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Nonparametric Identification and Estimation with Non-Classical Errors-in-Variables
\title{\textsc{Nonparametric Identification and Estimation with
Non-Classical Errors-in-Variables}}
\author{Kirill S. \textsc{Evdokimov}\thanks{
Universitat Pompeu Fabra and Barcelona School of Economics: \textsf{
[email removed]}.} \and Andrei \textsc{Zeleneev}\thanks{
University College London: \textsf{[email removed]}.} \thanks{
Evdokimov gratefully acknowledges the support from the Spanish MCIN/AEI via
grants RYC2020-030623-I, PID2019-107352GB-I00, and Severo Ochoa Programme
CEX2019-000915-S.}}
\date{This version: December, 2023}
\maketitle
\begin{abstract}
This paper considers nonparametric identification and estimation of the
regression function when a covariate is mismeasured. The measurement error
need not be classical. Employing the small measurement error approximation,
we establish nonparametric identification under weak and easy-to-interpret
conditions on the instrumental variable. The paper also provides
nonparametric estimators of the regression function and derives their rates
of convergence.
\end{abstract}
\newpage
\section{Introduction}
Regression is a fundamental tool for empirical analysis. Errors-in-Variables
(EIV) are a widespread problem in empirical applications. Mismeasurement of
a covariate, when not accounted for, may lead to biased estimates and
invalid inferences.
The goal of this paper is to study the nonparametric identification and
estimation of the regression function when a covariate is mismeasured.
Importantly, the measurement error need not be classical and can be
correlated with the mismeasured covariate. In this paper, we adopt the small
measurement error approximation, which allows us to provide a simple
nonparametric characterization of the problem. Then we provide transparent
and constructive identification analysis under weak and easy-to-interpret
conditions on the instrumental variable.
First, we focus on the Weakly Classical Measurement Error (WCME) model,
where the measurement error is uncorrelated with the true covariate but
generally is not independent from it. We show that the skedastic function of
the measurement errors, together with its derivative, plays a key role in
determining the bias of the naive regression estimator. We also show how the
EIV skedastic function can be recovered from the distribution of the
observables using a (possibly discrete) instrument, and how one can
construct a bias-corrected estimator of the regression function. We derive
its rate of convergence and provide conditions under which the approximation
error becomes negligible compared to the errors arising from the
nonparametric estimation of unknown functions in large samples.
Next, we consider the general Non-Classical Measurement Error (NCME) model,
which allows for a very broad form of EIV. In particular, the measurement
error can be correlated with the true covariate. Even though the NCME model
is much more general than the WCME model, we demonstrate how the results
from our analysis of the WCME model can be utilized to establish
identification of the general NCME model.
Importantly, our approach only requires an instrumental variable that can be
discrete. This allows for a broader range of applications compared to the
methods that require a continuously distributed instrument or the
availability of repeated (multiple) measurements of the true covariate.
{}
In Section~\ref{sec:WCME}, we discuss in detail the exclusion and relevance
conditions that the instrumental variable needs to satisfy, and consider
some examples.
Our paper contributes to the large literature studying models with
mismeasured data. \cite{CarrollEtAl2006Book-ME,ChenHongNekipelov2011JEL};
and \cite{Schennach2020HB-ME,Schennach2022JEcLit} provide excellent
literature overviews.
Our main focus is on the settings where the distribution of measurement
error is unknown. Nonparametric analysis of the EIV problem in such settings
requires additional information to separate the true covariate from the
measurement error. In Economics, most commonly instrumental variables are
used for this purpose (
\citealp{HINP1991JoE,HausmanNeweyPowell1995JoE,Newey2001REStat,Schennach2007Ecta,Hu2008JoE,HuSchennach2008Ecta,Wilhelm2019WP-TestingForME}
, among others). Repeated measurements can also be utilized (
\citealp{HINP1991JoE,LiVuong1998JoE,Schennach2004Ecta}, among others) but
are less frequently available. Note that repeated measurements can serve as
valid instruments in our analysis.
Small measurement error (SME) approximation has been widely employed in
Statistics and Econometrics to study the effect of EIV on various estimators
and to bias-correct them (e.g.,
\citealp{WolterFuller1982AS,CarrollStefanski1990JASA,Chesher1991Biomet,Chesher2000WP,CarrollEtAl2006Book-ME,ChesherSchluter2002ReStud}
, among others). \cite{BoundBrownMathiowetz2001HBoE} document that the EIV
in economic applications are typically relatively small although are often
non-classical, which suggests that our analysis should be useful in many
applied settings. This paper differs from the previous literature in two
ways. First, it appears to be the first paper to study the nonparametric SME
approximation with non-classical EIV. Second, previously developed SME bias
reduction techniques usually assume that the EIV variance is either known or
can be directly estimated from an available dataset, e.g., using repeated
measurements. In contrast, this paper demonstrates how the whole EIV
skedastic function can be identified and estimated using only a (possibly
discrete) instrumental variable.
The analysis of this paper complements the existing \textquotedblleft
large\textquotedblright\ measurement error literature. By focusing on a
narrower range of settings, the paper provides simpler characterizations of
the problem and estimators, which are valid under very weak and
easy-to-interpret conditions on the instrumental variable. In particular,
our identification results do not rely on the completeness conditions, and
estimation does not involve solving ill-posed inverse problems or
deconvolution.
The rest of this paper is organized as follows. Section~\ref{sec:WCME}
studies the Weakly Classical Measurement Error (WCME) model. Section~\ref
{sec:NCME} considers the general Non-Classical Measurement Error (NCME)
model. The proofs are collected in the Appendix.
\bigskip
\section{Weakly Classical Measurement Errors\label{sec:WCME}}
We consider the regression model
\begin{equation}
{\Greekmath 011A} \left( x\right) \equiv E\left[ Y_{i}|X_{i}^{\ast }=x\right] ,
\label{eq:reg rho x def}
\end{equation}
where $Y_{i}\in \mathbb{R}$ is the outcome variable, and $X_{i}^{\ast }\in
\mathbb{R}$ is the true value of the covariate for individual $i$. The
researcher observed a mismeasured version of $X_{i}^{\ast }$:
\begin{equation*}
X_{i}=X_{i}^{\ast }+{\Greekmath 0122} _{i}.
\end{equation*}
where ${\Greekmath 0122} _{i}$ is the measurement error. The researcher has a
random sample of $\left( Y_{i},X_{i},Z_{i}\right) $, where $Z_{i}$ are
instrumental variables that are used to identify the model and will be
discussed later. It is straightforward to also include correctly measured
covariates into the model, see Remark~\ref{rem:covars W} for details.
In this section we consider the Weakly Classical Measurement Error (WCME)\
model:
\begin{assumption}[WCME]
\namedlabel{ass:WCME}{WCME} $X_{i}=X_{i}^{\ast }+{\Greekmath 0122} _{i}$ and $E
\left[ {\Greekmath 0122} _{i}|X_{i}^{\ast }\right] =0$.
\end{assumption}
The measurement error ${\Greekmath 0122} _{i}$ is uncorrelated with the true
covariate $X_{i}^{\ast }$. Assumption~\ref{ass:WCME} is significantly weaker
than the (Strongly) Classical Measurement Error (CME) assumption, since $
{\Greekmath 0122} _{i}$ need not be independent from $X_{i}^{\ast }$. For example,
the measurement error can be conditionally heteroskedastic, i.e., its
conditional variance
\begin{equation*}
v\left( x \right) \equiv V\left[ {\Greekmath 0122} _{i}|X_{i}^{\ast }=x\right] .
\end{equation*}
may depend on $x$. Function $v\left( x^{\ast }\right) $ is usually unknown.
\begin{example*}[WCME-LIN-RC]
Suppose $X_{i}={\Greekmath 0120} _{i1}+{\Greekmath 0120} _{i2}X_{i}^{\ast }$, where $\left( {\Greekmath 0120}
_{i1},{\Greekmath 0120} _{i2}\right) \perp X_{i}^{\ast }$. Assumption~\ref{ass:WCME} is
satisfied if $E\left[ \left( {\Greekmath 0120} _{i1},{\Greekmath 0120} _{i2}\right) \right] =\left(
0,1\right) $. Here $v(x) ={\Greekmath 011B} _{{\Greekmath 0120} _{1}}^{2}+{\Greekmath 011B} _{{\Greekmath 0120} _{2}}^{2}
x^2 +2{\Greekmath 011B} _{{\Greekmath 0120} _{1}{\Greekmath 0120} _{2}}x$.
\end{example*}
In this paper we use the Small Measurement Error (SME)\ approximation (e.g.,
\citealp{WolterFuller1982AS}) for the analysis, i.e., we will consider the
approximations of the model when $v(x^*)$ and the higher conditional moments
of ${\Greekmath 0122}_i$ are small.
Specifically, we model the measurement error as ${\Greekmath 0122} _{i}={\Greekmath 011C} {\Greekmath 0118}
_{i}$ where the distribution of ${\Greekmath 0118} _{i}$ is fixed, and ${\Greekmath 011C} $ is a
non-stochastic parameter. Assumption~\ref{ass:WCME} requires $E[{\Greekmath 0118}
_{i}|X_{i}^{\ast }]=0$. The conditional variance of ${\Greekmath 0122} _{i}$ is
given by $v(x)={\Greekmath 011C} ^{2}V[{\Greekmath 0118} _{i}|X_{i}^{\ast }=x]=O({\Greekmath 011C} ^{2})$.
We study the properties of the model when ${\Greekmath 011C} \rightarrow 0$. Under some
smoothness conditions,
\begin{equation*}
E\left[ Y_{i}|X_{i}=x\right] ={\Greekmath 011A} \left( x\right) +O\left( {\Greekmath 011C} ^{2}\right)
.
\end{equation*}
Thus, a naive regression estimator of ${\Greekmath 011A} $ that ignores the presence of
the measurement errors in $X_{i}$ has a bias of order $O\left( {\Greekmath 011C}
^{2}\right) $, e.g., see \cite{Chesher1991Biomet}.
The goal of the small measurement error analysis is to provide a function $
\widetilde{{\Greekmath 011A} }\left( x\right) $ that {}has a smaller bias, i.e., satisfies
\begin{equation}
\widetilde{{\Greekmath 011A} }\left( x\right) ={\Greekmath 011A} \left( x\right) +O\left( {\Greekmath 011C}
^{p}\right) , \label{eq:intro:rho-tilde goal}
\end{equation}
for some $p\geq 3$.
To identify the model we will rely on an observed instrumental variable
(instrument) $Z_{i}$ that satisfies the following exogeneity assumption.
\setcounter{assumption}{0}
\begin{assumption}
\label{ass:NPID:exclusion} $E\left[ Y_{i}|X_{i}^{\ast },Z_{i}\right] =E\left[
Y_{i}|X_{i}^{\ast }\right] $.
\end{assumption}
This assumption states that $Z_{i}$ is an \textquotedblleft
excluded\textquotedblright\ variable: given $X_{i}^{\ast }$, instrument $
Z_{i}$ has no effect on the conditional mean of $Y_{i}$. Without loss of
generality, we can assume that $Z_{i}$ is discrete. (The instrument also
needs to satisfy a \textquotedblleft relevance\textquotedblright condition:
it needs to affect the conditional distribution $f_{X^{\ast }|Z}\left(
x|z\right) $. This condition will appear in Theorem~\ref{thm:NPID-non-cl:IV}
.)
\begin{assumption}
\label{ass:NPID:Nondiffer ME} $E\left[ Y_{i}|X_{i}^{\ast },Z_{i},X_{i}\right]
=E\left[ Y_{i}|X_{i}^{\ast },Z_{i}\right] $.
\end{assumption}
Assumption~\ref{ass:NPID:Nondiffer ME} says that the measurement error $
{\Greekmath 0122} _{i}$ is nondifferential: conditional on $\left( X_{i}^{\ast
},Z_{i}\right) $, $X_{i}$ provides no additional information about (the
conditional mean of) $Y_{i}$. This assumption can be equivalently stated as $
E\left[ Y_{i}|X_{i}^{\ast },Z_{i},{\Greekmath 0122} _{i}\right] =E\left[
Y_{i}|X_{i}^{\ast },Z_{i}\right] =E\left[ Y_{i}|X_{i}^{\ast }\right] $.
\begin{assumption}
\label{ass:NCME} ${\Greekmath 0122} _{i}={\Greekmath 011C} {\Greekmath 0118} _{i}$, where ${\Greekmath 011C} \geq 0$ is
non-random, $E[{\Greekmath 0118} _{i}|X_{i}^{\ast }]=0$, and $f_{{\Greekmath 0118} |X^{\ast }Z}\left(
u|x,z\right) =f_{{\Greekmath 0118} |X^{\ast }}\left( u|x\right) $ $\forall u,x,z$.
{}
\end{assumption}
Assumption~\ref{ass:NCME} and the smoothness conditions below are stated
using the auxiliary variable ${\Greekmath 0118} _{i}$, whose variance does not shrink.
This allows formulating the smoothness conditions in the conventional form.
For example, we will assume that the density $f_{{\Greekmath 0118} |X^{\ast }}\left(
u|x\right) $ is bounded. In contrast, the density of ${\Greekmath 0122} _{i}={\Greekmath 011C}
{\Greekmath 0118} _{i}$ is $f_{{\Greekmath 0122} |X^{\ast }}\left( e|x\right) =\frac{1}{{\Greekmath 011C} }
f_{{\Greekmath 0118} |X^{\ast }}\left( \left. \frac{e}{{\Greekmath 011C} }\right\vert x\right) $ and is
not bounded as ${\Greekmath 011C} \rightarrow 0$. {} Assumption~\ref{ass:NCME} also implies that ${\Greekmath 0122}
_{i}\perp Z_{i}|X_{i}^{\ast }$.
Finally, the following two assumptions are smoothness conditions.
\begin{assumption}
\label{ass:NPID:smoothness} Function ${\Greekmath 011A} \left( x\right) $ and the
conditional densities $f_{X^{\ast }|Z}(x|z)$ and $f_{{\Greekmath 0118} |X^{\ast }}\left( u
|x\right) $ are bounded functions with $m\geq p$ bounded derivatives with
respect to $x$, for some integer $p\geq 3$.
\end{assumption}
\begin{assumption}
\label{ass:NPID:dominance} $\int |u|^{m}\sup_{\tilde{x}\in \mathcal{S}_X}\left\vert
{\Greekmath 0272}_{x}^{\ell }f_{{\Greekmath 0118} |X^{\ast }}(u|\tilde{x})\right\vert du< \infty$ for
$\ell \in \{0,\ldots,m\}$ for some closed convex set $\mathcal{S}_X \subseteq
\mathbb{R}$ containing the supports of $X_i^*$ and $X_i$.
\end{assumption}
Assumption \ref{ass:NPID:dominance} is a weak restriction imposed on the
conditional moments of ${\Greekmath 0118} _{i}$.
Appendix \ref{sec: verification of dominance} provides a set of primitive
conditions that guarantee that Assumption \ref{ass:NPID:dominance} holds.
Also notice that Assumption \ref{ass:NPID:dominance} would automatically
hold if
the support of ${\Greekmath 0118} _{i}$ is bounded, since $f_{{\Greekmath 0118} |X^{\ast }}(u |x )$ and
its derivatives are uniformly bounded under Assumption \ref
{ass:NPID:smoothness}.
We can now state the first main result of the paper. Let
\begin{eqnarray}
q\left( x,z\right) &\equiv &E\left[ Y_{i}|X_{i}=x,Z_{i}=z\right] ,\qquad
q\left( x\right) \equiv E\left[ Y_{i}|X_{i}=x\right] , \notag \\
s_{X|Z}\left( x|z\right) &\equiv &\frac{f_{X|Z}^{\prime }\left( x|z\right) }{
f_{X|Z}\left( x|z\right) },\qquad s_{X^{\ast }|Z}\left( x|z\right) \equiv
\frac{f_{X^{\ast }|Z}^{\prime }\left( x|z\right) }{f_{X^{\ast }|Z}\left(
x|z\right) }, \notag \\
\widetilde{v}\left( x\right) &\equiv &\frac{q\left( x,z_{1}\right) -q\left(
x,z_{2}\right) }{q^{\prime }\left( x\right) \left[ s_{X|Z}\left(
x|z_{1}\right) -s_{X|Z}\left( x|z_{2}\right) \right] }, \label{eq: v tilde}
\\
\widetilde{{\Greekmath 011A} }\left( x,z\right) &\equiv &q\left( x,z\right) -\widetilde{v}
\left( x\right) \left[ q^{\prime }\left( x\right) s_{X|Z}\left( x|z\right) +
\tfrac{1}{2}q^{\prime \prime }\left( x\right) \right] -q^{\prime }\left(
x\right) \widetilde{v}^{\prime }\left( x\right) . \label{eq: rho tilde}
\end{eqnarray}
Let $\mathcal{S}_{X^{\ast }}(z)$ denote the conditional support of $
X_{i}^{\ast }|Z_{i}=z$. Consider any two values $z_{1}$ and $z_{2}$ the
instrument can take.
\begin{theorem}
\label{thm:NPID-non-cl:IV}Suppose that Assumptions~\ref{ass:WCME} and \ref
{ass:NPID:exclusion}-\ref{ass:NPID:dominance} are satisfied. Suppose either
(i) $p=3$, or (ii) $E\left[ {\Greekmath 0118} _{i}^{3}|X_{i}^{\ast }\right] =0$ and $p=4$.
Consider any point $x\in \mathcal{S}_{X^{\ast }}(z_{1})\cap \mathcal{S}
_{X^{\ast }}(z_{2})$ such that
\begin{equation}
{\Greekmath 011A} ^{\prime }\left( x\right) \left[ s_{X^{\ast }|Z}\left( x|z_{1}\right)
-s_{X^{\ast }|Z}\left( x|z_{2}\right) \right] \neq 0.
\label{eq:thm:rank cond}
\end{equation}
Then, as ${\Greekmath 011C} \rightarrow 0$,
\begin{eqnarray*}
\widetilde{v}\left( x\right) &=&v\left( x\right) +O\left( {\Greekmath 011C} ^{p}\right)
,\quad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and} \\
\widetilde{{\Greekmath 011A} }\left( x,z_{1}\right) &=&{\Greekmath 011A} \left( x\right) +O\left( {\Greekmath 011C}
^{p}\right) .
\end{eqnarray*}
\end{theorem}
Theorem \ref{thm:NPID-non-cl:IV} demonstrates that $\widetilde {\Greekmath 011A} (x,z_1)$
identifies ${\Greekmath 011A}(x)$ up to an error of order $O({\Greekmath 011C}^p)$ when ${\Greekmath 011C}
\rightarrow 0$. This is a substantial improvement over naive regression $
q(x) $ which has a bias of order $O({\Greekmath 011C}^2)$. The improvement in the
magnitude of the approximation error (from $O({\Greekmath 011C}^2)$ to $O({\Greekmath 011C}^4)$) is
especially noticeable when $E[{\Greekmath 0118}_i^3|X_i^*] = 0$, e.g., when the
measurement error is symmetric.
To establish the desired result, we first characterize the bias of $q(x,z)$
up to an error of order $O({\Greekmath 011C}^p)$. The bias of $q(x,z)$ is of order $
O({\Greekmath 011C}^2)$ and determined by the conditional variance of the measurement
error $v(x)$ and its derivative $v^{\prime }(x)$, which are unknown. Then,
we show that $\widetilde v (x)$ identifies $v(x)$ up to an error of order $
O({\Greekmath 011C}^p)$.\footnote{
We also demonstrate that $\widetilde v^{\prime }(x) = v^{\prime }(x) +
O({\Greekmath 011C}^p)$. This is an important step of the proof.} This allows us to
approximate the bias of $q(x,z)$ with a sufficient precising using $
\widetilde v(x)$ in place of $v(x)$. Finally, we construct $\widetilde {\Greekmath 011A}
(x,z_1)$ by bias correcting $q(x,z_1)$ and demonstrate that it approximates
the true regression function ${\Greekmath 011A}(x)$ up to an error of order $O({\Greekmath 011C}^p)$.
The idea behind nonparametric identification is that although function $E
\left[ Y_{i}|X_{i}^{\ast }=x,Z_{i}=z\right] $ does not depend on $z$,
function $q\left( x,z\right) \equiv E\left[ Y_{i}|X_{i}=x,Z_{i}=z\right] $
does vary with $z$. The theorem shows how this variation allows recovering $
v\left( x\right) $. Specifically, the proof of the Theorem shows that
\begin{equation}
q\left( x,z\right) ={\Greekmath 011A} \left( x\right) +v\left( x\right) {\Greekmath 011A} ^{\prime
}\left( x\right) s_{X^{\ast }|Z}\left( x|z\right) +\frac{1}{2}v\left(
x\right) {\Greekmath 011A} ^{\prime \prime }\left( x\right) +{\Greekmath 011A} ^{\prime }\left(
x\right) v^{\prime }\left( x\right) +O\left( {\Greekmath 011C} ^{p}\right) .
\label{eq:NPID:non-cl:key expr for q(x,z)}
\end{equation}
Then it is shown that replacing the derivatives of ${\Greekmath 011A} $ with those of $q$
and $s_{X^{\ast }|Z}$ with $s_{X|Z}$ on the right-hand side in the above
equation does not increase the magnitude of the approximation error, i.e.,
that
\begin{equation}
q\left( x,z\right) ={\Greekmath 011A} \left( x\right) +v\left( x\right) q^{\prime }\left(
x\right) s_{X|Z}\left( x|z\right) +\frac{1}{2}v\left( x\right) q^{\prime
\prime }\left( x\right) +q^{\prime }\left( x\right) v^{\prime }\left(
x\right) +O\left( {\Greekmath 011C} ^{p}\right) .
\label{eq:NPID:non-cl:key expr for q(x,z) - feasible}
\end{equation}
Note that only the second term on the right-hand side depends on $z$. Since $
q$, $q^{\prime }$, and $s_{X|Z}$ are directly identified from the joint
distribution of the observables, considering the differences $q\left(
x,z_{1}\right) -q\left( x,z_{2}\right) $ then allows identification of $
v\left( x\right) $ by $\widetilde{v}\left( x\right) $ up to an error of order
$\ O\left( {\Greekmath 011C} ^{p}\right) $. This identification approach requires the
rank condition that $s_{X|Z}\left( x|z\right) $ depends on $z$, which is
ensured by equation~(\ref{eq:thm:rank cond}). In addition, it is necessary
that $q^{\prime }\left( x\right) \neq 0$, which is also ensured by equation~(
\ref{eq:thm:rank cond}). The latter condition is weak: ${\Greekmath 011A} ^{\prime
}\left( x\right) =0$ for all $x$ only if ${\Greekmath 011A} \left( x\right) $ is a
constant.\footnote{
For an analysis of the role of the conditions such as ${\Greekmath 011A} ^{\prime }\left(
x\right) \neq 0$ in the measurement error literature see \cite
{EvdokimovZeleneev2018WP-Inference}.} {}
\begin{remark}
It is easy to check that $\widetilde{{\Greekmath 011A} }\left( x,z_{1}\right) =\widetilde{
{\Greekmath 011A} }\left( x,z_{2}\right) $.{}
\end{remark}
\begin{corollary}[Classical Measurement Error]
\label{cor:NPID-CME}Suppose the hypotheses of Theorem~\ref
{thm:NPID-non-cl:IV} hold, and the measurement error is classical, i.e., $
{\Greekmath 0122} _{i}\perp \left( X_{i}^{\ast },Z_{i},Y_{i}\right) $. Suppose
condition~(\ref{eq:thm:rank cond}) holds for some point $\dot{x}$. Then for
all $x$ and $z$ such that $x \in \mathcal{S}_{\mathcal{X}^*}(z)$, as ${\Greekmath 011C}
\rightarrow 0$,
\begin{equation*}
\widetilde{{\Greekmath 011A} }_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{CME}}\left( x,z\right) ={\Greekmath 011A} \left( x\right)
+O\left( {\Greekmath 011C} ^{p}\right),
\end{equation*}
where
\begin{equation*}
\widetilde{{\Greekmath 011A} }_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{CME}}\left( x,z\right) \equiv q\left( x,z\right) -
\widetilde{v}\left( \dot{x}\right) \left[ q^{\prime }\left( x\right)
s_{X|Z}\left( x|z\right) +\tfrac{1}{2}q^{\prime \prime }\left( x\right)
\right] .
\end{equation*}
\end{corollary}
When the measurement error is classical, $v\left( x\right) $ is constant,
and hence the term containing $v^{\prime }\left( x\right) $ is absent from
equation~(\ref{eq:NPID:non-cl:key expr for q(x,z)}). In addition, Corollary~
\ref{cor:NPID-CME} requires only a single point $\dot{x}$ satisfying the
rank condition (\ref{eq:thm:rank cond}) for identification of ${\Greekmath 011A} \left(
x\right) $ for all $x$, since $v\left( x\right) =v\left( \dot{x}\right) $
for all $x$.\footnote{
For the result of Corollary \ref{cor:NPID-CME} to hold, it is sufficient to
require $v(x)$ to be constant, i.e., to assume that ${\Greekmath 0122}_i$ is
homoskedastic, instead of requiring ${\Greekmath 0122} _{i}\perp \left(
X_{i}^{\ast },Z_{i},Y_{i}\right)$.} In contrast, in the general case of
Theorem~\ref{thm:NPID-non-cl:IV}, nonparametric identification of $v\left(
x\right) $ and ${\Greekmath 011A} \left( x\right) $ for a given $x$ requires condition~(
\ref{eq:thm:rank cond}) to hold at that point $x$. Identification of the
Classical Measurement Error model has been previously established in \cite
{EvdokimovZeleneev-Estimation-WP-2022}.{}
\begin{remark}[Examples of IVs]
{} First, variable $X_{i}^{\ast }$ can be
caused by $Z_{i}$; for example, $X_{i}^{\ast }=q\left( Z_{i},{\Greekmath 0111}
_{i}\right) $ for some unobserved (vector) ${\Greekmath 0111} _{i}$ and function $q$.
Assumptions \ref{ass:NPID:exclusion} and \ref{ass:NPID:Nondiffer ME} will be
satisfied if $E\left[ U_{i}|Z_{i},{\Greekmath 0111} _{i},{\Greekmath 0122} _{i}\right] =0$.
Second, variable $Z_{i}$ can be caused by $X_{i}^{\ast }$, for example be a
second measurement or proxy for $X_{i}^{\ast }$: $Z_{i}={\Greekmath 011F} \left(
X_{i}^{\ast },{\Greekmath 0117} _{i}\right) $. For example, $Z_{i}$ can be a second
measurement: $Z_{i}={\Greekmath 010B} _{1}+{\Greekmath 010B} _{2}X_{i}^{\ast }+{\Greekmath 0117} _{i}$.
Assumptions \ref{ass:NPID:exclusion} and \ref{ass:NPID:Nondiffer ME} will be
satisfied if $E\left[ U_{i}|X_{i}^{\ast },{\Greekmath 0117} _{i},{\Greekmath 0122} _{i}\right]
=0 $.{}
\end{remark}
\begin{remark}
If the skedastic function $v\left( x\right) $ is known, there is no need in
having the instrumental variable $Z_{i}$. In this case one can use $
\widetilde{{\Greekmath 011A} }\left( x\right) $ from equation~(\ref{eq: rho tilde}) with $
q\left( x,z\right) $ and $\widetilde{v}\left( x\right) $ replaced by $
q\left( x\right) $ and $v\left( x\right) $, and the conclusion of Theorem~
\ref{thm:NPID-non-cl:IV} will continue to hold, i.e., $\widetilde{{\Greekmath 011A} }
\left( x\right) ={\Greekmath 011A} \left( x\right) +O\left( {\Greekmath 011C} ^{p}\right) $.
\end{remark}
\begin{remark}
\label{rem:covars W}It is straightforward to include additional correctly
measured covariates $W_{i}$ into the model, and to consider regression
function ${\Greekmath 011A} \left( x,w\right) \equiv E\left[ Y_{i}|X_{i}^{\ast }=x,W_{i}=w
\right] $. The correctly measured covariates $W_{i}$ play no special role,
and all of the analysis can be thought of as applying conditionally on $
W_{i}=w$ for any given $w$, i.e., for the stratum with $W_{i}=w$. Thus, we
omit $W_{i}$ for simplicity of exposition.
\end{remark}
\bigskip
\paragraph{Nonparametric Estimation}
Theorem \ref{thm:NPID-non-cl:IV} suggests an analogue estimator of $
\widetilde{{\Greekmath 011A} }$ by replacing functions that appear in equations~(\ref{eq:
v tilde})-(\ref{eq: rho tilde}) with their standard nonparametric estimators
(e.g., kernel or sieve), with optimally chosen tuning parameters. Let $
\widehat{{\Greekmath 011A} }\left( x\right) $ denote this estimator. As an alternative to
$\widehat{{\Greekmath 011A} }\left( x\right)$, we also consider $\widehat{{\Greekmath 011A} }^{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{
Naive}}\left( x\right)$, a naive nonparametric estimator of ${\Greekmath 011A}(x)$
ignoring the presence of the measurement error.
To approximate the finite sample properties of the studied estimators when $
{\Greekmath 011C}$ is small, we consider a triangular asymptotic framework with drifting $
{\Greekmath 011C} = {\Greekmath 011C}_n$ converging to zero as the sample size $n \rightarrow \infty$.
\begin{lemma}
\label{lem:NCME:IV:rates}Suppose the hypotheses of Theorem~\ref
{thm:NPID-non-cl:IV} hold and $Z_{i}$ is discrete. Also, suppose ${\Greekmath 011C}
_{n}=o(1)$, then
\begin{eqnarray*}
\widehat{{\Greekmath 011A} }\left( x\right) -{\Greekmath 011A} \left( x\right) &=&O_{p}\left( n^{-
\frac{m-1}{2m+1}}+{\Greekmath 011C} _{n}^{p}\right) , \\
\widehat{{\Greekmath 011A} }^{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Naive}}\left( x\right) -{\Greekmath 011A} \left( x\right)
&=&O_{p}\left( n^{-\frac{m}{2m+1}}+{\Greekmath 011C} _{n}^{2}\right) .
\end{eqnarray*}
\end{lemma}
Lemma \ref{lem:NCME:IV:rates} establishes the rates of convergence for the
proposed and naive estimators. For each estimator, the rate of convergence
is determined by two components: the standard nonparametric learning rate
and the EIV (errors-in-variables) bias due to the presence of the
measurement error.
The nonparametric learning rate for $\widehat {\Greekmath 011A} (x)$ is slower than for
the naive estimator because it involves nonparametric estimation of
derivatives such as $q^{\prime }(x)$ and $s_{X|Z}(x|z)$. However, as Theorem
\ref{thm:NPID-non-cl:IV} suggests, the EIV bias of the proposed estimator $
\widehat {\Greekmath 011A} (x)$ is of order $O({\Greekmath 011C}_n^p)$, whereas the naive estimator
has a much larger bias of order $O({\Greekmath 011C}_n^2)$. Thus, despite the slower
nonparametric learning rate, $\widehat {\Greekmath 011A} (x)$ has a faster rate of
convergence than $\widehat{{\Greekmath 011A} }^{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Naive}}\left( x\right)$ unless $
{\Greekmath 011C}_n$ is very small, i.e., the measurement error is negligible.
To illustrate this result, suppose the conditions of Theorem~\ref
{thm:NPID-non-cl:IV}(ii) hold, $m=p=4$, and ${\Greekmath 011C} _{n}={}O\left( n^{-\frac{1
}{12}}\right) $. Then $\widehat{{\Greekmath 011A} }\left( x\right) -{\Greekmath 011A} \left( x\right)
=O_{p}\left( n^{-\frac{1}{3}}\right) $, but the naive estimator has a much
slower rate of convergence: $\widehat{{\Greekmath 011A} }^{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Naive}}\left( x\right)
-{\Greekmath 011A} \left( x\right) =O_{p}\left( n^{-\frac{1}{6}}\right) $, because of the
EIV bias.
\bigskip
\bigskip
\section{General Non-Classical Measurement Error\label{sec:NCME}}
In this section, we will use notation $\mathcal{X}_{i}^{\ast }$ for the true
mismeasured covariate and consider the general measurement model
\begin{equation}
X_{i}= \mathscr{m} \left( \mathcal{X}_{i}^{\ast },{\Greekmath 0120} _{i}\right) ,
\label{eq:NC ME def 1}
\end{equation}
where $\mathscr{m}$ is an unknown function and ${\Greekmath 0120} _{i}$ is a random vector
independent from $\left( Y_{i},\mathcal{X}_{i}^{\ast },Z_{i}\right) $.
Function $\mathscr{m}$ need not be monotone in any of the arguments. The
measurement error is non-classical: $X_{i}- \mathcal{X}_{i}^{\ast }$ and $
\mathcal{X}_{i}^{\ast }$ are generally correlated.
As before, we want to identify and estimate the regression function
\begin{equation}
{\Greekmath 011A} _{\mathcal{X}^{\ast }}\left( \varkappa \right) \equiv E\left[ Y_{i}|
\mathcal{X}_{i}^{\ast }=\varkappa \right] . \label{eq: NC ME rho def}
\end{equation}
We will assume that the measurement is sufficiently informative about the
true $\mathcal{X}_{i}^{\ast }$. Define
\begin{equation*}
{\Greekmath 0116} \left( \varkappa ^{\ast }\right) \equiv E\left[ \left. X_{i}\right\vert
\mathcal{X}_{i}^{\ast }=\varkappa ^{\ast }\right].
\end{equation*}
\begin{assumption}[MONOT-MEAS]
\namedlabel{ass:MONOT-MEAS}{MONOT-MEAS} ${\Greekmath 0116} \left( \varkappa ^{\ast
}\right) $ is a strictly increasing function.
\end{assumption}
\begin{example*}[NCME-LIN-RC, continued]
For $X_{i}={\Greekmath 0120} _{i1}+{\Greekmath 0120} _{i2}\mathcal{X}_{i}^{\ast }$, we have ${\Greekmath 0116}
\left( \varkappa ^{\ast }\right) =c_{{\Greekmath 0120} 1}+c_{{\Greekmath 0120} 2}\varkappa ^{\ast }$,
and Assumption~\ref{ass:MONOT-MEAS} is satisfied if $c_{{\Greekmath 0120} 2}>0$. Note
that ${\Greekmath 0120} _{i2}$ is allowed to take negative values, which makes $\mathscr{m}$ a
decreasing function of $\mathcal{X}_{i}^{\ast }$ for such observations.
\end{example*}
Note that since functions $\mathscr{m}$ and ${\Greekmath 011A} _{\mathcal{X}^{\ast }}$ are
unrestricted, the model \eqref{eq:NC ME def 1}-\eqref{eq: NC ME rho def}
cannot be identified without some normalization or additional information.
Specifically, for any strictly increasing function ${\Greekmath 0115} $, we can define
an observationally equivalent model with $\mathcal{\tilde{X}}_{i}^{\ast
}\equiv {\Greekmath 0115} \left( \mathcal{X}_{i}^{\ast }\right) $, ${\Greekmath 011A} _{\mathcal{
\tilde{X}}^{\ast }}\left( \tilde{\varkappa}^{\ast }\right) \equiv {\Greekmath 011A} _{
\mathcal{X}^{\ast }}\left( {\Greekmath 0115} ^{-1}\left( \tilde{\varkappa}^{\ast
}\right) \right) $ and $\mathscr{m}_{\mathcal{\tilde{X}}_{i}^{\ast }}\left( \tilde{
\varkappa}^{\ast },{\Greekmath 0120} \right) \equiv \mathscr{m}\left( {\Greekmath 0115} ^{-1}\left( \tilde{
\varkappa}^{\ast }\right) ,{\Greekmath 0120} \right) $. {}
Let us \emph{define} random variable
\begin{equation}
X_{i}^{\ast }\equiv {\Greekmath 0116} \left( \mathcal{X}_{i}^{\ast }\right) .
\label{eq:def:Xis}
\end{equation}
Then,
\begin{equation*}
E\left[ X_{i}|X_{i}^{\ast }\right] =E\left[ X_{i}|{\Greekmath 0116} \left( \mathcal{X}
_{i}^{\ast }\right) \right] =E\left[ X_{i}|\mathcal{X}_{i}^{\ast }\right]
={\Greekmath 0116} \left( \mathcal{X}_{i}^{\ast }\right) =X_{i}^{\ast },
\end{equation*}
where the first equality follows by~(\ref{eq:def:Xis}), the second equality
follows from the strict monotonicity of ${\Greekmath 0116} (\cdot)$, the third equality is
the definition of ${\Greekmath 0116} \left( \cdot \right) $, and the last equality follows
from~(\ref{eq:def:Xis}).
Thus, for the general measurement error model we can consider an
observationally equivalent model that defines $X_{i}^{\ast }$ as in
equation~(\ref{eq:def:Xis}):
\begin{eqnarray*}
{\Greekmath 011A} _{X^{\ast }}\left( x\right) &\equiv &E\left[ Y_{i}|X_{i}^{\ast }=x
\right] , \\
X_{i} &=&X_{i}^{\ast }+{\Greekmath 0122} _{i},\qquad E[{\Greekmath 0122}
_{i}|X_{i}^{\ast }]=0.
\end{eqnarray*}
Since in this model Assumption~\ref{ass:WCME} holds, we can apply the result
of Theorem~\ref{thm:NPID-non-cl:IV} to identify ${\Greekmath 011A} _{X^{\ast }}\left(
x\right) $ and $v\left( x\right) \equiv E\left[ {\Greekmath 0122}
_{i}^{2}|X_{i}^{\ast }=x\right] $ (up to an error of order $O({\Greekmath 011C} ^{p})$)
using $\widetilde{v}(x)$ and $\widetilde{{\Greekmath 011A} }(x)$ defined in equations~
\eqref{eq: v tilde} and \eqref{eq: rho tilde}, respectively. Notice that $
{\Greekmath 011A} _{\mathcal{X}^{\ast }}\left( \varkappa \right) ={\Greekmath 011A} _{X^{\ast }}\left(
{\Greekmath 0116} \left( \varkappa \right) \right) $. However, since $\mathcal{X}
_{i}^{\ast }$ is not observed, one cannot identify ${\Greekmath 0116} \left( \varkappa
\right) $ and ${\Greekmath 011A} _{\mathcal{X}^{\ast }}(\varkappa )$ without some
sadditional information.
Suppose for a moment that the marginal distribution $F_{\mathcal{X}^{\ast }}$
of $\mathcal{X}_{i}^{\ast }$ is known (for example, from a separate dataset,
e.g., administrative records). In this case, we can identify ${\Greekmath 011A} _{
\mathcal{X}^{\ast }}(\varkappa )$ up to an error of order $O({\Greekmath 011C} ^{p})$
using
\begin{equation}
\widetilde{{\Greekmath 011A} }_{\mathcal{X}^{\ast }}(\varkappa ,z)\equiv \widetilde{{\Greekmath 011A} }
_{X^{\ast }}\left( \widetilde{Q}_{X^{\ast }}(F_{\mathcal{X}^{\ast
}}(\varkappa )),z\right) , \label{eq:def:rho tilde NC}
\end{equation}
where
\begin{equation}
\widetilde{Q}_{X^{\ast }}\left( s\right) \equiv Q_{X}\left( s\right) +\frac{1
}{2}\left\{ s_{X}\left( Q_{X}\left( s\right) \right) \widetilde{v}\left(
Q_{X}\left( s\right) \right) +{\Greekmath 0272} _{x}\widetilde{v}\left( Q_{X}\left(
s\right) \right) \right\} . \label{eq:def:Q tilde NC}
\end{equation}
Here $F_{\mathcal{X}^{\ast }}(\cdot )$ denotes the CDF of $\mathcal{X}
_{i}^{\ast }$, and $Q_{X}(\cdot )$ denotes the quantile function (QF) of $
X_{i}$. Note that all functions on the right-hand side of equation~(\ref
{eq:def:Q tilde NC}) are identified directly from the observed data.
First, we demonstrate that $\widetilde Q_{X^*} (s) = Q_{X^*}(s) + O({\Greekmath 011C}^p)$
, where $Q_{X^*}(\cdot)$ denotes the quantile function of $X_i^*$. Combining
this with the result of Theorem \ref{thm:NPID-non-cl:IV} allows us to
establish the desired result formalized by the theorem below.
\setcounter{assumption}{0}
\begin{assumption}
\label{ass:NPID:integrable derivatives} $\int \left \vert {\Greekmath 0272}_x^\ell
f_{X^*}(x) \right \vert dx < \infty$ for $\ell \in \{1, \ldots, p\}$.
\end{assumption}
\begin{theorem}
\label{thm:NCME rho tilde} Suppose that the hypotheses of Theorem \ref
{thm:NPID-non-cl:IV} are satisfied for $x = Q_{X^* } \left(F_{\mathcal{X}^*}
(\varkappa)\right)$. Also, suppose Assumptions \ref{ass:MONOT-MEAS} and \ref
{ass:NPID:integrable derivatives} hold. Then, as ${\Greekmath 011C} \rightarrow 0$,
\begin{align*}
\widetilde {\Greekmath 011A}_{\mathcal{X}^*} (\varkappa, z_1) = {\Greekmath 011A}_{\mathcal{X}
^*}(\varkappa) + O({\Greekmath 011C}^p).
\end{align*}
\end{theorem}
Theorem \ref{thm:NCME rho tilde} demonstrates that, if the marginal
distribution of $\mathcal{X}_{i}^{\ast }$ is given, it is possible to
identify ${\Greekmath 011A} (\varkappa )$ up to an error of order $O({\Greekmath 011C} ^{p})$ in the
general NCME model \eqref{eq:NC ME def 1} building on the identification
results for the WCME model.
Note that obtaining (an estimate of) the marginal distribution $F_{\mathcal{X
}^{\ast }}$ is a much simpler task than obtaining a validation sample, i.e.,
the data on $\left( X_{i},\mathcal{X}_{i}^{\ast }\right) $ jointly. For
example, suppose $\mathcal{X}_{i}^{\ast }$ are individual wages, and $X_{i}$
are self-reported wages in a survey. The marginal distribution $F_{\mathcal{X
}^{\ast }}$ can be provided by the Social Security Administration or similar
tax authorities in other countries. Providing such marginal distribution
does not pose any privacy risks. In contrast, obtaining a validation sample
that links individual's responses $X_{i}$ to the individual's social
security records $\mathcal{X}_{i}^{\ast }$ is a difficult task that in
particular faces major challenges concerning privacy.
If the distribution of $\mathcal{X}_{i}^{\ast }$ is unknown we can still
apply Theorem \ref{thm:NCME rho tilde} at $\varkappa =Q_{\mathcal{X}^{\ast
}}(q)$ for any quantile $q\in \left( 0,1\right) $ to identify $E[Y_{i}|
\mathcal{X}_{i}^{\ast }=Q_{\mathcal{X}^{\ast }}(q)]={\Greekmath 011A} _{\mathcal{X}^{\ast
}}(Q_{\mathcal{X}^{\ast }}(q))$, where $Q_{\mathcal{X}^{\ast }}(\cdot )$ is
the (unknown) quantile function of $\mathcal{X}_{i}^{\ast }$:
\begin{corollary}
\label{cor: NCME quantiles} Suppose that the hypotheses of Theorem \ref
{thm:NCME rho tilde} are satisfied for $x = Q_{X^*} (q)$. Then, as ${\Greekmath 011C}
\rightarrow 0$,
\begin{align*}
\widetilde {\Greekmath 011A}_{X^*} \left(\widetilde Q_{X^*} (q), z_1\right) = \widetilde
{\Greekmath 011A}_{\mathcal{X}^*} (Q_{\mathcal{X}^*} (q), z_1) = {\Greekmath 011A}_{\mathcal{X}^*} (Q_{
\mathcal{X}^*} (q)) + O ({\Greekmath 011C}^p).
\end{align*}
\end{corollary}
Corollary \ref{cor: NCME quantiles} demonstrates that even if $F_{\mathcal{X}
^*}$ is unknown, we can still identify the conditional expectation of $Y_i^*$
given the $q$'th quantile of $\mathcal{X}_{i}^{\ast }$. Notice that in some
applications, the unobserved variable $\mathcal{X}_i^*$, for example an
individual's ability, might not even have well-defined economic units. In
such settings, identification of $E[Y_i | \mathcal{X}_i^* = Q_{\mathcal{X}
^*} (q)]$ is fully exhaustive.
\bigskip
\bigskip
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