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Local Indirect Least Squares and Average Marginal

E¤ects in Nonseparable Structural Systems

Susanne Schennach

University of Chicago

Halbert White?

UC San Diego

Karim Chalak

Boston College

December 26, 2009

Abstract

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: drhalwhite@yahoo.com

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1 Introduction

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

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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

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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

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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

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