# Partially Linear Models under Data Combination††thanks: We thank the Editor,
Francesca Molinari, three anonymous referees, Federico Bugni, Nathael Gozlan,
Jinyong Hahn, Jim Heckman, Matt Masten, David Pacini, Adam Rosen, Andres
Santos, Jörg Stoye, Martin Weidner and conference and seminar participants at
Duke, Oxford, Tilburg, UCLA, the 2021 European Winter Meeting of the
Econometric Society and the 2021 Bristol Econometric Study Group for useful
comments and suggestions. We also thank Hongchang Guo, Zhangchi Ma and Frank
Yan for capable research assistance.

Xavier D’Haultfoeuille CREST-ENSAE<EMAIL_ADDRESS>Xavier
D’Haultfoeuille thanks the hospitality of PSE where part of this research was
conducted. Christophe Gaillac Nuffield College and the University of Oxford,
<EMAIL_ADDRESS>Arnaud Maurel Duke University, NBER and
IZA<EMAIL_ADDRESS>Arnaud Maurel thanks the hospitality of the
University of Pennsylvania where part of this research was conducted.

###### Abstract

We study partially linear models when the outcome of interest and some of the
covariates are observed in two different datasets that cannot be linked. This
type of data combination problem arises very frequently in empirical
microeconomics. Using recent tools from optimal transport theory, we derive a
constructive characterization of the sharp identified set. We then build on
this result and develop a novel inference method that exploits the specific
geometric properties of the identified set. Our method exhibits good
performances in finite samples, while remaining very tractable. We apply our
approach to study intergenerational income mobility over the period 1850-1930
in the United States. Our method allows us to relax the exclusion restrictions
used in earlier work, while delivering confidence regions that are
informative.

Keywords: Partially Linear Model; Data combination; Partial Identification;
Intergenerational Mobility.

##  1 Introduction

In this paper, we derive partial identification and inference results for a
partially linear model, in a context where the outcome of interest and some of
the covariates are observed in two different datasets that cannot be merged.
Relevant situations include cases where the researcher is interested in the
effect of a particular variable that is not observed jointly with the outcome
variable, as well as cases where the outcome and covariates of interest are
jointly observed but some of the potential confounders are observed in a
different dataset.

Our analysis focuses on a partially linear model of the following form:

$E(Y|X)=f(X_{c})+X_{nc}^{\prime}\beta_{0},\quad X=(X_{nc},X_{c}),$ (1)

in a data combination environment where $F_{Y,X_{c}}$ and $F_{X_{nc},X_{c}}$
are supposed to be identified, but the joint distribution $F_{Y,X}$ is not.
The variable $X_{c}$ is thus common to both datasets, whereas the variable
$X_{nc}\in\mathbb{R}^{p}$ is only observed in one of the two datasets. In this
setup, $\beta_{0}=(\beta_{01},...,\beta_{0p})^{\prime}$ is generally not
point-identified, and as a result we focus on the identified set of either
$\beta_{0}$ or $\beta_{0k}$ for some $k\in\\{1,...,p\\}$; the identified set
of $f$ can then be deduced from that of $\beta_{0}$.

We first derive a tractable characterization of the identified set of
$\beta_{0}$. Unlike many other models considered in the partial identification
literature, our setup does not deliver a tractable characterization of the
identified set through the support function. However, using Strassen’s theorem
(Strassen, 1965), a recent result in optimal transport by Backhoff-Veraguas et
al. (2019), and a convenient characterization of second-order stochastic
dominance, we show that this set is convex, compact, includes the origin and
can be simply constructed from its radial function.111The radial function $S$
of a closed, compact convex set $\mathcal{C}$ including the origin is defined,
for any $q$ on the unit sphere, by $S(q)=\max_{\lambda
q\in\mathcal{C}}\lambda$. The identified set of $\beta_{0k}$, then, can also
be computed at low computational cost by solving an unconstrained convex
minimization problem.

The characterization of the identified set also implies that point
identification may be achieved if $\beta_{0}=0$, or under a restriction on the
unobserved term $Y-f(X_{c})-X_{nc}^{\prime}\beta_{0}$. While the latter
condition is not directly testable, we show how to assess its plausibility
when one has access to a validation sample in which the outcome and covariates
are jointly observed.

In the partially identified case, the identification region may be reduced by
adding restrictions on $f(\cdot)$. The two-sample two-stage least squares
estimator (TSTSLS) relies on the assumption
$f(X_{c})=X_{c,i}^{\prime}\gamma_{0}$ for some $\gamma_{0}$ and
$X_{c}=(X_{c,e}^{\prime},X_{c,i}^{\prime})^{\prime}$. In this context,
$X_{c,e}$ (resp. $X_{c,i}$) corresponds to the excluded (resp. included)
instruments. This is a leading example that results in point identification.
But the exclusion restriction that $E(Y|X)$ does not depend on $X_{c,e}$ may
not be credible. We show that alternative restrictions, such as imposing a
lower bound on the $R^{2}$ of the “long regression” of $Y$ on $X_{nc}$ and
$X_{c}$ (in a similar spirit as Oster, 2019) or shape restrictions such as
monotonicity or convexity of $f$, may in practice dramatically reduce the
identified set, and allow to, e.g., identify the sign of $\beta_{0k}$.

Our identification result is constructive, and readily leads to a simple,
plug-in estimator of the identified sets for $\beta_{0}$ or $\beta_{0k}$. A
difficulty arises, however, as the estimator of the radial function is
generally not asymptotically normal. To construct asymptotically valid
confidence regions on $\beta_{0}$ or confidence intervals on $\beta_{0k}$, we
propose to use subsampling (Politis et al., 1999).

Our method is based on a specific characterization of the identified set, and
one may wonder whether alternative characterizations would be more convenient.
In particular, the identified set can also be expressed through an infinite
collection of moment inequalities. Therefore, general approaches for such
problems such as that developed by Andrews and Shi (2017) could in principle
be used instead. We show through simulations the key computational advantage
of relying on the method we propose. With a univariate $X_{nc}$, confidence
regions are typically computed in seconds, whereas they take up to 30 seconds
with a bivariate $X_{nc}$. Compared to the method of Andrews and Shi (2017),
this corresponds to a dramatic reduction by a factor of more than 1,000 in
computational time.

We apply our method to study intergenerational income mobility over the period
1850 to 1930 in the United States, revisiting the analysis of Olivetti and
Paserman (2015). In this context where the main variable and outcome of
interest are observed in two different datasets that cannot be linked, we show
that the confidence sets obtained using our method are quite informative in
practice, while allowing us to relax the exclusion restrictions underlying the
TSTSLS approach used in Olivetti and Paserman (2015). In the appendix, we
consider another application where a key control variable is observed in a
separate database. When incorporating sign constraints, our bounds are again
very informative.

#### Related literatures

The method we develop in this paper can be used in a broad set of data
combination environments. Two such contexts have attracted much attention in
the empirical literature.

One can use our method to conduct inference on the relationship between a
particular covariate and an outcome variable, in situations where both
variables are not jointly observed. A large literature on intergenerational
income mobility often faces the unavailability of linked income data across
generations and relies on exclusion restrictions, as in the application we
revisit (see Santavirta and Stuhler, 2022, for a recent survey). More
generally, this type of data combination environment frequently arises in
various subfields of empirical microeconomics, including in education and
returns to skill estimation (Rothstein and Wozny, 2013; Piatek and Pinger,
2016; Garcia et al., 2020; Hanushek et al., 2021), health (Manski, 2018;
Robbins et al., 2022) and labor (Athey et al., 2020). A leading example that
has attracted much interest in the literature is one where the researcher
seeks to combine experimental data with another observational dataset, in
particular situations where data on long-term outcomes is not available in the
experimental data.

Our approach can also be used to conduct inference on the causal effect of a
variable of interest, in a setup where some of the confounders are observed in
an auxiliary dataset. As such, our paper expands the range of data
environments in which unconfoundedness is a credible assumption, complementing
a literature that focuses on evaluating its reasonableness in the absence of
data combination (see, e.g., Altonji et al., 2005; Oster, 2019; Diegert et
al., 2022).

From a methodological standpoint, our paper is connected to the seminal
article of Cross and Manski (2002) and subsequent work by Molinari and Peski
(2006). They consider the issue of identifying the “long regression”, in our
context $E(Y|X_{c},X_{nc})$, in the same data combination set-up as here.
Importantly though, these two papers focus on deriving the identification
region for $E(Y|X_{c},X_{nc})$, but do not address the issue of inference.
They also consider a setup where the covariates $X_{nc}$ have a discrete
distribution with finite support, while we allow $X_{nc}$ to be continuously
distributed. On the other hand their setup is entirely nonparametric, whereas
we focus on a model that is linear in the covariates $X_{nc}$ and without
interaction terms with $X_{c}$. The linearity assumption plays an important
role in our ability to derive a tractable inference method. The absence of
interaction further implies that in our set-up, and in contrast with these two
papers, the identified set shrinks as one considers different values of
$X_{c}$.

Our paper is also related to Pacini (2019) and Hwang (2022). Both papers
construct bounds on the best linear predictor of $Y$ on $X$ in a similar data
combination framework as here. We show that if one is ready to impose the
usual assumption that the model is partially linear, large identification
gains may be achieved, possibly up to point identification. Hwang (2022) also
considers a set-up where some of the $X$’s are only observed with $Y$ but not
with $X_{nc}$, a case we do not study in this paper.

More generally speaking, our paper relates to the broader literature on data
combination problems in econometrics and statistics. We refer the reader to
Ridder and Moffitt (2007) for a survey of this literature and to Fan et al.
(2014), Fan et al. (2016), Buchinsky et al. (2022), and Athey et al. (2020)
for recent contributions. Contrary to ours, most of these papers impose
restrictions that entail point identification.

Within the data combination literature, our paper is technically closest to
D’Haultfoeuille et al. (2021). Though that paper considered the entirely
different context of rational expectation testing, we also relied therein on
Strassen’s theorem to obtain a characterization of the null hypothesis of
rational expectations. Importantly, we extend here our previous main result in
a highly non-trivial way, by relying in particular on recent results from
Backhoff-Veraguas et al. (2019) to handle the case where $X_{nc}$ is
multivariate. Also, we previously based our inference on Andrews and Shi
(2017). In contrast, a key contribution of our paper lies in the novel and
tractable inference method that we derive.

Finally, by developing in this data combination context a feasible inference
method that can be implemented at a very limited computational cost, our paper
also adds to the growing set of papers that propose tractable computational
methods for partially identified models (see Bontemps and Magnac, 2017 and
Molinari, 2020 for recent surveys). In particular, our paper fits into the
strand of the literature that uses tools from optimal transport to devise
computationally tractable identification and inference methods for partially
identified models (Galichon and Henry, 2011; Galichon, 2016). By
characterizing the sharp identified set based on the radial function, a novel
approach in the partial identification literature, we show that it is possible
to achieve very substantial tractability gains in this context, relative to a
more standard characterization in terms of many moment inequalities.

#### Organization of the paper

The remainder of the paper is organized as follows. In Section 2 we present
our main identification results for the two-sample partially linear model
described above. Section 3 studies estimation and inference for this model. In
Section 4, we apply our method to intergenerational income mobility in the
United States. Section 5 concludes. The Appendix of the paper gathers
additional results on robustness to measurement errors, identification in
models with heterogeneous effects of $X_{nc}$ on $Y$, and a test for point-
identification. It also presents our second application to the black-white
wage gap in the United States. Monte Carlo simulation results, additional
material on the application, and the proofs are collected in the online
Appendix. Some complements of the proofs appear in supplementary material
available in our working paper version (see D’Haultfœuille et al., 2023).
Finally, our inference method can be implemented using our companion R
package, RegCombin, available at CRAN.R-project.org/package=RegCombin.

##  2 Identification

Before presenting our main identification results, we introduce some notation
that will be used throughout the paper. We let $\|\cdot\|$, $0_{p}$ and
$\mathcal{S}_{p}$ denote respectively the usual Euclidean norm in
$\mathbb{R}^{p}$, the vector $0$ and the unit sphere in $\mathbb{R}^{p}$; we
may omit the index $p$ in the absence of ambiguity. For any cumulative
distribution function (cdf) $F$ defined on $\mathbb{R}$, we let
$F^{-1}(t)=\inf\\{x:F(x)\geq t\\}$ denote its generalized inverse and
$\overline{F}=1-F$ be the corresponding survival function. For any random
variable $A$, we let $\text{Supp}(A)$ be its support, $F_{A}$ denote its cdf.
and $V(A)$ its variance, if defined. We also let $\succ_{\\!\text{cv}}$ denote
the convex ordering, namely, for two random variables $A$ and $B$ with
$E[|A|]<\infty$ and $E[|B|]<\infty$, $A\succ_{\\!\text{cv}}B$ if
$E[\phi(A)]\geq E[\phi(B)]$ for all convex functions $\phi$.222Even though we
may have $E[|\phi(A)|]=\infty$, $E[\phi(A)]$ is always well-defined because
$E[\max(0,-\phi(A))]<\infty$, since there exists $a,b$ such that for all $x$,
$\phi(x)\geq a+bx$. We write $A\not\succ_{\\!\text{cv}}B$ when
$A\succ_{\\!\text{cv}}B$ does not hold. Finally, for any sets $C$ and
$C^{\prime}$, we denote by $\partial C$ the boundary of $C$ and by
$d_{H}(C,C^{\prime})$ the Hausdorff distance between $C$ and $C^{\prime}$,
defined by

$d_{H}(C,C^{\prime})=\max\left(\sup_{c^{\prime}\in C^{\prime}}\inf_{c\in
C}||c-c^{\prime}||,\;\sup_{c\in C}\inf_{c^{\prime}\in
C^{\prime}}||c-c^{\prime}||\right).$

###  2.1 Identification without common regressors

####  2.1.1 A tractable characterization of the identified set

We first consider a linear model and derive the sharp identified set of
$\beta_{0}$ in the absence of common regressors observed in both datasets. We
suppose that we observe from two samples that can not be merged the
distributions of the outcome, $F_{Y}$, and covariates, $F_{X}$. We maintain
the following assumption:

######  Assumption 1.

We have $E(Y^{2})<\infty$, $E(\|X\|^{2})<\infty$, $V(Y)>0$ and $V(X)$ is non-
singular. Moreover, $E(Y|X)=\alpha_{0}+X^{\prime}\beta_{0}$ for some
$(\alpha_{0},\beta_{0})\in\mathbb{R}\times\mathbb{R}^{p}$.

We focus hereafter on the identified set $\mathcal{B}$ of $\beta_{0}$. Since
$\mathcal{B}$ is the set of all vectors in $\mathbb{R}^{p}$ that are
compatible with the model and the marginal distributions of $Y$ and $X$, we
have

$\mathcal{B}=\left\\{\beta\in\mathbb{R}^{p}:\exists\;\text{r.v.
}(\widetilde{X},\widetilde{Y}):\
E(\widetilde{Y}_{0}|\widetilde{X}_{0})=\widetilde{X}_{0}^{\prime}\beta,\
\widetilde{X}\stackrel{{\scriptstyle d}}{{=}}X,\
\widetilde{Y}\stackrel{{\scriptstyle d}}{{=}}Y\right\\},$ (2)

where, for any random variable $A$ with $E[|A|]<\infty$, we let $A_{0}=A-E(A)$
and we have used that $E(Y|X)=\alpha_{0}+X^{\prime}\beta_{0}$ for some
$\alpha_{0}$ is equivalent to $E(Y_{0}|X_{0})=X_{0}^{\prime}\beta_{0}$. Now,
our goal is to express $\mathcal{B}$ to make it amenable to (simple)
estimation. To this end, we define, for any $\alpha\in(0,1)$, $F$ and $G$ cdfs
with expectation 0, the following functions:

$\displaystyle R(\alpha,F,G)$
$\displaystyle=\frac{\int_{\alpha}^{1}F^{-1}(t)dt}{\int_{\alpha}^{1}G^{-1}(t)dt},$
(3) $\displaystyle S(F,G)$ $\displaystyle=\inf_{\alpha\in(0,1)}R(\alpha,F,G).$

These two functions play an important role in our analysis. Remark that, since
$F$ and $G$ are cdfs of mean zero distributions,
$\int_{\alpha}^{1}F^{-1}(t)dt$ and $\int_{\alpha}^{1}G^{-1}(t)dt$ are both
positive, so that the ratio of superquantiles $R(\alpha,F,G)$ is well-defined,
with $R(\alpha,F,G)>0$ and $S(F,G)\geq 0$. Theorem 1 is our main
identification result.

######  Theorem 1.

Suppose that Assumption 1 holds. Then

$\mathcal{B}=\left\\{\lambda q:\;q\in\mathcal{S},\;0\leq\lambda\leq
S(F_{Y_{0}},F_{X^{\prime}_{0}q})\right\\}.$ (4)

$\mathcal{B}$ includes $0_{p}$ and is a convex, compact subset of
$\mathcal{B}^{V}=\left\\{\beta\in\mathbb{R}^{p}:\ \beta^{\prime}V(X)\beta\leq
V(Y)\right\\}$.

We now give a sketch of the proof of (4). Let $\mathcal{B}^{\prime}$ denote
the set on the right-hand side of (4). First, by definition of
$S(F_{Y_{0}},F_{X_{0}^{\prime}q})$,

$\mathcal{B}^{\prime}=\left\\{\beta\in\mathbb{R}^{p}:\forall\alpha\in(0,1),\,\int_{\alpha}^{1}F^{-1}_{X_{0}^{\prime}\beta}(t)dt\leq\int_{\alpha}^{1}F^{-1}_{Y_{0}}(t)dt\right\\}.$

This, in turn, is equivalent to $F_{X_{0}^{\prime}\beta}$ dominating
$F_{Y_{0}}$ at the second order (see, e.g. De la Cal and Cárcamo, 2006),
implying that

$\mathcal{B}^{\prime}=\left\\{\beta\in\mathbb{R}^{p}:Y_{0}\succ_{\\!\text{cv}}X_{0}^{\prime}\beta\right\\}.$

The inclusion $\mathcal{B}\subset\mathcal{B}^{\prime}$ then follows
essentially from Jensen’s inequality. As a side remark, note that we can also
express $\mathcal{B}^{\prime}$ through infinitely many moment inequality
restrictions:

$\mathcal{B}^{\prime}=\left\\{\beta\in\mathbb{R}^{p}:\,E\left[\max(0,Y_{0}-t)\right]\geq
E\left[\max(0,X_{0}^{\prime}\beta-t)\right]\;\forall t\in\mathbb{R}\right\\}.$
(5)

This equality directly follows from Fubini-Tonelli, applied to the standard
characterization of the second-order stochastic dominance condition, namely
$\int_{-\infty}^{y}F_{Y_{0}}(t)dt\geq\int_{-\infty}^{y}F_{X_{0}^{\prime}\beta}(t)dt$
$\forall y\in\mathbb{R}$. We return to this alternative characterization of
the identified set in Subsections C.1 and C.4 of the online appendix, where we
document the computational advantages of using our characterization instead.

The inclusion $\mathcal{B}^{\prime}\subset\mathcal{B}$ is more intricate to
prove. First, if $Y_{0}\succ_{\\!\text{cv}}X_{0}^{\prime}\beta$, we have, by
Strassen’s theorem (Theorem 8 in Strassen, 1965),

$\inf_{(\widetilde{Y},\widetilde{X}^{\beta}):\widetilde{Y}\stackrel{{\scriptstyle
d}}{{=}}Y,\,\widetilde{X}^{\beta}\stackrel{{\scriptstyle
d}}{{=}}X^{\prime}\beta}E\left[\left|\widetilde{X}^{\beta}_{0}-E[\widetilde{Y}_{0}|\widetilde{X}^{\beta}_{0}]\right|\right]=0.$
(6)

This result was already used in D’Haultfoeuille et al. (2021) to characterize
the restrictions on $F_{Y}$ and $F_{\psi}$ entailed by the rational
expectation hypothesis $E(Y|\psi)=\psi$, where $\psi$ denotes the subjective
expectations on an outcome $Y$. Importantly though, when $X$ is multivariate,
(6) is not sufficient to conclude that
$\mathcal{B}^{\prime}\subset\mathcal{B}$, as the $\sigma$-algebras generated
by $X$ and $X^{\prime}\beta$ are not equal in general. Nonetheless, we
prove,333We thank Nathael Gozlan for his help in obtaining (7). using a recent
result in optimal transport (Theorem 1.3 in Backhoff-Veraguas et al., 2019),
that

$\displaystyle\inf_{(\widetilde{Y},\widetilde{X}):\widetilde{Y}\stackrel{{\scriptstyle
d}}{{=}}Y,\,\widetilde{X}\stackrel{{\scriptstyle
d}}{{=}}X}E\left[\left|\widetilde{X}_{0}^{\prime}\beta-E[\widetilde{Y}_{0}|\widetilde{X}_{0}]\right|\right]\leq$
$\displaystyle\inf_{(\widetilde{Y},\widetilde{X}^{\beta}):\widetilde{Y}\stackrel{{\scriptstyle
d}}{{=}}Y,\,\widetilde{X}^{\beta}\stackrel{{\scriptstyle
d}}{{=}}X^{\prime}\beta}E\left[\left|\widetilde{X}_{0}^{\beta}-E[\widetilde{Y}_{0}|\widetilde{X}_{0}^{\beta}]\right|\right].$
(7)

Together, (6), (7), and the existence of a minimizer on the left-hand side of
(7) (Theorem 1.2 in Backhoff-Veraguas et al., 2019), imply that we can find
random variables $\widetilde{Y}$ and $\widetilde{X}$ such that
$E[\widetilde{Y}_{0}|\widetilde{X}_{0}]=\widetilde{X}_{0}^{\prime}\beta$,
$\widetilde{Y}\stackrel{{\scriptstyle d}}{{=}}Y$ and
$\widetilde{X}\stackrel{{\scriptstyle d}}{{=}}X$. Thus, $\beta\in\mathcal{B}$.

Turning to the second part of the theorem, $0_{p}\in\mathcal{B}$ follows by
noting that one can always rationalize, from the sole knowledge of their
marginal distributions, that $X$ and $Y$ are independent. That
$\mathcal{B}\subset\mathcal{B}^{V}$ comes from the inclusion
$\mathcal{B}\subset\mathcal{B}^{\prime}$, combined with the fact that
$Y_{0}\succ_{\\!\text{cv}}X_{0}^{\prime}\beta$ implies $V(Y)\geq
V(X^{\prime}\beta)$. Hence, $\mathcal{B}$ is included in a bounded ellipsoid.
The equality $\mathcal{B}=\mathcal{B}^{V}$ occurs for instance when $Y$ and
$X$ are normally distributed. Otherwise, $\mathcal{B}$ may be substantially
smaller than $\mathcal{B}^{V}$, as we illustrate below. In such cases,
$\mathcal{B}^{V}$ remains a natural benchmark as it is very simple to
characterize using $V(Y)$ and $V(X)$ only, and straightforward to estimate.

######  Remark 2.1.

Using the exact same reasoning as above, one can prove that without any linear
restriction on the conditional expectation, the identified set for
$E[Y_{0}|X_{0}]$ is $\\{g:Y_{0}\succ_{\\!\text{cv}}g(X_{0})\\}$. Similarly, if
we only impose that $m(x):=E[Y_{0}|X_{0}=x]$ belongs to a linear space
$\mathcal{Z}$ of functions, then the identified set for $m$ is $\\{\lambda
q:q\in\mathcal{Z}:\,E[|q(X_{0})|]=1,E[q(X_{0})]=0,\,0\leq\lambda\leq
S(F_{Y_{0}},F_{q(X_{0})})\\}$.444We thank a referee for pointing out this
extension.

##### Radial vs. support function characterization of the identified set.

A key takeaway from Equation (4) is that the identified set admits a very
simple expression as a function of $S$, which is the inverse of the Minkowski
gauge function of $\mathcal{B}$ (see, e.g., Definition 1.2.4 p.137 and
Proposition 3.2.4 p.157 Hiriart-Urruty and Lemaréchal, 2012), also known as
the radial function of $\mathcal{B}$. This function differs from the support
function $\sigma$ of $\mathcal{B}$, defined by
$\sigma(q,F_{Y_{0}},F_{X_{0}})=\sup_{b\in\mathcal{B}}\ q^{\prime}b$. The
difference between these two functions is illustrated in Figure 1.

Figure 1: Two characterizations of a closed convex set including the origin,
either through its support function $\sigma$ (green), or through the radial
function $S$ (red).

The partial identification literature has largely relied on support functions,
as these are powerful tools that uniquely characterize their convex sets. But
the radial function also uniquely characterizes convex sets if, as is the case
here, these sets include the origin. Importantly, this approach allows us to
characterize the sharp identified set by minimizing a simple function over the
interval $(0,1)$. In contrast, the support function approach will generally be
significantly less tractable in our context as it would require solving a
high-dimensional constrained optimization problem. Namely, using the
characterization of the identified set given in Equation (5) above, the
support function can be obtained by solving the following program:

$\displaystyle\sigma(q,F_{Y_{0}},F_{X_{0}})$
$\displaystyle=\sup_{b\in\mathbb{R}^{p}}q^{\prime}b\quad\textrm{s.t.}\inf_{t\in\mathbb{R}}E\left[\max(0,Y_{0}-t)\right]-E\left[\max(0,X_{0}^{\prime}b-t)\right]\geq
0,$ (8)

where the constraint itself involves an optimization problem. Simulation
results indicate that using the radial function rather than the support
function approach does result in very large computational gains, see Online
Appendix C.4 for details on this.

##### Partial identification of subcomponents of $\beta_{0}$.

The support function still plays a key role in our context when one is
interested in a component of
$\beta_{0}=(\beta_{0,1},...,\beta_{0,p})^{\prime}$, say $\beta_{0,k}$. The
following result shows that we can actually recover this function at a low
computational cost once $S$ is known. Hereafter, we let $e_{k}$ denotes the
$k$-th element of the canonical basis in $\mathbb{R}^{p}$ and use the
convention $1/0=\infty$ and $1/\infty=0$.

######  Corollary 1.

Suppose that Assumption 1 holds. Then, the identified set $\mathcal{B}_{k}$ of
$\beta_{0,k}$ satisfies
$\mathcal{B}_{k}=[-\sigma(-e_{k},F_{Y_{0}},F_{X_{0}}),\sigma(e_{k},F_{Y_{0}},F_{X_{0}})]$.
Moreover,

$\sigma(e_{k},F_{Y_{0}},F_{X_{0}})=\frac{1}{\inf_{q\in\mathbb{R}^{p}:q_{k}=1}1/S(F_{Y_{0}},F_{X_{0}^{\prime}q})}.$
(9)

The same holds with $\sigma(-e_{k},F_{Y_{0}},F_{X_{0}})$, after replacing
$q_{k}=1$ by $q_{k}=-1$.

We use the expression (9) of the support function, rather than the simpler
expression
$\sigma(e_{k},F_{Y_{0}},F_{X_{0}})=\sup_{q\in\mathbb{R}^{p}:q_{k}=1}S(F_{Y_{0}},F_{X_{0}^{\prime}q})$,
because $q\mapsto 1/S(F_{Y_{0}},F_{X_{0}^{\prime}q})$ is convex (see the proof
of Proposition 6, which also applies when $\varepsilon=0$), whereas $q\mapsto
S(F_{Y_{0}},F_{X_{0}^{\prime}q})$ may not be concave. It follows that one can
recover the support function $\sigma$, and in turn the sharp bounds on
$\beta_{0,k}$, by simply minimizing a convex function over $\mathbb{R}^{p-1}$.

####  2.1.2 Point identification

In some cases, our approach yields point identification of the parameters of
interest, or subcomponents of it. Proposition 1 below presents two such cases
under which the identified sets $\mathcal{B}$ and $\mathcal{B}_{1}$,
respectively, boil down to a singleton.

######  Proposition 1.

Suppose that Assumption 1 holds and let $\phi$ be a convex function such that
$E[\phi(Y)]<\infty$. Then:

  1. 1.

If for all $\beta\neq 0_{p}$, $E[\phi(X^{\prime}\beta)]=\infty$, then
$\mathcal{B}=\\{\beta_{0}\\}=\\{0_{p}\\}$.

  2. 2.

If $E[\phi(X_{1}\beta_{1})]=\infty$ for all $\beta_{1}\neq 0$ and
$E[\phi(X^{\prime}_{-1}\beta_{-1})]$ $<\infty$ for all
$\beta_{-1}\in\mathbb{R}^{p-1}$, then
$\mathcal{B}_{1}=\\{\beta_{0,1}\\}=\\{0\\}$.

Recall from our main identification result above that the identified set
$\mathcal{B}$ always includes the origin. The first point of Proposition 1
further establishes point identification of $\beta_{0}=0_{p}$ when, basically,
$Y$ has lighter tails than any linear index of $X$. The second point is
similar but focuses on a subcomponent instead: if $Y$ and
$X_{-1}^{\prime}\beta_{-1}$ have lighter tails than $X_{1}$, then
$\beta_{0,1}=0$ is point identified. As an example of function $\phi$ for
which Proposition 1 holds, one might consider for instance $\phi(x)=|x|^{a}$
for some $a>2$ (in which case $X^{\prime}\beta$ or $X_{1}$ have heavy tails),
or $\phi(x)=\exp(a|x|^{b})$ for some $a,b>0$ (in which case $X^{\prime}\beta$
or $X_{1}$ have exponential tails).

To illustrate Point 1 of Proposition 1, suppose that $p=1$, $X$ follows a
Laplace distribution (with density $\exp(-|x|)/2$ on $\mathbb{R}$) and
$Y\sim\mathcal{N}(0,1)$. Then, by using $\phi(x)=\exp(|x|^{3/2})$, it follows
from Point 1 of Proposition 1 that $\beta_{0}=0$ is point identified in this
case. On the other hand, the variance restrictions only set identify
$\beta_{0}$, with an identified set given by
$\mathcal{B}^{V}=[-1/\sqrt{2},1/\sqrt{2}]\simeq[-0.707,\,0.707]$. This example
illustrates the (in this case point-) identifying power of higher-order
moments of the distributions of $X$ and $Y$.

###  2.2 Identification with common regressors

We now turn to the frequent situation where some regressors are observed in
both datasets. Namely, suppose we observe regressors $X_{c}$ that are common
to both datasets, and assume that the partially linear model (1) holds:

$E(Y|X)=f(X_{c})+X_{nc}^{\prime}\beta_{0},\quad X=(X_{nc},X_{c}),$

The key here is to note, following Robinson (1988), that this case is
equivalent to the previous setup without common regressors once we compute the
following residuals, for all $x$ in the support of $X_{c}$:

$\displaystyle X^{x}$ $\displaystyle=X_{nc}-E(X_{nc}|X_{c}=x),$ $\displaystyle
Y^{x}$ $\displaystyle=Y-E(Y|X_{c}=x).$

It directly follows that $\beta_{0}$ satisfies
$E(Y^{x}|X^{x})=X^{x}{}^{\prime}\beta_{0}$, which allows us to use the
characterization of the identified set without common regressors obtained in
Section 2.1.

Let $\mathcal{B}^{c}$ and $\mathcal{F}$ denote the identified sets of
$\beta_{0}$ and $f$, respectively. We have the following characterization of
$\mathcal{B}^{c}$ and $\mathcal{F}$:

######  Proposition 2.

Suppose that $E(Y^{2})<\infty$, for all $x\in\text{Supp}(X_{c})$,
$E(X^{x}X^{x}{}^{\prime}|X_{c}=x)$ is nonsingular and (1) holds. Then:

$\displaystyle\mathcal{B}^{c}$ $\displaystyle=\left\\{\lambda
q:q\in\mathcal{S},\;0\leq\lambda\leq\overline{S}(F_{Y,X_{c}},F_{X_{nc}^{\prime}q,X_{c}})\right\\},$
$\displaystyle\mathcal{F}$ $\displaystyle=\left\\{x\mapsto
E(Y|X_{c}=x)-E(X_{nc}|X_{c}=x)^{\prime}\beta:\;\beta\in\mathcal{B}^{c}\right\\},$

where
$\overline{S}(F_{Y,X_{c}},F_{X_{nc}^{\prime}q,X_{c}})=\inf_{x\in\text{Supp}(X_{c})}S(F_{Y^{x}|X_{c}=x},F_{X^{x}{}^{\prime}q|X_{c}=x})$.
$\mathcal{B}^{c}$ includes $0_{p}$, is compact and convex.

It is possible to extend (1) by including interaction terms. Notably, such
specification allows for heterogeneous effects of $X_{nc}$ on $Y$, which can
be important in practice. We consider this extension in Appendix A.2. Another
interesting extension corresponds to cases where
$E(Y|X)=f(X_{c})+X_{nc}^{\prime}\beta_{0}+X_{a}^{\prime}\delta_{0}$ and we
observe in a first dataset $(Y,X_{a},X_{c})$ and in a second dataset,
$(X_{c},X_{nc})$. This setup leads to qualitatively different results. For
instance, if there is no common regressors and $(Y,X_{a})$ and $X_{nc}$ are
Gaussian, one can show that the sharp identified set of
$(\beta_{0},\delta_{0})$ is not convex and does not include $0_{p+r}$ (with
$r$ the dimension of $X_{a}$). We refer the reader to Hwang (2022) for outer
bounds on the best linear predictor in this setup and leave its study for
future research.

###  2.3 Identifying power of additional restrictions

We now consider additional restrictions that may reduce the identified set, in
some cases resulting in point identification of the parameters of interest.

####  2.3.1 Lower bound on the $R^{2}$ of the long regression

A first way to reduce the identified set is to use a lower bound on the
predictive power of $X_{nc}$ and $X_{c}$ with respect to $Y$. To formalize
this idea, we assume that $R^{2}_{\ell}$, the coefficient of determination of
the “long” regression of $Y$ on $X_{nc}$ and $X_{c}$ is higher than a certain
threshold. This threshold may be absolute (e.g., 0.1) or relative to
$R^{2}_{s}:=V(E(Y|X_{c}))/V(Y)$, the $R^{2}$ of the “short” regression of $Y$
on $X_{c}$, which is directly identified from the data. This is in the same
spirit as Oster (2019), who suggests fixing $R^{2}_{\ell}/R^{2}_{s}$ to 1.3.
Note that

$f(X_{c})+X_{nc}^{\prime}\beta=E(Y|X_{c})+(X_{nc}-E(X_{nc}|X_{c}))^{\prime}\beta,$

and the two components on the right-hand side are uncorrelated. Thus,

$R^{2}_{\ell}=\frac{V(E(Y|X_{c}))+\beta^{\prime}E(V(X_{nc}|X_{c}))\beta}{V(Y)}=R^{2}_{s}+\frac{\beta^{\prime}E(V(X_{nc}|X_{c}))\beta}{V(Y)}.$

Then, if one imposes a lower bound $\underline{R}^{2}$ on $R^{2}_{\ell}$ such
that $\underline{R}^{2}\geq R^{2}_{s}$, the identified set on $\beta$ becomes

$\left\\{\lambda
q:q\in\mathcal{S},\,\left(\frac{(\underline{R}^{2}-R^{2}_{s})V(Y)}{q^{\prime}E(V(X_{nc}|X_{c}))q}\right)^{1/2}\leq\lambda\leq\overline{S}(F_{Y,X_{c}},F_{X_{nc}^{\prime}q,X_{c}})\right\\},$

provided that $E(V(X_{nc}|X_{c}))$ is nonsingular. This restriction has three
key attractive features. First, one can in practice motivate this restriction
based on a “validation sample”, namely a subset of the population or another
population (e.g., a different country than that under investigation), for
which we identify the joint distribution of the outcome and covariates, and
thus the $R^{2}$ of the “long” regression. Second, imposing a lower bound such
that $\underline{R}^{2}>R^{2}_{s}$ allows one to exclude $0_{p}$ from the
identified set. Third, the identified set still admits a very simple
expression.

####  2.3.2 Shape restrictions

Another way to narrow the identified set $\mathcal{B}^{c}$ with common
regressors is to impose some constraints on $f(\cdot)$. Shape restrictions
such as monotonicity or convexity often follow from economic theory; see
Matzkin (1994) and Chetverikov et al. (2018) for econometric reviews, and
Tripathi (2000) and Abrevaya and Jiang (2005) for their use and testability
with partially linear models. We characterize here the identified set when we
impose such restrictions on $f$.

We model these restrictions by $[Rf](r)\geq\underline{c}(r)$ for all
$r\in\mathcal{R}$, with $R$ a known linear operator, $\underline{c}$ a known,
real function and $\mathcal{R}$ the domain of $[Rf]$ and $\underline{c}$. For
instance, if $X_{c}$ is discrete such that
$\text{Supp}(X_{c})=\\{x_{c,1},...,x_{c,K}\\}\subset\mathbb{R}$, with $K>1$
and $x_{c,1}<...<x_{c,K}$, considering $[Rf](r)=f(x_{c,r+1})-f(x_{c,r})$ for
$r\in\mathcal{R}=\\{1,...,K-1\\}$ (resp.
$[Rf](r)=(f(x_{c,r+2})-f(x_{c,r+1}))/(x_{c,r+2}-x_{c,r+1})-(f(x_{c,r+1})-f(x_{c,r}))/(x_{c,r+1}-x_{c,r})$
for $r\in\mathcal{R}=\\{1,...,K-2\\}$ with $K>2$) and $\underline{c}(r)=0$
corresponds to imposing that $f$ is non-decreasing (resp. convex). When
$X_{c}$ is continuous, the same two constraints can be imposed by considering
$[Rf](r)=f^{\prime}(r)$ and $[Rf](r)=f^{\prime\prime}(r)$, with
$\mathcal{R}=\text{Supp}(X_{c})$.

This framework also accommodates restrictions on the magnitude of the effect
of $X_{c}$ on $Y$. Namely, suppose for simplicity that $X_{c}$ is binary and
consider $[Rf](1)=-[Rf](2)=f(x_{c,2})-f(x_{c,1})$ with $\mathcal{R}=\\{1,2\\}$
and $\underline{c}(1)=\underline{c}(2)=\underline{c}\geq 0$. The extreme case
$\underline{c}=0$ corresponds to $X_{c}$ having no effect on $Y$, as in the
two-sample two-stage least squares strategy (see the next subsection for a
related, more general point identification result in this context). More
generally, this corresponds to the constraint that the magnitude of the effect
of $X_{c}$ is bounded by the cutoff $\underline{c}$,
$|f(x_{c,2})-f(x_{c,1})|\leq\underline{c}$.555If $X_{c}$ has $K>2$ points of
support, the same idea can be generalized by imposing restrictions on
$|f(x_{c,k})-f(x_{c,j})|$ for specific pairs $(j,k)\in\\{1,...,K\\}^{2}$,
$j\neq k$. By increasing $\underline{c}$, one can therefore study how the
identified set varies when relaxing the exclusion restriction, in a similar
spirit to, e.g., Masten and Poirier (2018).

Hereafter, we denote by $m_{Y}(\cdot)=E[Y|X_{c}=\cdot]$,
$m_{X_{nc}}(\cdot)=E[X_{nc}|X_{c}=\cdot]$ and

$\displaystyle\underline{S}^{c}(m_{Y},m_{X_{nc}},q)$
$\displaystyle=\sup_{\begin{subarray}{c}r\in\mathcal{R}:\\\
[Rm^{\prime}_{X_{nc}}q](r)\leq 0\end{subarray}}\lim_{u\downarrow
0}\frac{[Rm_{Y}-\underline{c}](r)+u}{[Rm^{\prime}_{X_{nc}}q](r)-u^{2}},$
$\displaystyle\overline{S}^{c}(m_{Y},m_{X_{nc}},q)$
$\displaystyle=\inf_{\begin{subarray}{c}r\in\mathcal{R}:\\\
[Rm^{\prime}_{X_{nc}}q](r)\geq 0\end{subarray}}\lim_{u\downarrow
0}\frac{[Rm_{Y}-\underline{c}](r)+u}{[Rm^{\prime}_{X_{nc}}q](r)+u^{2}},$

where we let $\sup\emptyset=-\inf\emptyset=-\infty$ and we note that the two
functions above may be infinite. We introduce limits to deal with the cases
where $[Rm^{\prime}_{X_{nc}}q](r)=0$. Proposition 3 characterizes the
identified sets of $\beta_{0}$ and $f$ under such shape restrictions.

######  Proposition 3.

Suppose that the conditions of Proposition 2 hold and
$[Rf](r)\geq\underline{c}(r)$ for all $r\in\mathcal{R}$. Then, the identified
sets $\mathcal{B}^{\text{\text{con}}}$ and $\mathcal{F}^{\text{con}}$ of
$\beta_{0}$ and $f$ satisfy

$\displaystyle\mathcal{B}^{\text{\text{con}}}$ $\displaystyle=\left\\{\lambda
q:q\in\mathcal{S}^{+},\;\underline{S}^{\text{con}}(q,F_{Y,X_{c}},F_{X_{nc},X_{c}})\leq\lambda\leq\overline{S}^{\text{con}}(q,F_{Y,X_{c}},F_{X_{nc},X_{c}})\right\\},$
$\displaystyle\mathcal{F}^{\text{con}}$ $\displaystyle=\left\\{x\mapsto
E(Y|X_{c}=x)-E(X_{nc}|X_{c}=x)^{\prime}\beta:\;\beta\in\mathcal{B}^{\text{\text{con}}}\right\\},$

where
$\mathcal{S}^{+}=\mathcal{S}\cap\\{(x_{1},...,x_{p})\in\mathbb{R}^{p}:x_{1}\geq
0\\}$ and

$\displaystyle\underline{S}^{\text{con}}(q,F_{Y,X_{c}},F_{X_{nc},X_{c}})$
$\displaystyle=\max\bigg{(}-\overline{S}(F_{Y,X_{c}},F_{-X^{\prime}_{nc}q,X_{c}}),\underline{S}^{c}(m_{Y},m_{X_{n}c},q)\bigg{)},$
$\displaystyle\overline{S}^{\text{con}}(q,F_{Y,X_{c}},F_{X_{nc},X_{c}})$
$\displaystyle=\min\bigg{(}\overline{S}(F_{Y,X_{c}},F_{X^{\prime}_{nc}q,X_{c}}),\overline{S}^{c}(m_{Y},m_{X_{n}c},q)\bigg{)}.$

$\mathcal{B}^{\text{\text{con}}}$ is compact, convex but does not include
$0_{p}$ if for some $r\in\mathcal{R}$, $[Rm_{Y}-\underline{c}](r)<0$.

In contrast to our baseline identification results in the absence of
additional restrictions, the resulting identified set may exclude the origin.
This illustrates the practical importance of imposing these types of shape
restrictions in contexts where these are likely to hold. Suppose for instance
that $p=1$, $X_{c}$ is binary ($\text{Supp}(X_{c})=\\{0,1\\}$),
$\mathcal{R}=\\{1\\}$ and $[Rf](1)=f(1)-f(0)$, namely we impose that $f$ is
non-decreasing. If $f(1)-f(0)<(m_{X_{nc}}(0)-m_{X_{nc}}(1))\beta_{0}$, then
$m_{Y}(1)<m_{Y}(0)$. As a result, $0\not\in\mathcal{B}^{\text{\text{con}}}$.
The condition $f(1)-f(0)<(m_{X_{nc}}(0)-m_{X_{nc}}(1))\beta_{0}$ holds for
instance if $m_{X_{nc}}$ is decreasing and $\beta_{0}$ is positive and large
enough.

######  Remark 2.2.

While we focus here on the identifying power of each type of restrictions
considered separately, researchers may in some contexts want to jointly impose
several of these restrictions and consider the intersection of the associated
identified sets. In the particular cases of the shape restrictions and the
restrictions on the $R^{2}$ considered above, the identified sets share the
same structure. Thus, the identified set resulting from both types of
constraints can be simply computed by replacing the lower bound on $\lambda$
by the maximum of the lower bounds of the initial sets, and proceeding
symmetrically for the upper bound.

####  2.3.3 Functional form restrictions involving common regressors

One may alternatively be willing to impose functional form restrictions on
$f$. The following proposition shows that this may yield point identification.

######  Proposition 4.

Suppose that $E(Y^{2})<\infty$, $E[\|X\|^{2}]<\infty$ and $f$ belongs to a
vector space $\mathcal{G}$. Then, if for all $\gamma\neq 0$,
$m_{X_{nc}}^{\prime}\gamma\not\in\mathcal{G}$, $\beta_{0}$ and $f$ are point
identified.

This proposition encompasses several popular restrictions. We consider in
particular three such restrictions, for which the key point-identifying
condition $m_{X_{nc}}^{\prime}\gamma\not\in\mathcal{G}$ has a simple
interpretation:

  1. 1.

$f(X_{c})=f_{1}(X_{i,c})$, with
$X_{c}=(X_{i,c}^{\prime},X_{e,c}^{\prime})^{\prime}$. This restriction, which
is implicit in, and central to the two-sample two-stage least squares
strategy, states that conditional on $(X_{nc},X_{i,c})$, $Y$ is mean-
independent of $X_{e,c}$. In such a case,
$m_{X_{nc}}^{\prime}\gamma\not\in\mathcal{G}$ for all $\gamma\neq 0$ basically
means that $m_{X_{nc}}(X_{c})$ varies with $X_{e,c}$. To see this, consider
the simple case where $m_{X_{nc}}(X_{c})=m_{i}(X_{i,c})+\Pi X_{e,c}$, for some
function $m_{i}$ and a $p\times q$ matrix $\Pi$. Then,
$m_{X_{nc}}^{\prime}\gamma\not\in\mathcal{G}$ is equivalent to $\Pi$ having
rank $p$, which is the usual rank condition in linear instrumental variable
models.

  2. 2.

$f(X_{c})=X_{c}^{\prime}\gamma_{0}$. Under this linearity restriction on
$f(\cdot)$, $m_{X_{nc}}^{\prime}\gamma\not\in\mathcal{G}$ for all $\gamma\neq
0$ basically means that $m_{X_{nc}}(X_{c})$ is nonlinear in $X_{c}$ (the two
notions are actually equivalent if $X_{nc}\in\mathbb{R}$). Note that this
point identification result fully relies on the linearity of $f(\cdot)$
combined with the nonlinearity of $E(X_{nc}|X_{c})$, and is thus akin to,
e.g., the identification of sample selection models without instruments
exploiting the nonlinearity of the inverse Mill’s ratio. Also, this result
does not apply when $X_{c}$ is binary, since in this case $m_{X_{nc}}(X_{c})$
is necessarily linear in $X_{c}$.

  3. 3.

$f(X_{c})=\sum_{j=1}^{J}f_{j}(X_{j,c})$, with
$X_{c}=(X_{1,c},...,X_{J,c})^{\prime}$. Under this additivity restriction on
$f(\cdot)$, $m_{X_{nc}}^{\prime}\gamma\not\in\mathcal{G}$ for all $\gamma\neq
0$ means that $m_{X_{nc}}(X_{c})$ is not additive in $X_{c}$. If for instance
$X_{c}=(X_{1,c},X_{2,c})$ with $X_{1,c},X_{2,c}$ both binary,
$m_{X_{nc}}^{\prime}\gamma\not\in\mathcal{G}$ for all $\gamma\neq 0$ holds if
in the regression of $X_{nc}$ on $X_{1,c},X_{2,c}$ and $X_{1,c}\times
X_{2,c}$, the coefficient of $X_{1,c}\times X_{2,c}$ is not zero.

####  2.3.4 Tail conditions

Finally, if one is ready to impose a relative tail condition between the error
term $U:=Y_{0}-X_{0}^{\prime}\beta_{0}$ and $X_{0}^{\prime}\beta_{0}$, the
identified set is considerably reduced. For simplicity, we assume here that
there are no common regressors but Proposition 5 readily extends to accomodate
such regressors.

######  Proposition 5.

Suppose that Assumption 1 holds. Then:

  1. 1.

If there exists a convex function $\phi$ such that
$E[\phi(U\lambda)]<E[\phi(X_{0}^{\prime}\beta_{0}\lambda)]=\infty$ for all
$\lambda>1$, the identified set of $\beta_{0}$ is included in
$\partial\mathcal{B}$;

  2. 2.

$X\in\mathbb{R}$, $E[\phi(U\lambda)]<E[\phi(X\lambda)]=\infty$ for all
$\lambda>0$ and it is known that $\beta_{0}>0$, $\beta_{0}$ is point
identified.

With $X\in\mathbb{R}$, the condition
$E[\phi(U\lambda)]<E[\phi(X\lambda)]=\infty$ for all $\lambda>0$ holds for
instance if $E[|U|^{a}]<E[|X|^{a}]=\infty$ for some $a>2$. More generally, the
condition $E[\phi(U\lambda)]<E[\phi(X_{0}^{\prime}\beta_{0}\lambda)]=\infty$
basically imposes that $X_{0}^{\prime}\beta_{0}$ has fatter tails than $U$. In
this sense, this condition is similar to those in Proposition 1 above.

##### Testability.

Note that we cannot test the condition
$E[\phi(U\lambda)]<E[\phi(X_{0}^{\prime}\beta_{0}\lambda)]=\infty$ for some
convex function $\phi$ and all $\lambda>1$, simply because $U$ is not
identified. On the other hand, we can assess the plausibility of
$\beta_{0}\in\partial\mathcal{B}$ using a validation sample, as defined above.
Denoting by $(Y_{v},X_{v})$ the variables corresponding to this validation
sample, it becomes possible to test whether the corresponding parameter
$\beta_{v}=V(X_{v})^{-1}\text{cov}(X_{v},Y_{v})$ is at the boundary of the
identified set one would get from the sole knowledge of $F_{Y_{v}}$ and
$F_{X_{v}}$. Provided that $\beta_{v}\neq 0$, this condition is indeed
equivalent to
$\left\|\beta_{v}\right\|=S(F_{Y_{v0}},F_{X^{\prime}_{v0}\beta_{v}/\left\|\beta_{v}\right\|})$
or, in simpler terms,

$S(F_{Y_{v0}},F_{X^{\prime}_{v0}\beta_{v}})=1.$

We consider a statistical test of this condition in Appendix A.3, and apply it
in Section 4 below.

###  2.4 Numerical illustration

We illustrate the previous results by considering the following model:

$Y=\gamma_{0,0}+X_{c}^{1.3}\gamma_{0,1}+X_{nc,1}\beta_{nc,1}+X_{nc,2}\beta_{nc,2}+U,\;U|X\sim\mathcal{N}(0,9).$

We set the coefficients as follows: $\gamma_{0,0}=-0.1$, $\gamma_{0,1}=0.3$,
$\beta_{nc,1}=1$ and $\beta_{nc,2}=1$. The variables $X$ are transformations
of $(N_{1},N_{2},N_{3})^{\prime}$, which is supposed to follow a multivariate
normal distribution with mean 0 and covariance matrix

$\Sigma=\left(\begin{array}[]{ccc}1&-0.3&-0.8\\\ -0.3&1&-0.1\\\
-0.8&-0.1&1\end{array}\right).$

Specifically, the common regressor is given by
$X_{c}=\sum_{k=1}^{K}(k-1)1\\{c_{k-1}\leq N_{1}\leq c_{k}\\}$, $K=4$,
$c_{0}=-\infty$, $c_{1},\dots,c_{K-1}$, are respectively the quantiles of
order 0.1, 0.37, 0.67 and 0.9 of the standard normal, and $c_{K}=\infty$. We
consider two cases for the regressors that are observed in one of the datasets
only, $X_{nc}$. In the first case, $(X_{nc,1},X_{nc,2})=(N_{2},\exp(N_{3}))$
and in the second, $(X_{nc,1},X_{nc,2})=(\exp(N_{2}),\exp(N_{3}))$.

Figure 2 displays several identified sets for each of the two data-generating
processes (DGPs) described above, each of them being associated with
particular restrictions. Namely, the set in red, denoted by $\mathcal{B}^{V}$,
is obtained from the variance restrictions only:

$\mathcal{B}^{V}=\left\\{\beta:\beta^{\prime}V(X^{0})\beta\leq
V(Y^{0})\right\\}\cap\left\\{\beta:\beta^{\prime}V(X^{1})\beta\leq
V(Y^{1})\right\\},$

where $X^{x}$ and $Y^{x}$ are defined as in Section 2.2. Hence,
$\mathcal{B}^{V}$ is the intersection of two ellipses. The set in green,
$\mathcal{B}^{c}$, is obtained as in Proposition 2 and relies on the
restrictions $E(Y^{x}|X_{nc},X_{c}=x)=X^{x}{}^{\prime}\beta_{0}$ for
$x\in\\{0,1\\}$. Finally, the set in blue, $\mathcal{B}^{\text{\text{con}}}$,
is a subset of $\mathcal{B}^{c}$ that imposes both convexity and monotonicity
constraints on $X_{c}$.

(a) $X_{nc,1}$ and $\ln(X_{nc,2})$ normal

(b) $X_{nc,1}$ and $X_{nc,2}$ lognormal

Note: the sets are obtained using a sample of size 100,000 and taking the
convex hull of the set obtained from a uniform grid of 1,000 directions on the
2 dimensional sphere. $\mathcal{B}^{\text{\text{con}}}$ uses both convexity
and monotonicity constraints on $X_{c}$. Figure 2: Identification regions for
different distributions of $(X_{nc,1},X_{nc,2})$

A couple of comments are in order. In case (a) the restrictions implied by the
model are much more informative than the variance restrictions, because of the
non-normality of $X_{nc,2}$, and in particular the fact that it has fatter
tails than the residuals $U$. The true point is at the boundary of
$\mathcal{B}^{c}$, illustrating Proposition 1 applied conditional on $X_{c}=0$
and $X_{c}=1$. In this case, the shape restrictions are sufficient to imply
that $0_{2}\not\in\mathcal{B}^{\text{\text{con}}}$ but also that
$\beta_{nc,1}=0$ and $\beta_{nc,2}=0$. The identified set $\mathcal{B}^{c}$ is
reduced further in case (b), as a result of the fatter tails of both
$X_{nc,1}$ and $X_{nc,2}$. Like in case (a), the shape constraints on
$f(X_{c})$ allow to reduce dramatically the identified set.

Figure 3 presents convexity constraints and different constraints on the
$R^{2}$ on the first DGP. While unlike case (a) of Figure 2, convexity
constraints alone fail to reject
$0_{2}\not\in\mathcal{B}^{\text{\text{con}}}$, imposing a constraint of the
form $\underline{R}^{2}\geq rR_{s}^{2}$ with $r>1$ rejects it by definition.
In the latter case, the identified set is no longer convex, allowing to
exclude some directions from the identified set and providing an informative
lower bound on $|\beta_{nc,1}|$. Overall, that the sharp identified sets
$\mathcal{B}^{c}$ are much more informative than the identified set
$\mathcal{B}^{V}$ based on the variance restrictions highlights the importance
of using all of the restrictions implied by the model. Another takeaway from
these numerical illustrations is that sign constraints can be very informative
in practice, resulting in significant shrinkage of the identified set.

(a) Convexity constraint on $f(\cdot)$

(b) $R_{l}^{2}\geq rR_{s}^{2}$ constraints

Note: For Panel 3(a) and $\mathcal{B}^{V}$ and $\mathcal{B}^{c}$ in Panel
3(b), the sets are obtained as in Figure 2. For the constraints $R_{l}^{2}\geq
rR_{s}^{2}$ in Panel 3(b), we use 1,500 directions and no convexification. For
this DGP, the true values of the $R^{2}$ of the long and short regressions are
$R_{l}^{2}=0.307$ and $R_{s}^{2}=0.107$.  Figure 3: Identification regions for
different shape restrictions

###  2.5 Regularization

An issue for estimation and inference on $\mathcal{B}$ is that when $\alpha\to
0$ or $\alpha\to 1$, $R(\alpha,F,G)$ is a ratio of two terms tending to 0. It
follows that its plug-in estimator may become very unstable. To regularize the
problem, we consider an outer set of $\mathcal{B}$ based on the removal of
extreme values of $\alpha$. We will focus on this outer set when we turn to
estimation and inference in Section 3. Specifically, we define, for any
$\varepsilon\in(0,1/2)$,

$\displaystyle S_{\varepsilon}(F,G)$
$\displaystyle=\min_{\alpha\in[\varepsilon,1-\varepsilon]}R(\alpha,F,G),$ (10)
$\displaystyle\mathcal{B}_{\varepsilon}$ $\displaystyle=\left\\{\lambda
q:q\in\mathcal{S},0\leq\lambda\leq
S_{\varepsilon}(F_{Y_{0}},F_{X_{0}^{\prime}q})\right\\}.$

Note that for all $F,G$, $\alpha\mapsto R(\alpha,F,G)$ is continuous on
$[\varepsilon,1-\varepsilon]$. Thus, the minimum in (10) is well-defined.
Proposition 6 below describes some properties of $\mathcal{B}_{\varepsilon}$
and relates it to the sharp identified set $\mathcal{B}$.

######  Proposition 6.

Suppose that Assumption 1 holds. Then:

  1. 1.

For all $\varepsilon\in(0,1/2)$, $\mathcal{B}_{\varepsilon}$ includes $0_{p}$,
is compact and convex;

  2. 2.

For all $0<\varepsilon<\varepsilon^{\prime}<1/2$,
$\mathcal{B}\subset\mathcal{B}_{\varepsilon}\subset\mathcal{B}_{\varepsilon^{\prime}}$
and $\cap_{\varepsilon\in(0,1/2)}\mathcal{B}_{\varepsilon}=\mathcal{B}$;

  3. 3.

Suppose that $F_{Y}$ is continuous and $U:=Y_{0}-X_{0}^{\prime}\beta_{0}$
satisfies

$\forall\lambda>0,\quad\lim_{t\to\infty}\sup_{s}\frac{\overline{F}_{\|X_{0}\|}(\lambda
t)}{\overline{F}_{U|X_{0}^{\prime}\beta_{0}=s}(t)}=0,\;\lim_{t\to\infty}\sup_{s}\frac{\overline{F}_{\|X_{0}\|}(\lambda
t)}{\overline{F}_{-U|X_{0}^{\prime}\beta_{0}=s}(t)}=0.$ (11)

Then, there exists $\varepsilon_{0}\in(0,1/2)$ such that for all
$\varepsilon\in(0,\varepsilon_{0}]$, $\mathcal{B}=\mathcal{B}_{\varepsilon}$.

The first part of Proposition 6 states that the regularized set
$\mathcal{B}_{\varepsilon}$, for all $\varepsilon\in(0,1/2)$, preserves the
compactness and convexity of the sharp identified set $\mathcal{B}$. The
second part states that $\mathcal{B}_{\varepsilon}$ is always a superset of
$\mathcal{B}$, which is arbitrarily close to $\mathcal{B}$ as
$\varepsilon\downarrow 0$. The third part states that if, basically, the tails
of $\|X_{0}\|$ are thinner than those of $U$ (Condition (11)), the set
$\mathcal{B}_{\varepsilon}$ coincides with the sharp set $\mathcal{B}$ for
$\varepsilon$ small enough. Condition (11) holds in particular if $X$ has a
bounded support and $\text{Supp}(U)=\mathbb{R}$, or if $U$ is symmetric and
has a tail index larger than that of $\|X_{0}\|$. Note that
$\mathcal{B}=\mathcal{B}_{\varepsilon}$ may hold even without (11). For
instance, if both $U$ and $X$ are normally distributed, it is easy to check
that $\mathcal{B}_{\varepsilon}=\mathcal{B}$ for all $\varepsilon\in(0,1/2)$.

On the other hand, when $U$ has thinner tails than $X^{\prime}\beta_{0}$,
$\mathcal{B}_{\varepsilon}$ will be a strict superset of $\mathcal{B}$ for
$\varepsilon$ large enough. In such cases, and under additional restrictions,
we provide upper bounds on the Hausdorff distance between $\mathcal{B}$ and
$\mathcal{B}_{\varepsilon}$ in Proposition 7. Intuitively, these bounds inform
us about the maximal possible loss, in terms of identification, that is due to
regularization.

######  Proposition 7.

Suppose that Assumption 1 holds and let $U:=Y_{0}-X_{0}^{\prime}\beta_{0}$.
Then:

  1. 1.

Assume that $X$ has an elliptical distribution with nonsingular variance
matrix $\Sigma$, a density with respect to the Lebesgue measure and
$\lim\inf_{|x|\to\infty}|x|^{1+c}f_{X_{0}^{\prime}\beta_{0}}(x)>0$ for some
$c>1$. Suppose also that $\lim\sup_{x\to\infty}$
$x^{d}\overline{F}_{|U|}(x)<\infty$ for some $d>c$. Then, there exists
$K_{1}>0$ such that for $\varepsilon$ small enough,

$d_{H}(\mathcal{B},\mathcal{B}_{\varepsilon})\leq
K_{1}\varepsilon^{\frac{1/c-1/d}{1+1/d}}.$

  2. 2.

Assume that $\beta_{0}=0_{p}$,
$\lim\inf_{x\to\infty}\inf_{q\in\mathcal{S}}x^{c}\overline{F}_{X_{0}^{\prime}q}(x)>0$
and $\lim\sup_{x\to\infty}x^{d}\overline{F}_{|U|}(x)$ $<\infty$ for some
$d>c$. Then, there exists $K_{2}>0$ such that for $\varepsilon$ small enough,

$d_{H}(\mathcal{B},\mathcal{B}_{\varepsilon})\leq K_{2}\varepsilon^{1/c-1/d}.$

The tail conditions imposed in Proposition 7 are basically the opposite as in
Point 3 of Proposition 6, as they imply that $\|X\|$ has fatter tails than
$U$. The assumption that $X$ has an elliptical distribution in Point 1 allows
us to relate
$S_{\varepsilon}(F_{Y_{0}},F_{X_{0}^{\prime}q})-S(F_{Y_{0}},F_{X_{0}^{\prime}q})$,
for any $q\in\mathcal{S}$, with
$S_{\varepsilon}(F_{Y_{0}},F_{X_{0}^{\prime}\beta_{0}})-S(F_{Y_{0}},F_{X_{0}^{\prime}\beta_{0}})$,
but it is not necessary to obtain an upper bound on
$S_{\varepsilon}(F_{Y_{0}},F_{X_{0}^{\prime}\beta_{0}})-S(F_{Y_{0}},F_{X_{0}^{\prime}\beta_{0}})$.

In the two cases of Proposition 7, we produce upper bounds on the Hausdorff
distance between $\mathcal{B}$ and $\mathcal{B}_{\varepsilon}$ that are, up to
some constants, power of the regularization parameter $\varepsilon$. The upper
bounds are close to 0 when $\varepsilon$ is small, in line with Point 2 of
Proposition 6. They are also closer to 0 the smaller $c$ is, i.e. the fatter
the tails of $X^{\prime}\beta_{0}$ (or $X^{\prime}q$) are, or the larger $d$
is, i.e. the thinner the tails of $U$ are.

##  3 Inference

We now consider the estimation of the identified set, and how to conduct
inference on the parameters of interest $\beta_{0}$. As in the previous
section, we first consider the case without common regressors before showing
how to incorporate such regressors and combine them with additional
constraints. We conclude this section by discussing some computational aspects
of our procedure. We illustrate the finite sample performances of our
inference method in Online Appendix C.

###  3.1 No common regressors

####  3.1.1 Estimation of the identification region and confidence region

We rely on random samples from the distributions of $Y$ and $X$.

######  Assumption 2.

We observe $(Y_{1},...,Y_{n_{Y}})$ and $(X_{1},...,X_{n_{X}})$, two
independent samples of i.i.d. variables with the same distribution as $Y$ and
$X$, respectively.

For any $q\in\mathcal{S}$, let $\widehat{F}_{Y}$ and
$\widehat{F}_{X^{\prime}q}$ denote the empirical cdf of $Y$ and $X^{\prime}q$
and let $\widehat{F}_{Y_{0}}(t)=\widehat{F}_{Y}(t+\overline{Y})$ and
$\widehat{F}_{X^{\prime}_{0}q}(t)=\widehat{F}_{X^{\prime}q}(t+\overline{X}^{\prime}q)$.
We simply estimate $R(\alpha,F_{Y_{0}},F_{X_{0}^{\prime}q})$ and
$S_{\varepsilon}(F_{Y_{0}},F_{X_{0}^{\prime}q})$ by their empirical
counterpart $R(\alpha,\widehat{F}_{Y_{0}},\widehat{F}_{X^{\prime}_{0}q})$ and
$S_{\varepsilon}(\widehat{F}_{Y_{0}},\widehat{F}_{X^{\prime}_{0}q})$. It turns
out that these functions can be computed quickly, as detailed in Section 3.3
below. We then also simply estimate the identified set
$\mathcal{B}_{\varepsilon}$ by plug-in:

$\widehat{\mathcal{B}}_{\varepsilon}:=\left\\{\lambda q:q\in\mathcal{S},\
0\leq\lambda\leq
S_{\varepsilon}\left(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q}\right)\right\\}.$

Next, we build confidence regions on $\beta_{0}$. The asymptotic distribution
of
$S_{\varepsilon}\left(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q}\right)$
is not Gaussian in general, so we rely on subsampling (Politis et al., 1999).
One could alternatively use the numerical bootstrap, see the discussion pp.
18-19 in the first version of D’Haultfœuille et al. (2023).

Let $n=(n_{X}n_{Y})/(n_{X}+n_{Y})$ and let $b_{n}$ denote the size of the
subsample. For any estimator $\widehat{\theta}$, let $\widehat{\theta}^{*}$
denotes its subsampling counterpart. For a nominal coverage of $1-\alpha$, the
confidence region on $\beta_{0}$ we consider is given by

$\text{CR}_{1-\alpha}(\beta_{0})=\left\\{\lambda q:\ q\in\mathcal{S},\
0\leq\lambda\leq
S_{\varepsilon}\left(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q}\right)-\widehat{c}_{\alpha,\varepsilon}(q)n^{-1/2}\right\\},$

where $\widehat{c}_{\alpha,\varepsilon}(q)$ is the quantile of order $\alpha$
of the distribution of
$b_{n}^{1/2}[S_{\varepsilon}(\widehat{F}^{*}_{Y_{0}},\widehat{F}^{*}_{X_{0}^{\prime}q})-S_{\varepsilon}(\widehat{F}_{Y_{0}},$
$\widehat{F}_{X_{0}^{\prime}q})]$, conditional on the data.

##### Inference on subcomponents of $\beta_{0}$.

In practice, one is often interested in conducting inference on subcomponents
of $\beta_{0}$. In view of (9), the identified (outer) set
$\mathcal{B}_{k,\varepsilon}$ of $\beta_{0,k}$ corresponding to
$\mathcal{B}_{\varepsilon}$ satisfies

$\mathcal{B}_{k,\varepsilon}=[-\sigma_{\varepsilon}(-e_{k},F_{Y_{0}},F_{X_{0}}),\,\sigma_{\varepsilon}(e_{k},F_{Y_{0}},F_{X_{0}})],$
(12)

where $\sigma_{\varepsilon}(\cdot,F_{Y_{0}},F_{X_{0}})$ denotes the support
function associated to $q\mapsto
S_{\varepsilon}(F_{Y_{0}},F_{X_{0}^{\prime}q})$ and $e_{k}$ is the $k$-th
element of the canonical basis of $\mathbb{R}^{p}$. To construct confidence
intervals on $\beta_{0k}$, we first estimate
$\sigma_{\varepsilon}(\cdot,F_{Y_{0}},F_{X_{0}})$ by

$\sigma_{\varepsilon}(e,\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}})=\frac{1}{\inf_{q\in\mathbb{R}^{p}:q^{\prime}e=1}1/S_{\varepsilon}\left(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q}\right)},$
(13)

see Corollary 1. Then, denoting by $\widetilde{c}_{\beta,\varepsilon}(e)$ the
quantile of order $\beta\in(0,1)$ of the distribution of
$b_{n}^{1/2}(\sigma_{\varepsilon}(e,\widehat{F}^{*}_{Y_{0}},\widehat{F}^{*}_{X_{0}})-\sigma_{\varepsilon}(e,\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}}))$,
conditional on the data, the confidence interval we consider for $\beta_{0,k}$
is

$\text{CI}_{1-\alpha}(\beta_{0,k})=\left[\left(-\sigma_{\varepsilon}(-e_{k},\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}})+\frac{\widetilde{c}_{\alpha,\varepsilon}(-e_{k})}{n^{1/2}}\right)^{-},\left(\sigma_{\varepsilon}(e_{k},\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}})-\frac{\widetilde{c}_{\alpha,\varepsilon}(e_{k})}{n^{1/2}}\right)^{+}\right],$

where $x^{-}=\min(0,x)$ and $x^{+}=\max(0,x)$. The rationale for using
$(\cdot)^{-}$ and $(\cdot)^{+}$ is to ensure that
$0\in\text{CI}_{1-\alpha}(\beta_{0,k})$: recall that without constraints,
$0\in\mathcal{B}_{k,\varepsilon}$. The advantage, then, is that we can still
use the quantiles of order $\alpha$ while maintaining coverage even under
point identification, as formally shown in Theorem 3 below.

##### Choice of the regularization parameter $\varepsilon$.

Because $S_{\varepsilon}(F_{Y_{0}},F_{X_{0}^{\prime}q})\geq
S(F_{Y_{0}},F_{X_{0}^{\prime}q})$, the confidence regions and intervals above
are conservative in general. To gain in efficiency, we suggest using several
$\varepsilon$, and, basically, keep the one leading to the smallest confidence
regions or intervals. We distinguish the cases $p=1$, where we can adapt the
choice to the direction $q\in\mathcal{S}$ while preserving the convexity of
$\widehat{\mathcal{B}}_{\varepsilon}$, from the case $p>1$. When $p=1$, let us
define, for $q\in\mathcal{S}=\\{-1,1\\}$,

$\varepsilon(q)=\underset{\varepsilon\in\mathcal{E}}{{\rm argmin}}\
S_{\varepsilon}(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q})-\widehat{c}_{\alpha,\varepsilon}(q)n^{-1/2},$
(14)

where $\mathcal{E}$ is a finite grid in $(0,1/2]$. Hence, $\varepsilon(q)$
simply minimizes the boundary value of the confidence region in the direction
$q\in\mathcal{S}$. This idea is similar to that of Chernozhukov et al. (2013)
in the context of intersection bounds.

Now consider the case $p>1$. If one focuses on confidence intervals on
$\beta_{0k}$, we need to choose the parameter $\varepsilon$ that appears in
$\sigma_{\varepsilon}(\pm e_{k},F_{Y_{0}},F_{X_{0}})$. To this end, we simply
use $\varepsilon(q)$ as given above, with $q=\pm e_{k}$. If we are interested
instead in the set $\mathcal{B}$ itself, we recommend using
$\underline{\varepsilon}=\min_{q\in\mathcal{Q}}\varepsilon(q)$, where
$\mathcal{Q}$ is a finite subset of $\mathcal{S}$.

####  3.1.2 Consistency and validity of the confidence region

The following theorem shows that $\widehat{\mathcal{B}}_{\varepsilon}$ is
consistent for $\mathcal{B}_{\varepsilon}$, in the sense of the Hausdorff
distance, under mild regularity conditions.

######  Theorem 2.

Suppose that Assumptions 1-2 hold. Then, as $n\to\infty$,

$d_{H}\left(\widehat{\mathcal{B}}_{\varepsilon},\mathcal{B}_{\varepsilon}\right)\stackrel{{\scriptstyle\mathbb{P}}}{{\longrightarrow}}0.$

Next, we establish the asymptotic validity of
$\text{CR}_{1-\alpha}(\beta_{0})$ and $\text{CI}_{1-\alpha}(\beta_{0,k})$,
under Assumptions 3 and 4 respectively. Assumption 5 (resp. 6) is used to
establish the asymptotic validity of $\text{CR}_{1-\alpha}(\beta_{0})$ (resp.
$\text{CI}_{1-\alpha}(\beta_{0,k})$) using $\varepsilon(q)$ or
$\underline{\varepsilon}$ (resp. $\varepsilon(\pm e_{k})$), as defined above,
instead of a fixed $\varepsilon$.

######  Assumption 3.

(Regularity conditions for $\text{CR}_{1-\alpha}(\beta_{0})$)
$E[\|X\|^{2}]<\infty$, $E[Y^{2}]<\infty$. Also, for all $q\in\mathcal{S}$,
there exists $\varepsilon^{\prime}\in(0,\varepsilon)$ such that
$F_{X^{\prime}q}$ and $F_{Y}$ are continuous and strictly increasing on
$[F_{X^{\prime}q}^{-1}(\varepsilon^{\prime}),F_{X^{\prime}q}^{-1}(1-\varepsilon^{\prime})]$
and $[F_{Y}^{-1}(\varepsilon^{\prime}),F_{Y}^{-1}(1-\varepsilon^{\prime})]$
respectively.

######  Assumption 4.

(Regularity conditions for $\text{CI}_{1-\alpha}(\beta_{0,k})$)
$E[\|X\|^{2}]<\infty$, $E[Y^{2}]<\infty$. Also, there exists
$\varepsilon^{\prime}\in(0,\varepsilon)$ such that for all
$(\alpha,\alpha^{\prime})\in[\varepsilon^{\prime},1-\varepsilon^{\prime}]^{2}$,
there exists a strictly increasing and continuous function $m$ such that
$m(0)=0$ and

$\displaystyle\sup_{q\in\mathcal{S}}\left|F_{X^{\prime}q}^{-1}(\alpha^{\prime})-F_{X^{\prime}q}^{-1}(\alpha)\right|$
$\displaystyle<m(|\alpha^{\prime}-\alpha|),$ (15)
$\displaystyle\left|F_{Y}^{-1}(\alpha^{\prime})-F_{Y}^{-1}(\alpha)\right|$
$\displaystyle<m(|\alpha^{\prime}-\alpha|).$

Finally, for all $e=\pm e_{k}$ $(k=1,...,p)$, either (i)
$\sigma_{\varepsilon}(e,F_{Y_{0}},F_{X_{0}})>\sigma(e,F_{Y_{0}},F_{X_{0}})$,
(ii) $q\mapsto[qS_{\varepsilon}(F_{Y_{0}},F_{X_{0}^{\prime}q})]^{\prime}e$
admits a unique maximizer on $\mathcal{S}$, or (iii) for all
$q_{m}\in\arg\max_{q\in\mathcal{S}}[qS_{\varepsilon}(F_{Y_{0}},F_{X_{0}^{\prime}q})]^{\prime}e$,
$a\mapsto R(a,F_{Y},F_{X^{\prime}q_{m}})$ admits a unique minimizer on
$[\varepsilon,1-\varepsilon]$.

######  Assumption 5.

(Regularity conditions for the validity of $\text{CR}_{1-\alpha}(\beta_{0})$
based on data-dependent $\varepsilon$) For all $q\in\mathcal{S}$, we either
have (i)
$S_{\varepsilon}(F_{Y_{0}},F_{X_{0}^{\prime}q})>S(F_{Y_{0}},F_{X_{0}^{\prime}q})$
for all $\varepsilon\in\mathcal{E}$, or (ii) $a\mapsto
R(a,F_{Y_{0}},F_{X_{0}^{\prime}q})$ admits a unique minimizer on $(0,1)$.

######  Assumption 6.

(Regularity conditions for the validity of $\text{CI}_{1-\alpha}(\beta_{0,k})$
based on data-dependent $\varepsilon$) For all $e=\pm e_{k}$ $(k=1,...,p)$, we
either have (i)
$\sigma_{\varepsilon}(e,F_{Y_{0}},F_{X_{0}})>\sigma(F_{Y_{0}},F_{X_{0}})$ for
all $\varepsilon\in\mathcal{E}$ or (ii) for all
$q_{m}\in\arg\max_{q\in\mathcal{S}}[qS_{\varepsilon_{j_{0}}}(F_{Y_{0}},F_{X_{0}^{\prime}q})]^{\prime}e$,
$a\mapsto R(a,F_{Y_{0}},F_{X_{0}^{\prime}q_{m}})$ admits a unique minimizer
$a(q_{m})$ on $(0,1)$, with
$a(q_{m})\in[\varepsilon_{j_{0}},1-\varepsilon_{j_{0}}]$ and
$\varepsilon_{j_{0}}:=\max\\{\varepsilon\in\mathcal{E}:\sigma_{\varepsilon}(F_{Y_{0}},F_{X_{0}^{\prime}q})=\sigma(F_{Y_{0}},F_{X_{0}^{\prime}q})\\}$.

The second part of Assumption 3 holds if for all $q\in\mathcal{S}$, the
distributions of $X^{\prime}q$ and $Y$ are continuous with respect to the
Lebesgue distribution and their support is a (possibly unbounded) interval.
The first part of Assumption 4 is basically a reinforcement of Assumption 3 to
ensure that some of our results hold uniformly over $q$. This is needed when
we consider the support function, as this function implies an optimization
over $q$. A sufficient condition for (15) is that, for all $q\in\mathcal{S}$,
$X^{\prime}q$ admits a density $f_{X^{\prime}q}$ with respect to the Lebesgue
measure and
$\inf_{(q,\alpha)\in\mathcal{S}\times[\varepsilon,1-\varepsilon]}f_{X^{\prime}q}(F_{X^{\prime}q}^{-1}(\alpha))>0$.
The conditions (ii) and (iii) in Assumption 4 are sufficient conditions for
the continuity of the asymptotic distribution of
$n^{1/2}\left(\sigma_{\varepsilon}(e,\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}})-\sigma_{\varepsilon}(e,F_{Y_{0}},F_{X_{0}})\right)$,
which is necessary for the validity of subsampling.

In Assumption 5, we exclude cases where
$S_{\varepsilon_{0}}(F_{Y_{0}},F_{X_{0}^{\prime}q})=S(F_{Y_{0}},F_{X_{0}^{\prime}q})$
for some $\varepsilon_{0}\in(0,1/2)$ and there are multiple minima for
$a\mapsto R(a,F_{Y_{0}},F_{X_{0}^{\prime}q})$. This occurs when both $X$ and
$Y$ are Gaussian. Assumption 6 is basically similar to Assumption 5 but
somewhat more complicated, as we consider therein the support function instead
of the radial function.

######  Theorem 3.

Fix $(\varepsilon,\alpha)\in(0,1/2)^{2}$ and suppose that
$n_{X}/(n_{X}+n_{Y})\to\mu\in(0,1)$, $b_{n}\to\infty$, $b_{n}/n\to 0$ and
Assumptions 1-2 hold. Then:

  1. 1.

If Assumption 3 also holds,

$\underset{\beta\in\mathcal{B}}{\inf}\
\underset{n\to\infty}{\liminf}P\left(\beta\in\text{CR}_{1-\alpha}(\beta_{0})\right)\geq
1-\alpha,$ (16)

with equality if $\mathcal{B}=\mathcal{B}_{\varepsilon}$. Moreover, if
Assumption 5 also holds, (16) is still true if we use $\varepsilon(q)$ (when
$p=1)$ or $\underline{\varepsilon}$ (when $p>1$) instead of $\varepsilon$.

  2. 2.

If Assumption 4 also holds,

$\underset{n\to\infty}{\liminf}\underset{\beta_{k}\in\mathcal{B}_{k}}{\inf}\
P\left(\beta_{k}\in\text{CI}_{1-\alpha}(\beta_{0,k})\right)\geq 1-\alpha,$
(17)

with equality if $\mathcal{B}_{k}=\mathcal{B}_{k,\varepsilon}$. Moreover, if
Assumption 6 also holds, (17) is still true if we use $\varepsilon(e_{k})$ and
$\varepsilon(-e_{k})$ instead of $\varepsilon$.

To prove (16)-(17), we first show the weak convergence of

$\sqrt{n}\left(R(\alpha,\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q})-R(\alpha,F_{Y_{0}},F_{X_{0}^{\prime}q})\right),$

seen as a process indexed by either $\alpha$ or $(\alpha,q)$. The convergence
in distribution of
$S_{\varepsilon}(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q})$ and
$\sigma_{\varepsilon}(e,\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}})$, and in turn
(16)-(17), then essentially follows by the Hadamard directional
differentiability of the minimum and maximin maps, shown respectively by
Cárcamo et al. (2020) and Firpo et al. (2023).

Our results for a fixed $\varepsilon>0$ extend to the data-dependent
$\varepsilon(q)$ and $\underline{\varepsilon}$, under the additional
conditions provided above. Note that one could avoid these conditions by using
sample splitting, with one subsample used to choose $\varepsilon(q)$ or
$\underline{\varepsilon}$ and the other to construct the confidence
regions/intervals. One drawback of this alternative solution, though, is that
it increases the size of confidence regions/intervals, to a point that we may
lose the benefits of using a data-dependent rather than a fixed $\varepsilon$.

###  3.2 Common regressors and possible constraints

We now turn to inference on $\beta_{0}$ with common regressors $X_{c}$. Recall
from Proposition 2 that the identified set on $\beta_{0}$ is

$\mathcal{B}^{c}=\left\\{\lambda
q:q\in\mathcal{S},\;0\leq\lambda\leq\overline{S}(F_{Y,X_{c}},F_{X_{nc}^{\prime}q,X_{c}})\right\\},$

with
$\overline{S}(F_{Y,X_{c}},F_{X_{nc}^{\prime}q,X_{c}})=\inf_{x\in\text{Supp}(X_{c})}S(F_{Y^{x}|X_{c}=x},F_{X^{x}{}^{\prime}q|X_{c}=x})$.

Let us first assume that $X_{c}$ has a finite support. Let
$\widehat{F}_{Y^{x}|X_{c}=x}$ and $\widehat{F}_{X^{x}{}^{\prime}q|X_{c}=x}$
denote the empirical estimators of $F_{Y^{x}|X_{c}=x}$ and
$F_{X^{x}{}^{\prime}q|X_{c}=x}$, respectively. Following the same logic as
above, we estimate $\overline{S}(F_{Y,X_{c}},F_{X_{nc}^{\prime}q,X_{c}})$ by

$\widehat{\overline{S}}(q,F_{Y,X_{c}},F_{X_{nc}^{\prime}q,X_{c}})=\min_{x\in\text{Supp}(X_{c})}S_{\varepsilon}(\widehat{F}_{Y^{x}|X_{c}=x},\widehat{F}_{X^{x}{}^{\prime}q|X_{c}=x}).$

Let $\widehat{c}^{c}_{\alpha,\varepsilon}(q)$ be the quantile of order
$\alpha\in(0,1)$ of the distribution of
$b_{n}^{1/2}(\widehat{\overline{S}}{}^{*}(q,F_{Y,X_{c}},F_{X_{nc},X_{c}})$
$-\widehat{\overline{S}}(q,F_{Y,X_{c}},F_{X_{nc},X_{c}}))$, conditional on the
data. For a nominal coverage of $1-\alpha$, the confidence region on
$\beta_{0}$ we consider is

$\text{CR}_{1-\alpha}^{c}(\beta_{0})=\left\\{\lambda q:q\in\mathcal{S},\
0\leq\lambda\leq\widehat{\overline{S}}(q,F_{Y,X_{c}},F_{X_{nc},X_{c}})-\widehat{c}^{c}_{\alpha,\varepsilon}(q)n^{-1/2}\right\\}.$

With continuous common regressors, one can adapt the earlier arguments using
sieve estimation. Specifically, suppose that Model (1) holds and consider a
linear sieve approximation of $f(\cdot)$ by a step function
$x_{c}\mapsto\sum_{k=1}^{K_{n}}\mathds{1}\left\\{x_{c}\in
I_{n,k}\right\\}\gamma_{k}$ for some partition $(I_{n,k})_{k=1...K_{n}}$ of
the support of $X_{c}$ and with $K_{n}$ tending to infinity at an appropriate
rate. Then, one can construct a confidence region on $\beta_{0}$ by following
a similar logic as above.666Establishing the asymptotic validity of such a
confidence region would require to handle both the bias stemming from the
approximation of $f(\cdot)$ and the increasing complexity of the
approximation. We leave this analysis for future research.

We now discuss how to conduct inference under constraints on the $R^{2}$ or
shape restrictions, as considered in Subsections 2.3.1 and 2.3.2 respectively.
The main difference with above is that for a given direction
$q\in\mathcal{S}$, both the lower and upper bounds on the identified set need
to be estimated. As before, we estimate them with plug-in estimators.777With a
finite number of constraints and $X_{c}$ finitely supported, one can prove
that the plug-in estimators $\underline{S}^{c}(m_{Y},m_{X_{nc}},q)$ and
$\overline{S}^{c}(m_{Y},m_{X_{nc}},q)$ are consistent if there are no
$r\in\mathcal{R}$ such that
$[Rm_{X_{nc}}^{\prime}q](r)=[Rm_{y}-\underline{c}](r)=0$. In the latter case,
one may need to replace the plug-in estimator by removing the limit and using
$u_{n}$ instead of $u$, with $u_{n}\to 0$ at an appropriate rate. We leave
this issue for future research. The only substantive difference is that in the
confidence regions, we then account for the variability of both bounds. For
instance, with shape restrictions, we consider the following confidence
region:

$\displaystyle\text{CR}_{1-\alpha}^{con}(\beta_{0})=\bigg{\\{}$
$\displaystyle\lambda q:q\in\mathcal{S},\
\widehat{\underline{S}}{}^{con}(q,F_{Y,X_{c}},F_{X_{nc},X_{c}})+\widehat{\underline{c}}{}^{con}_{1-\alpha/2,\varepsilon}(q)n^{-1/2}\leq\lambda$
$\displaystyle\leq\widehat{\overline{S}}{}^{con}(q,F_{Y,X_{c}},F_{X_{nc},X_{c}})-\widehat{\overline{c}}{}^{con}_{\alpha/2,\varepsilon}(q)n^{-1/2}\bigg{\\}},$

where $\widehat{\underline{c}}^{con}_{\delta,\varepsilon}(q)$ is the quantile
of order $\delta$ of
$b_{n}^{1/2}(\widehat{\underline{S}}{}^{con*}(q,F_{Y,X_{c}},F_{X_{nc},X_{c}})$
$-\widehat{\underline{S}}{}^{con}(q,F_{Y,X_{c}},$ $F_{X_{nc},X_{c}}))$,
conditional on the data and similarly for
$\widehat{\overline{c}}{}^{con}_{\delta,\varepsilon}$. Although to conserve on
space we do not establish any formal result, using quantiles of order
$\alpha/2$ and $1-\alpha/2$, depending on whether
$\underline{S}{}^{con}(q,F_{Y,X_{c}},F_{X_{nc},X_{c}})=\overline{S}{}^{con}(q,F_{Y,X_{c}},F_{X_{nc},X_{c}})$,
should ensure that the confidence region is asymptotically conservative.

###  3.3 Computational aspects

We first discuss how to efficiently compute
$S_{\varepsilon}(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q})$. Let
$Y_{(1)}<...<Y_{(m_{y})}$ represent the $m_{y}\leq n_{y}$ distinct, ordered
values of the $(Y_{i})_{i=1,...,n_{y}}$ and let
$W_{(j)}^{Y}=\\#\\{i:Y_{i}=Y_{(j)}\\}/n_{y}$. Let us also define
$I^{Y}=\\{\sum_{j=1}^{i}W_{(j)}^{Y}:i=1,...,m_{y}-1\\}$. We define similarly
$W_{(j)}^{X^{\prime}q}$ and $I^{X^{\prime}q}$. By construction, the numerator
$\widehat{f}^{Y}(\alpha):=\int_{\alpha}^{1}\widehat{F}^{-1}_{Y_{0}}(t)dt$ of
$R(\alpha,\widehat{F}_{Y_{0}},\widehat{F}_{X^{\prime}_{0}q})$ is linear on all
intervals $[\sum_{j=1}^{i}W_{(j)}^{Y},\sum_{j=1}^{i+1}W_{(j)}^{Y}]$
($i=0,...,m_{y}-1$). Moreover, for any $\alpha=\sum_{j=1}^{i}W_{(j)}^{Y}\in
I^{Y}$,

$\widehat{f}^{Y}(\alpha)=\sum_{j=i+1}^{m_{y}}W^{Y}_{(j)}\left(Y_{(j)}-\overline{Y}\right).$
(18)

The same holds for the denominator $\widehat{f}^{X^{\prime}q}(\alpha)$ of
$R(\alpha,\widehat{F}_{Y_{0}},\widehat{F}_{X^{\prime}_{0}q})$. As a result,
$R(\alpha,\widehat{F}_{Y_{0}},$ $\widehat{F}_{X^{\prime}_{0}q})$ is of the
form $(a\alpha+b)/(c\alpha+d)$ on intervals between two consecutive values of
$I^{Y}\cup I^{X^{\prime}q}$. Now, observe that the minimum of such a function
is reached at one of the endpoints of the interval. As a result, we can
compute $S_{\varepsilon}(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q})$
using the following algorithm:

  1. 1.

Compute $\widehat{f}^{Y}(\cdot)$ on $I^{Y}$ using (18) and let
$\widehat{f}^{Y}(0)=\widehat{f}^{Y}(1)=0$. Proceed similarly with
$\widehat{f}^{X^{\prime}q}(\cdot)$;

  2. 2.

Interpolate linearly $\widehat{f}^{Y}(\cdot)$ (resp.
$\widehat{f}^{X^{\prime}q}(\cdot)$) on $\\{\varepsilon,1-\varepsilon\\}\cup
I^{X^{\prime}q}$ (resp. $\\{\varepsilon,1-\varepsilon\\}\cup I^{Y}$).

  3. 3.

Compute
$S_{\varepsilon}(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q})=\min_{\alpha\in\\{\varepsilon,1-\varepsilon\\}\cup
I^{Y}\cup
I^{X^{\prime}q}}\widehat{f}^{Y}(\alpha)/\widehat{f}^{X^{\prime}q}(\alpha)$.

To compute $\sigma_{\varepsilon}(\pm e_{k},F_{Y_{0}},F_{X_{0}})$, we solve
(13), in which $q\mapsto
1/S_{\varepsilon}(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q})$ is also
convex. In practice, we use the BFGS quasi-Newton method implemented in the R
package optim, using as a starting point the considered direction $e$.

Finally, the exact computation of $\widehat{\mathcal{B}}_{\varepsilon}$ and
$\text{CR}_{1-\alpha}(\beta_{0})$ requires the computation of
$S_{\varepsilon}(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q})$ and
$\widehat{c}_{\alpha,\varepsilon}(q)$ for all $q\in\mathcal{S}$, which is in
practice infeasible if $p>1$ as $\mathcal{S}$ is infinite. Instead, we suggest
to (i) fix a grid $\widetilde{\mathcal{S}}\subset\mathcal{S}$; (ii) compute
$S_{\varepsilon}(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q})$ and
$\widehat{c}_{\alpha,\varepsilon}(q)$ for each $q\in\widetilde{\mathcal{S}}$;
(iii) construct an approximation of $\widehat{\mathcal{B}}_{\varepsilon}$ and
$\text{CI}_{1-\alpha}$ by computing the convex hulls of
$\\{S_{\varepsilon}(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q})q:\
q\in\widetilde{\mathcal{S}}\\}$ and
$\left\\{\left(S_{\varepsilon}(\widehat{F}_{Y_{0}},\widehat{F}_{X_{0}^{\prime}q})-\widehat{c}_{\alpha,\varepsilon}(q)n^{-1/2}\right)q:\right.$
$\left.q\in\widetilde{\mathcal{S}}\right\\}$, respectively.888The convex hull
of $n$ points in $\mathbb{R}^{p}$ can be computed efficiently by the quickhull
algorithm (Barber et al., 1996), which requires around $n^{p/2}$ operations.
The resulting sets, $\widetilde{\mathcal{B}}_{\varepsilon}$ and
$\widetilde{\text{CR}}_{1-\alpha}(\beta_{0})$ say, are convex, inner
approximations of $\widehat{\mathcal{B}}_{\varepsilon}$ and
$\text{CR}_{1-\alpha}(\beta_{0})$, and satisfy, as
$d_{H}(\mathcal{S},\widetilde{\mathcal{S}})\to 0$,
$d_{H}(\widetilde{\mathcal{B}}_{\varepsilon},\widehat{\mathcal{B}}_{\varepsilon})\to
0$ and
$d_{H}(\widetilde{\text{CR}}_{1-\alpha}(\beta_{0}),\text{CR}_{1-\alpha}(\beta_{0}))\to
0$.

The computation of the estimated set, the confidence regions on $\beta_{0}$
and $\gamma_{0}$ in the specification $f(X_{c})=X_{c}^{\prime}\gamma_{0}$
(where $X_{c}$ is the vector of all dummy variables associated with a finitely
supported variable) and the confidence intervals on the corresponding
subcomponents are implemented in our companion R package RegCombin. The
package also handles shape restrictions and lower bound on the $R^{2}$ of the
long regression, as well as combinations of these. The RegCombin vignette,
available through the description of the package on CRAN, provides additional
details about the implementation, including the choice of the tuning
parameters $\mathcal{E}$ and $b_{n}$.

##  4 Application to intergenerational mobility in the United States

We now apply our method to conduct inference on the intergenerational income
mobility over the period 1850 to 1930 in the United States, revisiting the
influential analysis of Olivetti and Paserman (2015) on this question. We
follow their paper and focus on the father-son and father-son-in-law
intergenerational income elasticities. We conduct our analysis using 1 percent
extracts from the decennial censuses of the United States, over the period
1850 to 1930 (1850-1930 IPUMS).999We refer the reader to Section 2 of Olivetti
and Paserman (2015) for a detailed discussion of the data used in the
analysis. Note that they estimate the evolution of the intergenerational
income mobility over a longer time window (1850 to 1940) than we do. We
confine our analysis to the period 1850-1930 as the 1940 portion of the data
(1% extract of the IPUMS Restricted Complete Count Data) is not publicly
available.

An important feature of the historical Census data used in this analysis is
that father’s and son’s (as well as son-in-law’s) incomes are not jointly
observed. Olivetti and Paserman (2015) address this measurement issue by
predicting, for any given child (John, say) observed in one of the Census
datasets, their father’s log earnings using the mean log earnings of fathers
whose children have the same first name (namely, John). Olivetti and Paserman
then estimate in a second step the intergenerational elasticity by regressing
son’s log earnings on the predicted father’s log earnings computed from the
previous step. This procedure boils down to a two-sample two-stage least
squares estimator (TSTSLS).101010Another limitation of the data used in
Olivetti and Paserman (2015) and in this application is that it does not allow
us to directly calculate the intergenerational elasticity in income. Instead,
we follow the baseline specification of Olivetti and Paserman (2015) and proxy
income using an index of occupational standing available from IPUMS
(OCCSCORE), which is constructed as the median total income of the persons in
each occupation in 1950. The corresponding exclusion restriction that the
son’s first name does not predict his log earnings, once we control for his
father’s log earnings, may nonetheless be problematic; see Santavirta and
Stuhler (2022) for a critical review of the empirical literature using TSTSLS
in this context of intergenerational mobility. For the periods 1860-1880 and
1880-1900 only, the IPUMS Linked Representative Samples link fathers and sons
using information on first and last names, which allows us to estimate more
directly the father-son elasticity using OLS.

Using our notation and consistent with Olivetti and Paserman (2015), the
population parameter of interest here is given by

$\theta_{0}:=\frac{\text{Cov}(Y,X_{nc})}{V(X_{nc})}=\beta_{0}+\left(\frac{\text{Cov}(X_{c},X_{nc})}{V(X_{nc})}\right)^{\prime}\gamma_{0},$

where $Y$ denotes the son’s (or son-in-law’s) log-income, $X_{nc}$ the
father’s log-income and $X_{c}$ the vector of indicators corresponding to the
son’s (or son-in-law’s) first names observed in both datasets. The second
equality follows from (1), since $X_{c}$ is discrete and thus
$f(X_{c})=X_{c}^{\prime}\gamma_{0}$ for some $\gamma_{0}$. In what follows, we
report the upper bound of the estimated identified set and confidence interval
on $\theta_{0}$.

Even though the sample sizes as well as the number of common regressors
$X_{c}$ are quite large, our method can still be implemented at a very
reasonable computational cost. For instance, for the sample of sons over the
first period (1850-1870), the computation of the confidence intervals only
takes less than 4 minutes with our R package. As expected, computational time
is highest for the period 1910-1930 associated with the largest number of
observations, with $n>100,000$ for both samples of $Y$ and $X_{nc}$.
Nonetheless, our inference procedure remains tractable in this case too, with
a computational time of about 11 minutes.111111These CPU times are obtained
using our companion R package, parallelized on 20 CPUs on an Intel Xeon Gold
6130 CPU 2.10GHz with 382Gb of RAM. Overall, this illustrates the
applicability of our method, which can be easily implemented even in this type
of rich and high-dimensional data environment.

Figures 4(a)-4(b) and Table 1 below display the results, for the father-son as
well as father-son-in-law elasticities, obtained using our approach, the
TSTSLS and, for the sample of sons over the years 1860-1880 and 1880-1900, the
OLS.121212In practice we need to restrict the set of first names included in
$X_{c}$ to avoid very uncommon occurrences that are perfect predictors of the
outcome variable $Y$. In our baseline specification, we implement this by
restricting $X_{c}$ to the set of first names that account for at least 0.01%
of the observations in the pooled sample, and appear at least 10 times in
either of the samples. We discuss in the following the robustness of our
results to alternative cutoffs. Specifically, we report in Figures 4(a)-4(b)
the estimated upper bounds of the identified sets (in solid red) and the
confidence intervals (dashed red) obtained with our method, the TSTSLS
estimates and confidence intervals (solid and dashed blue, resp.) as well as,
for 1860-1880 and 1880-1900 and the sample of sons only, the OLS estimates and
confidence intervals (solid and dashed green, resp.).

(a) For sons

(b) For sons-in-law Figure 4: Intergenerational income correlation using
different methods.

Note: for readability and because 0 is a natural lower bound, the y-axis
starts at 0, even though the lower bounds of our confidence intervals without
restrictions are negative (see Table 1).

Sample: | 1850-1870 |  | 1860-1880 |  | 1880-1900 |  | 1900-1920 |  | 1910-1930
---|---|---|---|---|---|---|---|---|---
Sons |  |  |  |  |  |  |  |  |
DGM, set | [-0.555,0.555] |  | [-0.465,0.465] |  | [-0.473,0.473] |  | [-0.430,0.430] |  | [-0.443,0.443]
DGM, CI. | [-0.614,0.614] |  | [-0.517,0.517] |  | [-0.532,0.532] |  | [-0.483,0.483] |  | [-0.499,0.499]
DGM, $\underline{R}^{2}\geq 1.3R_{s}^{2}$, set | [0.081,0.555] |  | [0.075,0.465] |  | [0.075,0.473] |  | [0.076,0.430] |  | [0.071,0.443]
DGM, $\underline{R}^{2}\geq 1.3R_{s}^{2}$, CI. | [0.033,0.617] |  | [0.034,0.519] |  | [0.039,0.527] |  | [0.044,0.477] |  | [0.047,0.491]
DGM, $\underline{R}^{2}\geq 2R_{s}^{2}$, set | [0.163,0.555] |  | [0.151,0.465] |  | [0.153,0.473] |  | [0.164,0.430] |  | [0.159,0.443]
DGM, $\underline{R}^{2}\geq 2R_{s}^{2}$, CI. | [0.095,0.601] |  | [0.093,0.506] |  | [0.102,0.513] |  | [0.127,0.459] |  | [0.132,0.477]
TSTSLS, pt. | 0.350 |  | 0.310 |  | 0.344 |  | 0.495 |  | 0.476
TSTSLS, CI. | [0.305,0.395] |  | [0.272,0.348] |  | [0.313,0.376] |  | [0.468,0.523] |  | [0.454,0.497]
Test of equality, p-value |  $<$0.001 |  | 0.001 |  | 0.001 |  | 0.999 |  | 0.014
(Stat.; critical val. 95%) | (28.52; 15.16) |  | (25.21; 12.37) |  | (25.92; 28.30) |  | (15.09; 17.69) |  | (8.18; 6.79)
OLS, pt. |  |  | 0.455 |  | 0.472 |  |  |  |
OLS, CI. |  |  | [0.414,0.497] |  | [0.443,0.501] |  |  |  |
Test pt identification, p-value |  |  | 0.147 |  | 0.003 |  |  |  |
(Stat.; critical val. 95%) |  |  | (9.21,17.03) |  | (23.06,6.33) |  |  |  |
Number of names $X_{c}$ | 225 |  | 261 |  | 382 |  | 514 |  | 598
Sample sizes $Y$ and $X_{nc}$ | (39,734; 34,603) |  | (55,728; 47,014) |  | (85,340; 73,999) |  | (116,986; 102,053) |  | (131,089; 116,328)
Sample: | 1850-1870 |  | 1860-1880 |  | 1880-1900 |  | 1900-1920 |  | 1910-1930
Sons-in-law |  |  |  |  |  |  |  |  |
DGM, set | [-0.531,0.531] |  | [-0.442,0.442] |  | [-0.481,0.481] |  | [-0.454,0.454] |  | [-0.452,0.452]
DGM, CI. | [-0.601,0.600] |  | [-0.507,0.507] |  | [-0.554,0.555] |  | [-0.515,0.515] |  | [-0.513,0.513]
DGM, $\underline{R}^{2}\geq 1.3R_{s}^{2}$, set | [0.089,0.531] |  | [0.085,0.442] |  | [0.075,0.481] |  | [0.073,0.454] |  | [0.062,0.452]
DGM, $\underline{R}^{2}\geq 1.3R_{s}^{2}$, CI. | [0.030,0.605] |  | [0.036,0.505] |  | [0.029,0.552] |  | [0.037,0.513] |  | [0.033,0.509]
DGM, $\underline{R}^{2}\geq 2R_{s}^{2}$, set | [0.186,0.531] |  | [0.178,0.442] |  | [0.159,0.481] |  | [0.164,0.454] |  | [0.146,0.452]
DGM, $\underline{R}^{2}\geq 2R_{s}^{2}$, CI. | [0.115,0.596] |  | [0.114,0.490] |  | [0.105,0.534] |  | [0.122,0.499] |  | [0.113,0.496]
TSTSLS, pt. | 0.340 |  | 0.400 |  | 0.400 |  | 0.493 |  | 0.414
TSTSLS, CI. | [0.299,0.381] |  | [0.364,0.436] |  | [0.365,0.434] |  | [0.469,0.518] |  | [0.395,0.433]
Test of equality, p-value |  $<$0.001 |  | 0.998 |  | 0.012 |  | 1 |  | 1
(Stat.; critical val. 95%) | (23.12; 9.67) |  | (5.87; 18.53) |  | (13.08; 12.94) |  | (8.03; 13.28) |  | (8.33; 13.07)
Number of names $X_{c}$ | 155 |  | 212 |  | 323 |  | 468 |  | 545
Sample sizes $Y$ and $X_{nc}$ | (25,760; 33,256) |  | (32,970; 45,800) |  | (49,068; 71,141) |  | (73,425; 99,871) |  | (85,122; 112,763)
Notes: Dependent variable $Y$ is son’s (or son-in-law’s) log income. Common
regressors $X_{c}$ are dummies for the first names appearing more than 0.01%
in the pooled dataset and 10 times in both datasets. “DGM, set” and “DGM, CI.”
refer to the estimated identified set and 95% confidence interval,
respectively, obtained with our method. “TSTSLS, pt.” and “TSTSLS, CI.” refer
to the TSTSLS point estimate and 95% confidence interval, respectively. The
test of equality between the TSTSLS ($\beta_{TSTSLS}$) estimates and DGM
($\beta_{DGM}$) upper bound estimates is performed using subsampling with
1,000 replications. The statistic (“Stat.”) is $n^{1/2}\widehat{\theta}$,
where $\widehat{\theta}=\widehat{\beta}_{TSTSLS}-\widehat{\beta}_{DGM}$ and
$n=n_{y}n_{x}/(n_{y}+n_{x})$, $n_{y}$ and $n_{x}$ being the respective sample
sizes of $Y$ and $X_{nc}$. The critical value corresponds to the $1-\alpha$
quantile of the distribution of
$b_{n}^{1/2}|\widehat{\theta}^{*}-\widehat{\theta}|$, where
$\widehat{\theta}^{*}$ is a subsampled version of $\widehat{\theta}$ and
$b_{n}$ is the subsample size. The sample sizes where the joint distribution
is observed for both periods 1860-1880 and 1880-1900 are respectively 3,947
and 9,076. The $R^{2}$ on the short and long regressions are respectively 0.04
and 0.18 for 1860-1880, and 0.02 and 0.17 for 1880-1900. The test for point
identification is performed with the $\varepsilon$ selected in (14), however
this choice appears conservative on simulations. Taking $\varepsilon/2$ yields
p-values of 0.69 for the period 1860-1880 and 0.04 for 1880-1900.

Table 1: Intergenerational income correlation for sons using different
methods.

A first conclusion from these results is that the upper bounds of the
confidence intervals associated with our method range, depending on the
periods, between 0.48 and 0.61 (0.51 and 0.6) for the sample of sons (sons-in-
law). These values of the intergenerational correlation coefficient are all
well below the natural upper bound of 1. Also, even though the estimates vary
depending on the data and econometric specification being used, most of the
existing point estimates of the father-son income elasticity range between
0.40 and 0.50 (Olivetti and Paserman, 2015). Overall, this clearly indicates
that our method leads to informative inference on the parameter of interest.

Second, consider the two cases where the linked data is available (1860-1880
and 1880-1900 for the sample of sons). Results in Table 1 indicate that the
corresponding OLS estimates of the intergenerational income elasticities are
quantitatively very close to the estimated upper bound of our identified set.
Recall that, from Proposition 5 in Section 2.3.4, the upper bound of our
identified set ($\overline{\theta}_{0}$, say) plays a special role: under an
additional restriction on the distributions of $X_{nc}$ and the error term,
$\theta_{0}$ is actually point identified and equal to
$\overline{\theta}_{0}$.131313Proposition 5 is obtained without $X_{c}$. Yet,
it can be combined with Proposition 2 to show that $\beta_{0}$, and in turn
$\gamma_{0}$ (and thus $\theta_{0}$ here) are point identified with such
$X_{c}$. In other words, the results from these two periods support the
hypothesis that the restriction on the distributions of $X_{nc}$ and the error
term guaranteeing point identification of $\theta_{0}$ by
$\overline{\theta}_{0}$ hold. Besides, the fact that we do not reject at
standard levels the null hypothesis of point identification with our formal
test described in Section A.3 for the period 1860-1880 (p-value of 0.147)
provides suggestive evidence in this direction.141414Simulation results
available from the authors upon request indicate that our choice of
$\varepsilon$ tends to be conservative for the test of point identification.
One would not reject either at the 1% level the null hypothesis for the period
1880-1900 with a less conservative choice of $\varepsilon$ (e.g. we obtain a
p-value of 0.04 using $\varepsilon/2$). Under this assumption, our results are
informative not only on the maximal father-son elasticity coefficient for a
given period of time, but also on its evolution. It follows in particular that
our estimates point to a mild decrease in this elasticity coefficient for sons
between 1850 and 1930.

Third, the results from the test of equality reported in Table 1 indicate that
the TSTSLS estimates are in several cases statistically distinguishable from
the estimated upper bounds of our identified sets. This includes, for the
sample of sons, all periods with the exception of 1900-1920, and the periods
1850-1870 and 1880-1900 for the sample of sons-in-law. Besides, for the sample
of sons in particular, the TSTSLS estimates exhibit a sharp increase, while
our estimated upper bound decreases between the periods 1880-1900 and
1900-1920. In that sense, our results offer suggestive evidence that the
intergenerational income correlation might have been more stable at the
beginning of the 20th century than what one would infer from the TSTSLS
estimates.

Fourth, we also report in Table 1 the estimated identified set and confidence
intervals associated with our method when we impose a lower bound on the
$R^{2}$ of the long regression, namely $\underline{R}^{2}\geq 1.3R_{s}^{2}$ or
$\underline{R}^{2}\geq 2R_{s}^{2}$ . Imposing any of these restrictions, which
are satisfied for the periods 1860-1880 and 1880-1900 for which the linked
data is available, results in substantially tighter confidence intervals. In
particular, for the sample of sons, the confidence intervals obtained under
the restriction $\underline{R}^{2}\geq 2R_{s}^{2}$ allow us to reject values
of the intergenerational income elasticity coefficient smaller than 0.13 and
larger than 0.48 for the years 1910-1930.

Finally, we consider in Tables 8 and 9, and Figure 5(b) in online Appendix D
several robustness checks. They relate to the set of first names that we
include as controls in our estimation procedure (Panel A), the choice of
$\varepsilon$ (Panel B and Figure 5(b)), and restrictions of the sample to the
set of individuals whose first name is included in the set of controls $X_{c}$
(Panel C). Throughout the tables, we focus on the upper bound of the estimated
identified set (“DGM, set”) and of the confidence interval (“DGM, CI.”).

The main takeaway from Table 8 and Figure 5(b) is that, for the sample of
sons, the results from our inference procedure are qualitatively, and in most
cases quantitatively, robust to these different sensitivity analyses. The one
case that exhibits more sensitivity is the specification where we control for
the first names that account for at least 0.02% of the sample, instead of
0.01% in our baseline specification. The upper bound of our confidence
interval for the period 1900-1920 increases in this case from 0.48 to 0.58,
the results remaining, however, stable for the other periods. The results for
the sample of sons-in-law (Table 9) are also, for most periods at the
exception of the same limit for 1900-1920, qualitatively, and in some cases
quantitatively similar across specifications. The main difference with the
sample of sons is that the choice of $\varepsilon$ does appear to matter more
for the sons-in-law, a limitation that one should keep in mind when
interpreting the findings for this subgroup. Nonetheless, to the extent that
our baseline choice of $\varepsilon$ (see Section 3.3) is motivated by the
theory and is found to perform well in our Monte Carlo simulation exercises,
we do not view this as particularly worrisome.

##  5 Conclusion

We study the identification of and inference on partially linear models, in an
environment where the outcome of interest and some of the covariates are
observed in two different datasets that can not be matched. This setup arises
in particular when one is interested in the effect of a variable that is not
observed jointly with the outcome variable, or in cases where potential
confounders are observed in a different dataset from the one including the
outcome and regressor of interest. In such situations, researchers often rely
on strong assumptions to point identify their parameters of interest. Our
approach offers a useful alternative when such assumptions are debatable. The
application shows that in addition to its tractability, our method is able to
deliver informative bounds. Finally, beyond the model considered in this
paper, our analysis suggests that the radial function is an appealing tool in
partial identification problems where the support function proves difficult to
compute.

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##  Appendix A Additional theoretical results

###  A.1 Measurement errors

We have assumed so far that the outcome and covariates are perfectly observed.
However, measurement errors are pervasive in survey data. We now explore the
robustness of the identified set proposed earlier to measurement errors on the
outcome and covariates, which we denote by $Y^{*}$ and $X^{*}$. Specifically,
consider a situation where both the covariates and the outcome are measured
with error, such that:

$\left\\{\begin{array}[]{rl}X=X^{*}+\xi_{X},&\xi_{X}\perp\\!\\!\\!\perp
X^{*},\\\\[5.69054pt]
Y=Y^{*}+\xi_{Y},&\xi_{Y}\perp\\!\\!\\!\perp(X^{*},Y^{*}).\end{array}\right.$
(19)

We introduce a new set, $\mathcal{B}^{*}$, which is defined as the original
identified set $\mathcal{B}$ after replacing the observed measurement error-
ridden covariates and outcome $(X,Y)$ by their latent counterparts
$(X^{*},Y^{*})$.

######  Proposition 8.

If Assumption 1 is satisfied with $(X,Y)$ replaced by $(X^{*},Y^{*})$, (19)
holds and for all $\beta\in\mathcal{B}^{*}$,
$\xi_{Y_{0}}\succ_{\\!\text{cv}}\xi_{X_{0}}^{\prime}\beta$, then
$\mathcal{B}^{*}\subset\mathcal{B}$.

The proof is in our supplementary material. This proposition establishes that
the identified set is robust to measurement errors in the following sense: if
(centered) measurement errors on the outcome $Y^{*}$ second-order
stochastically dominate those on the linear index $X_{0}^{*}{}^{\prime}\beta$
for all $\beta\in\mathcal{B}^{*}$, the identified set $\mathcal{B}$ based on
the observed covariates $X$ and outcome $Y$ always contains the true value of
the parameter of interest.151515 This result and underlying assumptions are
closely related to the robustness to measurement errors on the beliefs of the
test of rational expectations proposed in D’Haultfoeuille et al. (2021)
(Subsection 2.2.4). To better understand the above domination condition,
suppose that $p=1$, $\xi_{Y}\sim\mathcal{N}(0,\sigma^{2}_{Y})$ and
$\xi_{X}\sim\mathcal{N}(0,\sigma^{2}_{X})$. Then, recalling that any
$\beta\in\mathcal{B}^{*}$ satisfies the variance restriction
$\beta^{2}V(X^{*})\leq V(Y^{*})$, a sufficient condition for the dominance
condition $\xi_{Y_{0}}\succ_{\\!\text{cv}}\xi_{X_{0}}\beta$ is
$\sigma^{2}_{Y}\geq[V(Y^{*})/V(X^{*})]\sigma^{2}_{X}$. In our application for
instance, $Y^{*}$ and $X^{*}$ are the log earnings of fathers and sons (or
sons-in-law), respectively, so $V(Y^{*})\simeq V(X^{*})$ and
$\sigma^{2}_{Y}\simeq\sigma^{2}_{X}$ seem credible. This suggests that the key
domination condition from Proposition 8 is likely to hold in this context.

###  A.2 Identification of a model with interaction terms

Let $X_{c}=(X_{1,c},X_{-1,c})$ and $X_{nc}=(X_{1,nc},X_{-1,nc})$. We consider
here the following model

$E(Y|X)=f(X_{c})+X_{nc}^{\prime}\beta_{0}+X_{1,nc}X_{1,c}\delta_{0}.$

First define, for $x\in\text{Supp}(X_{1c})$ and $q\in\mathbb{R}^{p}$ ($q\neq
0_{p}$),

$\displaystyle\overline{S}_{x}(q,F_{Y,X_{c}},F_{X})$
$\displaystyle=\inf_{x_{-1,c}\in\text{Supp}(X_{-1,c}|X_{1,c}=x)}S(F_{Y|X_{-1,c}=x_{-1,c},X_{1,c}=x},F_{X_{nc}^{\prime}q|X_{-1,c}=x_{-1,c},X_{1,c}=x})$
$\displaystyle\mathcal{B}_{x}$ $\displaystyle=\bigg{\\{}\lambda
q:\,q\in\mathcal{S},0\leq\lambda\leq\overline{S}_{x}(q,F_{Y,X_{c}},F_{X})\bigg{\\}}.$

Proposition 2 applied to the subpopulation $X_{1,c}=x$ implies that for all
$x\in\text{Supp}(X_{1,c})$, $\beta_{0}+x\delta_{0}e_{1}\in\mathcal{B}_{x}$,
where $e_{1}=(1,0,\dots,0)^{\prime}\in\mathbb{R}^{p}$. Because the converse
also holds, the identified set $\mathcal{B}^{\delta\beta}$ of
$(\delta_{0},\beta_{0})$ is

$\mathcal{B}^{\delta\beta}=\\{(\delta,\beta):\forall\,x\in\text{Supp}(X_{1,c}),\;\beta+x\delta
e_{1}\in\mathcal{B}_{x}\\}.$ (20)

The sets $\mathcal{B}_{x}$ are convex and include $0_{p}$. Hence,
$\mathcal{B}^{\delta\beta}$ is convex too, and also includes $0_{p+1}$.
Moreover, because $\mathcal{B}_{x}$ are compact, any
$(\delta,\beta)\in\mathcal{B}^{\delta\beta}$ satisfies, for any
$(x,x^{\prime})\in\text{Supp}(X_{1,c})^{2}$, $x\neq x^{\prime}$,

$|\delta||x-x^{\prime}|\leq\|\beta+x\delta e_{1}\|+\|\beta+x^{\prime}\delta
e_{1}\|\leq M_{x}+M_{x^{\prime}},$ (21)

for some $M_{x},M_{x^{\prime}}>0$. Moreover,

$\|\beta\|\leq\|\beta+x\delta e_{1}\|+|x||\delta|\leq
M_{x}+\frac{|x|}{|x-x^{\prime}|}(M_{x}+M_{x^{\prime}}),$

which implies that $\mathcal{B}^{\delta\beta}$ is also compact. Thus,
$\mathcal{B}^{\delta\beta}$ can also be described by its radial function,
which we denote by $S(q,F_{Y,X_{c}},F_{X})$. Moreover, it follows from (20)
that

$S(q,F_{Y,X_{c}},F_{X})=\inf_{x\in\text{Supp}(X_{1,c})}\overline{S}_{x}(q_{-1}+xq_{1}e_{1},F_{Y,X_{c}},F_{X}).$

###  A.3 Test for point-identification

We develop here a statistical test that can be used to check whether
$\beta_{0}\in\partial\mathcal{B}$. Following the discussion in Subsection
2.3.4, this boils down to testing for

$H_{0}:\;S(F_{Y_{v0}},F_{X^{\prime}_{v0}\beta_{v}})=1\quad\text{against}\quad
H_{1}:S(F_{Y_{v0}},F_{X^{\prime}_{v0}\beta_{v}})>1,$ (22)

where we recall that the joint distribution of the validation data
$(X_{v},Y_{v})$ is observed and
$\beta_{v}=V(X_{v})^{-1}\text{cov}(X_{v},Y_{v})$. We consider a statistical
test based on i.i.d. data $(X_{vi},Y_{vi})_{i=1,...,n}$. The test statistic is

$T=b_{n}^{1/2}\left(S_{\varepsilon}(\widehat{F}_{Y_{v0}},\widehat{F}_{X^{\prime}_{v0}\widehat{\beta}_{v}})-1\right),$

where $\widehat{\beta}_{v}$ is the OLS estimator of $\beta_{v}$. The critical
value is then $q_{1-\alpha}(T^{*})$, the quantile of order $1-\alpha$ (defined
conditional on the data) of

$T^{*}=n^{1/2}\left(S_{\varepsilon}(\widehat{F}^{*}_{Y_{v0}},\widehat{F}^{*}_{X^{\prime}_{v0}\widehat{\beta}^{*}_{v}})-S_{\varepsilon}(\widehat{F}_{Y_{v0}},\widehat{F}_{X^{\prime}_{v0}\widehat{\beta}_{v}})\right),$

where $\widehat{F}^{*}_{Y_{v0}}$, $\widehat{F}^{*}_{X_{v0}^{\prime}q}$ and
$\widehat{\beta}^{*}_{v}$ are the subsampling counterpart of
$\widehat{F}_{Y_{v0}}$, $\widehat{F}_{X_{v0}^{\prime}q}$ and
$\widehat{\beta}_{v}$, respectively. We establish the asymptotic properties of
the test under the following assumption.

######  Assumption 7.

$E[\|X_{v}\|^{2+\delta}]<\infty$ for some $\delta>0$, $E[Y^{2}_{v}]<\infty$,
$\beta_{v}\neq 0$ and
$S(F_{Y_{v0}},F_{X^{\prime}_{v0}\beta_{v}})=S_{\varepsilon}(F_{Y_{v0}},$
$F_{X^{\prime}_{v0}\beta_{v}})$. Also, there exists
$\mathcal{V}\subset\mathcal{S}$, compact and including a ball of positive
radius centered at $\beta_{v}/\left\|\beta_{v}\right\|$, and
$\varepsilon^{\prime}\in(0,\varepsilon)$ such that for all
$(\alpha,\alpha^{\prime})\in[\varepsilon^{\prime},1-\varepsilon^{\prime}]^{2}$,
there exists $c>0$ and a strictly increasing and continuous function $m$ such
that $m(0)=0$ and

$\displaystyle\inf_{q\in\mathcal{V}}\left|F_{X^{\prime}q}^{-1}(\alpha^{\prime})-F_{X^{\prime}q}^{-1}(\alpha)\right|$
$\displaystyle>c|\alpha^{\prime}-\alpha|,$
$\displaystyle\sup_{q\in\mathcal{V}}\left|F_{X^{\prime}q}^{-1}(\alpha^{\prime})-F_{X^{\prime}q}^{-1}(\alpha)\right|$
$\displaystyle<m(|\alpha^{\prime}-\alpha|),$
$\displaystyle\left|F_{Y}^{-1}(\alpha^{\prime})-F_{Y}^{-1}(\alpha)\right|$
$\displaystyle<m(|\alpha^{\prime}-\alpha|).$

Up to the condition
$S(F_{Y_{v0}},F_{X^{\prime}_{v0}\beta_{v}})=S_{\varepsilon}(F_{Y_{v0}},F_{X^{\prime}_{v0}\beta_{v}})$
on which we come back below, Assumption 7 is very close to the first part of
Assumption 4, but it is weaker as we require that it holds over $\mathcal{V}$
instead of $\mathcal{S}$.

######  Proposition 9.

Suppose that $b_{n}\to\infty$, $b_{n}/n\to 0$ and Assumptions 1-2 and 7 hold.
Then:

  1. 1.

If $H_{0}$ in (22) holds, $\lim_{n\to\infty}P(T>q_{1-\alpha}(T^{*}))=\alpha$.

  2. 2.

If $H_{1}$ in (22) holds, $\lim_{n\to\infty}P(T>q_{1-\alpha}(T^{*}))=1$.

The proof is in our supplementary material. Note that if $H_{0}$ holds but
$S(F_{Y_{v0}},F_{X^{\prime}_{v0}\beta_{v}})<S_{\varepsilon}(F_{Y_{v0}},F_{X^{\prime}_{v0}\beta_{v}})$,
$T$ will tend to infinity and $H_{0}$ will be rejected. Because we are testing
here the validation of the tail condition described above, failing to reject
$H_{0}$ under the alternative is more of an issue than wrongly rejecting
$H_{0}$. Thus, potential over-rejection is arguably not as problematic as in
other more standard contexts, such as testing the null of no effect of a
treatment.

##  Appendix B Application to the black-white wage gap

We apply our method to estimate the black-white wage gap among young males in
the United States using the 1979 panel of the National Longitudinal Survey of
Youth (NLSY79), revisiting the seminal work of Neal and Johnson (1996) on this
question. Considering the same restrictions as Neal and Johnson (1996) leads
to a sample of size $n=1,675$.161616We refer the reader to Neal and Johnson
(1996) for a detailed discussion on the data. We focus on the following model
:

$Y=\gamma_{c,0}+X_{c,1}\gamma_{c,1}+X_{c,2}\gamma_{c,2}+X_{nc}\beta_{nc}+\epsilon,\quad
E\left[\epsilon|X_{c},X_{nc}\right]=0,$

where $Y$ is the mean log wage in 1990-1991, $X_{nc}$ denotes the AFQT and
$X_{c,k}$, $k=1,2$ are dummy variables for being black or Hispanic. While
$(Y,X_{c},X_{nc})$ is jointly observed in the NLSY79 dataset, we proceed in
the following as if AFQT, which is used in Neal and Johnson (1996) to control
for pre-market factors, was not observed jointly with wages. This setup, which
mimics the data environments in several other countries, allows us to directly
compare the confidence intervals based on our partial identification approach
with the ones obtained from the oracle OLS specification.

Results in Table 2 below show the effect on our bounds when we impose
different sets of constraints, namely i) a negative sign constraint on the
coefficient $\gamma_{c,1}$ associated with the black indicator as well as a
positive sign constraint on the coefficient $\beta_{nc}$ associated with the
AFQT, ii) the latter constraints combined with the constraint
$\underline{R}^{2}\geq 1.3R^{2}_{s}$, and iii) the sign constraints i)
combined with a less conservative bound $\underline{R}^{2}\geq 2R^{2}_{s}$.
Focusing on the main coefficient of interest $\gamma_{c,1}$, these results
indicate that imposing these constraints on the $\underline{R}^{2}$ results in
an identified set and confidence interval that are quite informative. Notably,
the lower bound of the confidence interval is equal to $-.25$ and $-.2$
respectively in cases ii) and iii), against $-.17$ (i.e. a 17 log points wage
penalty) for the OLS estimator. Taken together, these results show that our
method is able to deliver confidence intervals that are very informative in
practice.

| OLS | DGM
---|---|---
Constraints |  | Without | With signs constraints
|  |  | Only | And $\underline{R}^{2}\geq 1.3R^{2}_{s}$ | And $\underline{R}^{2}\geq 2R^{2}_{s}$
| (1) | (2) | (3) | (4) | (5)
Omitted variable $X_{nc}$ |  |  |  |  |
AFQT | 0.150 | [-0.437,0.437] | [0,0.154] | [0.045,0.154] | [0.082,0.154]
CI | [0.11,0.19] | [-0.522,0.522] | [0,0.215] | [0.004,0.211] | [0.010,0.207]
Common variables $X_{c}$ |  |  |  |  |
Black | -0.076 | [-0.664,0.318] | [-0.173,0] | [-0.123,0] | [-0.081,0]
CI | [-0.171,0.02] | [-0.847,0.507] | [-0.304,0] | [-0.247,0] | [-0.199,0]
Hispanic | 0.016 | [-0.334,0.266] | [-0.034,0.071] | [-0.003,0.071] | [0.022,0.071]
CI | [-0.083,0.116] | [-0.506,0.450] | [-0.197,0.226] | [-0.186,0.225] | [-0.174,0.223]
Notes: $Y$ is average log wage in 1990 and 1991, $X_{nc}$ is the AFQT, $X_{c}$
are dummies for being Black or Hispanic. The sample size is $n=1,675$, which
is randomly split in two to artificially create a dataset where we observe
$(Y,X_{c})$ and another one with $(X_{nc},X_{c})$. The first column presents
the OLS estimates on the full dataset, where the 95% CI have been multiplied
by $\sqrt{2}$ to make it comparable with the DGM procedure using only half of
it. The second column (2) presents the DGM estimates without constraints.
Column (3) is the DGM estimates with a negative sign constraint on the
coefficient of Black and a positive one on the coefficient of AFQT. Column (4)
and (5) gather the DGM estimates with the latter constraints plus a lower
bound constraint $R^{2}$ of the long regression: $\underline{R}^{2}\geq
rR^{2}_{s}$, with respectively $r=1.3$ and $r=2$. The $R^{2}$ on the short and
long regressions are respectively 0.051 and 0.142.

Table 2: Bounds on the wage gap under different constraints for NLSY79

Online Appendix

##  Appendix C Monte Carlo simulations

In this section we study the finite sample performances of our inference
method through Monte Carlo simulations. We first consider the baseline case
where no common regressor is available, before evaluating the performance of
our method in the presence of a common regressor. Finally, we discuss the
computational time of our procedure compared to a many moment inequality-based
alternative.

###  C.1 Univariate case without common regressors

We first explore the finite sample performances of our inference method with
$p=1$ and no $X_{c}$, considering the following DGP:

$Y=X_{nc}\beta_{0}+U,\quad\beta_{0}=1,\;X_{nc}\perp\\!\\!\\!\perp U.$

Then, we either assume that $X_{nc}\sim\mathcal{N}(0,1.5)$ and
$U\sim\mathcal{N}(0,1)$, referred to in the following as the normal case, or
$X_{nc}\sim\Gamma(1,2)$ and $U\sim\Gamma(0.4,2)$, which we refer to as the
gamma case.

We compare the finite sample performances of our inference method with those
based on Andrews and Shi, 2017, henceforth AS. Specifically, recall from (5)
above that

$\mathcal{B}=\left\\{\beta\in\mathbb{R}^{p}:\,E\left[\max(0,Y_{0}-t)\right]\geq
E\left[\max(0,X_{nc0}^{\prime}\beta-t)\right]\;\forall
t\in\mathbb{R}\right\\}.$

Hence, $\mathcal{B}$ is characterized by infinitely many moment inequalities.
We then construct confidence regions for $\beta_{0}$ by inverting tests that
these moment inequalities hold.171717These tests involve several tuning
parameters. Following the recommendation of AS (and using their notation), we
fix $\epsilon=0.05$ and $\eta=10^{-6}$. To fix $b_{0}$ and $\kappa$, we follow
the same procedure as in D’Haultfoeuille et al. (2021), which yields
$b_{0}=0.5$ and $\kappa=10^{-4}$. To construct a confidence region on
$\beta_{0}$, we first fix a few directions $(q_{1},...,q_{n})$ in
$\mathcal{S}$. Then, for $q=q_{k}$, we compute by a bisection method the
maximal $\lambda\in\mathbb{R}^{+}$ such that the test of the moment
inequalities at $\beta=\lambda q$ is not rejected.

In Table 3 below, we report the average bounds, across all 500 simulations, of
the estimated identified sets and the 95% confidence intervals associated with
each of the five different sample sizes (Column “Bounds”) obtained with our
method (“DGM”) and by applying Andrews and Shi (2017) (“AS”). In order to
isolate sampling uncertainty, we report for each sample size and separately
for our method and AS what we call the excess length (“Ex. length”), namely
the mean difference between the length of the confidence sets and that of the
identified set. We also report the coverage rates across simulations
(“Coverage”). Finally, we report the average, across all simulations, of the
estimates of the identified set $\mathcal{B}_{\varepsilon(q)}$, where
$\varepsilon(q)$ is given by (14) and thus varies from one simulation to
another.

| DGM |  | AS
---|---|---|---
Sample size | Bounds | Ex. length | Coverage | $\widehat{\mathcal{B}}_{\varepsilon(q)}$ |  | Bounds | Ex. length | Coverage
Normal |  |  |  |  |  |  |  |
Identified set | [-1.202,1.202] |  |  |  |  | [-1.202,1.202] |  |
400 | [-1.305,1.307] | 0.208 | 0.938 | [-1.202,1.202] |  | [-1.374,1.367] | 0.337 | 0.983
800 | [-1.280,1.280] | 0.156 | 0.942 | [-1.202,1.202] |  | [-1.329,1.328] | 0.253 | 0.985
1,200 | [-1.266,1.267] | 0.129 | 0.940 | [-1.202,1.202] |  | [-1.301,1.301] | 0.198 | 0.978
2,400 | [-1.246,1.247] | 0.089 | 0.948 | [-1.202,1.202] |  | [-1.268,1.270] | 0.134 | 0.975
4,800 | [-1.234,1.235] | 0.065 | 0.936 | [-1.202,1.202] |  | [-1.251,1.250] | 0.097 | 0.980
Gamma |  |  |  |  |  |  |  |
Identified set | [-0.025,1.046] |  |  |  |  | [-0.025,1.046] |  |
400 | [-0.758,1.357] | 1.043 | 1 | [-0.464,1.287] |  | [-0.538,1.343] | 0.809 | 1
800 | [-0.603,1.302] | 0.834 | 0.996 | [-0.340,1.257] |  | [-0.466,1.313] | 0.707 | 1
1,200 | [-0.546,1.28] | 0.754 | 1 | [-0.293,1.247] |  | [-0.438,1.302] | 0.668 | 1
2,400 | [-0.458,1.243] | 0.629 | 1 | [-0.237,1.220] |  | [-0.391,1.277] | 0.596 | 1
4,800 | [-0.391,1.213] | 0.532 | 1 | [-0.199,1.197] |  | [-0.362,1.258] | 0.548 | 1

Notes: results obtained with 500 simulations. Column “Bounds” reports either
the identified set or the average of the bounds of the 95% confidence
intervals over simulations. “Ex. length” is the excess length, i.e. the
average length of the confidence region minus the length of the identified
set. Column “Coverage” displays the minimum, over $\beta\in\mathcal{B}$, of
the estimated probability that $\beta\in\text{CR}_{1-\alpha}(\beta_{0})$.
Column “$\widehat{\mathcal{B}}_{\varepsilon(q)}$” displays the average, across
all simulations, of the estimates of the identified set
$\mathcal{B}_{\varepsilon(q)}$, where $\varepsilon(q)$ is given by (14). We
use 1,000 subsampling (resp. bootstrap) replications to compute the confidence
intervals for the DGM (resp. AS) method. Table 3: Finite sample performances
for $p=1$

A couple of remarks are in order. First, as expected, the 95% confidence
intervals shrink with the sample sizes $n$. For both DGPs and all sample
sizes, comparing the identified set with the confidence intervals indicates
that identification uncertainty clearly dominates sampling uncertainty. This
is especially striking for the normal case, which yields a substantially wider
identified set, but also holds in the gamma case, where the regressor $X_{nc}$
has thicker tails. In particular, considering the excess length in the normal
case, the confidence set is only between 8.6% (for $n=400$) and 2.7% (for
$n=4,800$) wider than the identified set. In the gamma case, the confidence
set ranges between 20.7% and 14.8% larger than the (regularized) identified
set ($\mathcal{B}_{\varepsilon}$).

Second, the coverage of our confidence intervals is good: coverage rates are
always larger than 93.6%. Third, our inference method generally performs
similarly or better than AS, delivering consistently tighter confidence sets
for sufficiently large sample size. For example, in the normal case, the
excess length of the confidence set is reduced by around 30% to 39% depending
on the sample sizes. In the gamma case, the two methods are close, AS doing
slightly better only for sample sizes smaller than $n=4,800$. These results
are consistent with our inference method exploiting the specific geometric
structure of the identified set. This could also be due to the fact that we do
not need to bear the cost, in terms of statistical power, of incorporating
potentially many non-binding inequality constraints.

Finally, the good finite sample performances of our inference method offers
supporting evidence that our choice of the regularization parameter
$\varepsilon(q)$, given by (14) and motivated in Section 3.3 above, is
appropriate. In the normal case where $\mathcal{B}_{\varepsilon}=\mathcal{B}$
for all $\varepsilon$, $\varepsilon(q)$ remains close to 0.5 for all sample
sizes. In contrast, in the gamma case where the minimum of
$R(\cdot,F_{Y_{0}},F_{X_{0}^{\prime}q})$ is reached at $\varepsilon=0$ for
both $q=1$ and $q=-1$, $\varepsilon(q)$ tends to 0 as $n$ tends to infinity.
Overall, the results suggest that the chosen $\varepsilon(q)$ achieves a good
balance between identification (a large $\varepsilon$ leading to an increase
in $\mathcal{B}_{\varepsilon}$) and statistical uncertainty (a small
$\varepsilon$ leading to more volatility when estimating $S_{\varepsilon}$ and
thus larger quantiles $\widehat{c}_{\alpha,\varepsilon}$).

###  C.2 Multivariate case without common regressor

We now consider the multivariate case ($p=2$) with the following DGP:

$\displaystyle Y$ $\displaystyle=$
$\displaystyle\gamma_{0}+X_{nc}^{\prime}\beta_{0}+U,\;U|X_{nc}\sim\mathcal{N}(0,4).$
(23)

We set the coefficients as follows: $\gamma_{0}=-0.1$, $\beta_{0,1}=1$, and
$\beta_{0,2}=1$. The variables $X_{nc}$ follow a multivariate normal
distribution with mean 0 and covariance matrix

$\Sigma=\left(\begin{array}[]{cc}1&-0.2\\\ -0.2&1\end{array}\right).$

We report in Table 4 below the performances of our inference method, applied
to the first component of $\beta_{0}$, for the same sample sizes as above,
along with the identified set of the projection. These results were obtained
using 500 simulations. We restrict to the first component of $\beta_{0}$ as
the results are very similar for the second component. The main takeaway of
this table is that our inference method exhibits similar finite-sample
performances to the ones discussed in the univariate ($p=1$) normal case. In
particular, the excess length of the confidence sets relative to the
identified set tends to be quite small, and declines as $n$ gets larger.

Average | Bounds | Excess length | Coverage
---|---|---|---
Identified set | [-2.367, 2.367] |  |
Sample size |  |  |
400 | [-2.599, 2.599] | 0.465 | 0.94
800 | [-2.555, 2.554] | 0.376 | 0.962
1,200 | [-2.523, 2.522] | 0.312 | 0.96
2,400 | [-2.496, 2.497] | 0.26 | 0.982
4,800 | [-2.475, 2.474] | 0.217 | 0.986

Notes: results obtained with 500 simulations. Column “Bounds” reports either
the identified set or the average of the bounds of the 95% confidence
intervals over simulations. “Excess length” is the average length of the
confidence region minus the length of the identified set. Column “Coverage”
displays the minimum, over $\beta_{1}\in\mathcal{B}_{1}$, of the estimated
probability that $\beta_{1}\in\text{CI}_{1-\alpha}(\beta_{0,1})$. We use 200
subsampling replications to compute the confidence intervals. Table 4: Finite
sample performances for $\beta_{0,1}$ with $p=2$

###  C.3 Case with a common regressor and possible constraints

We now examine the performances of our inference method in the presence of a
common regressor. Namely, we consider the DGP:

$Y=X_{c}\gamma_{0}+X_{nc}\beta_{0}+U,\;U|X\sim\mathcal{N}(0,4).$

We set the coefficients as follows: $\gamma_{0}=0.3$ and $\beta_{0}=1$. The
covariates are transformations of $(N_{1},N_{2})^{\prime}$, which follows a
multivariate normal distribution with mean 0 and covariance matrix

$\Sigma=\left(\begin{array}[]{cc}1&0.8\\\ 0.8&1.5\\\ \end{array}\right).$

Specifically, the common regressor is given by
$X_{c}=\mathds{1}\left\\{N_{1}\leq 0.3\right\\}$, and the regressors observed
in one of the datasets only are such that $X_{nc}=N_{2}$.

We report in Table 5 the performances of our inference method applied to the
parameters $\beta_{0}$ and $\gamma_{0}$ along with the identified sets, with
or without imposing the sign constraint $\gamma_{0}\geq 0$. For $\beta_{0}$,