Measurement Errors in Quantile Regression Models∗

Sergio Firpo† Antonio F. Galvao‡ Suyong Song§

November 8, 2016

Abstract

This paper develops estimation and inference for quantile regression models withmeasurement errors. We propose an easily-implementable semiparametric two-step es-timator when repeated measures for the covariates are available. Building on recenttheory on Z-estimation with infinite-dimensional parameters, consistency and asymp-totic normality of the proposed estimator are established. We also develop statisticalinference procedures and show the validity of a bootstrap approach to implement themethods in practice. Monte Carlo simulations assess the finite-sample performance ofthe proposed methods. We apply the methods to the investment equation model usinga firm-level data with repeated measures of investment demand, Tobin’s q. We docu-ment strong heterogeneity in the sensitivity of investment to Tobin’s q and cash flowacross the conditional distribution of investment. The cash flow sensitivity is relativelylarger at the lower part of the distribution, providing evidence that these firms aremore exposed to and dependent on fluctuations in internal finance.

Key Words: Quantile regression; measurement errors, investment equation

JEL Classification: C14, C23, G31

∗The authors would like to express their appreciation to Stephane Bonhomme, Tim Conley, SilviaGoncalves, Stefan Hoderlein, Roger Koenker, Arthur Lewbel, Salvador Navarro, Yuya Sasaki, Susanne Schen-nach, Liang Wang, Zhijie Xiao, and seminar participants at Boston College, Syracuse University, Universityof Western Ontario, 2015 Midwest Econometrics Group, 2015 CMStatistics, 2016 Latin American Workshopin Econometrics of the Econometric Society, and 2016 North America Summer Meeting of the EconometricSociety for helpful comments and discussions. All the remaining errors are ours.†Insper, Rua Quata 300, Sao Paulo, SP 04546-042. E-mail: firpo@insper.edu.br‡Department of Economics, University of Iowa, W284 Pappajohn Business Building, 21 E. Market Street,

Iowa City, IA 52242. E-mail: antonio-galvao@uiowa.edu§Department of Economics, University of Iowa, W360 Pappajohn Business Building, 21 E. Market Street,

Iowa City, IA 52242. E-mail: suyong-song@uiowa.edu

1 Introduction

Quantile regression (QR) models have provided a valuable tool in economics and statistics as

a way of capturing heterogeneous effects of covariates on the outcome of interest, exposing

a wide variety of forms of conditional heterogeneity under weak distributional assumptions.

Under some assumptions on the unobservable factors, QR can also be interpreted as providing

a structural relationship between the outcome of interest and its observable and unobservable

determinants. Also importantly, QR provides a framework for robust inference.

Measurement errors (ME) have important implications for the reliability of general stan-

dard estimation and testing. Variables used in empirical analysis are frequently measured

with error, particularly if information is collected through one-time retrospective surveys,

which are notoriously susceptible to recall errors. Recently, the topic of ME in variables

has received considerable attention in the QR literature. The standard QR estimator suffers

from bias in the presence of ME (see, e.g., He and Liang (2000)). Chesher (2001) studies

the impact of covariate ME on quantile functions using a small variance approximation ar-

gument. Schennach (2008) investigates identification of a nonparametric quantile function

when there is an instrumental variable measured on all sampling units. Wei and Carroll

(2009) develop a QR method that corrects the ME bias by constructing joint estimating

equations that simultaneously hold for all quantile levels. The method makes use of an iter-

ative algorithm and requires parametric pre-specification of a conditional density to obtain a

consistent estimator for the parameters of interest. Hu and Schennach (2008) establish that

the availability of instruments enables the identification of nonclassical errors-in-variables

models. The resulting identification induces an estimating equation that encompasses QR

methods. Schennach (2014) studies a model defined by moment conditions with unobserv-

ables, which covers QR with ME. The parameters of interest are obtained by averaging

the moment functions over the unobservables using a least favorable distribution obtained

through an entropy maximization procedure.1,2

Thus, in the analysis of QR with mismeasured covariates, it has been common to employ

estimation methods that either impose parametric restrictions on nuisance functionals or use

1Notice that the conditions used in this paper for identification of the conditional density in the first stageare different from those in Hu and Schennach (2008) and Schennach (2014), and non-nested.

2Recently, Torres-Saavedra (2013) and Hausman, Luo, and Palmer (2014) study ME in the dependentvariable of QR models. We refer to Ma and Yin (2011), Wang, Stefanski, and Zhu (2012), and Wu, Ma, andYin (2015) for other recent developments in QR models with ME.

1

exogenous information as those provided by instrumental variables (see, e.g., Wei and Carroll

(2009), Schennach (2008), and Chernozhukov and Hansen (2006)). Nevertheless, methods

relying on parametric assumptions are very sensitive to misspecification of such conditions

in practical inference. In addition, finding exogenous instrumental variables is known to be

a nontrivial task in most empirical applications.

This paper contributes to both the QR and ME branches of the literature by develop-

ing estimation and inference methods for QR models in the presence of ME in the covari-

ates. This is achieved by exploring repeated measures of the true regressor.3 The main

contributions are the following. First, we propose a simple and easily-implementable two-

step semiparametric estimation procedure for QR models that preserves the semiparametric

distribution-free and heteroscedastic features of the model. The first step employs a general

nonparametric estimation of the density function. The second step uses the estimated densi-

ties as weights in a weighted QR estimation. For the first step, we propose a nonparametric

estimator for the conditional density without imposing distributional assumptions on the

ME. Specifically, we show that two mismeasured covariates are sufficient to identify the con-

ditional density of interest in the presence of ME. In turn, this result guarantees consistent

estimation of the structural parameters of interest.

In the second main contribution we establish the asymptotic properties of the two-step

estimator, assuming that the conditional densities satisfy smoothness conditions and can

be estimated at an appropriate nonparametric rate. The third contribution is to develop

practical statistical inference and testing procedures for general linear hypotheses based on

the Wald statistic. To implement these tests in practice the critical values are computed

using a bootstrap method. We provide sufficient conditions under which the bootstrap is

theoretically valid, and discuss an algorithm for its practical implementation. Our method

leads to a simple algorithm that can be conveniently implemented in empirical applications.

Compared to the existing procedures for QR models with ME, our approach has several

distinctive advantages. First, the method employs a nonparametric estimator in the first

step and does not assume global linearity at all quantile levels for the estimation of the

conditional density function. Such feature makes the procedure applicable to any τ -quantile

3Identification and estimation of conditional average regression models and conditional density withrepeated measures of the true regressor have been studied in Li and Vuong (1998), Li (2002), Schennach(2004a), Delaigle, Hall, and Meister (2008), and Hu and Sasaki (2015a) among others. We extend thisliterature to estimation and inference for QR models using repeated measures of the true regressor.

2

of interest, thus relaxing the requirement of a joint estimation and providing more flexibility.

In contrast, Wei and Carroll (2009) require a parametric pre-specification of a conditional

density for implementation of their estimator. Second, our algorithm is computationally

simple and easy to implement in practice because estimation of the weights does not require

recursive algorithms allowing the weights for all observations to be obtained from one single

step. As a result, the quantile estimation in the second step is attained by minimizing only

one single convex objective function at the quantile of interest. On the other hand, Wei

and Carroll (2009) make use of an iterative algorithm and require the estimating equations

to be solved jointly for all quantiles, which increases the dimensionality of the problem

substantially. Finally, the estimated weights exhibit a property of uniform consistency,

implying that it is feasible to establish both the consistency and asymptotic normality of

the resulting estimators of the parameters of interest. Hence, the method provides standard

inference and testing procedures.

Monte Carlo simulations assess the finite-sample properties of the proposed methods.

We evaluate the estimator in terms of empirical bias, standard deviation, and mean squared

error, and compare its performance with methods that are not designed for dealing with

ME issues. The experiments suggest that the proposed approach performs relatively well in

finite samples and effectively removes the bias induced by ME.

Our procedure should be useful for those empirical settings based on QR models in which

ME in the independent variables is a concern. To motivate and illustrate the applicability

of the methods, we apply the developed methods to Fazzari, Hubbard, and Petersen (1988)

investment equation model, where a firm’s investment is regressed on a proxy for investment

demand (Tobin’s q) and cash flows. This a well-known model in the corporate investment

literature.4 The QR approach is a useful tool in this example because it allows us to cap-

ture the heterogeneity in the Tobin’s q and cash flows along the conditional distribution of

investment. Concerns about measurement errors have been emphasized in the context of

the empirical investment model. Theory suggests that firm’s investment demand is captured

by marginal q, but this quantity is unobservable and researchers use instead its measur-

able proxy, average q. Because average q measures marginal q imperfectly, a measurement

problem naturally arises (see, e.g., Hayashi (1982), Poterba (1988), Erickson and Whited

4Following Fazzari, Hubbard, and Petersen (1988), investment-cash flow sensitivities became a standardmetric in the literature that examines the impact of financing imperfections on corporate investment (Stein(2003)).

3

(2000), and Almeida, Campello, and Galvao (2010)). Within that framework, finding valid

and strong instrumental variables to solve the endogeneity problem is not, in general, an

easy task.

In the empirical example we use a data set taken from Cummins, Hasset, and Oliner

(2006), where there are two measures of average q. The first measure conforms with prior

research and is constructed using the standard equity prices. The second proxy for the firm’s

intrinsic value is based on analysts’ earnings expectations. Thus, our method is a natural

alternative solution to the ME problem where repeated measures on Tobin’s q are available.

The results document strong evidence of substantial heterogeneity in the sensitivity of in-

vestment to Tobin’s q and cash flow across the conditional distribution of investment. The

empirical results show that larger cash flow sensitivity occur at the lower part of the invest-

ment distribution, showing evidence that these firms are more exposed to and dependent on

fluctuations in internal finance. Our empirical findings support the idea that the proposed

methods are a useful alternative to existing approaches in economic applications in which

ME is an important concern.

The rest of the paper is organized as follows. Section 2 presents the model and discusses

identification of the parameters of interest in presence of ME. Section 3 proposes the two-step

QR estimator. Section 4 establishes the asymptotic properties of the estimator. Inference

is discussed in Section 5. Section 6 presents the Monte Carlo experiments. In Section 7, we

illustrate empirical usefulness of the the new approach with an application to the investment

equation model. Finally, Section 8 concludes the paper.

2 Model and identification

2.1 Model

We first introduce the model studied in this paper. Given a quantile τ ∈ (0, 1), we define

the following quantile regression (QR) model,

Yi = Xiβ0(τ) + Z>i δ0(τ) + εi(τ), (1)

where Yi is the scalar dependent variable of interest, Xi is a potentially-mismeasured scalar

continuous covariate,5 Zi is a L-vector of correctly-observed covariates, and εi(τ) is the inno-

5Below we describe the extension to the multi-dimensional case.

4

vation term whose τ -th quantile is zero conditional on (Xi, Zi). The structural parameters

of interest are θ0(τ) = (β0(τ), δ0(τ)>)>. In general, each β0(τ) and δ0(τ) will depend on

τ , but we assume τ to be fixed throughout the paper and suppress such a dependence for

notational simplicity.

Suppose (Yi, Xi, Zi) are i.i.d. random variables defined on a complete probability space

(Ω,F, P ). Denote the support of a random variable by supp(·). Define the population

objective function for the τ -th conditional quantile as

Q(β0, δ0) := E[ψτ (Yi −Xiβ0 − Z>i δ0)[Xi Zi]

]= 0, (2)

where ψτ (u) := (τ − Iu < 0) with the indicator function I·. When the true covariates

(X,Z) are observed, β0 and δ0 in (1) can be consistently estimated from the standard quantile

regression model with sample analog of Q(β, δ) in (2) as

Qn(β, δ) :=1

n

n∑i=1

ψτ (Yi −Xiβ − Z>i δ)[Xi Zi] = 0. (3)

The presence of the indicator function in equation (3) implies that the solution may not be

an exact zero. It is usual to write the estimator as a minimization problem, and use linear

programming to solve the optimization. Thus, the above moment condition is a slight abuse

of notation, but since everything else involving observed data is an estimating equation that

will have a zero, we will use the estimating equation nomenclature. For more details on

Z-estimator with non-smooth objective functions, see e.g., He and Shao (1996, 2000).

2.2 Measurement error bias and its solution

Under the assumption of perfectly-measured regressors, the solution of (3) can be shown

to produce consistent estimates of (β0, δ0). Nevertheless, it is commonly observed that re-

searchers have to use the regressor X measured with error. Using mismeasured X in the

standard QR estimation in (3) induces bias in the estimates of the coefficients of interest

(see, e.g., He and Liang, 2000). Thus, estimation of the standard QR model under measure-

ment errors (ME) leads to inconsistent estimates. To overcome this drawback we propose a

methodology that makes use of repeated measures. Both variables are mismeasured observ-

ables of the true covariate.

Suppose that true covariate X is unobservable due to ME. Instead, a researcher observes

5

two error-laden measurements which are noisy measures of X and defined as follows

X1i = Xi + U1i

X2i = Xi + U2i,

where U1i and U2i are ME. Therefore, the observed random variables are (Yi, X1i, X2i, Zi),

and one seeks to estimate the parameters (β0, δ0).

We show how to use information from the measures X1 and X2 to obtain consistent

estimates of the parameters of interest. For that purpose, it is useful to rewrite Q(β, δ) as a

function of the density function as well as (β, δ):

Q(β0, δ0, f0) := E[ψτ (Y −Xβ0 − Z>δ0)[X Z]]

=

∫ψτ (y − xβ0 − z>δ0)[x z] · fY XZ(y, x, z)dydxdz

=

∫ψτ (y − xβ0 − z>δ0)[x z] · fX|Y Z(x | y, z)fY Z(y, z)dydxdz

= E

[∫x

ψτ (Y − xβ0 − Z>δ0)[x Z] · fX|Y Z(x | Y, Z)dx

](4)

= 0,

where fY XZ(y, x, z) and fY Z(y, z) are the joint density of (Y,X,Z) and (Y, Z), respectively,

and where fX|Y Z(x | y, z) ≡ f0 is the conditional density of X given (Y, Z). By replacing

the outer expectation with its empirical counterpart, we write the sample analog of the

population objective function (4) as:

Qn(β, δ, f) :=1

n

n∑i=1

∫x

ψτ (Yi − xβ − Z>i δ)[x Zi] · fX|Y Z(x | Yi, Zi)dx (5)

= 0.

The integration in (5) makes the function continuous in its argument. The summand of

(5) is Ex[ψτ (Yi−xβ−Z>i δ)[x Zi] | Yi, Zi], the conditional mean of the original score function

given the observed Y and Z. Moreover, (5) is an unbiased estimating function, that is, has

mean zero, and will be the basis for constructing estimating equations to obtain consistent

estimates of the parameters of interest.

Therefore, one would solve the new estimating equation (5) to estimate the parameters

of interest. In empirical applications, however, the true conditional density fX|Y Z(x | y, z)is unknown and to implement the estimator (5) in practice one needs to replace it with

6

fX|Y Z(x | y, z), a consistent estimate of fX|Y Z(x | y, z). Thus, a (feasible) estimator would

first estimate fX|Y Z(x | y, z). The fitted density function from this step would be used to

estimate the coefficients of interest in a second step. Finally, with a consistent estimate

of the conditional density, (β0, δ0) can be consistently estimated. However, in general, the

conditional density is not identified due to the unobservability of the true X.

In this paper, we make use of the repeated measures, X1 and X2, and show that two

mismeasured covariates are sufficient to identify the conditional density in the presence

of ME on the covariate. We also propose a nonparametric estimator for the conditional

density without imposing assumptions on known distributions of the ME. In turn, this result

guarantees consistent estimation of the structural parameters of interest.

In a related model, Wei and Carroll (2009) make use of an iterative algorithm to obtain a

consistent estimator of the conditional density fX|Y X1(x | y, x1) in the presence of ME on X.6

They focus on model with one measurement of true X (here X1) and with no other observed

covariates Z for simplicity. Although their approach can be useful in some applications, it has

important technical challenges. First, to implement the estimator, one needs to estimate the

conditional density fX|Y X1(x | y, x1) which requires pre-specified parametric form of fX|X1(x |x1). This suffers from potentially serious model misspecification. Second, and related to the

first problem, there is a problem to solve the estimating equations, since estimating the

conditional density fX|Y X1(x | y, x1) involves estimation of the entire process β0(τ) over

quantiles τ . In other words, the estimating equations in Wei and Carroll (2009) need to be

solved jointly for all the τ ’s, which increases the dimensionality of the problem substantially

and makes implementation considerably difficult. This is reflected in the tractability of

inference for their method.

In the next section we propose a procedure that yields a consistent estimator of (β0, δ0)

in (5). We develop a method for QR with measurement errors, which relies on estimating

the conditional density function nonparametrically. The method is a two-step estimator,

where the first step estimates the density nonparametrically, and the second step employs a

standard weighted QR procedure. Before proceeding to estimation, we show an identification

result for the density function which is essential in the estimation. For expositional ease, we

use fX|Y Z(x | y, z) and f(x | y, z) synonymously.

6We note that their conditional density is slightly different from ours since there is a mismeasured covari-ate, X1, in their conditioning set.

7

2.3 Conditional density

As described above, f(x | y, z) is an important element for the identification of the pa-

rameters of interest in the QR with ME. This section describes the identification of the

conditional density function f(x | y, z) which is required to compute the two-step estimator.

The identification is based on the assumption that repeated measures of the true regressor

are observed. We state the following assumptions to obtain the main identification result.

Assumption A.I: (i) E[U1 | X,U2] = 0; (ii) U2 ⊥ (Y,X,Z).

Assumption A.II: (i) E[|X|] < ∞; (ii) E[|U1|] < ∞; (iii) |E[exp(iζX2)]| > 0 for any

finite ζ ∈ R.

Assumption A.III: (i) sup(x,y,z)∈supp(X,Y,Z) f(x | y, z) < ∞; (ii) f(x | y, z) is integrable

on R for each (y, z) ∈ supp(Y, Z).

Assumption A.I imposes restrictions on the repeated measures of X. Assumption A.I

(i) requires conditional mean zero of ME on X1, but allows dependence of the ME and

(X,U2). Assumption A.I (ii) requires that ME on X2 is independent of true X as well as

other variables. However, it does not necessarily require zero mean of U2. Thus, our setting

on the repeated measures can be useful for an example such that there is a drift or trend

in the mismeasured covariates. Assumption A.II imposes mild restrictions on the existence

of the first moments of X and U1, and nonvanishing characteristic function of X2. These

have been commonly assumed in the deconvolution literature (see, e.g., Fan (1991) and Fan

and Truong (1993)). Assumption A.III is trivially satisfied in commonly-used conditional

densities.

Let φ(ζ, y, z) ≡ E[eiζX | Y = y, Z = z] be the conditional characteristic function of X

given Y and Z. The following theorem presents the identification of f(x | y, z).

Theorem 1 Suppose Assumptions A.I–A.III hold. Then, for (x, y, z) ∈ supp(X, Y, Z),

f(x | y, z) =1

2π

∫φ(ζ, y, z) exp(−iζx)dζ, (6)

where for each real ζ,

φ(ζ, y, z) =E[eiζX2 | Y = y, Z = z]

E[eiζX2 ]exp

(∫ ζ

0

iE[X1eiξX2 ]

E[eiξX2 ]dξ

).

8

Proof. See Appendix.

The theorem implies that conditional density f(x | y, z) can be written as a function of

purely-observed variables. For this, we use useful properties of Fourier transform. Namely, we

write f(x | y, z) as the inverse Fourier transform of φ(ζ, y, z). This simplifies identification

since φ(ζ, y, z) is easily identified from Assumptions A.I–A.III by removing the ME, U1

and U2, in the frequency domains (ζ, ξ). It is worth noting that the identification result is

similar to Kotlarski (1967) who identifies the density of X from its repeated measurements by

assuming mutual independence of X, U1, and U2. Our approach rests on weaker assumptions

than their mutual independence, which is highlighted in condition A.I. In addition, the result

in Theorem 1 is related to that in Schennach (2004b) who identifies the conditional mean

of Y given X. Since we are interested in conditional quantile effects we need to identify the

conditional density of X, which requires a stronger independence condition (A.I(ii)).

We note that the current identification setup can be extended to a K-vector of X. In

this case, the integral in equation (5) becomes a multiple integral over the K-vector of X.

In addition, the integral in equation (6) should be an integral over a K-vector of ζ, while the

integral in the definition of φ(·) should be over a piecewise smooth path connecting 0 and ζ,

where ξ and ζ are K-dimensional vectors. The estimating equations are simply adjusted to

these corresponding changes.

3 Estimation

Given the identification condition in equation (6) of Theorem 1, we are able to estimate

the structural parameters of interest, (β0, δ0). We propose a semiparametric estimator that

involves two-step estimation. Implementation of the estimator is simple in practice. In the

first step, one estimates the nuisance parameter, the conditional density, using a nonpara-

metric method which requires no optimization. In the second step, by plugging in these

estimates, a general weighted quantile regression (QR) is performed.

Specifically in equation (6), we can rewrite Q(β, δ, f) as:

Q(β, δ, f) = E

[∫x

ψτ(Y − xβ − Z>δ

)[x Z] · fX|Y Z(x | Y, Z)dx

],

which does not depend on data on X. Thus, estimation of (β0, δ0) follows from solving a

9

feasible version of Qn(β, δ, f):

Qn(β, δ, f) =1

n

n∑i=1

∫x

ψτ(Yi − xβ − Z>i δ

)[x Zi] · fX|Y Z(x | Yi, Zi)dx,

where

fX|Y Z(x | Yi, Zi) =1

2π

∫ζ

φ(ζ, Yi, Zi) exp(−iζx)dζ,

and the only feature of this sample objective function that has not yet been presented is φ,

the estimate of φ, which is defined in the next section. In practice, as we discuss next, we

approximate integrals by sums, and end up with a double sum (on observations and on grid

values of X). Importantly on that representation is the fact that the estimates (β, δ) will be

obtained by a weighted QR, whose weights will be given by the estimate fX|Y Z .

3.1 Estimation of nuisance parameter

In this subsection we discuss the estimation of the nuisance parameter in the first step, i.e.,

the conditional density f(x | y, z). We propose a nonparametric method by adapting the

class of flat-top kernels of infinite order suggested by Politis and Romano (1999). Consider

the following assumption.

Assumption A.IV: The real-valued kernel x→ k(x) is measurable and symmetric with∫k(x)dx = 1, and its Fourier transform ξ → κ(ξ) is bounded, compactly supported, and

equal to one for |ξ| < ξ for some ξ > 0.

From Assumption A.IV, we allow for a kernel of the form (e.g., Li and Vuong (1998))

k(x) =sin(x)

πx,

with its Fourier transform such that

κ(hxζ) =

∫1

hxk( xhx

)exp(iζx)dx,

for a bandwidth hx. This flat-top kernel of infinite order has the property that its Fourier

transform is equal to one over the [−1, 1] interval and zero elsewhere, which guarantees

that the bias goes to zero faster than any power of the bandwidth. We note that the ill-

posed inverse problem occurs when one tries to invert a convolution operation. This is

10

true to our proposed estimator because it is divided by a quantity which converges to zero

as the frequency parameter goes to infinity by Riemann-Lebesgue lemma. By estimating

the numerator using the kernel whose Fourier transform is compactly supported, one can

guarantees that the ratio is under control. This is because that the numerator can decay

to zero before the denominator converges to zero. This compact support of the Fourier

transform of the kernel can be easily implemented by preserving most of the properties of

the original kernel. For instance, one can transform any given kernel k into a modified

kernel k with compact Fourier support by using a window function that is constant in the

neighborhood of the origin and vanishes beyond a given frequency.

The following theorem summarizes the result.

Theorem 2 Suppose Assumptions A.I–A.III hold, and let k(·) satisfy Assumption A.IV.

For (x, y, z) ∈ supp(X, Y, Z) and hx > 0, let

f(x | y, z;hx) ≡∫

1

hxk

(x− xhx

)f(x | y, z)dx.

Then we have

f(x | y, z;hx) =1

2π

∫κ(hxζ)φ(ζ, y, z) exp(−iζx)dζ.

Proof. See Appendix.

Let hn ≡ (hxn, h(2)n ) with h

(2)n ≡ (hyn, h

zn) be a set of smoothing parameters. Let E[·] denote

a sample average, i.e., 1n

∑ni=1[·]. Finally, we introduce a consistent nonparametric estimator

of f(x | y, z) motivated by Theorem 2.

Definition 2.3 The estimator of f(x | y, z) is defined as

f(x | y, z;hn) ≡ 1

2π

∫κ(hxnζ)φ(ζ, y, z, h(2)

n ) exp(−iζx)dζ, (7)

for hn → 0 as n→∞, where

φ(ζ, y, z, h(2)n ) ≡ E[eiζX2 | Y = y, Z = z]

E[eiζX2 ]exp

(∫ ζ

0

iE[X1eiξX2 ]

E[eiξX2 ]dξ

).

The above estimator is useful to compute the structural parameters of interest. Since it

has an explicit closed form, it requires no optimization routine unlike other likelihood-based

11

approaches. Estimation of conditional mean, E[eiζX2 | Y = y, Z = z], can be achieved via

any nonparametric method. For instance, one might use popular kernel estimation with

khn(·) ≡ h−1n k (·/hn) (e.g., Epanechnikov kernel) defined as

E[eiζX2 | Y = y, Z = z] ≡E[eiζX2khyn(Y − y)khzn(Z − z)]

E[khyn(Y − y)khzn(Z − z)]],

where khyn(Y − y) is a univariate kernel function and khzn(Z − z) is a L-multivariate kernel

function with corresponding bandwidths hyn and hzn, respectively.

3.2 Estimation of the structural parameters

This section describes the general estimator for QR models with ME. We propose a Z-

estimator that involves two-step estimation. Given the identification condition in equation

(5) and the estimator of the density function described in the previous section, we are able to

estimate the structural parameters of interest. We estimate the parameters, θ0 = (β0, δ>0 )>,

for a selected τ of interest, from the following two steps:

Step 1. Estimate f(xj | Yi, Zi;h) for each i-th observation and j-th grid as in equation

(5) where j ∈ J ≡ 1, 2, . . . ,m with m number of grids for approximating the numer-

ical integral. The choice of kernels and bandwidths are provided in Definition 2.3 above.

The integrals in equation (7) are performed using the fast Fourier transforms (FFT) algo-

rithm. Well-behaving performance of the algorithm is guaranteed by the smoothness of the

characteristic function φ(·) and the finiteness of the moments.

Step 2. To compute equation (5) in practice, we perform a numerical approximation to

the integral over x. We do this via translating the problem into a weighted QR problem. Let

x = (x1, x2, ..., xm) be a fine grid of possible xj values, akin to a set of abscissas in Gaussian

quadrature. For each τ , θ(τ) = (β(τ), δ(τ)>)> can be computed by solving

n∑i=1

m∑j=1

ψτ (Yi − xjβ − Z>i δ)[xj Zi] · f(xj | Yi, Zi;h) = 0, (8)

where f(xj | Yi, Zi;h) is obtained from Step 1. The weighted QR of Yi on xj and Zi with

corresponding weights f(xj | Yi, Zi;h) can be readily computed using the function called

“rq” in R package quantreg.

12

4 Asymptotic properties

This section investigates the large sample properties of the proposed two-step estimator.

While these methods seem similar to the ones discussed by Wei and Carroll (2009), the

nonparametric estimation of the conditional density function raises some new issues for the

asymptotic analysis of the estimator. First, we establish the asymptotic results for the

estimator of the conditional density function given in (7). Second, we establish consistency

and asymptotic normality of the second step estimator in (8).

4.1 Asymptotic properties of the density estimator

In this subsection we establish the asymptotic properties of the density function estimator

in equation (7). Let µ(ζ) ≡ E[eiζX ], ω1(ζ) ≡ E[eiζX2

], and χ(ζ, y, z) ≡ E[eiζX2 | Y = y, Z =

z]f(y, z). We denote the kernel density estimator of f(y, z) by f(y, z) ≡ E[khyn(Y −y)khzn(Z−z)]]. Let X ≡ R be the support of X, and Y ×Z be a compact set contained in the support

of (Y, Z), and Dζ denote a partial derivative with respect to ζ. We impose the following

assumptions.

Assumption B.I: (i) There exist constants C1 > 0 and γµ ≥ 0 such that

|Dζ lnµ(ζ)| =∣∣∣∣Dζµ(ζ)

µ(ζ)

∣∣∣∣ ≤ C1(1 + |ζ|)γµ ;

(ii) There exist constants Cφ > 0, αφ ≤ 0, νφ ≥ 0, and γφ ∈ R such that νφγφ ≥ 0 and

sup(y,z)∈Y×Z

|φ(ζ, y, z)| ≤ Cφ(1 + |ζ|)γφ exp(αφ|ζ|νφ),

and if αφ = 0, then γφ < −1;

(iii) There exist constants Cω > 0,αω ≤ 0, νω ≥ νφ ≥ 0, and γω ∈ R such that νωγω ≥ 0 and

min inf(y,z)∈Y×Z

|χ(ζ, y, z)|, |ω1(ζ)| ≥ Cω(1 + |ζ|)γω exp(αω|ζ|νω).

Assumption B.II: (i) E[|X1|2] <∞; (ii) E[|X1||X2|] <∞; (i) E[|X2|] <∞.

Assumption B.III: sup(y,z)∈Y×Z |f(y, z)− f(y, z)| = Op

((lnn)1/2

(nhyhz)1/2+∑

s∈y,z(hs)2)

.

Assumptions B.I–B.III are standard for nonparametric deconvolution estimators, since

the rates of convergence depend on the tails of the Fourier transforms (see, e.g., Fan (1991)

and Fan and Truong (1993)).

13

Assumption B.I concerns the smoothness of the (conditional) characteristic functions of

the true regressor X and the observed measure X2. The literature commonly adopts two

types of smoothness assumptions: ordinary and super smoothness. Ordinary smoothness

admits a Fourier transform whose tail decays to zero at a geometric rate |ζ|γ, γ < 0 whereas

super smoothness admits a Fourier transform whose tail decays to zero at an exponential

rate exp (α |ζ|γ), α < 0, γ > 0.7 Assumption B.I simultaneously imposes ordinary and super

smoothness conditions for notational ease. Assumption B.I (i) imposes a restriction on the

tail behavior of µ(ζ), the characteristic function of X. A term exp (α1 |ζ|ν1) is omitted with

merely a small loss of generality since lnµ (ζ) is indeed a power of ζ for all common ordinary

and super smoothness functions. Assumption B.I (ii) concerns φ(ζ, y, z), the conditional

characteristic function of X given Y = y and Z = z. It states that the rate of decay

of φ(ζ, y, z) is governed by the conditional density of X given Y = y and Z = z (i.e.,

f(X | Y = y, Z = z)), the parameter of interest in the first step. Assumption B.I (iii)

jointly imposes a restriction on χ(ζ, y, z), the conditional characteristic function of X2 given

Y = y and Z = z weighted by the joint density of (Y, Z) (i.e., E[eiζX2 | Y = y, Z = z]f(y, z)),

and ω1(ζ), the characteristic function of X2. Since ω1(ζ) = E[eiζX2 ] = E[eiζX ]E[eiζU2 ], the

smoothness of ω1(ζ) is determined by the combination of X and U2. As commonly imposed

in deconvolution-type estimators, χ(ζ, y, z) and ω1(ζ) need to be bounded below because

they appear in the linearization of the estimator. The restriction on χ(ζ, y, z) typically

requires the joint density of (Y,X2, Z) to be bounded away from zero on its support. This

is generally satisfied by common continuous distributions supported on the real line.

Assumption B.II imposes mild moment restrictions required for consistency. Assumption

B.III imposes a standard condition on nonparametric estimator of the joint density of f(y, z).

The next result establishes the asymptotic properties of the density function estimator.

Theorem 3 Let Assumptions A.I–A.IV and B.I–B.III hold. Then for (x, y, z) ∈ X ×Y × Z and h > 0 satisfying max(hyn)−1, (hzn)−1 = O (nη) and

(hxn)−1 = O((lnn)1/νω−η

)if νω 6= 0,

(hxn)−1 = O(n(1−20η)/2(γµ−γω)

)if νω = 0,

7The typical examples of ordinarily smooth functions are uniform, gamma, symmetric gamma, Laplace(or double exponential), and their mixtures. Normal, Cauchy, and their mixtures are super smooth functions.

14

for some η > 0, we have

sup(x,y,z)∈X×Y×Z

|f(x | y, z;h)− f(x | y, z)|

= O((

(hx)−1)γB exp

(αB((hx)−1

)νB))+Op

(n−1/2((hx)−1)γL exp

(αL((hx)−1

)νL)) ,with αB ≡ αφξ

νφ, νB ≡ νφ, γB ≡ γφ + 1, αL ≡ αφ1νφ=νω − αω, νL ≡ νω, γL ≡ 1 + γφ − γω,

and δL ≡ γµ + γφ − γω + 2.

Proof. See Appendix.

The theorem above establishes consistency and the uniform convergence rate of the pro-

posed estimator. The conditions on the bandwidths are imposed to guarantee that asymp-

totic behavior of the linear approximation of the expression f(x | y, z;h) − f(x | y, z) is

essentially determined by a variance term since a nonlinear remainder term is asymptoti-

cally negligible. The result also shows that convergence rate depends on the tail behaviors of

the associated quantities. For instance, consider the case that φ(ζ, y, z) is ordinarily smooth.

When χ(ζ, y, z) and ω1(ζ) in Assumption B.I is also ordinarily smooth (i.e., νω = 0), one

can choose smaller bandwidth so that resulting convergence rate of the estimator is faster

than when one of them is super smooth.

We note that the uniform convergence rate for the first stage is obtained over the compact

set Y ×Z contained in the support of (Y, Z). However, we highlight that this does not nec-

essarily imply that the random variables must be compactly supported (see, e.g., Schennach

(2008)). Alternatively, it is possible to obtain a uniform convergence rate and establish the

asymptotic properties over an expanding set (see, e.g., Newey (1994b), Andrews (1995), and

Hansen (2008)). We consider uniformity over expanding sets that diverge slowly to infinity.

Detailed derivation of the uniform convergence rate over expanding sets is provided in the

Online Supplementary Appendix, and the results in Theorem 3 remain the same.

4.2 Asymptotic properties of the two-step estimator

In this subsection, we derive the asymptotic properties of the second step estimator of

parameters of interest in (8). We establish consistency and asymptotic normality for a

given quantile τ ∈ (0, 1).

15

4.2.1 Consistency

Consistency is a desirable property for most estimators. We wish to establish consistency of

the estimator θ = (β, δ>)> defined in equation (8), where f , given in (7), is an estimator of

f0 = f(x | y, z).

First, notice that from the estimating equation in (5) we have

Qn(β, δ, f) =1

n

n∑i=1

∫ψ(Yi − xβ − Z>i δ)(x Zi) · f(x | Yi, Zi) dx,

and its expectation is

Q(β, δ, f) = E

∫ψ(Yi − xβ − Z>i δ)(x Zi) · f(x | Yi, Zi) dx.

The estimator θ = (β, δ>)> is obtained by equating Qn(β, δ, f) to zero. Note that Q(β, δ, f0) =

0 if and only if (β, δ>)> = (β0, δ>0 )> ∈ Θ.

Now we state the following sufficient conditions for the estimator to be consistent.

Assumption C.I: For any δ > 0, there exists εδ > 0 such that

inf ||θ−θ0||>δ ||E[∫xψτ (Y − xβ − Z>δ)[x Z] · f0 dx

]|| > εδ.

Assumption C.II: (i) E[|X|2+ε] <∞; (ii) E[‖Z‖2+ε] <∞.

Assumption C.III: E[|X| | Y, Z] <∞.

Assumption C.I is a standard identification condition in the QR literature (see, e.g.,

Chen, Linton, and Van Keilegom (2003) and Kato, Galvao, and Montes-Rojas (2012)). C.II

is also standard in QR, and requires the second moment of latent variable and well-measured

regressor to be finite, see e.g., Angrist, Chernozhukov, and Fernandez-Val (2006) and Koenker

(2005). C.III requires the first conditional moment of the latent variable to be finite.

The following theorem derives consistency of the proposed estimator, θ = (β, δ>)>.

Theorem 4 Suppose that θ0 is the unique solution of E[∫xψτ (Y − xβ0 − Z>δ0)[x Z] · f0 dx

]=

0, and assumptions C.I–C.III, and conditions of Theorem 3 are satisfied. Then, as n→∞

θp→ θ0.

16

Proof. See Appendix.

A uniform law of large numbers for the first step estimator, f(x | y, z), is a standard

requirement in two-step estimation literature to establish consistency of the second step

estimator; see, e.g., Newey and McFadden (1994). We note that, for the proposed estimator,

this requirement is satisfied by Theorem 3.

4.2.2 Asymptotic normality

Now we derive the limiting distribution of the two-step estimator in (8). Let || · ||∞ be

the supremum norm over the argument. We impose the following assumptions to establish

asymptotic normality.

Assumption G.I: The distribution function GY (y | X = x, Z = z) is absolutely contin-

uous, with continuous densities gY (y | X = x, Z = z) uniformly bounded away from 0 and

∞.

Assumption G.II: Let Γ1 := E[gY (Xβ0 + Z>δ0 | X,Z)(X,Z)(X,Z>)>] be positive

definite and Vn := var[Qn(θ0)]. There exists a nonnegative definite matrix V such that

Vn → V as n→∞.

Assumption G.III: ||f − f0||∞ = op(n−1/4).

Assumption G.IV: The function f(x | y, z) ∈ F is a uniformly smooth function with

the uniform smoothness order δ > dim(x, y, z)/2, and Lipschitz.

Conditions G.I and G.II are standard in the QR literature; see, e.g., Koenker (2005).

Condition G.III imposes that the estimator of the nuisance parameter converges at a

rate faster than n−1/4. A similar condition appears in condition (2.4) in Theorem 2 of Chen,

Linton, and Van Keilegom (2003). We note that Assumption G.III is verifiable for particular

examples through Theorem 3. As shown in Theorem 3, the convergence rate is controlled

by the smoothness of quantities such as φ(ζ, y, z), χ(ζ, y, z), and ω1(ζ). Thus, the rate of

convergence depends on the possible combinations of smoothness of these quantities. For

instance, if φ(ζ, y, z) is ordinarily smooth and if χ(ζ, y, z) and ω1(ζ) are super smooth, a

convergence rate of the form (lnn)−υ for some υ > 0 is achieved. This case illustrates a very

17

slow rate of convergence. On the other hand, a faster convergence rate, n−υ for some υ > 0,

which satisfies Assumption G.III, can be achieved when φ(ζ, y, z) is also super smooth. In

addition, if all three quantities, φ(ζ, y, z), χ(ζ, y, z), and ω1(ζ), are ordinarily smooth, the

slow convergence problem is easily avoided. To illustrate this in a specific example, consider

the case that φ(ζ, y, z) and ω1(ζ) are supersmooth and both follow a normal distribution with

variance σ2. Then, νφ = νω 6= 0 in Assumption B.I. From Theorem 3, one can show that

the convergence rate is nαB

2(αL−αB)+η

for some small η > 0. Because the normal distribution

has the characteristic function whose the tail is of the form exp (−(σ2/2)|ζ|2), we have

αφ = αω = −σ2/2 and νφ = νω = 2. As a result, the convergence rate becomes n−12

+η

which is fast enough to achieve n−14 in Assumption G.III.8 Therefore, the required rate of

convergence in G.III is attainable under proper combination of smoothness conditions (see,

e.g., Fan (1991), Fan and Truong (1993), and Schennach (2004b)).

Assumption G.IV is a smoothness condition on the conditional density function. A

similar assumption appears in Newey (1994a) and Chernozhukov and Hansen (2006). Con-

dition G.IV allows for a wide variety of nonparametric estimators, including the estimator

described in Section 4.1 above. The role of G.IV is to allow for an estimated density, which

is the weight in the estimating equation in (8). This condition together with Theorem 3

ensures that the weight is asymptotically well behaved to obtain the limiting distribution of

the estimator of the structural parameters.

Asymptotic normality of the estimator, θ = (β, δ>)>, is established in the following result.

Theorem 5 Suppose that θ0 is the unique solution of E[∫xψτ (Y − xβ0 − Z>δ0)[x Z] · f0 dx

]=

0, and assumptions C.I–C.III, G.I–G.IV, and conditions of Theorem 3 are satisfied. Then,

as n→∞

√n(θ − θ0)

d→ N(0,Λ)

for some positive definite matrix Λ = Γ−11 V Γ−1

1 .

Proof. See Appendix.

8Similarly, the convergence rate satisfying Assumption G.III can be proved for the case of an ordinarysmooth distribution. For instance, consider the Laplace distribution with mean µ and variance σ2. The tailof the characteristic function of a Laplace distribution is of the form |ζ|−2. To establish the results, first,the convergence rate can be pinned down, with νω = 0. Second, the condition can be derived by plugging inγφ = γω = −2.

18

It is worth noting that the density estimator in equation (7) requires multivariate kernel

functions when X and Z are multi-dimensional. This generates slower rate of convergence

of the density estimator. As in the conventional kernel density estimation, it would be

difficult to resolve the problem associated with large dimensionality of X. On the other

hand, one could use semiparametric single index models to mitigate the problem from large

dimensionality of Z. The convergence rate in the second stage would not be affected by the

multi-dimensionality of X and Z as long as Assumption G.III still holds.

5 Inference

In this section, we turn our attention to inference in the quantile regression (QR) with mea-

surement errors (ME) model. Important questions posed in the econometric and statistical

literatures concern the nature of the impact of a policy intervention or treatment on the

outcome distributions of interest; for example, whether a policy exerts a significant effect,

a constant versus heterogeneous effect, or a non-decreasing effect. It is possible to formu-

late a wide variety of tests using variants of the proposed method, from simple tests on a

single QR coefficient to joint tests involving many covariates simultaneously. We suggest a

bootstrap-based inference procedure to test general linear hypotheses.

5.1 Test statistic

General hypotheses on the vector θ(τ) = (β(τ), δ(τ)>)> can be accommodated by standard

tests. The proposed statistic and the associated limiting theory provide a natural foundation

for the hypothesis Rθ(τ) = r. Consider the following null hypothesis for a given τ of interest

H0 : Rθ(τ)− r = 0,

where R is a full-rank matrix imposing q number of restrictions on the parameters, and r is

a column vector of q elements.

The following are examples of hypotheses that may be considered in this framework.

Example 1 (No effect of the mismeasured variable). For a given τ , if there is no mis-

measured variable effect in the model, then under H0 : β(τ) = 0, θ(τ) = (β(τ), δ(τ)>)>,

R = [1 0] and r = 0.

19

Example 2 (Location shifts). The hypotheses of location shifts for β(τ) and δ(τ) can

be accommodated in the model. For the first case, H0 : β(τ) = β, so θ(τ) = (β(τ), δ(τ)>)>,

R = [1 0] and r = β. For the latter case, H0 : δ(τ) = δ, so that R = [0 1] and r = δ.

Practical implementation of testing procedures for the null hypothesis can be carried out

based on the following test statistic

Wn(τ) = Rθ(τ)− r. (9)

From Theorem 5, at given τ , and under the null hypothesis, it follows√n(Rθ(τ)− r) d→

N(0, RΛR>). If we are interested in testing H0, a Chi-square test could be conducted based

on the statistic in equation (9). However, to carry out practical inference procedures, even for

a fixed quantile of interest, and construct a Wald statistic, one would need to first estimate

Λ consistently, and consequently nuisance parameters which depend on both the unknown θ0

and f0 in a complicated way. The estimation of Λ is potentially difficult because it contains

additional terms from the effect of θ on the objective function indirectly through f0. An

alternative method is to use the statistic Wn directly and the bootstrap to compute critical

values and also form confidence regions. Therefore, to make practical inference we suggest

the use of bootstrap techniques to approximate the limiting distribution.

5.2 Implementation of testing procedures

Practical implementation of the proposed tests is simple. To test H0, one needs to compute

the test statistics Wn(τ) for a given τ of interest. The steps for implementing the tests are

as following:

First, the estimates of θ(τ) are computed by solving the problem in equation (8). Second,

Wn(τ) is calculated by centralizing θ(τ) at r. Third, after obtaining the test statistic, it is

necessary to compute the critical values. We propose the following scheme. Take B as a

large integer. For each b = 1, . . . , B:

(i) Obtain the resampled data (Y bi , X

b1i, X

b2i, Z

bi ), i = 1, . . . , n.

(ii) Estimate θb(τ) and set W bn(τ) := R(θb(τ)− θ(τ)).

(iii) Go back to step (i) and repeat the procedure B times.

20

Let cB1−α denote the empirical (1 − α)-quantile of the simulated sample W 1n , . . . ,W

Bn ,

where α ∈ (0, 1) is the nominal size. We reject the null hypothesis if Wn is larger than cB1−α.

Confidence intervals for the parameters of interest can be easily constructed by inverting the

tests described above.

We provide a formal justification of the simulation method. Let f ∗ be the bootstrap

version of the estimator f . Consider the following condition.

Assumption G.IB: With P ∗-probability tending to one, ||f ∗ − f ||∞ = op∗(n−1/4).

Condition G.IB is a condition on the density estimation with the bootstrapped sample

and could be verified under the same assumptions implying condition G.III.

Lemma 1 Under Assumptions C.I–C.III, G.I–G.IV, G.IB with “in probability” replaced

by “almost surely”, and conditions of Theorem 3, the bootstrap estimator of the θ0 is√n-

consistent and√n(θ∗ − θ) d→ N(0,Λ) in P ∗-probability.

Proof. See Appendix.

Lemma 1 establishes the consistency of the bootstrap procedure. It is important to

highlight the connection between this result and the previous section. Lemma 1 shows that

the limiting distribution of the bootstrap estimator is the same as that of Theorem 5, and

hence the above resample scheme is able to mimic the asymptotic distribution of interest.

Thus, computation of critical values and practical inference are feasible.

6 Monte Carlo simulations

6.1 Monte Carlo design

In this section, we describe the design of a small simulation experiment that has been con-

ducted to assess the finite-sample performance of the proposed two-step estimator discussed

in the previous sections. We consider the following model as a data generating process:

Yi = β1 + β2Xi + εi,

21

where ε ∼ N(0, 0.25), and β1 and β2 are the parameters of interest.9 We set them as

(β1, β2) = (0.5,−0.5). The true variable X is not observed by the researcher, and we use

additive forms of measurement errors (ME) to generate the mismeasured X as follows:

X1i = Xi + U1i,

X2i = Xi + U2i,

where we generate X ∼ N(0, 1), and we use a Laplace distribution density as L(0, 0.25)

to generate both measurement errors, U1 and U2. We compute and report results for the

proposed QR estimator. For comparison, we compute the density fX|Y using different pro-

cedures. First, we construct our proposed estimator to control for ME, using the variables

(Y,X1, X2), where the density is estimated by the Fourier Estimator. Second, we use the

variables (Y,X) to construct an “infeasible” kernel estimator of fX|Y in the first step. Fi-

nally, the variables (Y,X1) are used for “naive” kernel estimator of fX|Y which still suffers

from ME. For all estimators, we consider fourth-order Gaussian kernel. We approximate the

inner summation in equation (8) using Gauss-Hermite quadrature which is useful for the

indefinite integral. We perform 1000 simulations with n = 500 and n = 1000. We scan a

set of bandwidths for X and Y in order to find empirical optimal bandwidths in terms of

minimizing mean squared error.

6.2 Monte Carlo results

We report results for the following statistics of the coefficient β2: bias (B), standard deviation

(SD), and mean squared error (MSE). First of all, in order to illustrate the problem of ME

in practice, we consider a model estimation where the researcher ignores the ME problem

and performs a parametric median regression of Y on X1 without correcting for the ME in

X. This simple regression provides the bias of 0.1686, the standard error 0.02655 and the

MSE of 0.02586. These results highlight the importance of correcting for the ME problem.

Now we discuss and present the results for the nonparametric estimators with(out) correc-

tion of ME. Tables 1–3 report finite-sample performance of three different two-step estimators

at the median: (i) our proposed estimator (Fourier estimator); (ii) infeasible kernel estima-

tor; (iii) naive kernel estimator. These results are for n = 500, but the results for n = 1000

are similar. At the bottom of each table, B, SD, and MSE from optimal bandwidth are

9For simplicity, the perfectly-observed covariate Z is absent here.

22

reported. In Table 4 we vary the quantiles and present results for the different estimators

across different deciles with n = 1000.

Tables 1 - 3 Simulation Results

[ABOUT HERE]

Table 1 shows that the proposed estimator is effective in reducing the bias when true X

is measured with errors and repeated measures of the mismeasured covariate are available.

These results are comparable to the infeasible kernel estimator in Table 2. On the other

hand, the results in Table 3 from the naive kernel estimator ignoring ME in X show much

larger bias over all selected bandwidths. Therefore, our estimator outperforms the naive

kernel estimator in terms of both bias and MSE. The minimum MSE for our proposed

method is 0.00674 while the minimum MSE from the naive kernel estimator is 0.01008. This

result confirms that the methods proposed in this paper are beneficial in finite samples when

repeated measures of the mismeasured regressor are available to the researcher.

Table 4 reports finite-sample performance of three estimators over various quantiles with

n = 1000. For simplicity, we use the optimal bandwidths obtained from the simulation

results above. The results confirm that our proposed estimator performs well over different

level of quantiles.

Table 4 - Simulation Results

[ABOUT HERE]

7 Empirical application

This section illustrates the usefulness of the proposed methods with an empirical exam-

ple. We study Fazzari, Hubbard, and Petersen’s (1988) investment equation model, where

a firm’s investment is regressed on a proxy for investment demand (Tobin’s q) and cash

flows. Theory suggests that a correct measure for firm’s investment demand is marginal

Tobin’s q. This measure stems from the relationship that equates firm’s marginal benefit

with marginal cost in equilibrium. Nevertheless, the presence of financial constraints may

distort this relationship by introducing other factors that influence the firm’s optimal in-

vestment level. More specifically, financial constraints create a wedge between internal and

23

external funding that invalidates theoretical arguments in the spirit of Modigliani and Miller

(1958) capital structure irrelevance proposition. In this scenario firm’s cash flows reflect the

presence of financial constraints and may contain information relevant for explaining the

differences in investment demand across firms. Following Fazzari, Hubbard, and Petersen

(1988), investment-cash flow sensitivities became a standard metric in the literature that

examines the impact of financing imperfections on corporate investment (Stein (2003)).

Fazzari, Hubbard, and Petersen (1988) develop estimators for the investment equation,

where a firm’s investment is the dependent variable, and independent variables are a proxy

for investment demand (Tobin’s q) and cash flow. Nevertheless, empirical models proposed

to assess the sensitivity of investment demand to firm characteristics are usually fraught

with the presence of measurement error (see e.g., Erickson and Whited (2000), and Almeida,

Campello, and Galvao (2010)). A typical example is the use of the average Tobin’s q for

describing the investment-capital ratio. Theory suggests that firm’s investment demand is

captured by marginal q, but this quantity is unobservable and researchers use instead its

measurable proxy, average q. Since the two variables are not the same, a measurement

problem naturally arises (Hayashi (1982), Poterba (1988)). The introduction of error when

measuring these variables causes bias in least squares estimators. This bias can lead to

erroneous interpretations of the effect of firm characteristics on investment demand. Thus,

Poterba (1988) introduces the idea that errors in measuring Tobin’s q may be responsible for

the observed investment-cash flow sensitivities. If cash flow were correlated with investment

opportunities not well measured by the proxy for the marginal Tobin’s q, investment-cash

flow sensitivities could arise. This argument minimizes the role of financing constraints in

determining the relationship between firms’ cash flows and investment.

A number of studies intend to control for the measurement error in Tobin’s q, while

analyzing the relationship between investment and cash flow. A common approach has been

to use the standard instrumental variables together with ordinary least squares (OLS) and

generalized method of moments (GMM) estimators to correct the measurement error problem

(see, e.g., Almeida, Campello, and Galvao (2010) and Lewellen and Lewellen (2014)). It has

been common to use lags of the observed Tobin’s q as instruments, by assuming that they

are uncorrelated with the error term in the regression equation. Almeida, Campello, and

Galvao (2010) use lagged Tobin’s q as instruments to resolve the measurement error problem.

They estimate standard conditional average models and the results show the importance of

both Tobin’s q and cash flow in investment equation models.

24

A related method in the literature is to use different proxies to the marginal Tobin’s q

as alternative instruments. Cummins, Hasset, and Oliner (2006) estimate the investment

models using two sources of firm-level data. They construct an analyst-based measure of

average q as well as a market-based measure of average q, and compare estimation results

based on each measure to assess the robustness of the findings in the literature. Estimating

conditional mean models, they find no evidence that the cash flow is a statistically significant

determinant of investment in U.S. companies.

More recently, Agca and Mozumdar (2016) propose using the two measures of q contem-

plated in Cummins, Hasset, and Oliner (2006) as instruments for controlling for measurement

errors in marginal Tobin’s q. Their approach consists of using the lags of one of the vari-

ables as instruments for the other in the linear error-in-variables model. This leads them to

conclude that cash flow is a statistically significant cause of investment and investment-cash

flow sensitivity is higher for financially constrained firms.10

We use quantile regression (QR) methods in the investment equation model. The pro-

posed QR estimator is designed to correct for the measurement error problem while exploring

heterogeneous covariate effects across the conditional quantile functions. The QR model has

two advantages. First, our QR method proposes a solution to the measurement error problem

in Tobin’s q by using repeated measures of this variable. Second, we accommodate possible

heterogeneity on the Tobin’s q and cash flow in the conditional distribution of investment.

Indeed, this heterogeneity is not revealed by conventional least squares procedures.

The objective is to estimate the following conditional quantile function:

QIKi(τ |qi, CFKi) = α(τ) + β(τ)qi + δ(τ)CFKi, (10)

where the quantity IK denotes the ratio of investment, I, to capital stock, K; CFK is the

ratio of cash flow, CF , to capital stock; and q is the observed measure of average q, which

is a measure of the (latent) true marginal Tobin’s q.

We use a data set taken from Cummins, Hasset, and Oliner (2006). In this data there

are two measures of average q. The first measure is constructed using the standard equity

prices (qe). The second proxy for the firm’s intrinsic value is based on analysts’ earnings

expectations (q). Thus, the data on investment, the capital stock, the market-based measure

of average q, and cash flow are standard from Compustat, while the data on expected earnings

10Erickson and Whited (2000) suggest another alternative solution which relies on the high-order moments.

25

are from I/B/E/S International Inc. The construction of qe and q are detailed in Appendix

B.2 of Cummins, Hasset, and Oliner (2006). The sample consists of 11,431 observations over

the 1982-1999 period. For each firm, in a given year, we have information on investment, cash

flow, and two measures of Tobin’s q. For practical estimation we standardize all variables by

subtracting the mean and dividing by the corresponding standard deviation of each variable.

The summary statistics (for the standardized data) are described in Table 5.

Table 5 - Summary Statistics

[ABOUT HERE]

We present results for the estimates, and corresponding 95% confidence intervals, using

our proposed methods (ME-QR). For completeness, we also provide results from the standard

QR and OLS, as well as the instrumental variables QR (IV-QR) and OLS (IV-OLS). The

QR estimation strategy follows Koenker and Bassett (1978), and does not correct for ME.

The IV-QR estimates use the method developed by Chernozhukov and Hansen (2006). For

the IV-QR and IV-OLS, we use the variable q as an instrument for qe. The IV strategy for

QR is based on the assumption that the q is strongly related to the qe (i.e., IV) but the IV is

independent of unobservable factors of investment as well as ME. We conjecture that the IV

approach delivers different estimates than our proposed ME estimator since both procedures

rely on different set of conditions. Our method is particularly useful for the data set where it

is unlikely that the IV is independent of the regression error which contains ME on q, since

the IV is also potentially mismeasured.11

To identify the effects of Tobins’ q and cash flow we work with the conditions given in

Section 2. Assumption A.I (i) requires the ME on the q (U1) to have zero conditional mean

given the true Tobin’s q (X) and the ME of qe (U2). This excludes correlation between

U1 and X or between U1 and U2. Hence, a nonclassical reporting error assumption (e.g.,

Hu and Sasaki (2015b)) is not allowed in our setting.12 Nevertheless, Assumption A.I (i)

allows nonlinear dependence of the ME (U1) and the true Tobin’s q (X) and the ME of qe

(U2). This assumption is particularly useful when the miscalculation is changing over the

11We note that the independence condition required by the IV method implies independence between MEon q and qe. Because the analysts’ earnings expectations (q) are dependent on the standard equity prices(qe), their ME are likely to be related. However, our approach requires a weaker assumption of conditionalmean zero as in Assumption A.I (i).

12We acknowledge that although our conditions partially relax the classical ME assumptions, it would beinteresting to extend the current results to a fully nonclassical ME model in QR.

26

0.2 0.4 0.6 0.8

0.1

0.2

0.3

0.4

0.5

0.6

quantiles

coefficients

Tobin's q

QROLS

0.2 0.4 0.6 0.8

-0.1

0.0

0.1

0.2

0.3

quantiles

coefficients

Cash Flow

QROLS

Figure 1: Quantile Regression and Ordinary Least Squares. Left box plots the coefficients on

Tobins’s q. Right box plots the coefficients on Cash Flow.

level of Tobin’s q. Assumption A.I (ii) requires that the ME of qe (U2) to be independent of

true Tobin’s q (X) as well as other variables (Y, Z). This does not necessarily require zero

mean of U2. Hence, this condition allows a possibility that, on average, analysts earnings

expectations on the Tobins’ q could be either larger or smaller than the standard equity

prices.

The results for both mean and quantile estimates of the sensitivity of investment to

Tobin’s q and cash flow are presented in the left panels and right panels, respectively, of

Figures 1-3. Figure 1 presents the results for QR and OLS. Figure 2 displays the results for

IV-QR and IV-OLS. Finally, Figure 3 collects the results for ME-QR. All figures contain point

estimates as well as the corresponding 95% pointwise confidence bands. In the nonparametric

estimation step in our proposed estimator, for the choice of the bandwidth (hxn in equation

(7)), we use an informal rule where the estimates are not sensitive to marginal changes in the

neighborhood of the optimal bandwidth. Other bandwidths are chosen based on Silverman’s

rule of thumb. The number of bootstrap replications to construct the confidence intervals is

250.

The left panels of Figures 1-3 show significant positive estimates for Tobin’s q estimates.

All the quantile estimates in Figures 1-3 display evidence of positive and increasing effect

of the investment demand, Tobin’s q, on the investment spending, across quantiles of the

27

0.2 0.4 0.6 0.8

0.1

0.2

0.3

0.4

0.5

0.6

quantiles

coefficients

Tobin's q

IV-QRIV-OLS

0.2 0.4 0.6 0.8

-0.1

0.0

0.1

0.2

0.3

quantiles

coefficients

Cash Flow

IV-QRIV-OLS

Figure 2: Quantile Regression Instrumental Variables and Two Stage Least Squares. Left box

plots the coefficients on Tobins’s q. Right box plots the coefficients on Cash Flow.

conditional distribution of investment. This result documents empirical evidence of strong

heterogeneity in the effects of investment across the distribution of investment. Relative to

the QR in Figure 1, the IV-QR in Figure 2 shows virtually no difference in the importance

of Tobin’s q. This result might be interpreted as the IV approach being ineffective. On

the other hand, the ME-QR in Figure 3 shows different results. We note that, according

to the theory of investment equation, see, e.g., Fazzari, Hubbard, and Petersen (1988), and

previous empirical studies, e.g., Kaplan and Zingales (1997), on average, the investment

demand has positive effect on the investment spending. Thus, this result has been verified

in our ME-QR estimations, although after correcting for ME the estimates decrease relative

to the standard QR. Moreover, we find evidence that firms have heterogeneous responses to

changes of investment demand across the conditional distribution of investment.

The results regarding the sensitivity of investment to cash flow are presented in the right

panels of the Figures 1-3. The right panel of Figure 1 presents the results for standard QR and

shows decreasing point estimates of cash flow effects on investment over the quantiles. The

mean regression estimate is represented by the horizontal straight line, which shows a positive

effect close to 0.1, and is statistically different from zero at usual levels of significance. The

results for IV-QR in Figure 2 are virtually the same as those in Figure 1. Thus, again these

results might be interpreted as the instruments being invalid to resolve the measurement

28

0.2 0.4 0.6 0.8

0.00

0.05

0.10

0.15

0.20

quantiles

coefficients

Tobin's q

ME-QR

0.2 0.4 0.6 0.8

0.15

0.20

0.25

0.30

0.35

0.40

quantiles

coefficients

Cash Flow

ME-QR

Figure 3: Quantile Regression Measurement Error. Left box plots the coefficients on Tobins’s q.

Right box plots the coefficients on Cash Flow.

error problem.

The right panel results in Figure 3 for the ME-QR estimates exhibit a distinct inverted

U-shape, implying larger cash flow sensitivity for those firms in lower quantiles, that is,

the lower part of the conditional distribution of investment. In particular, the ME-QR

results show positive and monotonically-increasing cash flow sensitivity at low quantiles of

the distribution; estimated coefficient is increasing from 0.30 to 0.36 up to approximately

45th quantile. The cash flow sensitivity starts to decease for larger quantiles, and becomes

relatively smaller at the very top of the conditional investment distribution. These findings

uncover several important features. First, they document important heterogeneity on the

response of investment spending to cash flow along the conditional quantile function. Firms

in different quantiles of the conditional distribution of investment respond differently to

marginal changes in the cash flow. Second, Figure 3 shows evidence strong heterogeneity

with an inverted U-shape. This result could be interpreted at the light of the effects of

financial constraints on corporate policies as in Almeida, Campello, and Weisbach (2004).

The investment spending is more sensitive to cash flow (large magnitude of the coefficients)

for firms at lower quantiles. The large coefficients for lower quantiles is an intuitive result.

The cash flow coefficient captures the potential sensitivity of investment to fluctuations in

available internal finance–after investment opportunities. Thus, the results show evidence

29

that, on the one hand, firms at low levels of investment spending, and thus likely financially

constrained, are in fact more exposed to and dependent on fluctuations in internal finance.

On the other had, firms at upper quantiles, which have higher levels of investment spending

and are financially unconstrained, are less exposed to and dependent on fluctuations in

internal finance. Third, this heterogeneous effect across quantiles also indicates that, for a

fixed level of q, the variability of the investment spending across the conditional distribution

increases as the level of cash flow increases. Intuitively, firms with larger cash flow are

entitled to invest in a larger range in contrast to the firms with smaller cash flow.

Overall, the application illustrates that QR method is an important tool to study in-

vestment equation models. It allows us to study the impacts of Tobins’ q and cash flow

at different quantiles of the conditional distribution of investment. The empirical results

document findings that larger cash flow sensitivity occur at the lower part of the invest-

ment distribution, showing evidence that these firms are more exposed to and dependent on

fluctuations in internal finance.

8 Conclusion

This paper develops estimation and inference for quantile regression models with measure-

ment errors. We propose a semiparametric two-step estimator assuming availability of re-

peated measures of the true covariate. The asymptotic properties of the estimator are

established. We also develop statistical inference procedures and establish the validity of a

bootstrap approach to implement the methods in practice. Monte Carlo simulations assess

the finite-sample performance of the proposed methods and show that the proposed methods

have good finite-sample performance. We apply the methods to an empirical application to

the investment equation model. The results document strong heterogeneity in the sensitivity

of investment to Tobin’s q and cash flow across the conditional distribution of investment,

and illustrate that our methods are useful in empirical models where measurement error is

an important issue..

Many issues remain to be investigated. In this paper the quantile of interest is fixed,

τ ∈ (0, 1). The extension of the results to the uniform case is desirable and important

for uniform inference over the entire conditional quantile function over τ . Such extension

would require generalizing the current results. One of the key steps for the derivations

would be to establish stochastic equicontinuity of the appropriate centralized scores uniformly

30

over the quantiles. In addition, the analysis of the quantile regression with nonclassical

measurement error is also a critical direction for future research. There are many potential

applications for the proposed methods. Examples as employer-employee matched sample

for wages, matched federal agency and firm-level data, bidder’s private signals in auction,

earnings data in earnings dynamics would appear to be a natural laboratory for further

development of quantile regression models with repeated measures.

31

A Mathematical Appendix

Proof of Theorem 1. Given Assumption A.III, we have

φ(ζ, y, z) ≡ E[eiζX | Y = y, Z = z]

=

∫E[eiζX | Y = y, Z = z,X = x]f(x | y, z)dx

=

∫f(x | y, z)eiζxdx

where the last expression is the Fourier transform of f(x | y, z). Note that for (x, y, z) ∈ supp(X,Y, Z),

1

2π

∫φ(ζ, y, z) exp(−iζx)dζ

is the inverse Fourier transform of φ(ζ, y, z). Thus we have

f(x | y, z) =1

2π

∫φ(ζ, y, z) exp(−iζx)dζ.

We now need to show that

φ(ζ, Y, Z) =E[eiζX2 | Y,Z]

E[eiζX2 ]exp

(∫ ζ

0

iE[X1eiξX2 ]

E[eiξX2 ]dξ

).

From Assumptions A.I–A.II

Dξ ln(E[eiξX ]) =iE[XeiξX ]

E[eiξX ]

=iE[XeiξX ]E[eiξU2 ]

E[eiξX ]E[eiξU2 ]

=iE[Xeiξ(X+U2)]

E[eiξ(X+U2)]

=iE[Xeiξ(X+U2)] + iE[E(U1 | X,U2)eiξ(X+U2)]

E[eiξX2 ]

=iE[Xeiξ(X+U2)] + iE[E(U1e

iξ(X+U2) | X,U2)]

E[eiξX2 ]

=iE[Xeiξ(X+U2)] + iE[U1e

iξ(X+U2)]

E[eiξX2 ]

=iE[X1e

iξX2 ]

E[eiξX2 ].

32

Therefore, for each real ζ,

φ(ζ, Y, Z) ≡ E[eiζX | Y,Z]

=E[eiζX | Y, Z]E[eiζU2 ]

E[eiζX ]E[eiζU2 ]E[eiζX ]

=E[eiζX2 | Y, Z]

E[eiζX2 ]E[eiζX ]

=E[eiζX2 | Y, Z]

E[eiζX2 ]exp

(ln(E[eiζX ])− ln 1

)=

E[eiζX2 | Y, Z]

E[eiζX2 ]exp

(∫ ζ

0Dξ ln(E[eiξX ])dξ

)=

E[eiζX2 | Y,Z]

E[eiζX2 ]exp

(∫ ζ

0

iE[X1eiξX2 ]

E[eiξX2 ]dξ

),

where the third equality is obtained by U2 ⊥ (Y,X,Z).

Proof of Theorem 2. Note that the inverse Fourier Transform of κ(hxζ) is k(x/hx)/hx, andthe inverse Fourier Transform of E[eiζX | Y = y, Z = z] is f(x | y, z) by equation (13). Also notethat from the convolution theorem, the inverse Fourier Transform of the product of κ(hxζ) andE[eiζX | Y = y, Z = z] is the convolution between the inverse Fourier Transform of κ(hxζ) andthe inverse Fourier Transform of E[eiζX | Y = y, Z = z]. Because Assumptions A.II (iii)–A.IVguarantee the existence of f(x | y, z;hx), we conclude that

f(x | y, z;hx) ≡∫

1

hxk

(x− xhx

)f(x | y, z)dx

=1

2π

∫κ(hxζ)E[eiζX | Y = y, Z = z] exp(−iζx)dζ

=1

2π

∫κ(hxζ)φ(ζ, y, z) exp(−iζx)dζ.

The following lemma is helpful to derive the result given in Theorem 3. Recall that X ≡ R isthe support of X, and Y × Z is a compact set contained in the support of (Y, Z).

Lemma A.1 For (x, y, z) ∈ X × Y × Z and hn > 0,

f(x | y, z;h)− f(x | y, z) = B(x, y, z;hx) + L(x, y, z;h) +R(x, y, z;h),

where B(x, y, z;hx) is a nonrandom “bias term” defined as

B(x, y, z;hx) ≡ f(x | y, z;hx)− f(x | y, z);

L(x, y, z;h) is a “variance term” admitting the linear representation

L(x, y, z;h) ≡ f(x | y, z;h)− f(x | y, z, hx) = E [`(x, y, z, h;Y,X1, X2, Z)]

33

where `(x, y, z, h;Y,X1, X2, Z) is defined in the proof of the lemma, and R(x, y, z;h) is a “remainderterm,”

R(x, y, z;h) ≡ f(x | y, z;h)− f(x | y, z;h).

Proof of Lemma A.1. Let ωA(ζ) ≡ E[AeiζX2

]where A = 1, X1 and

ω(ζ, y, z) ≡ E[eiζX2 | Y = y, Z = z

]=

∫eiζx2f(x2 | y, z)dx2

=χ(ζ, y, z)

f(y, z),

where χ(ζ, y, z) ≡∫eiζx2f(x2, y, z)dx2. Also let ωA(ζ) ≡ E

[AeiζX2

]and δωA(ζ) ≡ ωA(ζ)− ωA(ζ),

and let

ω(ζ, y, z) ≡ E[eiζX2 | Y = y, Z = z

]≡ χ(ζ, y, z)/f(y, z)

where

χ(ζ, y, z) =1

n

n∑j=1

eiζX2jkhy(Yj − y)khz(Zj − z) = E[eiζX2khy(Y − y)khz(Z − z)

]f(y, z) =

1

n

n∑j=1

khy(Yj − y)khz(Zj − z) = E [khy(Y − y)khz(Z − z)] ,

and δχ(ζ, y, z) ≡ χ(ζ, y, z) − χ(ζ, y, z) and δf(y, z) ≡ f(y, z) − f(y, z). We use the followingrepresentation

ωX1(ζ)

ω1(ζ)=ωX1(ζ) + δωX1(ζ)

ω1(ζ) + δω1(ζ)= qX1(ζ) + δqX1(ζ) (11)

where qX1(ζ) = ωX1(ζ)/ω1(ζ) and where δqX1(ζ) can be written as either

δqX1(ζ) =

(δωX1(ζ)

ω1(ζ)− ωX1(ζ)δω1(ζ)

(ω1(ζ))2

)(1 +

δω1(ζ)

ω1(ζ)

)−1

or δqX1(ζ) = δ1qX1(ζ) + δ2qX1(ζ) with

δ1qX1(ζ) ≡ δωX1(ζ)

ω1(ζ)− ωX1(ζ)δω1(ζ)

(ω1(ζ))2

δ2qX1(ζ) ≡ ωX1(ζ)

ω1(ζ)

(δω1(ζ)

ω1(ζ)

)2(1 +

δω1(ζ)

ω1(ζ)

)−1

− δωX1(ζ)

ω1(ζ)

δω1(ζ)

ω1(ζ)

(1 +

δω1(ζ)

ω1(ζ)

)−1

.

Similarly,

1

ω1(ζ)=

1

ω1(ζ) + δω1(ζ)= q1(ζ) + δq1(ζ) (12)

34

where q1(ζ) ≡ 1/ω1(ζ), and where

δq1(ζ) =

(− δω1(ζ)

(ω1(ζ))2

)(1 +

δω1(ζ)

ω1(ζ)

)−1

or δq1(ζ) = δ1q1(ζ) + δ2q1(ζ) with

δ1q1(ζ) ≡ − δω1(ζ)

(ω1(ζ))2

δ2q1(ζ) ≡ 1

ω1(ζ)

(δω1(ζ)

ω1(ζ)

)2(1 +

δω1(ζ)

ω1(ζ)

)−1

.

And also

χ(ζ, y, z)

f(y, z)=χ(ζ, y, z) + δχ(ζ, y, z)

f(y, z) + δf(y, z)= q2(ζ, y, z) + δq2(ζ, y, z) (13)

where q2(ζ, y, z) ≡ χ(ζ, y, z)/f(y, z), and where

δq2(ζ, y, z) =

(δχ(ζ, y, z)

f(y, z)− χ(ζ, y, z)δf(y, z)

(f(y, z))2

)(1 +

δf(y, z)

f(y, z)

)−1

or δq2(ζ, y, z) = δ1q2(ζ, y, z) + δ2q2(ζ, y, z) with

δ1q2(ζ, y, z) ≡ δχ(ζ, y, z)

f(y, z)−χ(ζ, y, z)δf(y, z)

(f(y, z))2

δ2q2(ζ, y, z) ≡ χ(ζ, y, z)

f(y, z)

(δf(y, z)

f(y, z)

)2(1 +

δf(y, z)

f(y, z)

)−1

− δχ(ζ, y, z)

f(y, z)

δf(y, z)

f(y, z)

(1 +

δf(y, z)

f(y, z)

)−1

.

Let QX1(ζ) ≡∫ ζ

0 (iωX1(ξ)/ω1(ξ))dξ and δQX1(ζ) ≡∫ ζ

0 (iωX1(ξ)/ω1(ξ))dξ − QX1(ζ). Note that for

some random function δQX1(ζ) such that |δQX1(ζ)| ≤ |δQX1(ζ)| for all ζ,

exp(QX1(ζ) + δQX1(ζ)

)= exp(QX1(ζ))

(1 + δQX1(ζ) +

1

2

[exp(δQX1(ζ))

] (δQX1(ζ)

)2). (14)

35

From equations (11)–(14), we have

f(x | y, z;h)− f(x | y, z;hx)

=1

2π

∫κ(hxζ)φ(ζ, y, z, h(2)) exp(−iζx)dζ − 1

2π

∫κ(hxζ)φ(ζ, y, z) exp(−iζx)dζ

=1

2π

∫κ(hxζ) exp(−iζx)

[ω(ζ, y, z)

ω1(ζ)exp

(∫ ζ

0

iωX1(ξ)

ω1(ξ)dξ

)− ω(ζ, y, z)

ω1(ζ)exp

(∫ ζ

0

iωX1(ξ)

ω1(ξ)dξ

)]dζ

=1

2π

∫κ(hxζ) exp(−iζx)

[− ω(ζ, y, z)

ω1(ζ)exp

(∫ ζ

0

iωX1(ξ)

ω1(ξ)dξ

)+

χ(ζ, y, z)

f(y, z)+δχ(ζ, y, z)

f(y, z)− χ(ζ, y, z)δf(y, z)

(f(y, z))2+ δ2q2(ζ, y, z)

×

1

ω1(ζ)− δω1(ζ)

(ω1(ζ))2+ δ2q1(ζ)

× exp(QX1(ζ))

×

1 +

∫ ζ

0iδ1qX1(ξ)dξ +

∫ ζ

0iδ2qX1(ξ)dξ +

1

2exp(δQX1(ζ))

(∫ ζ

0iδqX1(ξ)dξ

)2]dζ.

We denote the linearization of f(x | y, z;hx) by f(x | y, z;hx). Then

L(x, y, z;h)

≡f(x | y, z;h)− f(x | y, z;hx)

=1

2π

∫κ(hxζ) exp(−iζx) exp(QX1(ζ))

[− χ(ζ, y, z)

f(y, z)

δω1(ζ)

(ω1(ζ))2

+χ(ζ, y, z)

f(y, z)

1

ω1(ζ)

∫ ζ

0iδ1qX1(ξ)dξ

+1

ω1(ζ)

δχ(ζ, y, z)

f(y, z)− 1

ω1(ζ)

χ(ζ, y, z)δf(y, z)

(f(y, z))2

]dζ

=1

2π

∫κ(hxζ) exp(−iζx)φ(ζ, y, z)

[− δω1(ζ)

ω1(ζ)+δχ(ζ, y, z)

χ(ζ, y, z)− δf(y, z)

f(y, z)

+

∫ ζ

0

(iδωX1(ξ)

ω1(ξ)− iωX1(ξ)δω1(ξ)

(ω1(ξ))2

)dξ

]dζ

=1

2π

∫κ(hxζ) exp(−iζx)φ(ζ, y, z)

(−δω1(ζ)

ω1(ζ)+δχ(ζ, y, z)

χ(ζ, y, z)− δf(y, z)

f(y, z)

)dζ

+1

2π

∫ ∫ ±∞ξ

κ(hxζ) exp(−iζx)φ(ζ, y, z)dζ

(iδωX1(ξ)

ω1(ξ)− iωX1(ξ)δω1(ξ)

(ω1(ξ))2

)dξ

=1

2π

∫κ(hxζ) exp(−iζx)φ(ζ, y, z)

(−δω1(ζ)

ω1(ζ)+δχ(ζ, y, z)

χ(ζ, y, z)− δf(y, z)

f(y, z)

)dζ

+1

2π

∫ ∫ ±∞ζ

κ(hxξ) exp(−iξx)φ(ξ, y, z)dξ

(iδωX1(ζ)

ω1(ζ)− iωX1(ζ)δω1(ζ)

(ω1(ζ))2

)dζ

36

=

∫ [− 1

2π

1

ω1(ζ)κ(hxζ) exp(−iζx)φ(ζ, y, z)

− 1

2π

iωX1(ζ)

(ω1(ζ))2

∫ ±∞ζ

κ(hxξ) exp(−iξx)φ(ξ, y, z)dξ

δω1(ζ)

+

1

2π

i

ω1(ζ)

∫ ±∞ζ

κ(hxξ) exp(−iξx)φ(ξ, y, z)dξ

δωX1(ζ)

+

1

2π

1

χ(ζ, y, z)κ(hxζ) exp(−iζx)φ(ζ, y, z)

δχ(ζ, y, z)

+

− 1

2π

1

f(y, z)κ(hxζ) exp(−iζx)φ(ζ, y, z)

δf(y, z)

]dζ

=

∫ [Ψ1(ζ, x, y, z, hx)

(E[eiζX2 ]− E[eiζX2 ]

)+ Ψ2(ζ, x, y, z, hx)

(E[X1e

iζX2 ]− E[X1eiζX2 ]

)+ Ψ3(ζ, x, y, z, hx)

(E[eiζX2khy(Y − y)khz(Z − z)]− E[eiζX2khy(Y − y)khz(Z − z)]

)+ Ψ4(ζ, x, y, z, hx)

(E[khy(Y − y)khz(Z − z)]− E[khy(Y − y)khz(Z − z)]

)]dζ

=E

[ ∫Ψ1(ζ, x, y, z, hx)

(eiζX2 − E[eiζX2 ]

)+ Ψ2(ζ, x, y, z, hx)

(X1e

iζX2 − E[X1eiζX2 ]

)+ Ψ3(ζ, x, y, z, hx)

(eiζX2khy(Y − y)khz(Z − z)− E[eiζX2khy(Y − y)khz(Z − z)]

)+ Ψ4(ζ, x, y, z, hx) (khy(Y − y)khz(Z − z)− E[khy(Y − y)khz(Z − z)])

]dζ

≡E [`(x, y, z, h;Y,X1, X2, Z)] ,

where the following identity was used in the fourth equality: for any absolutely integrable functiong ∫ ∞

−∞

∫ ζ

0g(ζ, ξ)dξdζ =

∫ ∞0

∫ ∞ξ

g(ζ, ξ)dζdξ +

∫ 0

−∞

∫ −∞ξ

g(ζ, ξ)dζdξ ≡∫ ∫ ±∞

ξg(ζ, ξ)dζdξ,

and where

Ψ1(ζ, x, y, z, hx) ≡ − 1

2π

1

ω1(ζ)κ(hxζ) exp(−iζx)φ(ζ, y, z)

− 1

2π

iωX1(ζ)

(ω1(ζ))2

∫ ±∞ζ

κ(hxξ) exp(−iξx)φ(ξ, y, z)dξ

Ψ2(ζ, x, y, z, hx) ≡ 1

2π

i

ω1(ζ)

∫ ±∞ζ

κ(hxξ) exp(−iξx)φ(ξ, y, z)dξ

Ψ3(ζ, x, y, z, hx) ≡ 1

2π

1

χ(ζ, y, z)κ(hxζ) exp(−iζx)φ(ζ, y, z)

Ψ4(ζ, x, y, z, hx) ≡ − 1

2π

1

f(y, z)κ(hxζ) exp(−iζx)φ(ζ, y, z).

We use the following convenient notation for expositional simplicity.

37

Definition A.1 We write f(ζ) g(ζ) for f, g : R 7→ R when there exists a constant C > 0,independent of ζ, such that f(ζ) ≤ Cg(ζ) for all ζ ∈ R (and similarly for ). Analogously, wewrite an bn for two sequences an, bn when there exists a constant C independent of n such thatan ≤ Cbn for all n ∈ N.

Proof of Theorem 3. In order to obtain the uniform convergence rate of f(x | y, z;h), we deriveasymptotic convergence rate of the bias term, divergence rate of the variance term, and rely onnegligibility of the remainder term. First, from Parseval’s identity and Assumption A.IV, we have

|B(x, y, z, hx)| = |f(x | y, z;hx)− f(x | y, z)|= |f(x | y, z;hx)− f(x | y, z; 0)|

=

∣∣∣∣ 1

2π

∫κ(hxζ)φ(ζ, y, z) exp(−iζx)dζ − 1

2π

∫φ(ζ, y, z) exp(−iζx)dζ

∣∣∣∣=

∣∣∣∣ 1

2π

∫(κ(hxζ)− 1)φ(ζ, y, z) exp(−iζx)dζ

∣∣∣∣≤ 1

2π

∫|(κ(hxζ)− 1)| |φ(ζ, y, z)| dζ

=1

π

∫ ∞ξ/hx|(κ(hxζ)− 1)| |φ(ζ, y, z)| dζ

∫ ∞ξ/hx|φ(ζ, y, z)| dζ.

Then, by Assumption B.I (ii), we have

sup(x,y,z)∈X×Y×Z

|B(x, y, z, hx)| ∫ ∞ξ/hx

Cφ(1 + |ζ|)γφ exp(αφ|ζ|νφ)dζ

∫ ∞ξ/hx

(1 + |ζ|)γφ exp(αφ|ζ|νφ)dζ (15)

= O((ξ/hx

)γφ+1exp

(αφ(ξ/hx

)νφ))= O

((hx)−γB exp

(αB (hx)−νB

)).

For the asymptotic divergence rate of the variance term, let an = (lnn)1/2

(nhyhz)1/2+∑

s∈y,z(hs)2.

and define

Ψ+(h) ≡∫

Ψ+1 (ζ, hx)dζ +

∫Ψ+

2 (ζ, hx)dζ

+ an

∫Ψ+

3 (ζ, hx)dζ + an

∫Ψ+

4 (ζ, hx)dζ,

where Ψ+A(ζ, hx) ≡ sup(x,y,z)∈X×Y×Z |ΨA(ζ, x, y, z, hx)| for A ∈ 1, 2, 3, 4. From Assumptions

38

A.IV and B.II, and from similar arguments above, one can show that

E(n |δω1(ζ)|2

) 1, E

n ∣∣∣∣∣a−1n · sup

(y,z)∈Y×Z(δχ(ζ, y, z))

∣∣∣∣∣2 1,

E(n |δωX1(ζ)|2

) 1, E

n ∣∣∣∣∣a−1n · sup

(y,z)∈Y×Z

(δf(y, z)

)∣∣∣∣∣2 1,

and ∫Ψ+

1 (ζ, hx)dζ (1 + (hx)−1)γµ+γφ−γω+2 exp(−αω(hx)−1)νω

)exp

(αφ((hx)−1)νφ

),∫

Ψ+2 (ζ, hx)dζ (1 + (hx)−1)γφ−γω+2 exp

(−αω(hx)−1)νω

)exp

(αφ((hx)−1)νφ

),

an

∫Ψ+

3 (ζ, hx)dζ an(1 + (hx)−1)γφ−γω+1 exp(−αω(hx)−1)νω

)exp

(αφ((hx)−1)νφ

),

an

∫Ψ+

4 (ζ, hx)dζ an(1 + (hx)−1)γφ+1 exp(αφ((hx)−1)νφ

).

Then we have

Ψ+(h) = O(

((hx)−1)γµ+γφ−γω+2 exp(

(αφ1νφ=νω − αω)((hx)−1)νω))

.

Note that by Minkowski inequality,

E

[sup

(x,y,z)∈X×Y×Z|L(x, y, z, h)|

]

= E

[sup

(x,y,z)∈X×Y×Z|f(x | y, z;h)− f(x | y, z;hx)|

]

= E

[sup

(x,y,z)∈X×Y×Z

∣∣∣∣ ∫ [Ψ1(ζ, x, y, z, hx)δω1(ζ) + Ψ2(ζ, x, y, z, hx)δωX1(ζ)

+ Ψ3(ζ, x, y, z, hx)δχ(ζ, y, z) + Ψ4(ζ, x, y, z, hx)δf(y, z)]dζ

∣∣∣∣]≤ E

∫ [(sup

(x,y,z)∈X×Y×Z|Ψ1(ζ, x, y, z, hx)|

)|δω1(ζ)|

+

(sup

(x,y,z)∈X×Y×Z|Ψ2(ζ, x, y, z, hx)|

)|δωX1(ζ)|

+

(sup

(x,y,z)∈X×Y×Z|Ψ3(ζ, x, y, z, hx)|

)(sup

(y,z)∈Y×Z|δχ(ζ, y, z)|

)

+

(sup

(x,y,z)∈X×Y×Z|Ψ4(ζ, x, y, z, hx)|

)(sup

(y,z)∈Y×Z

∣∣∣δf(y, z)∣∣∣) ]dζ

39

≤∫ [

Ψ+1 (ζ, hx)

E(|δω1(ζ)|2

)1/2+ Ψ+

2 (ζ, hx)

E(|δωX1(ζ)|2

)1/2

+ anΨ+3 (ζ, hx)

E

∣∣∣∣∣a−1n ·

(sup

(y,z)∈Y×Zδχ(ζ, y, z)

)∣∣∣∣∣2

1/2

+ anΨ+4 (ζ, hx)

E

∣∣∣∣∣a−1n ·

(sup

(y,z)∈Y×Zδf(y, z)

)∣∣∣∣∣2

]dζ

≤ n−1/2

[ ∫Ψ+

1 (ζ, hx)

E(n |δω1(ζ)|2

)1/2dζ +

∫Ψ+

2 (ζ, hx)

E(n |δωX1(ζ)|2

)1/2dζ

+ an

∫Ψ+

3 (ζ, hx)

E

n ∣∣∣∣∣a−1n ·

(sup

(y,z)∈Y×Zδχ(ζ, y, z)

)∣∣∣∣∣2

1/2

dζ

+ an

∫Ψ+

4 (ζ, hx)

E

n ∣∣∣∣∣a−1n ·

(sup

(y,z)∈Y×Zδf(y, z)

)∣∣∣∣∣2 dζ

] n−1/2Ψ+(h).

Thus, we have that by Markov’s inequality

sup(x,y,z)∈X×Y×Z

|L(x, y, z, h)| (16)

= Op

(n−1/2((hx)−1)γµ+γφ−γω+2 exp

((αφ1νφ=νω − αω)((hx)−1)νω

)).

From Assumptions B.II–B.III, selection of the bandwidths in the statement of the theorem,and minor adjustment of the argument for the variance term above, one can show that the remainderterm is asymptotically negligible. So detailed proof is omitted here for brevity. Then puttingequations (15) and (16) together yields the result.

Proof of Theorem 4. To show consistency of the estimator, we apply Theorem 1 of Chen,Linton, and Van Keilegom (2003). Thus, we need to verify Conditions (1.1)–(1.5’) in Chen,Linton, and Van Keilegom (2003). Recall that

Qn(β, δ, f) =1

n

n∑i=1

∫ψ(Yi − xβ − Z>i δ)[x Zi] · f(x | Yi, Zi) dx,

and

Q(β, δ, f) = E

∫ψ(Yi − xβ − Z>i δ)[x Zi] · f(x | Yi, Zi) dx.

a) First, we verify Condition (1.1) by verifying that Qn(β, δ, f) = op(n−1/2), which implies

the desired property. This result is usual in the quantile regression literature, and generally follows

40

for the boundness of the score function and the computational property of quantile regression.Consider the following

||Qn(β, δ, f)|| = || 1n

n∑i=1

∫ψ(Yi − xβ − Z>i δ)[x Zi] · f(x | Yi, Zi) dx||

= || 1n

n∑i=1

∫ψ(Yi − xβ − Z>i δ)[x Zi] · (f(x | Yi, Zi)− f(x | Yi, Zi)) dx

+1

n

n∑i=1

∫ψ(Yi − xβ − Z>i δ)[x Zi] · f(x | Yi, Zi) dx||

≤ || 1n

n∑i=1

∫ψ(Yi − xβ − Z>i δ)[x Zi] · (f(x | Yi, Zi)− f(x | Yi, Zi)) dx||

+ || 1n

n∑i=1

∫ψ(Yi − xβ − Z>i δ)[x Zi] · f(x | Yi, Zi) dx||

≤ C ·∫f(x | Yi, Zi)− f(x | Yi, Zi)dx||∞ ·

1

nsupx,i||x Zi||

+ C ·∫f(x | Yi, Zi)dx||∞ ·

1

nsupx,i||x Zi||

= op(n−1/2),

where C is generic constant, the first inequality is given by the triangle inequality, and the lastequality is given by the fact that ψ(·) ≤ 1 and the computational property of quantile regression(Koenker and Bassett (1978)), Theorem 3, 1

n supx,i ||x Zi|| = op(n−1/2) by Assumption C.II, and

f(x|Yi, Zi) being integrable by assumption A.III(ii). The last assertion follows from C.II andapplication of Markov inequality as 1

n supx,i ||x Zi|| = op(n−1/2), since P (supx,i ||x Z|| > n1/2) ≤

nP (||x Z|| > n1/2) ≤ nE||x Z||2+ε/n2+ε2 = o(1).

b) Condition (1.2) holds directly by the identification condition C.I.

c) Now we show that Condition (1.3) is satisfied by verifying that Q(β, δ, f) is continuous inf uniformly for all (β, δ>)> ∈ Θ. For any ||f − f0||∞ ≤ ε,

||Q(β, δ, f)− Q(β, δ, f0)|| =||E∫ψ(Yi − xβ − Z>i δ)[x Zi] · f(x | Yi, Zi) dx

− E

∫ψ(Yi − xβ − Z>i δ)[x Zi] · f0(x | Yi, Zi) dx||

=||E∫ψ(Yi − xβ − Z>i δ)[x Zi] · [f(x | Yi, Zi)− f0(x | Yi, Zi)] dx||

≤E

∫||[x Zi]|| · ||f(x | Yi, Zi)− f0(x | Yi, Zi)||∞ dx

The inequality holds by the property of exchanging norms and integral, Cauchy inequality, andthe fact that ψ(·) ≤ 1. By Assumptions C.II and C.III, E

∫||[x Zi]|| · ||f(x | Yi, Zi) − f0(x |

Yi, Zi)||∞ dx < C, and given that ||f−f0||∞ ≤ ε, as ε→ 0 the result follows. Therefore, Condition(1.3) holds.

41

d) Condition (1.4) is satisfied by our Theorem 3.

e) It only remains to verify Condition (1.5’). For any εn = o(1),

sup(β,δ)∈Θ,||f−f0||∞≤εn

||Qn(β, δ, f)− Q(β, δ, f)|| = op(1).

By noting that

||Qn(β, δ, f)− Q(β, δ, f)||

=||∫ (

1

n

n∑i=1

ψ(Yi − xβ − Z>i δ)[x Zi] · f(x | Yi, Zi)− Eψ(Yi − xβ − Z>i δ)[x Zi] · f(x | Yi, Zi)

)dx||

≤||∫

supif(x | Yi, Zi)

(1

n

n∑i=1

ψ(Yi − xβ − Z>i δ)[x Zi]− Eψ(Yi − xβ − Z>i δ)[x Zi]

)dx||

≤||∫

supif(x | Yi, Zi) dx|| · || sup

x

(1

n

n∑i=1

ψ(Yi − xβ − Z>i δ)[x Zi]− Eψ(Yi − xβ − Z>i δ)[x Zi]

)||,

we have

sup(β,δ)∈Θ,||f−f0||∞≤εn

||Qn(β, δ, f)− Q(β, δ, f)||

≤C · sup(β,δ)∈Θ,x

|| 1n

n∑i=1

ψ(Yi − xβ − Z>i δ)[x Zi]− Eψ(Yi − xβ − Z>i δ)[x Zi]||,

since ||∫

supi f(x | Yi, Zi) dx|| < C by Assumption A.III. Denote φβ,δ(Yi, x, Zi) = ψ(Yi − xβ −Z>i δ)[x Zi]. We need to show that φβ,δ : (β, δ) ∈ Θ is G-C. Because ψ(Yi−xβ−Z>i δ) : (β, δ) ∈ Θis bounded and VC, it is G-C. Also, E[|X|] < ∞ and E[‖Z‖] < ∞ by Assumption C.II. Finally,Fεn = f : ||f − f0||∞ ≤ ε is G-C by Theorem 3. Those conditions and Corollary 9.27 (ii) ofKosorok (2008) lead to our conclusion.

Proof of Theorem 5. We now apply Theorem 2 of Chen, Linton, and Van Keilegom (2003) toestablish weak convergence. We need to check their Conditions (2.1)–(2.6).

a) Condition (2.1) was verified in the first part of Theorem 4 above.

b) To verify Condition (2.2), note that

Q(β, δ, f0) = E

∫ψ(Yi − xβ − Z>i δ)[x Zi] · f0(x | Yi, Zi) dx

= EE[ψ(Yi − xβ − Z>i δ)[x Zi] · |Yi, Zi]= Eψ(Yi − xβ − Z>i δ)[x Zi]= EE[ψ(Y − xβ − Z>δ)[x Z]|X,Z]

= EE[ψ(Y − xβ − Z>δ)|X,Z][x Z]

= E(τ −GY (xβ + Z>δ))[x Z].

42

The derivative with respect to (β, δ), denoted by Γ1(β, δ, f), is −EgY (xβ + Z>δ))[x Z][x Z>]>. Itis continuous in (β, δ) at (β0, δ0) and positive definite by Assumptions G.I and G.II.

c) Now we verify Condition (2.3). We first calculate the pathwise derivative of Q(β, δ, f) atf0:

Γ2(β, δ, f0)[f − f0] = [Q(β, δ, f0 + ζ(f − f0))− Q(β, δ, f0)]/ζ

=E

∫ψ(Yi − xβ − Z>i δ)[x Zi] · [f(x | Yi, Zi)− f0(x | Yi, Zi)] dx.

For any εn ↓ 0, such that ||(β, δ)− (β0, δ0)|| ≤ εn and ||f − f0||∞ ≤ εn:

||Q(β, δ, f)− Q(β, δ, f0)− Γ2(β, δ, f0)[f − f0]|| = 0

and

||Γ2(β, δ, f0)[f − f0]− Γ2(β0, δ0, f0)[f − f0]||

=||E∫

[ψ(Yi − xβ − Z>i δ)− ψ(Yi − xβ0 − Z>i δ0)][x Zi] · [f(x | Yi, Zi)− f0(x | Yi, Zi)] dx||

≤E

∫|ψ(Yi − xβ − Z>i δ)− ψ(Yi − xβ0 − Z>i δ0)| · ||[x Zi]|| ||f(x | Yi, Zi)− f0(x | Yi, Zi)||∞ dx

=o(1).

The first inequality holds by the property of exchanging norm and integrals. The last equalityholds because the domain for integration is O(1) by the indicator function, Assumptions C.II andC.III, and ||f − f0||∞ ≤ εn. The result follows as εn → 0.

d) Condition 2.4 holds by Assumption G.III.

e) Now we verify Condition (2.5’):

sup||β−β0||≤εn,||δ−δ0||≤εn,||f−f0||∞≤εn

||Qn(β, δ, f)− Q(β, δ, f)− Qn(β0, δ0, f0)|| = op(1/√n)

Note that

43

||Qn(β, δ, f)− Q(β, δ, f)− Qn(β0, δ0, f0)||

=||∫ (

1

n

n∑i=1

ψ(Yi − xβ − Z>i δ)[x Zi] · f(x | Yi, Zi)− Eψ(Yi − xβ − Z>i δ)[x Zi] · f(x | Yi, Zi)

− 1

n

n∑i=1

ψ(Yi − xβ0 − Z>i δ0)[x Zi] · f0(x | Yi, Zi)

)− Eψ(Yi − xβ0 − Z>i δ0)[x Zi] · f0(x | Yi, Zi) dx||

≤||∫

supif(x | Yi, Zi) ·

1

n

n∑i=1

ψ(Yi − xβ − Z>i δ)[x Zi]− Eψ(Yi − xβ − Z>i δ)[x Zi]

− supif0(x | Yi, Zi) ·

1

n

n∑i=1

ψ(Yi − xβ0 − Z>i δ0)[x Zi]− Eψ(Yi − xβ0 − Z>i δ0)[x Zi] dx||

≤C · supx|| 1n

n∑i=1

ψ(Yi − xβ − Z>i δ)[x Zi]− Eψ(Yi − xβ − Z>i δ)[x Zi]

− 1

n

n∑i=1

ψ(Yi − xβ0 − Z>i δ0)[x Zi]− Eψ(Yi − xβ0 − Z>i δ0)[x Zi]||,

for a generic C by Assumption A.III. So we need to show

sup||β−β0||≤εn,||δ−δ0||≤εn,x

|| 1n

n∑i=1

ψ(Yi − xβ − Z>i δ)[x Zi]− Eψ(Yi − xβ − Z>i δ)[x Zi]

− 1

n

n∑i=1

ψ(Yi − xβ0 − Z>i δ0)[x Zi] + Eψ(Yi − xβ0 − Z>i δ0)[x Zi]||

= op(1/√n).

We need to show that φβ,δ is Donsker. Because ψ(Yi−xβ−Z>i δ) : (β, δ) ∈ Θ is bounded and VC,it is Donsker with constant envelope. Also E[|X|] <∞ and E[‖Z‖] <∞ by assumption C.II. Theproduct of φβ,δ with [x Zi] also forms a Donsker class with a square integrable envelope. Finally, theclass F , as defined in assumption G.IV, is Donsker with a constant envelope. Given assumptionG.IV, the bracketing number of F by Corollary 2.7.4 in van der Vaart and Wellner (1996) satisfies

logN[·](ε,F , L2(P )) = O(ε−dim(x,y,z)

δ ) = O(ε−2−δ′), for some δ′ < 0. Those conditions and Corollary9.32 (iii) of Kosorok (2008) lead to our conclusion.

f) Finally, we verify Condition (2.6). Noting that√nQn(β0, δ0, f0) converges weakly and

Assumption G.II, we only verify that

√nΓ2(β0, δ0, f0)[f − f0] =

√nE

∫ψ(Y − xβ0 − Z>δ0)[x Z] · (f(x | Y,Z)− f0(x | Y,Z)) dx,

converges weakly. Also, since the bias of f is op(1/√n), we only need to verify

√nE∫ψ(Y −xβ0−

Zδ0)[x Z] · (f(x | Y,Z)− Ef(x | Y, Z)) dx converges weakly:

√nE

∫ψ(Y − xβ0 − Z>δ0)[x Z] · [ 1

2π

∫κ(hxnζ)(φ− Eφ)e−iζx dζ] dx. (17)

44

First,

supζ| 1n

n∑j=1

eiζX2j − EeiζX2j | p→ 0.

This is because eiζX2j = cos(ζX2j) + i sin(ζX2j) and those two terms are Lipschitz in ζ. Simi-

larly, 1n

∑nj=1X1e

iζX2jp→ EX1e

iζX2j . Therefore,1n

∑nj=1X1e

iζX2j

1n

∑nj=1 e

iζX2j

p→ EX1eiζX2j

EeiζX2j. By the continuous

mapping theorem,

exp

(∫ ζ

0

i 1n

∑nj=1X1e

iζX2j

1n

∑nj=1 e

iζX2j

)p→ exp

(∫ ζ

0

iEX1eiζX2j

EeiζX2j

)Also we have

E[eiζX2 | Y = y, Z = z] ≡1

hynhznn

∑[eiζX2khyn(Y − y)khzn(Z − z)]

1hynhznn

∑[khyn(Y − y)khzn(Z − z)]]

.

So (17) equals

√n

∫ ∫ψ(y − xβ0 − z>δ0)[x z] · [ 1

2π

∫κ(hxnζ)(φ− φ)e−iζx dζ] dx dydz

=√n

∫ ∫ψ(y − xβ0 − z>δ0)[x z] · [ 1

2π

∫κ(hxnζ)×

(1

hynhznn

∑[eiζX2khyn(Y − y)khzn(Z − z)]

exp

(∫ ζ

0

i 1n

∑nj=1X1e

iζX2j

1n

∑nj=1 e

iζX2j

)/

1

n

n∑j=1

eiζX2j1

hynhznn

∑[khyn(Y − y)khzn(Z − z)]− φ)

× e−iζx dζ] dx dydz

=√n

∫ ∫ψ(y − xβ0 − z>δ0)[x z] · [ 1

2π

∫κ(hxnζ)(

1

hynhznn

∑[eiζX2khyn(Y − y)khzn(Z − z)]

[exp

(∫ ζ

0

iEX1eiζX2j

EeiζX2j

)+ op(1)]/[EeiζX2jf(y, z) + op(1)]− φ)e−iζx dζ] dx dydz

=√n 1

n

∑∫ψ(Yj − xβ0 − Z>j δ0)[x Zj ] · [

1

2π

∫κ(hxnζ)×

[eiζX2j ][exp

(∫ ζ

0

iEX1eiζX2

EeiζX2

)+ op(1)]/[EeiζX2f(y, z) + op(1)]e−iζx dζ] dx− φ+ op(n

−1/2),

which converges weakly, and the result follows.

Proof of Lemma 1. The proof is a direct application of Theorem B in Chen, Linton, and VanKeilegom (2003) and parallel to that of convergence in distribution.

Condition (2.4B) in Chen, Linton, and Van Keilegom (2003) is directly implied by conditionG.IB. Let

sup||β−β0||≤εn,||δ−δ0||≤εn,||f−f0||∞≤εn

||Q∗n(β, δ, f)− Qn(β, δ, f)− [Q∗n(β0, δ0, f0)− Qn(β0, δ0, f0)]|| = op∗(1/√n).

45

The verification of Condition (2.5’B) in Chen, Linton, and Van Keilegom (2003) is parallel tothat of (2.5’) of Theorem 5. Finally, to verify Condition (2.6’B) in Chen, Linton, and VanKeilegom (2003), from Gine and Zinn (1990), the P ∗-distribution of

√nQ∗n(β, δ, f)− Qn(β, δ, f)

approximates the distribution of√nQ∗n(β, δ, f)− Q(β, δ, f), which is approximately the same as

the distribution of√nQn(β0, δ0, f0) by the verification of condition (2.5’) of Theorem 5. For this

we need that f(·) possesses the same smoothness as f0(·), which is guaranteed by Theorem 3 andcondition G.IV.

46

References

Agca, S., and A. Mozumdar (2016): “Investment Cash Flow Sensitivity: Fact or Fiction?,”forthcoming in the Journal of Financial and Quantitative Analysis.

Almeida, H., M. Campello, and A. Galvao (2010): “Measurement Errors in InvestmentEquations,” Review of Financial Studies, 23, 3279–3328.

Almeida, H., M. Campello, and M. S. Weisbach (2004): “The Cash Flow Sensitivity ofCash,” Journal of Finance, 59, 1777–1804.

Andrews, D. W. K. (1995): “Nonparametric Kernel Estimation for Semiparametric Models,”Econometric Theory, 11, 560–596.

Angrist, J., V. Chernozhukov, and I. Fernandez-Val (2006): “Quantile Regression UnderMisspecification, with an Application to the U.S. Wage Structure,” Econometrica, 74, 539–573.

Chen, X., O. Linton, and I. Van Keilegom (2003): “Estimation of Semiparametric ModelsWhen the Criterion Function is not Smooth,” Econometrica, 71, 1591–1608.

Chernozhukov, V., and C. Hansen (2006): “Instrumental Quantile Regression Inference forStructural and Treatment Effects Models,” Journal of Econometrics, 132, 491–525.

Chesher, A. (2001): “Parameter Approximations for Quantile Regressions with MeasurementError,” Working Paper CWP02/01, Department of Economics, University College London.

Cummins, J. G., K. A. Hasset, and S. D. Oliner (2006): “Investment Behavior, ObservableExpectations, and Internal Funds,” American Economic Review, 96, 796–810.

Delaigle, A., P. Hall, and A. Meister (2008): “On Deconvolution With Repeated Measure-ments,” The Annals of Statistics, 36, 665–685.

Erickson, T., and T. Whited (2000): “Measurement Error and the Relationship betweenInvestment and Q,” Journal of Political Economy, 108, 1027–1057.

Fan, J. (1991): “On the Optimal Rates of Convergence for Nonparametric Deconvolution Prob-lems,” The Annals of Statistics, 19, 1257–1272.

Fan, J., and Y. K. Truong (1993): “Nonparametric Regression with Errors in Variables,” TheAnnals of Statistics, 21, 1900–1925.

Fazzari, S., R. G. Hubbard, and B. Petersen (1988): “Financing Constraints and CorporateInvestment,” Brooking Papers on Economic Activity, 1, 141–195.

Gine, E., and J. Zinn (1990): “Bootstrapping General Empirical Measures,” Annals of Probabil-ity, 18, 851–869.

Hansen, B. E. (2008): “Uniform Convergence Rates for Kernel Estimation with Dependent Data,”Econometric Theory, 24, 726–748.

Hausman, J., Y. Luo, and C. Palmer (2014): “Errors in the Dependent Variable of QuantileRegression Models,” Working Paper, MIT.

47

Hayashi, F. (1982): “Tobin’s Marginal q and Average q: A Neoclassical Interpretation,” Econo-metrica, 50, 213–224.

He, X., and H. Liang (2000): “Quantile Regression Estimates for a Class of Linear and PartiallyLinear Errors-in-Variables Models,” Statistica Sinica, 10, 129–140.

He, X., and Q.-M. Shao (1996): “A General Bahadur Representation of M-estimators andIts Application to Linear Regression with Nonstochastic Designs,” The Annals of Statistics, 6,2608–2630.

(2000): “On Parameters of Increasing Dimensions,” Journal of Multivariate Analysis, 73,120–135.

Hu, Y., and Y. Sasaki (2015a): “Closed-form Estimation of Nonparametric Models with Non-classical Measurement Errors,” Journal of Econometrics, 185, 392–408.

(2015b): “Identification of Paired Nonseparable Measurement Error Models,” WorkingPaper, Johns Hopkins University.

Hu, Y., and S. M. Schennach (2008): “Instrumental Variable Treatment of Nonclassical Mea-surement Error Models,” Econometrica, 76, 195–216.

Kaplan, S., and L. Zingales (1997): “Do Financing Constraints Explain Why Investment IsCorrelated with Cash Flow?,” Quarterly Journal of Economics, 112, 169–215.

Kato, K., A. F. Galvao, and G. Montes-Rojas (2012): “Asymptotics for Panel QuantileRegression Models with Individual Effects,” Journal of Econometrics, 170, 76–91.

Koenker, R. (2005): Quantile Regression. Cambridge University Press, New York, New York.

Koenker, R., and G. W. Bassett (1978): “Regression Quantiles,” Econometrica, 46, 33–49.

Kosorok, M. (2008): Introduction to Empirical Processes and Semiparametric Inference. Springer.

Kotlarski, I. (1967): “On Characterizing the Gamma and the Normal Distribution,” PacificJournal of Mathematics, 20, 69–76.

Lewellen, J., and K. Lewellen (2014): “Investment and Cashflow: New Evidence,” Journalof Financial and Quantitative Analysis, Forthcoming.

Li, T. (2002): “Robust and Consistent Estimation of Nonlinear Errors-in-variables Models,” Jour-nal of Econometrics, 110, 1–26.

Li, T., and Q. Vuong (1998): “Nonparametric Estimation of the Measurement Error ModelUsing Multiple Indicators,” Journal of Multivariate Analysis, 65, 139–165.

Ma, Y., and G. Yin (2011): “Censored Quantile Regression with Covariate Measurement Errors,”Statistica Sinica, 21, 949–971.

Modigliani, F., and M. Miller (1958): “The Cost of Capital, Corporation Finance and theTheory of Investment,” American Economic Review, 48, 261–297.

48

Newey, W. K. (1994a): “The Asymptotic Variance of Semiparametric Estimators,” Econometrica,62, 1349–1382.

(1994b): “Kernel Estimation of Partial Means and a Generalized Variance Estimator,”Econometric Theory, 10, 233–253.

Newey, W. K., and D. L. McFadden (1994): “Large Sample Estimation and HypothesisTesting,” in Handbook of Econometrics, Vol. 4, ed. by R. F. Engle, and D. L. McFadden. NorthHolland, Elsevier, Amsterdam.

Politis, D. N., and J. P. Romano (1999): “Multivariate Density Estimation with GeneralFlat-Top Kernels of Infinite Order,” Journal of Multivariate Analysis, 68, 1–25.

Poterba, J. (1988): “Financing Constraints and Corporate Investment: Comment,” BrookingsPapers on Economic Activity, 1, 200–204.

Schennach, S. M. (2004a): “Estimation of Nonlinear Models with Measurement Error,” Econo-metrica, 72, 33–75.

(2004b): “Nonparametric Regression in the Presence of Measurement Error,” EconometricTheory, 20, 1046–1093.

(2008): “Quantile Regression with Mismeasured Covariates,” Econometric Theory, 24,1010–1043.

(2014): “Entropic Latent Variable Integration via Simulation,” Econometrica, 82, 345–385.

Stein, J. (2003): “Agency Information and Corporate Investment,” in Handbook of the Economicsof Finance, ed. by M. Harris, and R. Stulz. Elsevier/North-Holland, Amsterdam.

Torres-Saavedra, P. (2013): “Quantile Regression for Repeated Responses Measured with Er-ror,” North Carolina State University, mimeo.

van der Vaart, A., and J. A. Wellner (1996): Weak Convergence and Empirical Processes.Springer-Verlag Press, New York, New York.

Wang, H. J., L. A. Stefanski, and Z. Zhu (2012): “Corrected-Loss Estimation for QuantileRegression with Covariate Measurement Errors,” Biometrika, 99, 405–421.

Wei, Y., and R. J. Carroll (2009): “Quantile Regression with Measurement Error,” Journalof the American Statistical Association, 104, 1129–1143.

Wu, Y., Y. Ma, and G. Yin (2015): “Smoothed and Corrected Score Approach to CensoredQuantile Regression With Measurement Errors,” Journal of the American Statistical Association,110, 1670–1683.

49

Table 1: Fourier Estimator

hx\hy 0.2 0.3 0.4 0.5 0.6

B 0.09592 0.07751 0.09131 0.06610 0.093731.5 SD 0.09765 0.08403 0.11718 0.06085 0.08298

MSE 0.01874 0.01307 0.02207 0.00807 0.01567

B 0.09770 0.07553 0.08439 0.08457 0.096961.6 SD 0.09206 0.07425 0.08380 0.07819 0.07446

MSE 0.01802 0.01122 0.01414 0.01327 0.01495

B 0.10225 0.06963 0.07425 0.07394 0.099681.7 SD 0.10788 0.04347 0.09206 0.08228 0.08720

MSE 0.02209 0.00674 0.01399 0.01224 0.01754

B 0.09728 0.07464 0.07571 0.08745 0.104261.8 SD 0.08762 0.05453 0.06794 0.07288 0.07359

MSE 0.01714 0.00855 0.01035 0.01296 0.01629

B 0.08473 0.09663 0.08869 0.10269 0.104581.9 SD 0.02795 0.09039 0.08893 0.11265 0.08798

MSE 0.007961 0.017507 0.015774 0.023235 0.018679

optimal hx hy B SD MSE1.7 0.3 0.06963 0.04347 0.00674

50

Table 2: Infeasible Kernel Estimator

hx\hy 0.1 0.2 0.3 0.4 0.5

B 0.00706 0.01670 0.01124 0.01097 0.033310.1 SD 0.04271 0.04013 0.04270 0.04468 0.04288

MSE 0.00187 0.00189 0.00195 0.00212 0.00295

B 0.01065 0.00200 0.00537 0.01181 0.030370.2 SD 0.03394 0.03158 0.03454 0.02594 0.02880

MSE 0.00127 0.00100 0.00122 0.00081 0.00175

B 0.00513 0.00737 0.00777 0.01121 0.018900.3 SD 0.02867 0.02620 0.02737 0.02755 0.02652

MSE 0.00085 0.00074 0.00081 0.00088 0.00106

B 0.01320 0.00935 0.01242 0.01729 0.028020.4 SD 0.02411 0.02800 0.02527 0.02289 0.02464

MSE 0.00076 0.00087 0.00079 0.00082 0.00139

B 0.02175 0.01778 0.01631 0.02742 0.033470.5 SD 0.02270 0.02195 0.02541 0.02582 0.02875

MSE 0.00099 0.00080 0.00091 0.00142 0.00195

optimal hx hy B SD MSE0.3 0.2 0.00737 0.02620 0.00074

51

Table 3: Naive Kernel Estimator

hx\hy 0.1 0.2 0.3 0.4 0.5

B 0.11254 0.11132 0.10409 0.10296 0.132490.1 SD 0.04776 0.04581 0.04199 0.04449 0.04084

MSE 0.01495 0.01449 0.01260 0.01258 0.01922

B 0.09695 0.09553 0.10231 0.10989 0.126300.2 SD 0.03399 0.03093 0.03381 0.02886 0.03153

MSE 0.01055 0.01008 0.01161 0.01291 0.01695

B 0.10133 0.09800 0.10012 0.10388 0.116630.3 SD 0.02957 0.02854 0.02820 0.02557 0.02600

MSE 0.01114 0.01042 0.01082 0.01145 0.01428

B 0.10243 0.09939 0.10476 0.10609 0.118840.4 SD 0.02715 0.03008 0.02441 0.02245 0.02630

MSE 0.01123 0.01078 0.01157 0.01176 0.01481

B 0.10757 0.10313 0.10299 0.11340 0.117820.5 SD 0.02196 0.02276 0.02703 0.02525 0.02700

MSE 0.01205 0.01115 0.01134 0.01350 0.01461

optimal hx hy B SD MSE0.2 0.2 0.09553 0.03093 0.01008

52

Table 4: Simulation Results over Various Quantiles

τ\estimator Fourier Infeasible Naive

B 0.07068 0.00510 0.09976τ = 0.2 SD 0.06196 0.02002 0.02746

MSE 0.00883 0.00043 0.01070

B 0.06372 0.00515 0.09785τ = 0.3 SD 0.05685 0.01775 0.02568

MSE 0.00729 0.00034 0.01023

B 0.06014 0.00480 0.09737τ = 0.4 SD 0.05530 0.01734 0.02507

MSE 0.00667 0.00032 0.01011

B 0.05943 0.00457 0.09778τ = 0.5 SD 0.05487 0.01770 0.02383

MSE 0.00654 0.00033 0.01013

B 0.06029 0.00479 0.09818τ = 0.6 SD 0.05546 0.01792 0.02391

MSE 0.00671 0.00034 0.01021

B 0.06326 0.00542 0.09915τ = 0.7 SD 0.05681 0.02003 0.02422

MSE 0.00723 0.00043 0.01042

B 0.07213 0.00458 0.10130τ = 0.8 SD 0.05947 0.02265 0.02564

MSE 0.00874 0.00053 0.01092

53

Table 5: Summary Statistics (standardized data)

First ThirdMin. Quartile Median Mean Quartile Max.

IK −1.7590 −0.7140 −0.1595 0.0000 0.4972 6.0110qe −1.1130 −0.6819 −0.3478 0.0000 0.3019 7.9500q −1.2590 −0.6898 −0.3018 0.0000 0.3591 9.2710

CFK −7.6380 −0.7284 −0.1864 0.0000 0.4988 5.4140Number of observations 11,431

54