Local Indirect Least Squares and Average Marginal
E¤ects in Nonseparable Structural Systems
University of Chicago
UC San Diego
December 26, 2009
We study the scope of local indirect least squares (LILS) methods for nonparamet-
rically estimating average marginal e¤ects of an endogenous cause X on a response
Y in triangular structural systems that need not exhibit linearity, separability, or
monotonicity in scalar unobservables. One main …nding is negative: in the fully
nonseparable case, LILS methods cannot recover the average marginal e¤ect. LILS
methods can nevertheless test the hypothesis of no e¤ect in the general nonseparable
case. We provide new nonparametric asymptotic theory, treating both the tradi-
tional case of observed exogenous instruments Z and the case where one observes
only error-laden proxies for Z.
Acknowledgement 0.1 We thank Stefan Hoderlein, Xun Lu, Andres Santos, and
Suyong Song for helpful comments and suggestions. Any errors are the authors’ re-
sponsibility. S. M. Schennach acknowledges support from the National Science Foun-
dation via grants SES-0452089 and SES-0752699. This is a revised version of a paper
titled "Estimating Average Marginal E¤ects in Nonseparable Structural Systems."
JEL Classi…cation Numbers: C13, C14, C31.
Keywords: indirect least squares, instrumental variables, measurement error, non-
parametric estimator, nonseparable structural equations.
?Corresponding author. Address: Dept. of Economics 0508, UCSD, La Jolla, CA 92093-0508. Tele-
phone: 858 534-3502; fax 858 523-2151; email address: email@example.com
This paper studies the scope of indirect least squares-like methods for the identi…cation
and nonparametric estimation of marginal e¤ects of an endogenous cause X on a response
of interest Y without assuming linearity, separability, monotonicity, or the presence of
solely scalar disturbances for the structural equations. As we show, control variables need
not be available in such circumstances, so we rely only on the availability of exogenous
instruments, Z; which may or may not be perfectly observed.
We follow the literature in distinguishing the “instrumental variable” (IV) and “control
variable” approaches for identifying and estimating structural e¤ects of endogenous causes
(see e.g. Blundell and Powell, 2003; Darolles, Florens, and Renault, 2003; and Hahn and
Ridder, 2009). Correspondingly, Chalak and White (2009) (CW) emphasize the structural
origins of instruments yielding (conditional) independence relationships that serve to iden-
tify e¤ects of interest. Classical IV methods make use of exogenous instruments that are
independent of the unobserved causes. On the other hand, control variable methods make
use of conditioning instruments that, once conditioned on, ensure the conditional indepen-
dence of the observed causes of interest and the unobserved causes. In general, neither of
these (conditional) independence relations is su¢cient for the other.
Using a control variable approach, Altonji and Matzkin (2005) and Hoderlein and Mam-
men (2007) study identifying and estimating local average structural derivatives (marginal
e¤ects) in general structures without specifying how the endogenous cause of interest or con-
ditioning instruments are generated. Hoderlein (2005, 2007) and Imbens and Newey (2009)
derive useful control variables in nonlinear structures where the cause of interest is deter-
mined by exogenous instruments and a scalar unobserved term and is strictly monotonic
(or even additively separable) in this scalar. Chalak and White (2007) and White and
Chalak (2008) discuss identifying and estimating causal e¤ects in structures nonseparable
between observables and multiple unobservables, providing structural conditions ensuring
the availability of useful conditioning instruments more generally.
In the absence of control variables, methods based on classical IVs may provide a way
to conduct structural inference in nonlinear systems. Two extensions of IV to nonlinear
systems have been studied in the literature. The …rst is based on what Darolles, Florens,
and Renault (2003) call "instrumental regression" (IR), where Y is separably determined
as, say, Y = r(X)+"; with E(" j Z) = 0: Blundell and Powell (2003), Darolles, Florens, and
Renault (2003), Newey and Powell (2003), and Santos (2006), among others, show that IR
methods can reliably identify speci…c e¤ect measures in separable structures. But they lose
their structural interpretation in the nonseparable case unless X is separably determined
(see e.g. Blundell and Powell, 2003; Hahn and Ridder, 2009).
A second extension of IV makes use of exogenous instruments to study e¤ect measures
constructed as ratios of certain derivatives, derivative ratio (DR) e¤ect measures, for short.
In classical linear structural systems with exogenous instruments, these e¤ects motivate and
underlie Haavelmo’s (1943) classical method of indirect least squares (ILS). In the treatment
e¤ects literature, Angrist and Imbens (1994) and Angrist, Imbens, and Rubin (1996) show
that DR e¤ect measures have causal interpretations for speci…c subgroups of the population
of interest. In selection models, such as the generalized Roy model, Heckman (1997),
Heckman and Vytlacil (1999, 2001, 2005), and Heckman, Urzua, and Vytlacil (2006), among
others, show that DR e¤ect measures correspond to a variety of structurally informative
weighted averages of e¤ects of interest; the corresponding estimators are "local IV" or local
ILS (LILS) estimators (see Heckman and Vytlacil, 2005; Carneiro, Heckman, and Vytlacil,
2009). A common feature of the treatment e¤ects and selection papers just mentioned is
their focus on speci…c triangular structures with binary or discrete treatment variables.
Although the work just cited establishes the usefulness of DR e¤ect measures and their
associated LILS estimators in speci…c contexts, an important open question is whether
these methods can be used to learn about the e¤ects of an endogenous cause on a response
of interest in more general triangular structures. We address this question here, studying
general structural equations that need not obey linearity, monotonicity, or separability. Nor
do we restrict the unobserved drivers to be scalar; these can be countably dimensioned.
Our analysis delivers contributions in a number of inter-related areas. The …rst is
a detailed analysis of the properties of DR/LILS methods that a¤ords clear insight into
their limitations and advantages, both inherently and relative to IR and control variable
methods. Our …ndings are a mixture of bad news and good news. One main …nding is
negative: in the fully nonseparable case, DR methods, like IR methods, cannot recover the
average marginal e¤ect of the endogenous cause on the response of interest. Nor can DR
methods identify local average marginal e¤ects of X on Y of the type recovered by control
variable methods. On the other hand, and also like IR methods, when X is separably
determined, DR methods do recover an instrument-conditioned average marginal e¤ect
more informative than the unconditional average marginal e¤ect.
We also …nd that, despite their failure to recover average marginal e¤ects in the fully
nonseparable case, DR/LILS methods can nevertheless generally be used to test the hy-
pothesis of no e¤ect. This is because DR methods identify a speci…c weighted average
marginal e¤ect that is always zero when the true marginal e¤ect is zero, and that is zero
only if a true average marginal e¤ect is zero given often plausible economic structure.
Thus, DR/LILS methods provide generally viable inference.
In the control variable literature, Imbens and Newey (2009) (see also Chesher (2003)
and Matzkin (2003)) study nonseparable structures in which although X is nonseparably
determined, it is strictly monotonic in a scalar unobserved cause. As we show, this structure
also enables suitably constructed DR ratios to measure average marginal e¤ects based on
IVs rather than control variables. Nevertheless, control variable methods, when available,
are more informative, as these provide local e¤ect measures, whereas DR methods do not.
IV methods based on restrictive functional form assumptions are typical in applications.
But economic theory is often uninformative about the validity of these restrictions, and all
methods (IR, control variable, and DR) are vulnerable to speci…c failures of these assump-
tions. Accordingly, it is important to develop speci…cation tests for critical functional form
assumptions. Thus, a second contribution is to show how DR methods can be used to
test the key hypothesis that X is separably determined. The results of this test inform
the interpretation of results, as a failure to reject implies that not only do LILS estimates
support inference about the absence of e¤ects, but the LILS estimates can be interpreted
as instrument-conditioned average marginal e¤ects. Given space limitations, however, we
leave to future work developing the statistical properties of these tests.
Our third area of contribution is to provide new nonparametric methods for DR/LILS
estimation and inference. We pay particular attention to the fact that in practice, one
may not be able to observe the true exogenous instruments. Instead, as in Butcher and
Case (1994) or Hausman (1997), one may use proxies for such unobserved instruments. In
linear structures, this poses no problem for structural inference despite the inconsistency
of the associated reduced form estimator, as CW discuss. As we show here, however, the
unobservability of instruments creates signi…cant obstacles to structural inference using DR
IV methods more generally. We introduce new methods that resolve this di¢culty.
In particular, we study two cases elucidated by CW: the traditional observed exogenous
instrument (OXI) case, where the exogenous instrument is observed without error; and the
proxies for unobserved exogenous instrument (PXI) case, where the exogenous instrument
is not directly observable, but error-contaminated measurements are available to serve
as proxy instruments. Standard IV su¢ces for both OXI and PXI in the linear case,
but otherwise OXI and PXI generally require fundamentally di¤erent estimation methods.
Generally, straightforward kernel or sieve methods su¢ce for OXI. The PXI case demands
a novel approach, however. Our PXI results are the …rst to cover the use of instrument
proxies in the general nonlinear nonparametric context.
For the OXI case, we apply in…nite order ("‡at-top") kernels (Politis and Romano, 1999)
to estimate functionals of the distributions of the observable variables that we then com-
bine to obtain new estimators of the average marginal e¤ect represented by the DR e¤ect
measure. We obtain new uniform convergence rates and asymptotic normality results for
estimators of instrument-conditioned average marginal e¤ects as well as root-n consistency
and asymptotic normality results for estimators of their unconditional weighted averages.
For the PXI case, we build on recent results of Schennach (2004a, 2004b) to obtain a
variety of new results. Speci…cally, we show that two error-contaminated measurements
of the unobserved exogenous instrument are su¢cient to identify objects of interest and
to deliver consistent estimators. The proxies need not be valid instruments. Our general
estimation theory covers densities of mismeasured variables and expectations conditional on
mismeasured variables, as well as their derivatives with respect to the mismeasured variable.
We provide new uniform convergence rates over expanding intervals (and, in some cases,
over the whole real line) as well as new asymptotic normality results in fully nonparametric
settings. We also consider nonlinear functionals of such nonparametric quantities and prove
root-n consistency and asymptotic normality. We thus provide numerous general-purpose
asymptotic results of independent interest, beyond the PXI case.
The plan of the paper is as follows. In Section 2 we specify a triangular structural sys-
tem that generates the data, and we de…ne the DR e¤ect measures of interest. We study
the structural objects identi…ed by DR e¤ect measures, devoting particular attention to
the interpretation of these DR e¤ect measures in a range of special cases. We also show
how DR measures can be used to test the hypothesis of no causal e¤ect and for struc-
tural separability. We then provide new results establishing consistency and asymptotic
normality for our nonparametric local ILS estimators of DR e¤ects. Section 3 treats the
OXI case. Section 4 develops new general results for estimation of densities and functionals
of densities of mismeasured variables. As an application, we treat the PXI case, ensuring
the identi…cation of the objects of interest and providing estimation results analogous to
those of Section 3. Section 5 contains a discussion of the results, and Section 6 provides a
summary and discussion of directions for future research. All proofs are gathered into the
2 Data Generation and Structural Identi…cation
2.1 Data Generation and Marginal E¤ects
We begin by specifying a triangular structural system that generates the data. In such
systems, there is an inherent ordering of the variables: "predecessor" variables may deter-
mine "successor" variables, but not vice versa. For example, when X determines Y , then
Y cannot determine X. In such cases, we say for convenience that Y succeeds X, and we
write Y ( X as a shorthand notation.
Assumption 2.1 Let a triangular structural system generate the random vector U and
random variables fX;Y;Zg such that Y ( (U;X;Z), X ( (U;Z), and Z ( U. Further:
(i) Let ?x;?y; and ?z be measurable functions such that Ux ? ?x(U);Uy ? ?y(U);Uz ?
?z(U) are vectors of countable dimension; (ii) X;Y; and Z are structurally generated as
Z = p(Uz)
X = q(Z;Ux)
Y = r(X;Uy);
where p;q; and r are unknown measurable scalar-valued functions; (iii) E(X) and E(Y ) are
…nite; (iv) The realizations of X and Y are observed; those of U;Ux;Uy; and Uzare not.
We consider scalar X;Y; and Z for simplicity; extensions are straightforward.We
explicitly assume observability of X and Y and unobservability of the U’s. We separately
treat cases in which Z is observable (Section 3) or unobservable (Section 4). An important
feature here is that the unobserved causes U;Ux;Uy; and Uz may be multi-dimensional.
Indeed, the unobserved causes need not even be …nite dimensional.
The response functions p;q; and r embody the structural relations between the system
variables. (Here and throughout, we use the term "structural" to refer to the system of
Assumption 2.1 or to any of its components or properties.) Assuming only measurability for
p;q; and r permits but does not require linearity, monotonicity in variables, or separability
between observables and unobservables. Signi…cantly, separability prohibits unobservables
from interacting with observable causes to determine outcomes; nonseparability permits
this, a generalization of random coe¢cients structure.
The structure of Assumption 2.1 can arise in numerous economic applications. For
example, when X is schooling and Y represents wages, this structural system corresponds
to models for educational choices with heterogeneous returns, as discussed in Imbens and
Newey (2009), Chesher (2003), and Heckman and Vytlacil (2005), for example. When X is
input and Y is output, the system corresponds to models for the estimation of production
functions (see Imbens and Newey, 2009). When Y is a budget share and X represents
total expenditures, the system corresponds to a nonparametric demand system with a
heterogeneous population, as in Hoderlein (2005, 2007). In all these examples, Z serves as
a driver of X excluded from the structural equation for Y .
Our interest attaches to the e¤ect of X on Y (e.g., the return to education). Speci…-
cally, consider the marginal e¤ect of continuously distributed X on Y , i.e., the structural
derivative Dxr(X;Uy), where Dx? (@=@x). If r were linear and separable, say,
r(X;Uy) = X?0+ U0
then Dxr(X;Uy) = ?0. Generally we will not require linearity or separability, so Dxr(X;Uy)
is no longer constant but generally depends on both X and Uy. To handle dependence on
the unobservable Uy, we consider certain average marginal e¤ects, de…ned below.
Generally, X and Uy may be correlated or otherwise dependent, in which case X is
“endogenous.” In the linear separable case, when X is endogenous, the availability of suit-
able instrumental variables permits identi…cation and estimation of e¤ects of interest. In
what follows, we study how the DR IV approach performs when linearity and separability
are relaxed. For this, we note that the structure above permits Z to play the role of an
instrument, given a suitable exogeneity condition. To specify this, we follow Dawid (1979)
and write X ? Y when random variables X and Y are independent and X 6? Y otherwise.
Assumption 2.2 Uz? (Ux;Uy).
Assumption 2.2 permits Ux6? Uy, which, given Assumption 2.1, implies that X may be
endogenous: X 6? Uy. On the other hand, Assumptions 2.1 and 2.2 imply Z ? (Ux;Uy), so
Z is exogenous with respect to both Uxand Uyin the classical sense.
2.2Absence of Control Variables
At the heart of the control variable approach are control variables, say W, such that X ? Uy
j W; as in Altonji and Matzkin (2005), Hoderlein and Mammen (2007), White and Chalak
(2008), and Imbens and Newey (2009). This conditional independence is neither neces-
sary nor su¢cient for Assumption 2.2; moreover, as will be apparent from our derivations
below, the structural e¤ects identi…ed under the various exogeneity conditions can easily
di¤er. Which exogeneity condition is appropriate in any particular instance depends on
the speci…cs of the economic structure, as extensively discussed by CW.
In particular, observe that under Assumptions 2.1 and 2.2, control variables ensuring the
conditional independence of X and Uyare generally not available. Assumptions 2.1 and 2.2
do imply Hoderlein’s (2005) assumption 2.3, which states that Z ? Uyj Ux: Assumption 2.1
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B Supplementary material
Proof of Lemma 3.1. This result holds by construction.
Lemma B.1 Suppose Assumption 3.4 holds. Then supz2R
1, 0 <R ??k(?)(z)??2dz < 1,R ??k(?)(z)??2+?dz < 1, and jzj??k(?)(z)??! 0 as jzj ! 1.
Proof. The Fourier transform of k(?)(z) is (?i?)??(?), which is bounded by assumption
and therefore absolutely integrable, given the assumed compact support of ?(?). Hence
polynomial, making it impossible to satisfyRk(z)dz = 1. Hence,R ??k(?)(z)??2dz > 0.
of ?(?), if ?(?) has two bounded derivatives then so does (?i?)??(?); and it follows that
inverse Fourier transform of i(d2=d?2)(?i?)??(?)
Hence, we know that there exists C such that
??k(?)(z)??< 1;R ??k(?)(z)??dz <
k(?)(z) is bounded, since
thatR ??k(?)(z)??2dz > 0 unless k(?)(z) = 0 for all z 2 R, which would imply that k (z) is a
The Fourier transform of z2k(?)(z) is ?(d2=d?2)
??? ?Rj?j?j?(?)jd? < 1. Note
. By the compact support
is absolutely integrable. By the Riemann-Lebesgue Lemma, the
is such that z2k(?)(z) ! 0 as jzj ! 1.
1 + z2;
and the function on the right-hand side satis…es all the remaining properties stated in the
Proof of Theorem 3.2. (i) The order of magnitude of the bias is derived in the proof of
Theorem 4.4 in the foregoing appendix. The convergence rate of BV;?(z;h) is also derived
in Theorem 4.4.
(ii) The facts that E [LV;?(z;h)] = 0 and E?L2
struction. Next, Assumptions 3.2(ii) and 3.4 ensure that
V;?(z;h)?= n?1?V;?(z;h) hold by con-
?V;?(z;h) = E(?1)?h???1V k(?)
?Z ? z
?Z ? z
?Z ? z
? E(?1)?h???1V k(?)
?Z ? z
??Z ? z
(by Assumption 3.2(ii) and Jensen’s inequality)
Z?k(?)(u)?2fZ(z + hu)du
(after a change of variable from ~ z to z + hu)
(by Lemma B.1)
?~ z ? z
fZ(~ z)d~ z
(by Assumption 3.1(i)
z2R?V;?(z;h) = O?h???1=2?:
We now establish the uniform convergence rate. Using Parseval’s identity, we have
Z?^E?V ei?Z?? E?V ei?Z??
?Z ? z
?Z ? z
so it follows that
Z???^E?V ei?Z?? E?V ei?Z????j?j?j?(h?)jd?;
E [jLV;?(z;h)j] ?
h???^E?V ei?Z?? E?V ei?Z????
?^E?V ei?Z?? E?V ei?Z??y])1=2j?j?j?(h?)jd?
?^E?V ei?Z?? E?V ei?Z??
Z?n?1E?V ei?ZV e?i?Z??1=2j?j?j?(h?)jd?
Hence, by the Markov inequality,
z2RjLV;?(z;h)j = Op
When hn! 0; lemma 1 in the appendix of Pagan and Ullah (1999, p.362) applies to yield:
! E?V2jZ = z?fZ(z)
By Assumptions 3.1 and 3.2(iii); E [V2jZ = z]fZ(z) > 0 for z 2 SZ and 3.4 ensures
R ?k(?)(z)?2dz > 0 by Lemma B.1, so that h2?+1
(iii) To show asymptotic normality, we verify that `V;?(z;hn;V;Z) satis…es the hypothe-
?Z ? z
?Z ? z
?Z ? z
?Z ? z
EE [V jZ]h?1k(?)
?V;?(z;hn) > 0 for all n su¢ciently large.
ses of the Lindeberg-Feller Central Limit Theorem for IID triangular arrays (indexed by
n). The Lindeberg condition is: For all " > 0,
n!1Qn;hn(z;") ! 0;
Qn;h(z;") ? (?V;?(z;h))?1E
Using the inequality E [1[W ? ?]W2] ? ???E?W2+??for any ? > 0, we have
Qn;h(z;") ? (?V;?(z;h))?1?
j`V;?(z;h;V;Z)j ? "(?V;?(z;h))1=2n1=2?
where Assumption 3.2(iv) ensures that
?Z ? z
?Z ? z
?Z ? z
The results above and Assumption 3.2(iv) ensure that for any given z there exist 0 <
A1;z;A2;z < 1 such that A1;zh?2??1
small. Hence, we have
< ?V;?(z;hn) < A2;zh?2??1
for all hn su¢ciently
= "??(nhn)??=2! 0
provided nhn! 1, which is implied by Assumption 3.6: hn! 0;nh2?+1
???~ gVj;?j? gVj;?j
Proof of Theorem 3.3.
remainder in eq.(24) can be dealt
with as in the proof above of Theorem 4.9. Next, we note that
s(z)(^ gV;?(z;h) ? gV;?(z))dz = L + Bh+ Rh;
^E?V s(?)(Z)?? E?V s(?)(Z)?=^E? V;?(s;V;Z)?
s(z)(gV;?(z;h) ? gV;?(z))dz
We then have, by Assumption 3.7,
js(z)jjBV;?(z;hn)jdz = op
Rh =s(z)(^ gV;?(z;h) ? gV;?(z;h))dz ?
?^E?V s(?)(Z)?? E?V s(?)(Z)??
s(z)(gV;?(z;hn) ? gV;?(z))dz
js(z)jjgV;?(z;hn) ? gV;?(z)jdz
js(z)jdz = op
s(z)(^ gV;?(z;h) ? gV;?(z;h))dz ?
^E?V?s(?)(z;h) ? s(?)(Z)?? E?V?s(?)(z;h) ? s(?)(Z)???
Hence, Rhnis a zero-mean sample average where the variance of each individual IID term
Proof of Theorem 3.4.
This proof is virtually identical to the proof of Theorem 4.10
?^E?V s(?)(Z)?? E?V s(?)(Z)??
?^E?V s(?)(Z)?? E?V s(?)(Z)??
?Z ? z
?Z ? z
?Z ? z
?Z ? z
?Z ? z
? V s(?)(Z)
?Z ? z
? V s(?)(Z)
s(?)(~ z;h) =s(?)(z)1
?~ z ? z
goes to zero, implying that Rhn= op
in the foregoing appendix, with "n= (h?1
Proof of Theorem 3.5.
This proof is virtually identical to the proof of Theorem 4.11,
invoking Theorem 3.2 instead of Corollary 4.8.