identied if the support of the random vector (X + ;V )
is contained in the support of (X;V ). For identication of the Unconditional Partial Eect, it
will be sucient that this condition holds for small values of  only. The role of this support
condition is to ensure that there is a sucient amount of variation in the endogenous regressors
induced by the instruments. To see this, assume for a moment that Z does not excert any
inuence on X. Then V = FXjZ(X;Z)  t(X) is simply a transformation of the endogenous
regressor. While the conditional independence condition X?"jV will still hold in this case, the
joint support of X and V is now given by f(x;t(x)) : x 2 supp(X)g, which is generally not a
subset of f(x + ;t(x)) : x 2 supp(X)g for any  6= 0.
In order to derive a general formula for the Unconditional Partial Eect of X on  (FY )
for some general functional  , we rst consider the simplest case where   = id is the identity
mapping, i.e.  (F) = F. Then
id(y) = lim
!0
FY;(y)   FY (y)

= lim
!0
E(FY jX;V (y;X + ;V ))   E(FY jX;V (y;X;V ))

= E(@xFY jX;V (y;X;V ))
where the last equality follows by dominated convergence. The Unconditional Partial Eect of
X on FY is thus simply the average derivative of the conditional CDF of Y given X and V ,
where the derivative is taken with respect to X. We formally state this preliminary nding in
the following lemma.
Lemma 2. Suppose that the conditions of Lemma 1 hold, and that for some c > 0 and  2
( c;c) the support of (X + ;V ) is contained in the support of (X;V ). Then
id() = E(@xFY jX;V (;X;V ))
and is thus identied.
5Using the last result, one can now easily extend the analysis of Unconditional Partial Eects
to more general quantities  (FY ), if  () is suciently "smooth". In particular, we consider
functionals that satisfy a Hadamard dierentiability condition, where   is called Hadamard
dierentiable at F if there exists a continuous linear functional  0
F such that
lim
!0

 

 (F + h)    (F)

   0
F(h)
 
  = 0 (3.1)
for all sequences of function h ! h such that F +h is contained in the domain of   for some
suciently small value of . See van der Vaart (2000, Chapter 20.2) for further details.
To derive a general representation of Unconditional Partial Eects on  (FY ), dene the
function h through h = (FY;   FY )=. We then obtain that
  = lim
!0
 (FY;)    (FY )

= lim
!0
 (FY + h)    (FY )

=  0
F(id);
where the last equality follows from the continuous mapping theorem since h ! id as shown
above. That is, we can identify general Unconditional Partial Eects by using the eect of X
on the unconditional CDF of Y as a building block. We formalize this nding in the following
Theorem.
Theorem 1. Suppose that the conditions of Lemma 2 hold, and that the functional   is
Hadamard dierentiable at FY with derivative  0
F. Then the Unconditional Partial Eect of
X on  (FY ) is given by   =  0
F(id).
This representation of the Unconditional Partial Eect of X on  (FY ) given in Theorem 1
is convenient for two reasons. First, results on Hadamard dierentiablity are widely available in
the literature for many functionals of interest. Under appropriate conditions, this smoothness
property is fullled for moments and quantiles, but also for inequality measures like the Gini
coecient and the Lorenz curve.
Second, the above representation is particularly useful when further parametric or semipara-
metric restrictions are imposed on the relationship of the outcome variable and the regressors.
In this case, the Unconditional Partial Eect of X on FY itself is usually still easy to compute,
and results for other statistics of interest follow immediately from Theorem 1. Our representa-
tion thus allows us to establish a tight link between the Unconditional Partial Eects and the
structural features of the model. This result is not specic for models with endogeneity, but
6applies analogously to the exogenous case where the control variable V is not present. On the
other hand, the general representation in Firpo, Fortin, and Lemieux (2009) for the exogenous
case, using a projection of the recentered inuence function of  (FY ) on the regressors, can be
much more dicult to evaluate for specic models.
We now illustrate this last point by considering the case where the model in equation (2.1)
is a standard linear quantile regression model (see Koenker (2005)). That is, suppose for the
moment that " is now a scalar random variable, normalized to be uniformly distributed on [0;1],
and that
g(X;") = 1(") + X2(");
where 1() and 2() are strictly monotonic functions. The form of (2.2) can remain unchanged.
Using standard arguments, one obtains that under this specication we have that
@xFY jXV (y;x;v) =  fY jXV (y;x;v)2(FY jXV (y;x;v))
and thus the Unconditional Partial Eect of X on FY is given by
id() =  E(fY jXV (;X;V )2(FY jXV (;X;V ))):
Now consider the Unconditional Partial Eect of X on  (FY ), where  (F)[] = F 1() =
inffy : F(y)  g is the functional that transfers a CDF into its quantile function. Then under
some standard restrictions (ensuring e.g. uniqueness of the quantiles) this map is Hadamard
dierentiable at FY with derivative
 7!  0
FY () =  


@yFY

 F 1
Y ;
which leads to the following expression for the Unconditional Partial Eect:
 () =  
id(F 1
Y ())
fY (F 1
Y ())
=
E(fY jXV (F 1
Y ();X;V )2(FY jXV (F 1
Y ();X;V )))
fY (F 1
Y ())
:
Note that this is a weighted average of the function 2 evaluated at FY jXV (F 1
Y ();X;V ),
which can be interpreted as the "rank" of F 1
Y () in the distribution of Y conditional on X
and V . Firpo, Fortin, and Lemieux (2009) obtain a similar result for the exogenous case
through somewhat more involved arguments (compare their Proposition 1). However, while
their arguments apply to the specic case where  (FY ) is the quantile function only, our analysis
can easily generalized to other statistics, such as the Lorenz curve or the Gini coecient, as
long as the Hadamard dierentiability condition holds.
74 Conclusions
In this paper, we established the identication of Unconditional Partial Eects introduced by
Firpo, Fortin, and Lemieux (2009) in general nonseparable models with endogenous regressors
using a control variable approach due to Imbens and Newey (2009). We also show that these
eects can be written in terms of an average derivative of the conditional CDF of the outcome
variable Y given the regressors X and the control variable V , where the derivative is taken
with respect to X. This representation is useful to give an explicit expression for Unconditional
Partial Eects in nonlinear parametric or semiparametric models.
References
Blundell, R., and J. Powell (2003): \Endogeneity in Nonparametric and Semiparametric
Regression Models," Advances in Economics and Econometrics: Theory and Applications,
Eighth World Congress, 2.
(2004): \Endogeneity in Semiparametric Binary Response Models," Review of Eco-
nomic Studies, 71(3), 655{679.
Blundell, R., and J. Powell (2007): \Censored Regression Quantiles with Endogenous
Regressors," Journal of Econometrics, 141(1), 65{83.
Chernozhukov, V., and C. Hansen (2005): \An IV Model of Quantile Treatment Eects,"
Econometrica, 73(1), 245{261.
Chernozhukov, V., G. Imbens, and W. Newey (2007): \Instrumental Variable Estimation
of Nonseparable Models," Journal of Econometrics, 139(1), 4{14.
Chesher, A. (2003): \Identication in Nonseparable Models," Econometrica, 71(5), 1405{
1441.
(2005): \Nonparametric Identication under Discrete Variation," Econometrica, 73(5),
1525{1550.
Firpo, S., N. Fortin, and T. Lemieux (2009): \Unconditional Quantile Regressions," Econo-
metrica, forthcoming.
8Florens, J., J. Heckman, C. Meghir, and E. Vytlacil (2008): \Identication of Treat-
ment Eects Using Control Functions in Models With Continuous, Endogenous Treatment
and Heterogeneous Eects," Econometrica, 76(5), 1191{1206.
Hoderlein, S., and E. Mammen (2007): \Identication of Marginal Eects in Nonseparable
Models Without Monotonicity," Econometrica, 75(5), 1513{1518.
Imbens, G., and W. Newey (2009): \Identication and Estimation of Triangular Simultane-
ous Equations Models Without Additivity," Econometrica, forthcoming.
Koenker, R. (2005): Quantile Regression. Cambridge University Press.
Matzkin, R. (2003): \Nonparametric Estimation of Nonadditive Random Functions," Econo-
metrica, 71(5), 1339{1375.
van der Vaart, A. (2000): Asymptotic Statistics. Cambridge University Press.
9

An IV Model of Quantile Treatment Eects,&quot;

Asymptotic Statistics.

Censored Regression Quantiles with Endogenous Regressors,&quot;

Endogeneity in Nonparametric and Semiparametric Regression Models,&quot;

Endogeneity in Semiparametric Binary Response Models,&quot;

Identi and Estimation of Triangular Simultaneous Equations Models Without Additivity,&quot;

Identi in Nonseparable Models,&quot;

Identi of Marginal Eects in Nonseparable Models Without Monotonicity,&quot;

Identi of Treatment Eects Using Control Functions in Models With Continuous, Endogenous Treatment and Heterogeneous Eects,&quot;

Identification of Unconditional Partial Effects in Non Separable Models.

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