monthyeardate[]

#  Informational Content of Factor Structures in
Simultaneous Binary Response Models111First version: October 2019. We thank
the Editor, Serena Ng, two anonymous referees, Arthur Lewbel, and seminar
participants at Arizona State University, Emory, Michigan State, Shanghai
University of Finance and Economics, University of Arizona, as well as
conference participants at the 2015 SEA meetings for helpful comments. We also
thank Zhangchi Ma and Qingsong Yao for excellent research assistance. Zhang
acknowledges the financial support from Singapore Ministry of Education Tier 2
grant under grant MOE2018-T2-2-169 and the Lee Kong Chian fellowship.

Shakeeb Khan
Boston College  Arnaud Maurel
Duke University, NBER and IZA  Yichong Zhang
Singapore Management University

(
)

###### Abstract

We study the informational content of factor structures in discrete triangular
systems. Factor structures have been employed in a variety of settings in
cross sectional and panel data models, and in this paper we formally quantify
their identifying power in a bivariate system often employed in the treatment
effects literature. Our main findings are that imposing a factor structure
yields point identification of parameters of interest, such as the coefficient
associated with the endogenous regressor in the outcome equation, under weaker
assumptions than usually required in these models. In particular, we show that
a “non-standard” exclusion restriction that requires an explanatory variable
in the outcome equation to be excluded from the treatment equation is no
longer necessary for identification, even in cases where all of the regressors
from the outcome equation are discrete. We also establish identification of
the coefficient of the endogenous regressor in models with more general factor
structures, in situations where one has access to at least two continuous
measurements of the common factor.

###### Abstract

This paper gathers the supplementary material to the original paper. In
Section A, we discuss the identification power of the factor structure. In
Section B, we propose an estimator based on our constructive identification
strategy and establish its asymptotic properties. Section C contains a
simulation study. In Sections D and E, we prove Theorems 2.1 and 3.1,
respectively. In Section F, we establish the asymptotic distribution for the
rank estimator. In Section G, we consider the identification of the model with
two idiosyncratic shocks but no continuous repeated measurements of the common
factor. In Sections H, I, J and K, we prove Theorems A.1, A.2, G.1 and G.2,
respectively. Section L considers the estimation of the model with a
nonparametric factor structure. Section M considers the distribution theory
for the closed form estimator.

Keywords: Factor Structures, Discrete Choice, Causal Effects.

JEL: C14, C31, C35.

##  1 Introduction

Factor models see widespread and increasing use in various areas of
econometrics. This type of structure has been employed in a variety of
settings in cross sectional, panel and time series models, and have proven to
be a flexible way to model the behavior of and relationship between unobserved
components of econometric models. The basic idea behind factor models is to
assume that the dependence across the unobservables is generated by a low-
dimensional set of mutually independent random factors. The applied and
theoretical research employing factor structures in econometrics is extensive.
In particular, these models are often used in the treatment effect literature
as a way to identify the joint distribution of potential outcomes from the
marginal distributions, and then recover the distribution of treatment effects
from this joint distribution.111See also Abbring and Heckman (2007) for an
extensive discussion of factor structures and prior studies using these models
in the context of treatment effect estimation. Factor models have been used in
a number of different contexts in applied microeconomics. These include, among
others, earnings dynamics (Abowd and Card, 1989; Bonhomme and Robin, 2010),
estimation of returns to schooling and work experiences (Ashworth, Hotz,
Maurel, and Ransom, 2020), as well as cognitive and non-cognitive skill
production technology (Cunha, Heckman, and Schennach, 2010). Heckman and
Vytlacil (2007a, b) provide various additional references. All of these
papers, with the notable exception of Cunha, Heckman, and Schennach (2010),
rely on linear factor models where the unobservables are assumed to be written
as the sum of a linear combination of mutually independent factors and an
idiosyncratic shock.

In this paper we bring together the literature on factor models with the
literature on the identification and estimation of triangular binary choice
models (Chesher (2005); Vytlacil and Yildiz (2007); Shaikh and Vytlacil
(2011); Han and Vytlacil (2017)) by exploring the informational content of
factor structures in this class of models.222See also recent work by Lewbel,
Schennach, and Zhang (2020), who study the identification of a triangular
linear model assuming that the disturbances are related through a factor
model. Focusing on this class can be well motivated from both an empirical and
theoretical perspective. From the former, many treatment effect models fit
into this framework as treatment is typically a binary and endogenous variable
in the system, whose effect on outcomes is often a parameter the
econometrician wishes to conduct inference on. From a theoretical perspective,
inference on this type of system can be complicated, if not impossible without
strong parametric assumptions, which may not be reflected in the observed
data. Imposing no restriction on the structure of endogeneity often fails to
achieve identification of parameter, or at best only do so in sparse regions
of the data, thus making inference impractical in practice. In this context,
modeling the endogeneity between the selection and the outcome by a factor
structure may be a useful “in-between” setting, which, at the very least, can
be used to gauge the sensitivity of the parametric approach to their stringent
assumptions.

We start our analysis by imposing a particular factor structure to the two
unobservables in this system and explore the informational content of this
assumption. We assume that the unobservables from the treatment equation ($V$)
and the outcome equation ($U$) are related through the following factor model:

$U=\gamma_{0}V+\Pi$ (1.1)

where $\Pi$ is an unobserved random variable assumed to be distributed
independently of $V$ and $\gamma_{0}$ is a scalar parameter. This structure
generalizes the canonical case where the unobservables $(U,V)$ are jointly
normally distributed, for which this relationship always holds. Our main
finding is that there is indeed informational content of factor structures in
the sense that, in contrast to prior literature - notably Vytlacil and Yildiz
(2007) \- one no longer requires an additional “non-standard” exclusion
restriction, nor the strong support conditions on the covariates entering the
outcome equation that are generally needed for identification in these models.
Our identification results are constructive and translate directly into a rank
based estimator of the coefficient associated with the binary endogenous
variable, which we provide and study in a supplement to this paper.

While an appealing feature of the structure considered in Equation (1.1) is
that it is a natural extension of the bivariate Probit specification that has
often been considered in the literature, this model does impose significant
restrictions on the nature of the dependence between the unobservables $U$ and
$V$. In the paper we extend this baseline specification by considering a
linear factor structure of the form:

$\displaystyle U$ $\displaystyle=$ $\displaystyle\gamma_{0}W+\eta_{1}$ (1.2)
$\displaystyle V$ $\displaystyle=$ $\displaystyle W+\eta_{2}$ (1.3)

where $(W,\eta_{1},\eta_{2})$ are mutually independent unobserved random
variables. We study the informational content of this extended factor
structure in the context of triangular binary choice models and establish
identification, assuming access to at least two continuous noisy measurements
of the unobserved factor $W$. This setup has been used in a number of
applications, in particular in labor economics. In these applications, the
unobserved factor is typically interpreted as latent individual ability, about
which several continuous noisy measurements are available from the data. This
is the case of, for instance, Carneiro, Hansen, and Heckman (2003), Cunha,
Heckman, and Schennach (2010), Heckman, Humphries, and Veramendi (2018) and
Ashworth, Hotz, Maurel, and Ransom (2020), who use components of the Armed
Services Vocational Aptitude Battery test as measurements of cognitive
ability. At a high level, it is interesting to note that these results
complement Bai and Ng (2010), who show that, in the context of a linear
regression model with endogenous regressors, factor models have identifying
power, in that they can be used to create instrumental variables even when
none of the observed variables are valid instruments.

The rest of the paper is organized as follows. In Section 2 we formally
describe the triangular system with our factor structure, and discuss our main
identification results for the parameters of interest in this model. Section 3
explores identification in more general factor structure models which involve
multiple idiosyncratic errors, in a context where one has access to two
continuous noisy measurements of the common unobserved factor. Finally,
Section 4 concludes. The Supplementary Material collects the proofs of our
results, derives and studies the asymptotic properties of a rank-based
estimator for $\alpha_{0}$, explores its finite sample properties through some
Monte Carlo simulation exercises, and shows the non-identification of the two-
factor model with a compact-supported common factor and no continuous repeated
measurements. Finally, we also establish the sharp identified set of
$\alpha_{0}$ when the support condition for point-identification is violated.

Notation: throughout the paper we write $\mathbf{1}\\{A\\}$ to denote the
usual indicator function that takes value 1 if event $A$ happens, and 0
otherwise. We also denote by $d(U)$ and $d(U|V)$ the lengths of the support of
random variable $U$, and the conditional support of $U$ given $V$,
respectively.

##  2 Triangular Binary Model with Factor Structure

###  2.1 Set-up and Main Identification Result

In this section we consider the identification of the following triangular
binary model:

$\displaystyle Y_{1}$ $\displaystyle=$ $\displaystyle{\bf
1}\\{Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}+\alpha_{0}Y_{2}-U>0\\}$
(2.4) $\displaystyle Y_{2}$ $\displaystyle=$ $\displaystyle{\bf
1}\\{Z^{\prime}\delta_{0}-V>0\\}$ (2.5)

where $Z\equiv(Z_{1},Z_{2})$ and $(U,V)$ is a pair of random shocks. $Z_{2}$
and $Z_{3}$ provide the exclusion restrictions in the model, and the
distribution of $(Z_{2},Z_{3})$ is required to be nondegenerate conditional on
$Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}$. We further assume that
the error terms $U$ and $V$ are jointly independent of $(Z_{1},Z_{2},Z_{3})$.
The endogeneity of $Y_{2}$ in (2.4) arises when $U$ and $V$ are not
independent.

The above model, or minor variations of it, have often been considered in the
recent literature. See for example, Vytlacil and Yildiz (2007), Abrevaya,
Hausman, and Khan (2010), Klein, Shan, and Vella (2015), Vuong and Xu (2017),
Khan and Nekipelov (2018) and references therein. A key parameter of interest
in our paper and in the rest of the literature is $\alpha_{0}$. In this paper
we provide conditions under which the parameters of interest are point-
identified. As such, our analysis complements alternative partial-
identification approaches that have been proposed in the context of triangular
binary models. See, in particular, Chiburis (2010), Shaikh and Vytlacil
(2011), and Mourifié (2015).333In Section A in the supplement, we establish
the sharp identified set of $\alpha_{0}$ when the support condition for point-
identification is violated. This result highlights that, except for the fact
that the sign of $\alpha_{0}$ is identified, we generally cannot say much
about the value of $|\alpha_{0}|$. Related work by Shaikh and Vytlacil (2011)
also provides partial identification results for a triangular binary model.
That the bounds for $\alpha_{0}$ are generally tighter in their analysis
reflects the identifying power of the additional support restrictions that
they impose. As discussed in the aforementioned papers, the parameter
$\alpha_{0}$ is difficult, if not impossible to identify and estimate without
imposing parametric restrictions on the unobserved variables in the model,
$(U,V)$.

The difficulty of identifying $\alpha_{0}$ in semi-parametric “distribution-
free” models, and the sensitivity of its identification to misspecification in
parametric models is what motivates the factor structure we add in this paper
to the above model. Specifically, to allow for endogeneity in the form of
possible non-zero correlation between $U$ and $V$, we augment the model with
the following equation:

$U=\gamma_{0}V+\Pi$ (2.6)

where $\Pi$ is an unobserved random variable, assumed to be distributed
independently of $(V,Z_{1},Z_{2},Z_{3})$, and $\gamma_{0}$ is an additional
unknown scalar parameter. Importantly, this type of factor structure always
holds when the residuals of both equations are jointly normally distributed.
Furthermore, this specification corresponds to the type of structure used in
Independent Component Analysis (ICA), where $V$ and $\Pi$ are two mutually
independent factors. This method has found many applications in various
fields, including signal processing and image extraction; applications in
economics include e.g., Hyvärinen and Oja (2000), Moneta, Hoyer, and Coad
(2013) and Gourieroux, Monfort, and Renne (2017). While, in contrast to the
ICA literature, the factors and the factor loadings are not the main objects
of interest in our analysis, this dimension-reducing structure plays a key
role in our identification results.

Our aim is to first explore identification of the parameters
$(\alpha_{0},\delta_{0},\gamma_{0},\beta_{0},\lambda_{0})$ under standard
nonparametric regularity conditions on $(V,\Pi)$. Note that the parameter
$\delta_{0}$ in the selection equation can be identified up to scale in
various ways. See, for example, Klein and Spady (1993) and Han (1987), among
others. We then impose the usual condition that one of $\delta_{0}$’s
coordinates is equal to one to fix the scale. For simplicity, for the rest of
the paper, we denote $X\equiv Z^{\prime}\delta_{0}$ and assume $X$ is
observed. We further define $X_{1}\equiv
Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}$. However, we cannot
identify $\lambda_{0}$ and $\beta_{0}$ beforehand. We propose instead to
identify them along with $\alpha_{0}$.

Our main identification result is based on the Assumptions A1-A4 we state
below:

A1

The first coefficient of $\lambda_{0}$ is normalized to one so that
$\lambda_{0}=(1,\lambda_{0,-1}^{T})^{T}$. The parameter
$\theta_{0}\equiv(\alpha_{0},\gamma_{0},\lambda_{0,-1},\beta_{0})$ is an
element of a compact subset of $\Re^{d_{1}+d_{3}+1}$, where $d_{1}$ and
$d_{3}$ are the dimensions of $Z_{1}$ and $Z_{3}$, respectively.

A2

The vector of unobserved variables, $(U,V,\Pi)$ is continuously distributed
with support on a subset of $\Re^{3}$ and independently distributed of the
vector $(Z_{1},Z_{2},Z_{3})$. Furthermore, we assume that the unobserved
random variables $\Pi,V$ are distributed independently of each other.

A3

$X$ is continuously distributed with absolute continuous density w.r.t.
Lebesgue measure. Its density is bounded and bounded away from zero on any
compact subset of its support.

A4

Let $Z_{1,-1}$ be all the coordinates of $Z_{1}$ except the first one, and
$d=d_{1}+d_{3}+1$. There exist $2d$ vectors
$\\{z_{1}^{(l)},z_{3}^{(l)},x^{(l)}\\}_{l=1}^{d}$ and
$\\{\tilde{z}_{1}^{(l)},\tilde{z}_{3}^{(l)},\tilde{x}^{(l)}\\}_{l=1}^{d}$ in
the joint support of $(Z_{1},Z_{3},X)$ such that

$\displaystyle\alpha_{0}+(z_{1,-1}^{(l)}-\tilde{z}_{1,-1}^{(l)})^{\prime}\lambda_{0,-1}+(z_{3}^{(l)}-\tilde{z}_{3}^{(l)})^{\prime}\beta_{0}-\gamma_{0}(x^{(l)}-\tilde{x}^{(l)})=\tilde{z}_{1,1}^{(l)}-z_{1,1}^{(l)},~{}l=1,\cdots,d$

and $\text{rank}(\mathcal{M})=d$, where

$\displaystyle\mathcal{M}=\begin{pmatrix}1&\cdots&1\\\
z_{1,-1}^{(1)}-\tilde{z}_{1,-1}^{(1)}&\cdots&z_{1,-1}^{(d)}-\tilde{z}_{1,-1}^{(d)}\\\
z_{3}^{(1)}-\tilde{z}_{3}^{(1)}&\cdots&z_{3}^{(d)}-\tilde{z}_{3}^{(d)}\\\
x^{(1)}-\tilde{x}^{(1)}&\cdots&x^{(d)}-\tilde{x}^{(d)}\\\ \end{pmatrix}.$

Before turning to our main identification result, a couple of remarks are in
order.

######  Remark 2.1.

The first part of Assumption A1 is a standard scale normalization. Assumption
A2 is also standard in this literature. The assumption that the instruments
are independent of the unobservables can also be found in, among others,
Abrevaya, Hausman, and Khan (2010), Vytlacil and Yildiz (2007), Klein, Shan,
and Vella (2015), and Khan and Nekipelov (2018). The assumption of
independence between $\Pi$ and $V$ is also made in Bai and Ng (2002) and
Carneiro, Hansen, and Heckman (2003).

######  Remark 2.2.

Assumptions A3 and A4 impose some restrictions on the distributions of the
covariates entering the selection and outcome equations, respectively.
Specifically, Assumption A3 requires one component of the covariates $Z$
entering the selection equation to be continuously distributed, which is often
required in models with discrete outcomes. In contrast, Assumption A4 only
requires some variation of $(Z_{1},Z_{3})$. In particular, the distribution of
$(Z_{1},Z_{3})$ cannot be degenerate but is allowed to be discrete. This
assumption can be interpreted as a full rank condition, which ensures that the
system of linear equations that delivers point identification has a unique
solution.

We now turn to our main identification result, Theorem 2.1, which concludes
that under our stated conditions and our factor structure we can attain point
identification of the vector of parameters $\theta_{0}$.

######  Theorem 2.1.

Under Assumptions A1-A4, $\theta_{0}$ is point identified.

An important takeaway from this result, which we discuss further in Subsection
2.2 below, is that imposing the factor structure (2.6) yields point-
identification under weaker support conditions when compared to the existing
literature, and does not require a second exclusion restriction either. In
particular, our model delivers point-identification of the parameters of
interest even in situations where all of the regressors from the outcome
equation are discrete. This indicates that, from the selection equation
combined with the factor structure that we impose here, we can overturn the
non-identification result of Bierens and Hartog (1988) which would apply to
the outcome equation alone.

The proof of Theorem 2.1, which is reported in Section D in the Supplementary
Appendix, relies on the fact that, for two observations $(Z_{1},Z_{3},X)$ and
$(\tilde{Z}_{1},\tilde{Z}_{3},\tilde{X})$,

$\displaystyle\partial_{x}P^{11}(Z_{1},Z_{3},X)/f_{V}(X)+\partial_{x}P^{10}(\tilde{Z}_{1},\tilde{Z}_{3},\tilde{X})/f_{V}(\tilde{X})=0$
$\displaystyle\iff$
$\displaystyle\alpha_{0}+(Z_{1}-\tilde{Z}_{1})^{\prime}\lambda_{0}+(Z_{3}-\tilde{Z}_{3})^{\prime}\beta_{0}-\gamma_{0}(X-\tilde{X})=0,$
(2.7)

where $f_{V}(\cdot)$ is the pdf. of $V$, which is identified over the support
of $X$, and $P^{ij}(z_{1},z_{3},x)\equiv
Prob(Y_{1}=i,Y_{2}=j|Z_{1}=z_{1},Z_{3}=z_{3},X=x)$
($\partial_{x}P^{ij}(z_{1},z_{3},x)$) denote the choice probability (partial
derivative of the $ij$-choice probability with respect to the third argument),
which are both identified from the data.

######  Remark 1.

This identification result can be extended to the case of a separable
nonparametric factor model. Namely, consider the following relationship
between unobserved components:

$U=g_{0}(V)+\tilde{\Pi}$ (2.8)

where $\tilde{\Pi}$ is an unobserved random variable assumed to be distributed
independently of $V$ and all instruments. $g_{0}(\cdot)$ is an unknown
function assumed to satisfy standard smoothness conditions. The parameter of
interest is $(\alpha_{0},\lambda_{0},\beta_{0})$, but now the unknown nuisance
parameter in the factor equation is infinite dimensional. By replacing
$\gamma_{0}X$ by $g_{0}(X)$ in (2.1), we have

$\displaystyle\partial_{x}P^{11}(Z_{1},Z_{3},X)/f_{V}(X)+\partial_{x}P^{10}(\tilde{Z}_{1},\tilde{Z}_{3},\tilde{X})/f_{V}(\tilde{X})=0$
$\displaystyle\iff$
$\displaystyle\alpha_{0}+(Z_{1}-\tilde{Z}_{1})^{\prime}\lambda_{0}+(Z_{3}-\tilde{Z}_{3})^{\prime}\beta_{0}-(g_{0}(X)-g_{0}(\tilde{X}))=0.$
(2.9)

One can then establish identification after modifying the rank condition A4 by
replacing $\gamma_{0}(x^{(l)}-\tilde{x}^{(l)})$ by
$g_{0}(x^{(l)})-g_{0}(\tilde{x}^{(l)})$.

###  2.2 Connection with Prior Literature

We now discuss in detail how our setup and main identification result relates
to the existing literature.

In a related work, Han and Vytlacil (2017) consider the identification of a
generalized bivariate Probit model.444See also recent work by Han and Lee
(2019) who study semiparametric estimation and inference in the framework
considered by Han and Vytlacil (2017). Our linear factor structure and the
one-parameter copula model considered in Han and Vytlacil (2017) are not
nested by each other. First, note that based on the factor structure, we can
recover $F_{\Pi}$, the distribution of $\Pi$, as a function of
$(F_{U},F_{V},\gamma_{0})$ by deconvolution. We can then write the copula of
$(U,V)$ as

$F_{U,V}(F_{U}^{-1}(u),F_{V}^{-1}(v))=\int_{-\infty}^{F_{V}^{-1}(v)}F_{\Pi}(F_{U}^{-1}(u)-\gamma_{0}w;F_{U},F_{V},\gamma_{0})f_{V}(w)dw=C(u,v;F_{U},F_{V},\gamma_{0}).$

The copula depends not only on $\gamma_{0}$ but also on two infinite
dimensional parameters $(F_{U},F_{V})$. Thus, unlike Han and Vytlacil (2017),
our factor structure cannot be characterized by a one-parameter copula. In
addition, in order to achieve identification, Han and Vytlacil (2017) first
nonparametrically identify the two marginals by assuming the existence of a
full support regressor that is common to both equations.555Han and Vytlacil
(2017) establish their identification of the coefficient on the endogeneous
regressor (Theorems 4.2 and 5.1) under the assumption that the marginal
distributions $F_{\varepsilon}$ and $F_{\nu}$ are known. Then, they verify
this condition by showing the identification of these two marginal
distributions using large support common regressors. In contrast, our approach
does not rely on the existence of such a regressor. Under the factor structure
assumed in our analysis, we bypass the nonparametric identification of the
marginals as a whole and directly consider the identification of the
structural parameters. It follows that our model cannot be nested by the one-
parameter copula model considered by Han and Vytlacil (2017). On the other
hand, there exist one-parameter copula models that cannot be decomposed into
linear factor structures.666For instance, suppose that $(U,V)$ has a Gaussian
copula with correlation $\rho$, and that the marginal distributions of $U$ and
$V$ are uniform $[0,1]$. It then follows that, denoting by $\Phi(.)$ the
standard normal cdf., $\left(\Phi^{-1}(U),\Phi^{-1}(V)\right)$ is bivariate
normal with correlation $\rho$, which in turn yields the following non-linear
relationship between $U$ and $V$: $U=\Phi\left(\rho\Phi^{-1}(V)+W\right)$,
where $W$ is normally distributed and independent from $V$. This implies that
our model does not nest Han and Vytlacil (2017) either.

Our analysis also relates to Vytlacil and Yildiz (2007) and Vuong and Xu
(2017), who consider the identification of $\alpha_{0}$ in a triangular binary
model. Our identification result, however, differs from theirs in important
ways. Namely, denote
$X=Z^{\prime}\delta_{0}=Z_{1}^{\prime}\delta_{1,0}+Z_{2}^{\prime}\delta_{2,0}$.
Then, Assumption A4 implies that we can find a pair of observations
$(z_{1},z_{2},z_{3})$ and $(\tilde{z}_{1},\tilde{z}_{2},\tilde{z}_{3})$ such
that

$\displaystyle
z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}+\alpha_{0}-\gamma_{0}(z_{1}^{\prime}\delta_{1,0}+z_{2}^{\prime}\delta_{2,0})=\tilde{z}_{1}^{\prime}\lambda_{0}+\tilde{z}_{3}^{\prime}\beta_{0}-\gamma_{0}(\tilde{z}_{1}^{\prime}\delta_{1,0}+\tilde{z}_{2}^{\prime}\delta_{2,0}).$
(2.10)

In contrast, using our notation, Vytlacil and Yildiz (2007) require that one
can find a pair of observations $(z_{1},z_{2},z_{3})$ and
$(\tilde{z}_{1},\tilde{z}_{2},\tilde{z}_{3})$ such that
$z^{\prime}\delta_{0}=\tilde{z}^{\prime}\delta_{0}$ and

$\displaystyle
z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}+\alpha_{0}=\tilde{z}_{1}^{\prime}\lambda_{0}+\tilde{z}_{3}^{\prime}\beta_{0}.$
(2.11)

Vuong and Xu (2017) do not assume the existence of $Z_{3}$. In our binary
outcome setup, the functions $h(0,x,\tau)$ and $h(1,x,\tau)$ defined in Vuong
and Xu (2017) are equal to $1\\{x+F_{-U}^{-1}(\tau)\geq 0\\}$ and
$1\\{x+\alpha+F_{-U}^{-1}(\tau)\geq 0\\}$, respectively, where
$x=z_{1}^{\prime}\lambda_{0}$ and $F_{-U}$ is the CDF of $-U$. Then, Vuong and
Xu (2017, Assumption C’(ii)) requires that we can find $z_{1}$ and
$\tilde{z}_{1}$ in the support of $Z_{1}$ so that for any $\tau_{1},\tau_{2}$,
if $1\\{\tilde{z}_{1}^{\prime}\lambda_{0}+F_{-U}^{-1}(\tau_{1})\geq
0\\}=1\\{\tilde{z}_{1}\lambda_{0}+F_{-U}^{-1}(\tau_{2})\geq 0\\}$, then
$1\\{z_{1}^{\prime}\lambda_{0}+\alpha_{0}+F_{-U}^{-1}(\tau_{1})\geq
0\\}=1\\{z_{1}\lambda_{0}+\alpha_{0}+F_{-U}^{-1}(\tau_{2})\geq 0\\}$. Provided
that the support of $U$ nests the supports of $Z_{1}^{\prime}\lambda_{0}$ and
$Z_{1}^{\prime}\lambda_{0}+\alpha_{0}$, Vuong and Xu (2017, Assumption C’(ii))
is then equivalent to:777To see this, note that if, say,
$z_{1}^{\prime}\lambda_{0}+\alpha_{0}>\tilde{z}_{1}^{\prime}\lambda_{0}$, then
we can find $\tau_{1},\tau_{2}$ such that
$-z_{1}^{\prime}\lambda_{0}-\alpha_{0}\leq
F_{-U}^{-1}(\tau_{1})<-\tilde{z}_{1}^{\prime}\lambda_{0}$ and
$F_{-U}^{-1}(\tau_{2})<-z_{1}^{\prime}\lambda-\alpha_{0}<-\tilde{z}_{1}^{\prime}\lambda_{0}$.
This violates the above requirement, and thus, shows that Vuong and Xu (2017,
Assumption C’(ii)) implies (2.12). On the other hand, if
$z_{1}^{\prime}\lambda_{0}+\alpha_{0}=\tilde{z}_{1}^{\prime}\lambda_{0}$, then
Vuong and Xu (2017, Assumption C’(ii)) holds trivially.

$\displaystyle
z_{1}^{\prime}\lambda_{0}+\alpha_{0}=\tilde{z}_{1}^{\prime}\lambda_{0}.$
(2.12)

Several remarks are in order. First, note that sufficient support conditions
for the restrictions (2.10)–(2.12) are
$d(Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}-Z^{\prime}\delta_{0}\gamma_{0})\geq|\alpha_{0}|$,
$d(Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}|Z^{\prime}\delta_{0})\geq|\alpha_{0}|$,
and $d(Z_{1}^{\prime}\lambda_{0}|Z^{\prime}\delta_{0})\geq|\alpha_{0}|$ with a
positive probability, respectively. These three support conditions are such
that

$d(Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}-Z^{\prime}\delta_{0}\gamma_{0})\geq
d(Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}|Z^{\prime}\delta_{0})\geq
d(Z_{1}^{\prime}\lambda_{0}|Z^{\prime}\delta_{0}),$

where the first and second inequalities are strict if $Z_{2}$ and $Z_{3}$ have
at least one continuous component, respectively. Importantly, we show in
Section A of the Supplement that for a version of the triangular binary model
with univariate $Z_{2}$ and $Z_{3}$ and no common regressor $Z_{1}$, the
support condition
$d(Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}|Z^{\prime}\delta_{0})\geq|\alpha_{0}|$
is actually also necessary to the identification of the model without factor
structure. This implies that by imposing our factor structure, one can
identify values of $\alpha_{0}$ in a region that cannot be identified in the
model considered by Vytlacil and Yildiz (2007). Such region is characterized
in Section A of the Supplement.

Second, it directly follows from these support conditions that, in the
presence of a factor model and in contrast to both Vytlacil and Yildiz (2007)
and Vuong and Xu (2017), variation in $Z_{2}$ helps in the identification of
$\alpha_{0}$. In that sense, the factor model allows to restore the intuition
from standard IV approaches in linear models that variation in the instrument
$Z_{2}$ is critical to the identification of the parameters of the outcome
equation. Related to this, the support of $Z_{2}$ plays an important role in
our identification analysis. In particular, if $Z_{2}$ is discrete, our
identification strategy requires sufficient variation in the variables in the
outcome equation, namely $Z_{1}$ and $Z_{3}$. In this case, our support
requirement is equivalent to that assumed by Vytlacil and Yildiz (2007).

Third, another important aspect of Assumption A4 is that it does not impose
any constraint on the variables from the outcome equation. Specifically,
consider a case where the outcome equation does not contain a variable that is
excluded from the selection equation (i.e., $\beta_{0}=0$), the regressor that
is common to both equations, $Z_{1}$, is scalar and binary, and where
$\lambda_{0}=1$. In this case, one can show that the identifying support
conditions associated with Vytlacil and Yildiz (2007) (2.11) and Vuong and Xu
(2017) (2.12) generally fail to hold, except for a finite set of values
$\alpha_{0}\in\\{-1,0,1\\}$. In contrast, our support restriction (2.10) holds
under more general conditions: without any restriction on $\alpha_{0}$ if one
element of $Z_{2}$ is continuous with large support, and on a continuum of
possible values for $\alpha_{0}$ if one element of $Z_{2}$ is continuous with
bounded support. In that sense, the factor structure replaces the need for a
continuous component in $(Z_{1},Z_{3})$ in the outcome equation.

Finally, at a high level, our identification strategy shares similarities with
the Local Instrumental Variable (LIV) approach that has been proposed by
Heckman and Vytlacil (2005) and further discussed by Carneiro and Lee (2009).
In particular, our identifying restriction (2.1) can be alternatively derived
from a local IV strategy applied to a potential outcomes model characterized
by $Y_{1}(y_{2})={\bf
1}\\{Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}+\alpha_{0}y_{2}-U>0\\}$,
with treatment given by $Y_{2}={\bf 1}\\{Z^{\prime}\delta_{0}-V>0\\}$. In
contrast to the LIV literature though, we focus in our analysis on the
structural parameter $\alpha_{0}$ rather than on the marginal treatment
effects. Our identification result shows that, by leveraging the identifying
power of the factor structure, one can identify $\alpha_{0}$ under weaker
support restrictions than in the prior literature. In particular, our strategy
makes it possible to use variation in $X=Z^{\prime}\delta_{0}$ to identify
$\alpha_{0}$, even when all the components of $Z_{1}$ and $Z_{3}$ are
discrete.888An alternative approach to identifying this parameter can be found
in Lewbel (2000). In his approach a second equation to model the endogenous
variable is not needed, nor is the factor structure we impose. However, he
imposes a strong support condition on a variable like $Z_{3}$ requiring that
it exceeds the length of the unobservable $U$.

##  3 Extended Factor Structure in the presence of Continuous Measurements

Up until now we have proposed identification and estimation results for a
triangular system with a particular factor structure. A disadvantage of this
structure is that it only includes one idiosyncratic shock ($\Pi$). We
consider below an extension that addresses this limitation.

Namely, we consider the following model:

$\displaystyle Y_{1}=\mathbf{1}\\{X_{1}+\alpha_{0}Y_{2}-U\geq 0\\}$ (3.1)
$\displaystyle Y_{2}=\mathbf{1}\\{X-V\geq 0\\},$

where $X_{1}=Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}$,
$X=Z^{\prime}\delta_{0}$, $U=\gamma_{0}W+\eta_{1}$, $V=W+\eta_{2}$, and
$(W,\eta_{1},\eta_{2})$ are mutually independent. Recall that, following the
arguments in Section 2.1 above, we assume that $X$ is observed. In addition,
we assume two auxiliary continuous measurements

$\displaystyle Y_{3}=\nu_{0}W+\eta_{3}$ (3.2) $\displaystyle
Y_{4}=\sigma_{0}W+\eta_{4},$ (3.3)

where $(W,\eta_{1},\eta_{2},\eta_{3},\eta_{4})$ are mutually independent, and
$\nu_{0}\neq 0.$

Our identification result is based on the following assumptions:

B0

The first coefficient of $\lambda_{0}$ is normalized to one so that
$\lambda_{0}=(1,\lambda_{0,-1}^{T})^{T}$. The parameter
$\theta_{0}\equiv(\alpha_{0},\gamma_{0},\lambda_{0,-1},\beta_{0},\nu_{0},\sigma_{0})$
is an element of a compact subset of $\Re^{d_{1}+d_{3}+3}$, where $d_{1}$ and
$d_{3}$ are the dimensions of $Z_{1}$ and $Z_{3}$, respectively. The vector of
unobservables in the outcome and selection equations
$(W,\eta_{1},\eta_{2},\eta_{3})$ are independently distributed of the vector
$(Z_{1},Z_{2},Z_{3})$. Both $\eta_{1}$ and $\eta_{2}$ are continuously
distributed.

B1

$\gamma_{0}\neq 0$. $X$ is continuously distributed with absolute continuous
density w.r.t. Lebesgue measure over the whole real line, conditionally on
$Z_{1}$ and $Z_{3}$. The unconditional density of $X$ is bounded and bounded
away from zero on any compact subset of its support.

B2

$W$ is not normally distributed or both $\eta_{3}$ and $\eta_{4}$ do not have
a Gaussian component.

B3

$E(\eta_{3})=E(\eta_{4})=0$, $E(|\eta_{3}|)<\infty$, and
$E(|\eta_{4}|)<\infty$.

B4

$E(\exp(i\zeta\eta_{2}))$, $E(\exp(i\zeta\eta_{3}))$, and
$E(\exp(i\zeta\eta_{4}))$ do not vanish for any $\zeta\in\Re$, where
$i=\sqrt{-1}$.

B5

$E(\exp(i\zeta W))\neq 0$ for all $\zeta$ in a dense subset of $\Re$.

B6

The distributions of $W$, $\eta_{2}$, and $\eta_{3}$ admit uniformly bounded
densities $f_{W}(\cdot)$, $f_{\eta_{2}}(\cdot)$, and $f_{\eta_{3}}(\cdot)$
with respect to the Lebesgue measure that are supported on an interval (which
may be infinite), respectively.

B7

Let $Z_{1,-1}$ be all the coordinates of $Z_{1}$ except the first one, and
$d=d_{1}+d_{3}+1$. There exist $2d$ vectors
$\\{z_{1}^{(l)},z_{3}^{(l)}\\}_{l=1}^{d}$ and
$\\{\tilde{z}_{1}^{(l)},\tilde{z}_{3}^{(l)}\\}_{l=1}^{d}$ in the joint support
of $(Z_{1},Z_{3})$ and
$\\{w^{(l)}\\}_{l=1}^{d},\\{\tilde{w}^{(l)}\\}_{l=1}^{d}$ such that

$\displaystyle\alpha_{0}+(z_{1,-1}^{(l)}-\tilde{z}_{1,-1}^{(l)})^{\prime}\lambda_{0,-1}+(z_{3}^{(l)}-\tilde{z}_{3}^{(l)})^{\prime}\beta_{0}-\gamma_{0}(w^{(l)}-\tilde{w}^{(l)})=\tilde{z}_{1,1}^{(l)}-z_{1,1}^{(l)},~{}l=1,\cdots,d$

and $\text{rank}(\mathcal{M})=d$, where

$\displaystyle\mathcal{M}=\begin{pmatrix}1&\cdots&1\\\
z_{1,-1}^{(1)}-\tilde{z}_{1,-1}^{(1)}&\cdots&z_{1,-1}^{(d)}-\tilde{z}_{1,-1}^{(d)}\\\
z_{3}^{(1)}-\tilde{z}_{3}^{(1)}&\cdots&z_{3}^{(d)}-\tilde{z}_{3}^{(d)}\\\
w^{(1)}-\tilde{w}^{(1)}&\cdots&w^{(d)}-\tilde{w}^{(d)}\\\ \end{pmatrix}.$

We now discuss these assumptions, before turning to the identification result.
First, Assumption B0 is similar to Assumptions A1 and A2. We only need one of
the idiosyncratic errors in the continuous measurements to be independent of
the covariates because the other one is used to identify the distribution of
the common factor $W$ only. Second, as we assume in Assumption B1 that
$\gamma_{0}\neq 0$ and $X$ has full support, the support condition

$d(Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}-\gamma_{0}X)\geq|\alpha_{0}|.$

holds automatically. The full support condition of $X$ is necessary to
identify the density of $V$, which is further used to identify the
distribution of $\eta_{2}$. Assumption B1 reinforces this condition by
supposing that $X$ has full support conditional on $Z_{1}$ and $Z_{3}$, which
is needed to identify the parameters from the outcome equation in a second
step. Since $X=Z^{\prime}\delta_{0}$ with $Z=(Z_{1},Z_{2})$, this is in turn
equivalent to $Z_{2}$ having full support conditional on $Z_{1}$ and $Z_{3}$.
Third, Assumptions B2–B6 imply Assumptions 1 to 4 in Hu and Schennach (2013).
In practice we add the condition that the characteristic function of
$\eta_{2}$ does not vanish, which is used for the deconvolution arguments in
the proof of Theorem 3.1. We refer the reader to Hu and Schennach (2013) for
more discussions of these assumptions.999Note that Hu and Schennach (2013,
Assumptions 5 and 6) hold automatically in our model with $\nu_{0}\neq 0.$

######  Theorem 3.1.

If (G.31)–(3.3) and Assumptions B0–B7 hold, then $\theta_{0}$ are identified.

The proof of Theorem 3.1 can be found in Section E of the Supplement. Several
remarks are in order. First, while we allow for a more general factor
structure on the unobservables $U$ and $V$, we also depart from our baseline
specification by supposing that we have access to two continuous noisy
measurements of the common factor $W$. This is a standard requirement in the
nonparametric measurement error literature (Hu and Schennach, 2008). Besides,
assuming access to a set of (selection-free) noisy measurements of the
unobserved factors is also very standard in the evaluation literature. See,
among many others, Carneiro, Hansen, and Heckman (2003), Heckman and Navarro
(2007), Heckman and Vytlacil (2007a), and Cunha, Heckman, and Schennach
(2010). For instance, in applications in labor economics, the unobserved
factor $W$ often captures individual ability. This would apply, for example,
to the evaluation of the effect of college employment ($Y_{2}$) on college
graduation ($Y_{1}$). In these cases, cognitive skill measurements, such as
the ASVAB test components that are available in the NLSY79 and NLSY97 surveys,
are natural and often used candidates for these types of continuous
measurements (Ashworth, Hotz, Maurel, and Ransom, 2020).

Second, as is clear from the proof of Theorem 3.1, the key purpose of the
continuous measurements is to identify the distribution of the common factor
$W$. While we assume in this section that the measurement equations are
linear, it is possible to identify $\theta_{0}$ with a more general nonlinear
system of continuous measurements, provided that the researcher has access to
at least three such measurements. One can then combine Theorem 2 in Cunha,
Heckman, and Schennach (2010) (Section 3.3, pp. 894-895), that yields
identification of the distribution of $W$, with the proof of Theorem 3.1 in
order to show identification of $\theta_{0}$ for the case of nonlinear
auxiliary measurements. Assuming access to a set of at least three
measurements also makes it possible to relax the non-normality requirement
imposed in Assumption B2.

Third, similar to the earlier discussions in Remark 2.2 and Section 2.2,
Assumption B7 may still hold even when $Z_{3}$ is an empty set and $Z_{1}$ is
discrete, since $W$ is assumed to have full support. In such a case,
identification primarily relies on the factor structure and the variation of
the covariates in the selection equation, rather than that in the outcome
equation. In this respect, this identification result is similar in spirit to
Theorem 2.1 and different from the existing identification results in the
literature for triangular binary models, e.g., Vytlacil and Yildiz (2007) and
Vuong and Xu (2017). More generally, in Section A.2 in the supplement we
establish that the factor model provides identification restrictions that are
not otherwise available.101010Specifically, we consider a version of the model
(G.31), where we do not impose the factor structure and allow for an arbitrary
(unknown to econometricians) dependence structure across the unobservables of
the model. In this case, we show non-identification of $\alpha_{0}$ as long as
$|\alpha_{0}|>b-a$, where $[a,b]$ denotes the conditional support of $X_{1}$
given $X$ and, consistent with our Assumption B1, $X$ has full support on the
real line. However, by imposing the factor structure (and other conditions
implied by B0–B7), Theorem 3.1 shows that $\alpha_{0}$ is identified for this
model even when $|\alpha_{0}|>b-a$.

##  4 Conclusion

In this paper, we explore the identifying power of linear factor structures in
the context of simultaneous binary response models. We impose two alternative
types of factor structures on the unobservables of the model. The first setup
is a natural distribution-free extension of the bivariate Probit model, while
the second model corresponds to a standard linear factor model with one common
factor and two equation-specific idiosyncratic shocks. We establish that both
factor models have identifying power in that they make it possible to relax
some of the exclusion and support conditions typically required for
identification in this class of models (Vytlacil and Yildiz, 2007). Overall,
our analysis complements results obtained by Bai and Ng (2010) in the context
of a linear regression model with endogenous regressors, and, more generally,
adds to our understanding of the identifying power of factor models, beyond
their well known usefulness to recover the joint distribution of potential
outcomes from the marginal distributions.

The work here opens areas for future research. The factor structure we assume
could prove useful in more general nonlinear models. For instance, non-
triangular discrete systems have shown to be an effective way to model entry
games in the empirical industrial organization literature- see, for example,
Tamer (2003). However, as shown in Khan and Nekipelov (2018), identification
of structural parameters in these models can be even more challenging than for
the triangular model considered in this paper. It would be useful to determine
if factor structures on the unobservables could alleviate this problem. We
leave this open question to future work.

Supplement to “Informational Content of Factor Structures in
Simultaneous Binary Response Models”

##  Appendix A Identification with and without Factor Structure

###  A.1 Identification Without Auxiliary Measurements

In this section, we discuss the information content of factor structure. For
illustration purpose, we focus on the “condensed” model:

$\displaystyle Y_{1}=\mathbf{1}\\{X_{1}+\alpha_{0}Y_{2}-U\geq 0\\}$ (A.1)
$\displaystyle Y_{2}=\mathbf{1}\\{X-V\geq 0\\}.$

######  Assumption 1.

1\. $(X_{1},X)\perp(U,V)$.
2\. $(X_{1},X)$ are continuously distributed with absolute continuous joint
density w.r.t. Lebesgue measure. The conditional support of $X_{1}$ given $X$
is $[a,b]$.
3\. $V$ is continuously distributed over $\Re$ and its density w.r.t. Lebesgue
measure exist.

######  Theorem A.1.

If Assumption 1 holds, then $|\alpha_{0}|\leq b-a$ is necessary and sufficient
for $\alpha_{0}$ to be identified.

We note that under Assumption 1, $|\alpha_{0}|\leq b-a$ is equivalent to the
fact that we can find $x_{1}$ and $\tilde{x}_{1}$ in the support of $X_{1}$
such that $\alpha_{0}=x_{1}-\tilde{x}_{1}$.

Next, we assume, in addition to Assumption 1, the factor structure, i.e.,
(2.6) in Section 2. Our rank estimator can be written as an M-estimator

$\hat{\theta}=\arg\max_{\theta}Q_{n}(\theta)\equiv\sum_{i\neq
j}\hat{g}_{i,j}(\theta)$

in which

$\displaystyle\hat{g}_{i,j}(\theta)$ $\displaystyle=$
$\displaystyle[\mathbf{1}\\{\partial_{2}\hat{P}^{11}(X_{1,i},X_{i})/\hat{f}_{V}(X_{i})+\partial_{2}\hat{P}^{10}(X_{1,j},X_{j})/\hat{f}_{V}(X_{j})\geq
0\\}\mathbf{1}\\{\Phi(X_{1,i},X_{i},X_{1,j},X_{j};\theta)\geq 0\\}$
$\displaystyle+$
$\displaystyle\mathbf{1}\\{\partial_{2}\hat{P}^{11}(X_{1,i},X_{i})/\hat{f}_{V}(X_{i})+\partial_{2}\hat{P}^{10}(X_{1,j},X_{j})/\hat{f}_{V}(X_{j})<0\\}\mathbf{1}\\{\Phi(X_{1,i},X_{i},X_{1,j},X_{j};\theta)<0\\}],$

with

$\Phi(x_{1},x,\tilde{x}_{1},\tilde{x};\theta)=x_{1}+\alpha-\gamma
x-(\tilde{x}_{1}-\gamma\tilde{x}).$

We will study the asymptotic properties of this estimator in Section B.

The information content explored by the M-estimator can be summarized as
follows:

$\displaystyle\mathcal{A}_{2}(\theta)=\\{(X_{1},\tilde{X}_{1},X,\tilde{X}),$
$\displaystyle\Phi(X_{1},X,\tilde{X}_{1},\tilde{X};\theta_{0})\geq
0>\Phi(X_{1},X,\tilde{X}_{1},\tilde{X};\theta)$ $\displaystyle\mbox{ or
}\Phi(X_{1},X,\tilde{X}_{1},\tilde{X};\theta_{0})<0\leq\Phi(X_{1},X,\tilde{X}_{1},\tilde{X};\theta)\\}.$

Then we cannot distinguish, from the true parameter $\theta_{0}$, all
impostors in

$\overline{\mathcal{A}}_{2}=\\{\theta:P(\mathcal{A}_{2}(\theta))=0\\}.$

In the condensed model, if $\text{Supp}(X_{1},X)=[a,b]\times[c,d]$, then
$\theta_{0}$ is identified if $|\alpha_{0}|<b-a+|\gamma_{0}|(d-c)$. Recall
Theorem A.1, without imposing factor structure, the necessary and sufficient
condition for achieving identification is $|\alpha_{0}|\leq b-a$. Therefore,
the blue area in the Figure below is the additional parts of parameter space
that are identified with factor structure but not otherwise.

$\alpha_{0}$$\gamma_{0}$$|\alpha_{0}|=b-a+|\gamma_{0}|(d-c)$$|\alpha_{0}|=b-a$
Figure 1: Identifying Power of Factor Structure

######  Theorem A.2.

Assumption 1 holds. When $|\alpha_{0}|>b-a$, the sharp identified set for
$\alpha_{0}$ is

$\mathcal{A}^{*}=\\{\alpha:\alpha>b-a\mbox{ if }\alpha_{0}>0\mbox{ and
}\alpha<a-b\mbox{ if }\alpha_{0}<0\\}.$

Theorem A.2 highlights that, in the case without the factor structure and
$\alpha_{0}$ does not satisfy the parameter restriction, except for the fact
that the sign of $\alpha_{0}$ is identified, we actually cannot say much about
the value of $|\alpha_{0}|$. When we assume the factor structure, the
parameter is still not identified if $|\alpha_{0}|>b-a+|\gamma_{0}|(d-c)$. In
addition, suppose $\alpha_{0}>0$. In this case, if we do not impose factor
structure, by Theorem A.2, the sharp identified set is
$\\{\alpha:\alpha>b-a\\}$ while with the factor structure, the identified set
(not necessarily sharp) is $\alpha>b-a+|\gamma|(d-c)$. This implies, when
identification fails in both cases, the blue area is also the extra
identifying power on the identified set given by the factor structure.

###  A.2 Identification with two auxiliary measurements

Next, we expand our condensed model to include two continuous measurements. We
show in this case, without the factor structure, $\alpha_{0}$ is not
identified. This is in contrast with the identification result established in
Theorem 3.1.

Suppose in addition to (A.1), we also observe two continuous measurements
denoted as $Y_{3}$ and $Y_{4}$.

######  Assumption 2.

1\. $(X_{1},X)\perp(U,V,Y_{3},Y_{4})$.
2\. $(X_{1},X)$ are continuously distributed with absolute continuous joint
density w.r.t. Lebesgue measure. The conditional support of $X_{1}$ given $X$
is $[a,b]$.
3\. $V$ is continuously distributed over $\Re$ and its density w.r.t. Lebesgue
measure exist.

######  Theorem A.3.

If Assumption 2 holds, then $|\alpha_{0}|\leq b-a$ is necessary and sufficient
for $\alpha_{0}$ to be identified.

The proof of Theorem A.3 is similar to that of Theorem A.1, and thus, is
omitted. In the proof of Theorem A.1, we show that when $|\alpha_{0}|>b-a$, we
can find an impostor $\alpha\neq\alpha_{0}$ and $\tilde{U}$ such that for any
$x_{1}\in[a,b]$ and any $v\in\text{Supp}(V)$, we have

$\displaystyle P(\tilde{U}\leq x_{1}+\alpha|V=v)=P(U\leq
x_{1}+\alpha_{0}|V=v)$ $\displaystyle P(\tilde{U}\leq x_{1}|V=v)=P(U\leq
x_{1}|V=v).$

This implies the conditional CDF of $(Y_{1},Y_{2})$ given $(X_{1},X)$ under
the DGPs $(U,V,\alpha_{0})$ and $(\tilde{U},V,\alpha)$ are the same, and thus,
$\alpha_{0}$ is observationally equivalent to the impostor $\alpha$.
Similarly, with the two continuous measurements, we can use the exact same
construction of $\tilde{U}$ and $\alpha$ to show that, for any $x_{1}\in[a,b]$
and $(v,y_{3},y_{4})\in\text{Supp}(V,Y_{3},Y_{4})$, we have

$\displaystyle P(\tilde{U}\leq
x_{1}+\alpha|V=v,Y_{3}=y_{3},Y_{4}=y_{4})=P(U\leq
x_{1}+\alpha_{0}|V=v,Y_{3}=y_{3},Y_{4}=y_{4})$ $\displaystyle P(\tilde{U}\leq
x_{1}|V=v,Y_{3}=y_{3},Y_{4}=y_{4})=P(U\leq
x_{1}|V=v,Y_{3}=y_{3},Y_{4}=y_{4}).$

This implies the conditional CDF of $(Y_{1},Y_{2},Y_{3},Y_{4})$ given
$(X_{1},X)$ under the DGPs $(U,V,Y_{3},Y_{4},\alpha_{0})$ and
$(\tilde{U},V,Y_{3},Y_{4},\alpha)$ are the same too. Such non-identification
result holds even when $X$ has full support.

##  Appendix B Estimation and Asymptotic Properties

Our identification result is constructive in the sense that it motivates an
estimator for the parameters of interest which we describe in detail here.

As we did in Section A, to simplify exposition, in the following we focus
exclusively on the parameters $\alpha_{0},\gamma_{0}$. Recall the choice
probabilities $P^{ij}(x_{1},x)=Prob(Y_{1}=i,Y_{2}=j|X_{1}=x_{1},X=x)$ and its
second derivative $\partial_{2}P^{ij}(x_{1},x)$, which can be estimated as we
describe below. Another function needed for our identification result is the
density function of the unobserved term $V$, denoted by $f_{V}(\cdot)$. This
is also unknown, but from the structure of our model can be recovered from the
derivative with respect to the instrument $X$ of $E[Y_{2}|X]$, and hence is
estimable from the data. Note that the proof of Theorem 2.1 shows that the
sign of the index evaluated at two different regressor values, which we denote
here by $(X_{1},X)$ and $(\tilde{X}_{1},\tilde{X})$ is determined by the
choice probabilities via

$\partial_{2}P^{11}(X_{1},X)/f_{V}(X)+\partial_{2}P^{10}(\tilde{X}_{1},\tilde{X})/f_{V}(\tilde{X})\geq
0\ \ \ \iff X_{1}+\alpha-\gamma X-(\tilde{X}_{1}-\gamma\tilde{X})\geq 0.$

This motivates us to use the maximum rank correlation estimator proposed by
Han (1987).

Implementation requires further details to pay attention to. The unknown
choice probabilities, their derivatives, and the density of $V$ will be
estimated using nonparametric methods, and for this we adopt locally linear
methods as they are particularly well suited for estimating derivatives of
functions.

With functions and their derivatives estimated in the first stage of our
procedure, the second stage plugs in these estimated values into an objective
function to be optimized. Specifically, letting $\hat{\theta}$ denote
$(\hat{\alpha},\hat{\gamma})$, our estimator is of the form:

$\hat{\theta}=\arg\max_{\theta}Q_{n}(\theta),\quad
Q_{n}(\theta)\equiv\sum_{i\neq j}\hat{g}_{i,j}(\theta)$ (B.2)

in which

$\displaystyle\hat{g}_{i,j}(\theta)$ $\displaystyle=$
$\displaystyle[\mathbf{1}\\{\partial_{2}\hat{P}^{11}(X_{1,i},X_{i})/\hat{f}_{V}(X_{i})+\partial_{2}\hat{P}^{10}(X_{1,j},X_{j})/\hat{f}_{V}(X_{j})\geq
0\\}\mathbf{1}\\{\Phi(X_{1,i},X_{i},X_{1,j},X_{j};\theta)\geq 0\\}$
$\displaystyle+$
$\displaystyle\mathbf{1}\\{\partial_{2}\hat{P}^{11}(X_{1,i},X_{i})/\hat{f}_{V}(X_{i})+\partial_{2}\hat{P}^{10}(X_{1,j},X_{j})/\hat{f}_{V}(X_{j})<0\\}\mathbf{1}\\{\Phi(X_{1,i},X_{i},X_{1,j},X_{j};\theta)<0\\}],$

with

$\Phi(x_{1},x,\tilde{x}_{1},\tilde{x};\theta)=x_{1}+\alpha-\gamma
x-(\tilde{x}_{1}-\gamma\tilde{x}).$

We note that this estimator falls into the class of those which optimize a
nonsmooth U-process involving components estimated nonparametrically in a
preliminary stage.111An alternative estimation procedure could be based on the
exact relationship in (2.1). Note the equality on the left-hand side of (2.1)
is a function of the data alone and not the unknown parameters. The right-hand
side equality can then be regarded as a moment condition to estimate the
unknown parameters. We describe this estimator and derive its asymptotic
properties in the Online Supplement to the paper. While the two estimation
approaches will have similar asymptotic properties (root-$n$ consistent,
asymptotically normal), we prefer the rank estimator in (B.2) which involves
fewer tuning parameters. Furthermore rank type estimators in general are more
robust to certain types of misspecification, as pointed out in Khan and Tamer
(2018). Examples of other estimators in this class can be found in Khan
(2001), Abrevaya, Hausman, and Khan (2010), Jochmans (2013), Chen, Khan, and
Tang (2016), and our approach to deriving the limiting distribution theory of
our estimator will follow along the steps used in those papers. Our limiting
distribution theory for this estimator is based on the following regularity
conditions:

RK1

$\theta_{0}$ lies in the interior of $\Theta$, a compact subset of $R^{2}$.

RK2

The index $X$ is continuously distributed with support on the real line, and
has a density function which is twice continuously differentiable.

RK3

(Order of smoothness of probability functions and regressor density functions)
The functions $P^{i,j}(\cdot)$ and $f_{X_{1},X}(\cdot.\cdot)$ (the density
function of the random vector $(X_{1},X)$) are continuously differentiable of
order $p_{2}$.

RK4

(First stage kernel function conditions) $K(\cdot)$, used to estimate the
choice probabilities and their derivatives is an even function, integrating to
1 and is of order $p_{2}$.

RK5

(Rate condition on first stage bandwidth sequence) The first stage bandwidth
sequence $H_{n}$ used in the nonparametric estimator of the choice probability
functions and their derivatives satisfies $\sqrt{n}H_{n}^{p_{2}-1}\rightarrow
0$ and $n^{-1/4}H_{n}^{-1}\rightarrow 0$.

The smoothness condition in Assumption RK4 and Assumption RK5 is due to the
fact that we need to nonparametrically estimate $\partial_{2}P^{ij}(X_{1},X)$
with sufficiently faster convergence rate. This will require a stronger
smoothness condition than that required for standard nonparametric estimation.
Assumption RK5 ensures that the bias of the first stage estimator of the
derivative function converges at the parametric rate and the RMSE of this
estimator (with two regressors) is fourth-root consistent, so results for two
step estimation in Newey and McFadden (1994) can be applied.

Based on these conditions, we have the following theorem, whose proof is in
Section F of the Supplementary Appendix which characterizes the rate of
convergence and asymptotic distribution of the proposed estimator:

######  Theorem B.1.

Under Assumptions RK1-RK5,

$\sqrt{n}(\hat{\theta}-\theta_{0})\Rightarrow N(0,V^{-1}\Delta V^{-1})$ (B.3)

where the forms of the Hessian term $V$ and outer score term $\Delta$ are
described in detail in Section F of the Supplementary Appendix.

##  Appendix C Finite Sample Properties

In this section we explore the finite sample properties of the proposed
estimation procedure via a simulation study. We will also see how sensitive
the performance of the proposed estimator is to the factor structure
assumption. As a base comparison, we also report results for the estimator
proposed in Vytlacil and Yildiz (2007) to see how sensitive it is to their
second instrument restriction.

Our data are simulated from base models of the form

$\displaystyle Y_{1}$ $\displaystyle={\bf 1}\\{X_{1}+\alpha_{0}Y_{2}-U\geq
0\\}$ (C.1) $\displaystyle Y_{2}$ $\displaystyle={\bf 1}\\{X-V>0\\},$ (C.2)

where $X_{1}$ is binary with success probability 0.6, $X$ has marginal
distribution $\mathcal{N}(0,1)$, $X_{1}$ and $X$ are mutually independent,
$(X_{1},X)\perp(V,\Pi)$, $U=\gamma_{0}V+\Pi$. $(V,\Pi)$ are distributed
independently of each other, where $\Pi$ is distributed following a standard
normal distribution, and $V$ is distributed either standard normal, Laplace,
or $T(3)$. The parameters $(\alpha_{0},\gamma_{0})=(-0.25,1.2)$ or
$(0.5,1.2)$.

Since $X_{1}$ is discrete, Vytlacil and Yildiz’s (2007) identification
condition does not hold. However, the identification condition in this paper
becomes

$|\alpha|\leq\mbox{length of the support of }X,$

which holds.

For each choice of sample size $n=100,200,400,800,1,600$, we simulate 280
samples and report the bias, standard deviation (std), root mean squared error
(RMSE), and median absolute deviation (MAD) for both Vytlacil and Yildiz’s
(2007) estimator (VY) and ours (KMZ). For implementation, we use the second
order local polynomial along with Gaussian kernels to nonparametrically
estimate the $\partial_{2}P^{11}(x_{1},x)$ and $\partial_{2}P^{10}(x_{1},x)$.
The bandwidth we use is $h_{1}=\sigma_{x}N^{-1/7}$ where $\sigma_{x}$ is the
standard deviation of $X$. $f_{V}(x)$ is nonparametrically estimated using a
local linear estimator with the tuning parameter $h_{2}=\sigma_{x}N^{-1/6}$.

As results from the table indicate, the finite sample performance of our
estimator generally agrees with the asymptotic theory. The RMSE for the
estimator proposed here is decreasing as the sample size increases, as one
could expect given the consistency property of our estimator. Besides, the
decay rate of the RMSE and MAD is about $\sqrt{2}$ when $n\geq 400$ as sample
sizes doubles, in line with the parametric rate of convergence of our
estimator.

Vytlacil and Yildiz’s (2007) estimator, which does not exploit the factor
structure, demonstrates inconsistency for certain parameter values, as
indicated by the bias and median bias not shrinking with the sample size. In
addition, the RMSE and MAD do not appear to decline at all, which also
suggests that Vytlacil and Yildiz’s (2007) estimator is inconsistent in these
designs.222Because $X_{1}$ is binary, Vytlacil and Yildiz’s (2007) estimator
can only take 3 possible values: 0, -1 or 1. In particular, when $\alpha=0.5$,
in most of the replications, the estimator takes values 0 or 1. When
$\alpha=-0.25$, in most of the replications, the estimator takes value -1. In
both of these cases, the MAD remains constant over the different sample sizes.

Table 1: Normal $V$, $\alpha=0.5$

$\Pi$ Normal Laplace T(3) kmz vy kmz vy kmz vy N Bias RMSE MAD Bias RMSE MAD
Bias RMSE MAD Bias RMSE MAD Bias RMSE MAD Bias RMSE MAD 100 -0.026 0.665 0.660
-0.246 0.658 0.500 0.032 0.634 0.560 -0.293 0.658 0.500 0.010 0.676 0.665
-0.225 0.662 0.500 200 0.004 0.591 0.475 -0.329 0.633 0.500 -0.015 0.568 0.400
-0.336 0.612 0.500 -0.003 0.616 0.495 -0.279 0.629 0.500 400 0.005 0.483 0.365
-0.341 0.573 0.500 0.030 0.459 0.310 -0.323 0.559 0.500 0.018 0.542 0.405
-0.314 0.589 0.500 800 0.065 0.456 0.300 -0.348 0.544 0.500 0.096 0.391 0.250
-0.357 0.511 0.500 0.046 0.462 0.295 -0.346 0.552 0.500 1,600 0.040 0.321
0.195 -0.413 0.503 0.500 0.017 0.294 0.190 -0.450 0.506 0.500 0.034 0.371
0.240 -0.368 0.506 0.500

Table 2: Normal $V$, $\alpha=-0.25$

$\Pi$ Normal Laplace T(3) kmz vy kmz vy kmz vy N Bias RMSE MAD Bias RMSE MAD
Bias RMSE MAD Bias RMSE MAD Bias RMSE MAD Bias RMSE MAD 100 -0.088 0.650 0.555
-0.466 0.710 0.750 0.092 0.614 0.530 -0.358 0.650 0.750 0.004 0.619 0.505
-0.430 0.681 0.750 200 -0.035 0.599 0.420 -0.446 0.681 0.750 0.012 0.552 0.385
-0.485 0.689 0.750 -0.008 0.583 0.425 -0.463 0.687 0.750 400 -0.016 0.467
0.325 -0.487 0.668 0.750 -0.010 0.388 0.200 -0.552 0.686 0.750 -0.003 0.496
0.340 -0.489 0.675 0.750 800 -0.028 0.324 0.165 -0.591 0.697 0.750 0.006 0.279
0.180 -0.599 0.701 0.750 0.032 0.399 0.230 -0.533 0.682 0.750 1,600 -0.006
0.244 0.150 -0.654 0.718 0.750 -0.028 0.204 0.130 -0.714 0.738 0.750 -0.021
0.279 0.190 -0.629 0.710 0.750

In the following, we also consider three DGPs (DGPs 1–3) such that the one-
factor model does not hold but the identification assumption in Vytlacil and
Yildiz (2007) does. In this case, our simulation results show that while, as
expected, the estimator VY is still valid, our estimator still performs
reasonably well. Interestingly, this offers suggestive evidence that our
estimator is robust to some degree of misspecification. As such, these results
complement previous work highlighting the robustness of rank type estimators
to misspecification Khan and Tamer (2018). In DGP 4, the identification
assumptions in both Vytlacil and Yildiz (2007) and our paper hold. In this
case, we found that our estimator has similar performance as that proposed by
Vytlacil and Yildiz (2007).

The outcome and selection equations are the same as (C.1) and (C.2),
respectively. Then,

  1. DGP 1

: $(X_{1},X)$ is jointly standard normally distributed. Let $(e_{1},e_{2})$
jointly Laplace distributed with mean zero and variance-covariance matrix
$\Sigma=\begin{pmatrix}1&-0.5\\\ -0.5&1\end{pmatrix}$, $e_{3}$ and $e_{4}$ are
uniformly distributed on $(0,1)$, independent of each other, and independent
of $(e_{1},e_{2})$, $V=e_{1}+e_{3}-0.5$, $U=e_{2}+e_{4}-0.5$, and
$\alpha=-0.25$.

  2. DGP 2

: $(X_{1},X)$ are the same as above, $U=e_{1}+e_{2}-0.5$, and
$V=e_{1}+e_{3}-0.5$, where $e_{1}$ is standard normally distributed,
$(e_{2},e_{3})$ are uniformly distributed on $(0,1)$, $(e_{1},e_{2},e_{3})$
are mutually independent, and $\alpha=-0.25$.

  3. DGP 3

: $(X_{1},X)$ are the same as above, $V=\frac{\exp(e_{1}+e_{2}-0.5)-1}{4}$,
$U=\frac{\exp(e_{1}+e_{3}-0.5)-1}{4}$, $(e_{1},e_{2},e_{3})$ are defined as
above, and $\alpha=-0.5$.

  4. DGP 4

: $(X_{1},X)$ are the same as above, $V$ is Laplace distributed with mean zero
and standard derivation $0.5$, $U=V+V^{\prime}-0.5$, where $V^{\prime}$ is
uniform distributed on $(0,1)$ and is independent of $V$, and $\alpha=-0.25$.

For DGPs 1, 2, and 4, when computing $\partial_{2}P^{11}(x_{1},x)$ and
$\partial_{2}P^{10}(x_{1},x)$, we use bandwidths $h_{1}=\sigma_{x1}N^{-1/7}$
and $h=\sigma_{x}N^{-1/7}$ for variables $X_{1}$ and $X$, respectively, where
$\sigma_{x1}$ and $\sigma_{x}$ are the standard errors of $X_{1}$ and $X$,
respectively. To estimate the density $f_{V}(x)$, we use bandwidth
$h_{2}=\sigma_{x}N^{-1/6}$. For DGP 3, we use
$h_{1}=h_{2}=h=\sigma_{x1}N^{-1/5}$. In all simulations, we use 280
replications.

Table 3: Alternative DGPs

DGP 1 DGP 2 kmz vy kmz vy N Bias RMSE MAD Bias RMSE MAD Bias RMSE MAD Bias
RMSE MAD 100 -0.065 0.678 0.600 -0.055 0.666 0.535 -0.058 0.621 0.505 -0.05
0.621 0.470 200 -0.118 0.543 0.370 -0.080 0.497 0.320 -0.122 0.523 0.350
-0.097 0.495 0.350 400 -0.117 0.413 0.280 -0.071 0.378 0.245 -0.062 0.335
0.215 -0.033 0.316 0.220 800 -0.102 0.287 0.170 -0.062 0.243 0.160 -0.031
0.242 0.150 -0.008 0.215 0.150 1,600 -0.071 0.193 0.140 -0.035 0.155 0.100
-0.038 0.167 0.100 -0.031 0.158 0.100 DGP 3 DGP 4 100 -0.012 0.583 0.480
-0.015 0.565 0.430 -0.057 0.401 0.240 -0.066 0.422 0.240 200 -0.061 0.425
0.275 -0.068 0.399 0.270 -0.041 0.282 0.180 -0.049 0.263 0.145 400 -0.041
0.259 0.170 -0.042 0.237 0.155 -0.062 0.184 0.135 -0.047 0.186 0.120 800
-0.061 0.219 0.140 -0.047 0.182 0.120 -0.029 0.119 0.080 -0.034 0.115 0.070
1,600 -0.038 0.130 0.080 -0.035 0.119 0.080 -0.024 0.090 0.060 -0.022 0.086
0.070

In the first three DGPs, we see that VY’s estimator has better performance in
terms of both bias and MSE. On the other hand, although the models do not have
a factor structure, our estimator still performs reasonably well. In the last
DGP, support conditions in both Vytlacil and Yildiz (2007) and our paper hold.
Table 3 shows that our and Vytlacil and Yildiz’s (2007) estimators have
similar performance in terms of bias and MSE. Although our estimator is
expected to be more efficient as we use the factor structure in estimation, it
is not. We conjecture that it is because our estimator does not necessarily
use all the information, or in other words, achieve the semiparametric
efficiency bound. To establish the semiparametric efficient estimator in the
model with and without the factor structure is an interesting yet challenging
task. We leave it as a topic for future research.

##  Appendix D Proof of Theorem 2.1

Proof: Note that

$\displaystyle P^{11}(z_{1},z_{3},x)$
$\displaystyle=\int_{-\infty}^{x}F_{\Pi}(z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}+\alpha_{0}-\gamma_{0}v)f_{V}(v)dv$
$\displaystyle P^{10}(\tilde{z}_{1},\tilde{z}_{3},\tilde{x})$
$\displaystyle=\int^{+\infty}_{\tilde{x}}F_{\Pi}(\tilde{z}_{1}^{\prime}\lambda_{0}+\tilde{z}_{3}^{\prime}\beta_{0}-\gamma_{0}v)f_{V}(v)dv.$

Taking derivatives w.r.t. the third argument of the LHS function, we obtain

$\displaystyle\partial_{x}P^{11}(z_{1},z_{3},x)/f_{V}(x)$
$\displaystyle=F_{\Pi}(z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}+\alpha_{0}-\gamma_{0}x)$
$\displaystyle-\partial_{x}P^{10}(\tilde{z}_{1},\tilde{z}_{3},\tilde{x})/f_{V}(\tilde{x})$
$\displaystyle=F_{\Pi}(\tilde{z}_{1}^{\prime}\lambda_{0}+\tilde{z}_{3}^{\prime}\beta_{0}-\gamma_{0}\tilde{x}).$

By Assumption A4, we know that there exists pairs such that

$Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}+\alpha_{0}-\gamma_{0}X=\tilde{Z}_{1}^{\prime}\lambda_{0}+\tilde{Z}_{3}^{\prime}\beta_{0}-\gamma_{0}\tilde{X}.$

Because $F_{\Pi}(\cdot)$ is monotone increasing, we have

$\displaystyle\partial_{x}P^{11}(Z_{1},Z_{3},X)/f_{V}(X)+\partial_{x}P^{10}(\tilde{Z}_{1},\tilde{Z}_{3},\tilde{X})/f_{V}(\tilde{X})=0$
$\displaystyle\iff$
$\displaystyle\alpha_{0}+(Z_{1}-\tilde{Z}_{1})^{\prime}\lambda_{0}+(Z_{3}-\tilde{Z}_{3})^{\prime}\beta_{0}-\gamma_{0}(X-\tilde{X})=0$

Note the LHS of the above display is identified from data. Denote $Z_{1,1}$ as
the first element of $Z_{1}$, whose coefficient is set to one. The rest of
$Z_{1}$ is denoted as $Z_{1,-1}$, whose coefficient is denoted as
$\lambda_{0,-1}$. Then, we have

$\displaystyle\alpha_{0}+(Z_{1,-1}-\tilde{Z}_{1,-1})^{\prime}\lambda_{0,-1}+(Z_{3}-\tilde{Z}_{3})^{\prime}\beta_{0}-\gamma_{0}(X-\tilde{X})=\tilde{Z}_{1,1}-Z_{1,1}.$

Then, by Assumption A4, we can find
$(z_{1}^{(l)},z_{3}^{(l)},x^{(l)})_{l=1}^{d}$ and
$(\tilde{z}_{1}^{(l)},\tilde{z}_{3}^{(l)},\tilde{x}^{(l)})_{l=1}^{d}$ such
that

$\displaystyle\text{rank}\begin{pmatrix}1&\cdots&1\\\
z_{1,-1}^{(1)}-\tilde{z}_{1,-1}^{(1)}&\cdots&z_{1,-1}^{(d)}-\tilde{z}_{1,-1}^{(d)}\\\
z_{3}^{(1)}-\tilde{z}_{3}^{(1)}&\cdots&z_{3}^{(d)}-\tilde{z}_{3}^{(d)}\\\
x^{(1)}-\tilde{x}^{(1)}&\cdots&x^{(d)}-\tilde{x}^{(d)}\\\ \end{pmatrix}=d.$

Then, we can identify $(\alpha_{0},\lambda_{0},\beta_{0},\gamma_{0})$ by
solving the linear system that

$\displaystyle\alpha_{0}+(z_{1,-1}^{(1)}-\tilde{z}_{1,-1}^{(1)})^{\prime}\lambda_{0,-1}+(z_{3}^{(1)}-\tilde{z}_{3}^{(1)})^{\prime}\beta_{0}-\gamma_{0}(x^{(1)}-\tilde{x}^{(1)})=$
$\displaystyle\tilde{z}_{1,1}^{(1)}-z_{1,1}^{(1)},$ $\displaystyle\vdots$
$\displaystyle\alpha_{0}+(z_{1,-1}^{(d)}-\tilde{z}_{1,-1}^{(d)})^{\prime}\lambda_{0,-1}+(z_{3}^{(d)}-\tilde{z}_{3}^{(d)})^{\prime}\beta_{0}-\gamma_{0}(x^{(d)}-\tilde{x}^{(d)})=$
$\displaystyle\tilde{z}_{1,1}^{(d)}-z_{1,1}^{(d)}.$

This concludes the proof.

##  Appendix E Proof of Theorem 3.1

For notation simplicity, we write $\tilde{W}=\nu_{0}W$,
$\tilde{\sigma}_{0}=\sigma_{0}/\nu_{0}$, $\tilde{\nu}_{0}=1/\nu_{0}$, and

$\displaystyle Y_{2}=\mathbf{1}\\{X\geq\tilde{\nu}_{0}\tilde{W}+\eta_{2}\\}$
$\displaystyle Y_{3}=\tilde{W}+\eta_{3}$ $\displaystyle
Y_{4}=\tilde{\sigma}_{0}\tilde{W}+\eta_{4}.$

Because Assumptions B2–B6 hold, by applying Hu and Schennach (2013, Theorem 1)
to $Y_{3}$ and $Y_{4}$, we can identify the densities for
$\nu_{0}W=\tilde{W}$, $\eta_{3}$, and $\eta_{4}$ as well as
$\sigma_{0}/\nu_{0}=\tilde{\sigma}_{0}$.

Then, we have

$\displaystyle\partial_{y_{3}}\mathbb{P}(Y_{2}=1,Y_{3}\leq y_{3}|X=x)=$
$\displaystyle\partial_{y_{3}}\int
F_{\eta_{2}}(x-\tilde{\nu}_{0}w)F_{\eta_{3}}(y_{3}-w)f_{\tilde{W}}(w)dw$
$\displaystyle=$ $\displaystyle\int
F_{\eta_{2}}(x-\tilde{\nu}_{0}w)f_{\eta_{3}}(y_{3}-w)f_{\tilde{W}}(w)dw.$

Applying Fourier transform w.r.t. $y_{3}$ on both sides, we have

$\displaystyle\mathcal{F}(\partial_{y_{3}}\mathbb{P}(Y_{2}=1,Y_{3}\leq\cdot|X=x))(t)=\mathcal{F}(F_{\eta_{2}}(x-\tilde{\nu}_{0}\cdot)f_{\tilde{W}}(\cdot))(t)\mathcal{F}(f_{\eta_{3}}(\cdot))(t),$

where for a generic function $g(w)$,

$\displaystyle\mathcal{F}(g(\cdot))(t)=\frac{1}{\sqrt{2\pi}}\int\exp(-2\pi
itw)g(w)dw.$

Therefore,

$\displaystyle\frac{\mathcal{F}^{-1}\left(\frac{\mathcal{F}(\partial_{y_{3}}\mathbb{P}(Y_{2}=1,Y_{3}\leq\cdot|X=x))(\cdot)}{\mathcal{F}(f_{\eta_{3}}(\cdot))(\cdot)}\right)(w)}{f_{\tilde{W}}(w)}=F_{\eta_{2}}(x-\tilde{\nu}_{0}w),$
(E.1)

where for a generic function $g(w)$,

$\displaystyle\mathcal{F}^{-1}(g(\cdot))(t)=\frac{1}{\sqrt{2\pi}}\int\exp(2\pi
itw)g(w)dw.$

Note the LHS of (E.1) can be identified from data. We choose two pairs $(x,w)$
and $(x^{\prime},w^{\prime})$ such that $w\neq w^{\prime}$ and

$\displaystyle\frac{\mathcal{F}^{-1}\left(\frac{\mathcal{F}(\partial_{y_{3}}\mathbb{P}(Y_{2}=1,Y_{3}\leq\cdot|X=x))(\cdot)}{\mathcal{F}(f_{\eta_{3}}(\cdot))(\cdot)}\right)(w)}{f_{\tilde{W}}(w)}=\frac{\mathcal{F}^{-1}\left(\frac{\mathcal{F}(\partial_{y_{3}}\mathbb{P}(Y_{2}=1,Y_{3}\leq\cdot|X=x^{\prime}))(\cdot)}{\mathcal{F}(f_{\eta_{3}}(\cdot))(\cdot)}\right)(w^{\prime})}{f_{\tilde{W}}(w^{\prime})}.$

Then, given the monotonicity of $F_{\eta_{2}}$, we have

$\displaystyle x-\tilde{\nu}_{0}w=x^{\prime}-\tilde{\nu}_{0}w^{\prime},$

or

$\displaystyle\tilde{\nu}_{0}=(x-x^{\prime})/(w-w^{\prime}),$

which is identified. Given the identification of $\tilde{\nu}_{0}$ and the
distribution of $\tilde{W}$, we can identify the distribution of
$W=\tilde{\nu}_{0}\tilde{W}$. Recall $F_{\eta_{1}}(\cdot)$ and
$f_{\eta_{2}}(\cdot)$ are the CDF and PDF of $\eta_{1}$ and $\eta_{2}$,
respectively. Then, we have

$\displaystyle P(Y_{2}=1|X=x)=P(W+\eta_{2}\leq x).$

Because $X$ has full support, we can identify the distribution of
$W+\eta_{2}$. Then, it follows from standard deconvolution argument and the
fact that the distribution of $W$ is identified that we can identify the
distribution of $\eta_{2}$. In addition, note that

$\displaystyle P^{11}(z_{1},z_{3},x)=$ $\displaystyle
P(Y_{1}=1,Y_{2}=1|Z_{1}=z_{1},Z_{3}=z_{3},X=x)$ $\displaystyle=$
$\displaystyle\int
F_{\eta_{1}}(z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}+\alpha_{0}-\gamma_{0}w)F_{\eta_{2}}(x-w)f_{W}(w)dw$

and

$\displaystyle P^{10}(z_{1},z_{3},x)=$ $\displaystyle
P(Y_{1}=1,Y_{2}=0|Z_{1}=z_{1},Z_{3}=z_{3},X=x)$ $\displaystyle=$
$\displaystyle\int
F_{\eta_{1}}(z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}-\gamma_{0}w)(1-F_{\eta_{2}}(x-w))f_{W}(w)dw.$

Taking derivatives of $P^{11}(z_{1},z_{3},x)$ and $P^{10}(z_{1},z_{3},x)$
w.r.t. $x$, we have

$\displaystyle\partial_{x}P^{11}(z_{1},z_{3},x)=\int
F_{\eta_{1}}(z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}+\alpha_{0}-w)f_{\eta_{2}}(x-w)f_{W}(w)dw$
(E.2)

and

$\displaystyle-\partial_{x}P^{10}(z_{1},z_{3},x)=\int
F_{\eta_{1}}(z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}-\gamma_{0}w)f_{\eta_{2}}(x-w)f_{W}(w)dw.$
(E.3)

Applying Fourier transform on both sides of (E.2) and (E.3), we have

$\displaystyle\mathcal{F}(\partial_{x}P^{11}(z_{1},z_{3},\cdot))=\mathcal{F}(F_{\eta_{1}}(z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}+\alpha_{0}-\cdot)f_{W}(\cdot))\mathcal{F}(f_{\eta_{2}}(\cdot))$
(E.4)

and

$\displaystyle\mathcal{F}(-\partial_{x}P^{10}(z_{1},z_{3},\cdot))=\mathcal{F}(F_{\eta_{1}}(z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}-\cdot)f_{W}(\cdot))\mathcal{F}(f_{\eta_{2}}(\cdot)).$

Then, by (E.4), we can identify
$F_{\eta_{1}}(z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}+\alpha_{0}-\cdot)$
by

$\displaystyle
F_{\eta_{1}}(z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}+\alpha_{0}-\gamma_{0}\cdot)=\mathcal{F}^{-1}\left(\frac{\mathcal{F}(\partial_{x}P^{11}(z_{1},z_{3},\cdot))}{\mathcal{F}(f_{\eta_{2}}(\cdot))}\right)(\cdot)/f_{W}(\cdot).$

Similarly, we can identify

$\displaystyle
F_{\eta_{1}}(z_{1}^{\prime}\lambda_{0}+z_{3}^{\prime}\beta_{0}-\gamma_{0}\cdot)=\mathcal{F}^{-1}\left(\frac{\mathcal{F}(-\partial_{x}P^{10}(z_{1},z_{3},\cdot))}{\mathcal{F}(f_{\eta_{2}}(\cdot))}\right)(\cdot)/f_{W}(\cdot).$

Because $F_{\eta_{1}}(\cdot)$ is monotone increasing, we have

$\displaystyle\mathcal{F}^{-1}\left(\frac{\mathcal{F}(\partial_{x}P^{11}(z_{1},z_{3},\cdot))}{\mathcal{F}(f_{\eta_{2}}(\cdot))}\right)(w)/f_{W}(w)=\mathcal{F}^{-1}\left(\frac{\mathcal{F}(-\partial_{x}P^{10}(\tilde{z}_{1},\tilde{z}_{3},\cdot))}{\mathcal{F}(f_{\eta_{2}}(\cdot))}\right)(\tilde{w})/f_{W}(\tilde{w})$
$\displaystyle\iff$
$\displaystyle\alpha_{0}+(z_{1}-\tilde{z}_{1})^{\prime}\lambda_{0}+(z_{3}-\tilde{z}_{3})^{\prime}\beta_{0}-\gamma_{0}(w-\tilde{w})=0$

Then, by Assumption B7, we can find
$(z_{1}^{(l)},z_{3}^{(l)},w^{(l)})_{l=1}^{d}$ and
$(\tilde{z}_{1}^{(l)},\tilde{z}_{3}^{(l)},\tilde{w}^{(l)})_{l=1}^{d}$ such
that

$\displaystyle\text{rank}\begin{pmatrix}1&\cdots&1\\\
z_{1,-1}^{(1)}-\tilde{z}_{1,-1}^{(1)}&\cdots&z_{1,-1}^{(d)}-\tilde{z}_{1,-1}^{(d)}\\\
z_{3}^{(1)}-\tilde{z}_{3}^{(1)}&\cdots&z_{3}^{(d)}-\tilde{z}_{3}^{(d)}\\\
w^{(1)}-\tilde{w}^{(1)}&\cdots&w^{(d)}-\tilde{w}^{(d)}\\\ \end{pmatrix}=d.$

Then, we can identify $(\alpha_{0},\lambda_{0},\beta_{0},\gamma_{0})$ by
solving the linear system that

$\displaystyle\alpha_{0}+(z_{1,-1}^{(1)}-\tilde{z}_{1,-1}^{(1)})^{\prime}\lambda_{0,-1}+(z_{3}^{(1)}-\tilde{z}_{3}^{(1)})^{\prime}\beta_{0}-\gamma_{0}(w^{(1)}-\tilde{w}^{(1)})=$
$\displaystyle\tilde{z}_{1,1}^{(1)}-z_{1,1}^{(1)},$ $\displaystyle\vdots$
$\displaystyle\alpha_{0}+(z_{1,-1}^{(d)}-\tilde{z}_{1,-1}^{(d)})^{\prime}\lambda_{0,-1}+(z_{3}^{(d)}-\tilde{z}_{3}^{(d)})^{\prime}\beta_{0}-\gamma_{0}(w^{(d)}-\tilde{w}^{(d)})=$
$\displaystyle\tilde{z}_{1,1}^{(d)}-z_{1,1}^{(d)}.$

This concludes the proof.

##  Appendix F Proof of Theorem B.1

Recall we defined our two step rank estimator as follows: Letting
$\hat{\theta}$ denote $(\hat{\alpha},\hat{\gamma})$, our estimator is of the
form:

$\hat{\theta}=\arg\max_{\theta}\hat{Q}_{n}(\theta)\equiv\sum_{i\neq
j}\hat{g}_{i,j}(\theta)$

in which

$\displaystyle\hat{g}_{i,j}(\theta)$ $\displaystyle=$
$\displaystyle[\mathbf{1}\\{\partial_{2}\hat{P}^{11}(X_{1,i},X_{i})/\hat{f}_{V}(X_{i})+\partial_{2}\hat{P}^{10}(X_{1,j},X_{j})/\hat{f}_{V}(X_{j})\geq
0\\}\mathbf{1}\\{\Phi(X_{1,i},X_{i},X_{1,j},X_{j};\theta)\geq 0\\}$
$\displaystyle+$
$\displaystyle\mathbf{1}\\{\partial_{2}\hat{P}^{11}(X_{1,i},X_{i})/\hat{f}_{V}(X_{i})+\partial_{2}\hat{P}^{10}(X_{1,j},X_{j})/\hat{f}_{V}(X_{j})<0\\}\mathbf{1}\\{\Phi(X_{1,i},X_{i},X_{1,j},X_{j};\theta)<0\\}],$

with

$\Phi(x_{1},x,\tilde{x}_{1},\tilde{x};\theta)=x_{1}+\alpha-\gamma
x-(\tilde{x}_{1}-\gamma\tilde{x})$

We first show consistency of the rank estimator. To do so we first define the
objective function $Q^{if}_{n,2}(\theta)$, defined as

$Q^{if}_{n,2}(\theta)\equiv\sum_{i\neq j}{g}_{i,j}(\theta)$

where

$\displaystyle{g}_{i,j}(\theta)$ $\displaystyle=$
$\displaystyle[\mathbf{1}\\{\partial_{2}{P}^{11}(X_{1,i},X_{i})/{f}_{V}(X_{i})+\partial_{2}{P}^{10}(X_{1,j},X_{j})/{f}_{V}(X_{j})\geq
0\\}\mathbf{1}\\{\Phi(X_{1,i},X_{i},X_{1,j},X_{j};\theta)\geq 0\\}$
$\displaystyle+$
$\displaystyle\mathbf{1}\\{\partial_{2}{P}^{11}(X_{1,i},X_{i})/{f}_{V}(X_{i})+\partial_{2}{P}^{10}(X_{1,j},X_{j})/{f}_{V}(X_{j})<0\\}\mathbf{1}\\{\Phi(X_{1,i},X_{i},X_{1,j},X_{j};\theta)<0\\}],$

Since $g_{i,j}$ is bounded by 1 $\forall i,j$, and our random sampling
assumption, we have for each $\theta$,

$Q^{if}_{n,2}(\theta)\stackrel{{\scriptstyle
p}}{{\rightarrow}}E[g_{i,j}(\theta)]\equiv\Gamma_{0}(\theta)$

Furthermore, by Assumptions RK2, RK3 we can extend this result to converging
uniformly over $\theta\in\Theta$ (see, e.g. Sherman (1994a), Sherman (1993).)
$\Gamma_{0}(\theta)$ is continuous in $\theta$ by Assumptions RK2,RK3, and
uniquely maximized at $\theta=\theta_{0}$ by our identification result in
Theorem 2.1. Along with Assumption RK1, the infeasible estimator, defined as
the maximizer of $Q^{if}_{n,2}(\theta)$ converges in probability to
$\theta_{0}$ by, for example Theorem 2.1 in Newey and McFadden (1994). To show
consistency of the feasible estimator, where we first estimate the choice
probability functions and their derivatives nonparametrically, we only now
need to show the two objective functions converged to each other uniformly in
$\theta\in\Theta$. Consistency of the first stage estimators follows from
Assumptions RK3-RK5, see for example Henderson, Li, Parmeter, and Yao (2015).
However, this does not immediately imply convergence of the difference in
feasible and infeasible objective functions since the nonparametric estimators
are inside indicator functions so the continuous mapping theorem does
immediately not apply. Nonetheless the desired result can still be attained in
one of two ways. One would be to replace indicator functions with smooth
distribution functions in a fashion analogous to Horowitz (1992). This would
have the disadvantage of introducing tuning parameters, but another approach
would be to replace the indicator functions with their conditional
expectations, and note that the conditional expectations are smooth functions
using Assumption RK2, RK3. To see why, let $\hat{m}(x_{i})$ be a nonparametric
estimator of a function $m(x_{i})$, which is assumed to be smooth. We evaluate
the plim of

$I[\hat{m}(x_{i})>0]-I[m(x_{i})>0]=I[\hat{m}(x_{i})>0,m(x_{i})<0]-I[\hat{m}(x_{i})<0,m(x_{i})>0]$

we show that the first term converges in probability to 0 as identical
arguments can be used for the second term. Let $\varepsilon>0$ be given;
$P(I[\hat{m}(x_{i})>0,m(x_{i})<0]>\varepsilon)\leq
E[I[\hat{m}(x_{i})>0,m(x_{i})<0]/\varepsilon$ by Markov’s inequality. But the
expectation in the numerator on the right hand side is

$P(\hat{m}(x_{i})>0,m(x_{i})<0)=P(\hat{m}(x_{i})>0,m(x_{i})\leq-\delta_{n})+P(\hat{m}(x_{i})>0,m(x_{i})\in(-\delta_{n},0))$

where $\delta_{n}$ is a sequence of positive numbers converging to 0, at a
slow rate, e.g.$(\log n^{-1})$. The first term on the right hand side is
bounded above by

$P(|\hat{m}(x_{i})-m(x_{i})|>\delta_{n})\leq
P(\|\hat{m}(\cdot)-m(\cdot)\|>\delta_{n})$

where the notation $\|\hat{m}(\cdot)-m(\cdot)\|$ above denotes the sup norm
over $x_{i}$. The right hand side probability above will be sufficiently small
for $n$ large enough by the rate of convergence of the nonparametric
estimator. The second term, $P(\hat{m}(x_{i})>0,m(x_{i})\in(-\delta_{n},0))$,
is bounded above by $P(m(x_{i})\in(-\delta_{n},0))$ which by the smoothness of
$m(x_{i})$ converges to 0, and hence can be made arbitrarily small.
$\hfill\square$

To derive the rate of convergence and limiting distribution theory for the
feasible estimator where we first estimate choice probability functions and
their derivatives nonparametrically, we expand the nonparametric estimators
around true functions that are inside the indicator function in $Q_{n2}$. Then
we can follow the approach in Sherman (1994b). Having already established
consistency of the estimator, we will first establish root-$n$ consistency and
then asymptotic normality. For root-$n$ consistency we will apply Theorem 1 of
Sherman (1994b) and so here we change notation to deliberately stay as close
as possible to his. We will actually apply this theorem twice, first
establishing a slower than root-$n$ consistency result and then root-$n$
consistency. Keeping our notation deliberately as close as possible to
Sherman(1994b), here replacing our second stage rank objective function
$\hat{Q}_{2,n}(\theta)$ with $\hat{\cal G}_{n}(\theta)$, our infeasible
objective function $Q^{if}_{n,2}(\theta)$ with ${\cal G}_{n}(\theta)$, and
denoting our limiting objective function, previously denoted by
${\Gamma_{0}}(\theta)$, by ${\cal G}(\theta)$. We have the following theorem:

######  Theorem F.1.

(From Theorem 1 in Sherman (1994b)).

If $\delta_{n}$ and $\varepsilon_{n}$ are sequences of positive numbers
converging to 0, and

  1. 1.

$\hat{\theta}-\theta_{0}=o_{p}(\delta_{n})$

  2. 2.

There exists a neighborhood of $\theta_{0}$ and a constant $\kappa>0$ such
that ${\cal G}(\theta)-{\cal
G}(\theta_{0})\geq\kappa\|\theta-\theta_{0}\|^{2}$ for all $\theta$ in this
neighborhood.

  3. 3.

Uniformly over $O_{p}(\delta_{n})$ neighborhoods of $\theta_{0}$

$\hat{\cal G}_{n}(\theta)={\cal
G}(\theta)+O_{p}(\|\theta-\theta_{0}\|/\sqrt{n})+o_{p}(\|\theta-\theta_{0}\|^{2})+O_{p}(\varepsilon_{n})$

then $\hat{\theta}-\theta_{0}=O_{p}(\max(\varepsilon^{1/2},n^{-1/2}))$.

Once we use this theorem to establish the rate of convergence of our rank
estimator, we can attain limiting distribution theory, which will follow from
the following theorem:

######  Theorem F.2.

(From Theorem 2 in Sherman (1994b)). Suppose $\hat{\theta}$ is
$\sqrt{n}$-consistent for $\theta_{0}$, an interior point of $\Theta.$ Suppose
also that uniformly over $O_{p}(n^{-1/2})$ neighborhoods of $\theta_{0}$,

$\hat{\cal
G}_{n}(\theta)=\frac{1}{2}(\theta-\theta_{0})^{\prime}V(\theta-\theta_{0})+\frac{1}{\sqrt{n}}(\theta-\theta_{0})^{\prime}W_{n}+o_{p}(1/n)$
(F.1)

where $V$ is a negative definite matrix, and $W_{n}$ converges in distribution
to a $N(0,\Delta)$ random vector. Then

$\sqrt{n}(\hat{\theta}-\theta_{0})\Rightarrow N(0,V^{-1}\Delta V^{-1})$ (F.2)

We first turn attention to applying Theorem F.1 to derive the rate of
convergence of our estimator. Having already established consistency of our
rank estimator, we turn attention to the second condition in Theorem F.1. To
show the second condition, we will first derive an expansion for ${\cal
G}(\theta)$ around ${\cal G}(\theta_{0})$. We denote that even though ${\cal
G}_{n}(\theta)$ is not differentiable in $\theta$, ${\cal G}(\theta)$ is
sufficiently smooth for Taylor expansions to apply as the expectation operator
is a smoothing operator and the smoothness conditions in Assumptions RK2, RK3.
Taking a second order expansion of ${\cal G}(\theta)$ around ${\cal
G}(\theta_{0})$, we obtain

${\cal G}(\theta)={\cal G}(\theta_{0})+\nabla_{\beta}{\cal
G}(\theta_{0})^{\prime}(\theta-\theta_{0})+\frac{1}{2}(\theta-\theta_{0})^{\prime}\nabla_{\theta\theta}{\cal
G}(\theta^{\ast})(\theta-\theta_{0})$ (F.3)

where $\nabla_{\theta}$ and $\nabla_{\theta\theta}$ denote first and second
derivative operators and $\theta^{\ast}$ denotes an intermediate value. We
note that the first two terms of the right hand side of the above equation are
0, the first by how we defined the objective function, and the second by our
identification result in Theorem 2.1. Define

$V\equiv\nabla_{\theta\theta}{\cal G}(\theta_{0})$ (F.4)

and $V$ is positive definite by Assumption A3, so we have

$(\theta-\theta_{0})^{\prime}\nabla_{\theta\theta}{\cal
G}(\theta_{0})(\theta-\theta_{0})>0$ (F.5)

$\nabla_{\theta\theta}{\cal G}(\theta)$ is also continuous at
$\theta=\theta_{0}$ by Assumptions RK2 and RK3, so there exists a neighborhood
of $\theta_{0}$ such that for all $\theta$ in this neighborhood, we have

$(\theta-\theta_{0})^{\prime}\nabla_{\theta\theta}{\cal
G}(\theta)(\theta-\theta_{0})>0$ (F.6)

which suffices for the second condition to hold.

To show the third condition in Theorem F.1, we next establish the form of the
remainder term when we replace nonparametric estimators with the true
functions they are estimating. Specifically we wish to evaluate the difference
between

$\displaystyle[\mathbf{1}\\{\partial_{2}\hat{P}^{11}(X_{1,i},X_{i})/\hat{f}_{V}(X_{i})+\partial_{2}\hat{P}^{10}(X_{1,j},X_{j})/\hat{f}_{V}(X_{j})\geq
0\\}\mathbf{1}\\{\Phi(X_{1,i},X_{i},X_{1,j},X_{j};\theta)\geq 0\\}$ (F.7)
$\displaystyle+$
$\displaystyle\mathbf{1}\\{\partial_{2}\hat{P}^{11}(X_{1,i},X_{i})/\hat{f}_{V}(X_{i})+\partial_{2}\hat{P}^{10}(X_{1,j},X_{j})/\hat{f}_{V}(X_{j})<0\\}\mathbf{1}\\{\Phi(X_{1,i},X_{i},X_{1,j},X_{j};\theta)<0\\}$
(F.8)

and

$\displaystyle[\mathbf{1}\\{\partial_{2}{P}^{11}(X_{1,i},X_{i})/{f}_{V}(X_{i})+\partial_{2}{P}^{10}(X_{1,j},X_{j})/{f}_{V}(X_{j})\geq
0\\}\mathbf{1}\\{\Phi(X_{1,i},X_{i},X_{1,j},X_{j};\theta)\geq 0\\}$ (F.9)
$\displaystyle+$
$\displaystyle\mathbf{1}\\{\partial_{2}{P}^{11}(X_{1,i},X_{i})/{f}_{V}(X_{i})+\partial_{2}{P}^{10}(X_{1,j},X_{j})/{f}_{V}(X_{j})<0\\}\mathbf{1}\\{\Phi(X_{1,i},X_{i},X_{1,j},X_{j};\theta)<0\\}$
(F.10)

To establish a representation for this difference, we first simplify notation
we write the expressions as:

$\displaystyle I[\hat{m}_{1}({\bf x}_{i})+\hat{m}_{2}({\bf x}_{j})\geq
0]I[\Delta{\bf x}_{ij}^{\prime}\theta\geq 0]$ (F.11) $\displaystyle+$
$\displaystyle I[\hat{m}_{1}({\bf x}_{i})+\hat{m}_{2}({\bf
x}_{j})<0]I[\Delta{\bf x}_{ij}^{\prime}\theta<0]$ (F.12)

and

$\displaystyle I[m_{1}({\bf x}_{i})+m_{2}({\bf x}_{j})\geq 0]I[\Delta{\bf
x}_{ij}^{\prime}\theta\geq 0]$ (F.13) $\displaystyle+$ $\displaystyle
I[m_{1}({\bf x}_{i})+m_{2}({\bf x}_{j})<0]I[\Delta{\bf
x}_{ij}^{\prime}\theta<0]$ (F.14)

respectively, where here ${\bf x}_{i}$ denotes the separate components of
$x_{1i},x_{i}$, and analogous for ${\bf x}_{j}$. We first explore

$(I[\hat{m}_{1}({\bf x}_{i})+\hat{m}_{2}({\bf x}_{j})\geq 0]-I[m_{1}({\bf
x}_{i})+m_{2}({\bf x}_{j})\geq 0])I[\Delta{\bf x}_{ij}^{\prime}\theta\geq 0]$

for each $i,j$ inside the double summation:

$\frac{1}{n(n-1)}\sum_{i\neq j}(I[\hat{m}_{1}({\bf x}_{i})+\hat{m}_{2}({\bf
x}_{j})\geq 0]-I[m_{1}({\bf x}_{i})+m_{2}({\bf x}_{j})\geq 0])I[\Delta{\bf
x}_{ij}^{\prime}\theta\geq 0]$ (F.15)

An immediate technical difficulty that arises with the above term is the
presence of a nonparametric estimator inside the indicator function above. A
simple approach to deal with this would be to replace the indicator function
with a smoothed indicator function in a fashion analogous to Horowitz (1992),
under appropriate conditions on the kernel function and smoothing parameter.
Such an approach is not necessary as long as the nonparametric estimator
$\hat{m}_{1}(x_{i})$ is asymptotically normal, and asymptotically centered at
$m_{1}(x_{i})$, which will be the case with our proposed kernel estimator of
the probability function and its derivative. In either approach (smoothed
indicator or not) we can show that (F.15) can be represented as:

$\frac{1}{n(n-1)}\sum_{i\neq j}\phi(0)f_{m_{ij}}(0)\left((\hat{m}_{1}({\bf
x}_{i})-m_{1}({\bf x}_{i}))+(\hat{m}_{2}({\bf x}_{j})-m_{2}({\bf
x}_{j}))\right)I[\Delta{\bf x}_{ij}^{\prime}\theta\geq 0]+o_{p}(n^{-1})$
(F.16)

where $\phi(0)$ denotes the standard normal pdf evaluated at 0,
$f_{m_{ij}}(0)$ denotes the density function of $m_{1}({\bf x}_{i})+m_{2}({\bf
x}_{j})$ evaluated at 0, and the $o_{p}(n^{-1})$ term is uniform in $\theta$
lying in $o_{p}(1)$ neighborhoods of $\theta_{0}$. Therefore, uniformly for
$\theta$ in an $o_{p}(1)$ neighborhood of $\theta_{0}$, this remainder term
converges to 0 at the rate of convergence of the first stage nonparametric
estimator, which under Assumptions RK3, RK4, RK5, is $o_{p}(n^{-1/4})$. Thus
by repeated application of Theorem F.1, we can conclude that the estimator is
root-$n$ consistent. To show that the estimator is also asymptotically normal,
we will first derive a linear representation for the term:

$\frac{1}{n(n-1)}\sum_{i\neq j}\phi(0)f_{m_{ij}}(0)(\hat{m}_{1}({\bf
x}_{i})-m_{1}({\bf x}_{i}))I[\Delta{\bf x}_{ij}^{\prime}\theta\geq 0]$ (F.17)

As this term is linear in the nonparametric estimator $\hat{m}_{1}(x_{i})$,
the desired linear representation follows from arguments used in Khan (2001).
One slight difference here compared to Khan (2001) is that here our
nonparametric estimators and estimands are each ratios of derivatives.
Nonetheless, after linearizing these ratios as done in, e.g. Newey and
McFadden (1994). Specifically, we have that F.17 can be expressed as:

$\frac{1}{n(n-1)}\sum_{i\neq j}\phi(0)f_{m_{ij}}(0)\frac{1}{m_{1den}({\bf
x}_{i})}(\hat{m}_{1num}({\bf x}_{i})-m_{1num}({\bf x}_{i}))I[\Delta{\bf
x}_{ij}^{\prime}\theta\geq 0]$ (F.18) $-\frac{1}{n(n-1)}\sum_{i\neq
j}\phi(0)f_{m_{ij}}(0)\frac{m_{1num}({\bf x}_{i})}{m_{1den}({\bf
x}_{i})^{2}}(\hat{m}_{1den}({\bf x}_{i})-m_{1den}({\bf x}_{i}))I[\Delta{\bf
x}_{ij}^{\prime}\theta\geq 0]$ (F.19)

where $\hat{m}_{1num}({\bf x}_{i})$ denotes the numerator
$\\{\partial_{2}\hat{P}^{11}(X_{1,i},X_{i})\\}$, the estimator of
$m_{1num}({\bf x}_{i})$ which denotes
$\\{\partial_{2}{P}^{11}(X_{1,i},X_{i})\\}$, and $\hat{m}_{1den}({\bf x}_{i})$
denotes the denominator $\hat{f}_{V}(X_{i})$, the estimator of $m_{1den}({\bf
x}_{i})$ which denotes ${f}_{V}(X_{i})$.

Plugging in the definitions of the kernel estimators of $\hat{m}_{1num}({\bf
x}_{i})$, and $\hat{m}_{1den}({\bf x}_{i})$, results in a third order process.
Using arguments in Khan (2001) and Powell, Stock, and Stoker (1989) we can
express the third order $U$ process as a second order $U$ process plus an
asymptotically negligible remainder term. This is of the form:

$\frac{1}{n}\sum_{i=1}^{n}\phi(0)\frac{\ell(x_{i})}{m_{1den}({\bf
x}_{i})}(y_{1i}-m_{1num}({\bf x}_{i}))E\left[I[f_{m_{ij}}(0)\Delta{\bf
x}_{ij}^{\prime}\theta\geq 0]|x_{i}\right]$ (F.20)

where $\ell(x_{i})\equiv\frac{-f_{X}^{\prime}(x_{i})}{f_{X}(x_{i})}$. We note
that the function $E\left[f_{m_{ij}}(0)I[\Delta{\bf x}_{ij}^{\prime}\theta\geq
0]|x_{i}\right]$, which we denote here by ${\cal H}(x_{i},\theta)$ is a smooth
function in $\theta$. We will use this feature to expand ${\cal
H}(x_{i},\theta)$ around ${\cal H}(x_{i},\theta_{0})$. Analogous arguments can
be used to attain a linear representation of (F.19), which is of the form:

$\frac{1}{n}\sum_{i=1}^{n}\phi(0)\frac{\ell_{2}(x_{1i})m_{1num}({\bf
x}_{i})}{m_{1den}({\bf x}_{i})^{2}}(y_{2i}-m_{1den}({\bf
x}_{i}))E\left[I[f_{m_{ij}}(0)\Delta{\bf x}_{ij}^{\prime}\theta\geq
0]|x_{i}\right]$ (F.21)

where
$\ell_{2}(x_{1i})\equiv\frac{-f_{X_{1}}^{\prime}(x_{1i})}{f_{X}(x_{1i})}$.
Grouping (F.20) and (F.21) we have

$\frac{1}{n}\sum_{i=1}^{n}\phi(0)\frac{1}{m_{1den}({\bf
x}_{i})}\left\\{\ell(x_{i})(y_{1i}-m_{1num}({\bf x}_{i}))-\frac{m_{1num}({\bf
x}_{i})}{m_{1den}({\bf x}_{i})}\ell_{2}(x_{1i})(y_{2i}-m_{1den}({\bf
x}_{i}))\right\\}{\cal H}(x_{i},\theta)$ (F.22)

Note that by Assumptions RK2, RK3, ${\cal H}(x_{i},\theta)$ is smooth in
$\theta$ implying the expansion

${\cal H}(x_{i},\theta)={\cal H}(x_{i},\theta_{0})+\nabla_{\theta}{\cal
H}(x_{i},\theta_{0})^{\prime}(\theta-\theta_{0})$

Thus we can express (F.22) as the which we note is a mean 0 sum

$\frac{1}{n}\sum_{i=1}^{n}\psi_{1rnki}(\theta-\theta_{0})$ (F.23)

where

$\psi_{1rnki}=\phi(0)\frac{1}{m_{1den}({\bf
x}_{i})}\left\\{\ell(x_{i})(y_{1i}-m_{1num}({\bf x}_{i}))-\frac{m_{1num}({\bf
x}_{i})}{m_{1den}({\bf x}_{i})}\ell_{2}(x_{1i})(y_{2i}-m_{1den}({\bf
x}_{i}))\right\\}\nabla_{\theta}{\cal H}(x_{i},\theta_{0})$ (F.24)

We can use identical arguments to attain a linear representation for the $U-$
process:

$\frac{1}{n(n-1)}\sum_{i\neq j}\phi(0)f_{m_{ij}}(0)\left(\hat{m}_{2}({\bf
x}_{j})-m_{2}({\bf x}_{j})\right)I[\Delta{\bf x}_{ij}^{\prime}\theta\geq 0]$
(F.25)

where $\hat{m}_{2}({\bf x}_{j})$ is also a ratio of nonparametric estimators
where here the numerator is $\hat{m}_{2n}({\bf x}_{j})$ denoting
$\\{\partial_{2}\hat{P}^{10}(X_{1,j},X_{j})\\}$, the estimator of $m_{2n}({\bf
x}_{2})$ which denotes $\\{\partial_{2}{P}^{10}(X_{1,j},X_{j})\\}$, and
$\hat{m}_{2d}({\bf x}_{j})$ denotes the denominator $\hat{f}_{V}(X_{j})$, the
estimator of $m_{1den}({\bf x}_{j})$ which denotes ${f}_{V}(X_{j})$.

and by using identical arguments it too can be represented as a mean 0 sum
denoted here by

$\frac{1}{n}\sum_{i=1}^{n}\psi_{2rnki}$ (F.26)

where $\psi_{2rnki}$ is defined as:

Finally after grouping the two terms and expanding ${\cal H}(x_{i},\theta)$
around ${\cal H}(x_{i},\theta_{0})$ we get that (F.16) can be represented as:

$\frac{1}{n}\sum_{i=1}^{n}(\psi_{1rnki}+\psi_{2rnki})^{\prime}(\theta-\theta_{0})+o_{p}(n^{-1})$
(F.27)

Combining our results, from Theorem F.2, we have that

$\sqrt{n}(\hat{\theta}-\theta_{0})\Rightarrow N(0,V^{-1}\Delta V^{-1})$ (F.28)

where

$V=\nabla_{\theta\theta}{\cal G}(\theta_{0})$ (F.29)

and

$\Delta=E\left[(\psi_{1rnki}+\psi_{2rnki})(\psi_{1rnki}+\psi_{2rnki})^{\prime}\right]$
(F.30)

##  Appendix G Model with Two Idiosyncratic Shocks

In this section, we focus on the identification of $(\alpha_{0},\gamma_{0})$
in the “condensed” model that
$X_{1}=Z_{1}^{\prime}\lambda_{0}+Z_{3}^{\prime}\beta_{0}$ is observed and

$\displaystyle Y_{1}=\mathbf{1}\\{X_{1}+\alpha_{0}Y_{2}-U\geq 0\\}$ (G.31)
$\displaystyle Y_{2}=\mathbf{1}\\{X-V\geq 0\\}.$

with the understanding that $(\lambda_{0},\beta_{0})$ can be identified
jointly with $\alpha_{0}$ and $\gamma_{0}$, as shown in Theorems 2.1 and 3.1.
We further impose $U=\gamma_{0}W+\eta_{1}$, $V=W+\eta_{2}$, and
$(W,\eta_{1},\eta_{2})$ are mutually independent. First we consider the case
$\gamma_{0}=1$ and $X_{1}$ is binary, because even in this context, for the
baseline case with one idiosyncratic shock, we can identify $\alpha_{0}$. But
identification of $\alpha_{0}$ becomes more difficult in this model without
the help of repeated measurements, as established in the following theorem.

######  Theorem G.1.

Suppose (G.31) holds, $\gamma_{0}$ is known to be one, $X_{1}$ is binary, and
$W$ has a bounded support $[-b,-a]$ such that $0.5>b-a$ and
$1-(b-a)>\alpha_{0}>b-a$, then $\alpha_{0}$ is not point identified.

This nonidentification result motivates imposing additional structure on $W$,
and we consider the following model

C1

$U=\gamma_{0}W+\eta_{1}\quad\text{and}\quad V=\sigma_{0}W+\eta_{2}$.

C2

$W$ is standard normally distributed.

C3

$W$, $\eta_{1}$ and $\eta_{2}$ are mutually independent.

C4

$X$ has full support.

C5

Denote the density of $\eta_{2}$ as $f_{\eta_{2}}$, then $f_{\eta_{2}}$ does
not have a Gaussian component in the sense that

$\displaystyle f_{\eta_{2}}\in\mathcal{G}=\\{g~{}\text{is a density on
}\Re~{}\text{s.t.}:g=g^{\prime}*\phi_{\sigma}~{}\text{for some density
$g^{\prime}$ implies that }\sigma=0\\},$

where $\phi_{\sigma}$ is the density for a normal distribution with zero mean
and $\sigma^{2}$ variance.

Assumption C5 effectively assumes that the distribution of $\eta_{2}$ has tail
properties different from those of a normal distribution. This type of
assumption is made in the deconvolution literature as it is necessary for
identification of the target density when the error distribution is not
completely known- see, e.g., Butucea and Matias (2005).333In fact, based on
the results in Butucea and Matias (2005), $W$ can belong to a more general
class of known distributions. Furthermore, we note that if $\sigma_{0}$ is
known, then Assumption C5 is not necessary. The importance of non-normality in
factor models goes back to Geary (1942) and Reiersol (1950), who have shown
that factor loadings are identified in a linear measurement error model if the
factor is not Gaussian. In our case, note $V=\sigma_{0}W+\eta_{2}$ where $W$
is standard normal and the density of $V$ is identified from data. Here we
want to identify $\sigma_{0}$ and the density of $\eta_{2}$. If $\eta_{2}$ has
a Gaussian component, then

$\displaystyle\eta_{2}=\eta_{2}^{\prime}+\tilde{\sigma}\tilde{W},$

where $\tilde{W}$ is a standard normal random variable that is independent of
$\eta_{2}^{\prime}$ and $W$ and $\tilde{\sigma}>0$. It implies

$\displaystyle V=(\sigma_{0}W+\tilde{\sigma}\tilde{W})+\eta_{2}^{\prime},$

where $\eta_{2}^{\prime}$ does not have a Gaussian component. In addition,
note that
$(\sigma_{0}W+\tilde{\sigma}\tilde{W})=\sqrt{\sigma_{0}^{2}+\tilde{\sigma}^{2}}G$,
for some standard normal random variable $G$. Therefore, without Assumption
B5, $\sigma_{0}$ is not identified.

######  Theorem G.2.

If Assumptions C1–C5 hold, then $\sigma_{0}$, $\gamma_{0}$ and $\alpha_{0}$
are identified.

Note that this identification result does not require any variation from
$X_{1}$, which is in spirit close to the one-factor model in our paper and is
different from the identification result in Vytlacil and Yildiz (2007). We
also note that this result does not contradict the counterexample in the
paper. In the counterexample, we only assume that we know the support of $W$
is bounded. Here we assume that the full density of $W$, and thus, the support
of $W$ is known.

##  Appendix H Proof of Theorem A.1

Denote $P^{ij}(x_{1},x)=Prob(Y_{1}=i,Y_{2}=j|X_{1}=x_{1},X=x)$. Then

$\displaystyle P^{11}(x_{1},x)$
$\displaystyle=\int_{-\infty}^{x}F_{U}(x_{1}+\alpha_{0}|V=v)f(v)dv$ (H.32)
$\displaystyle P^{10}(\tilde{x}_{1},x)$
$\displaystyle=\int^{+\infty}_{x}F_{U}(\tilde{x}_{1}|V=v)f(v)dv.$

Taking derivatives w.r.t. the second argument of the the LHS function, we have

$\displaystyle\partial_{2}P^{11}(x_{1},x)$
$\displaystyle=F_{U}(x_{1}+\alpha_{0}|V=x)f(x)$
$\displaystyle\partial_{2}P^{10}(\tilde{x}_{1},x)$
$\displaystyle=-F_{U}(\tilde{x}_{1}|V=x)f(x).$

If $|\alpha_{0}|\leq b-a$, then there exists a pair $(x_{1},\tilde{x}_{1})$
such that $x_{1}+\alpha_{0}=\tilde{x}_{1}$. This pair can be identified by
checking the equation below:

$\partial_{2}P^{11}(x_{1},x)/f(x)+\partial_{2}P^{10}(\tilde{x}_{1},x)/f(x)=0.$

This concludes the sufficient part.

When $\alpha_{0}<a-b$, for any $\alpha<\alpha_{0}$, we can define

$\displaystyle\tilde{U}=U+\alpha-\alpha_{0}$ if $\displaystyle U\leq
b+\alpha_{0}$ $\displaystyle\tilde{U}=U$ if $\displaystyle U>b+\alpha_{0}$

Then for any $x_{1}\in[a,b]$,

$\displaystyle P(\tilde{U}\leq x_{1}+\alpha|V=v)$
$\displaystyle=P(\tilde{U}\leq x_{1}+\alpha,U\leq
b+\alpha_{0}|V=v)+P(\tilde{U}\leq x_{1}+\alpha,U>b+\alpha_{0}|V=v)$
$\displaystyle=P(U\leq x_{1}+\alpha_{0}|V=v)$ $\displaystyle P(\tilde{U}\leq
x_{1}|V=v)$ $\displaystyle=P(\tilde{U}\leq x_{1},U\leq
b+\alpha_{0}|V=v)+P(\tilde{U}\leq x_{1},U>b+\alpha_{0}|V=v)$
$\displaystyle=P(U\leq b+\alpha_{0},U\leq
x_{1}+\alpha_{0}-\alpha|V=v)+P(b+\alpha_{0}<U\leq x_{1}|V=v)$
$\displaystyle=P(U\leq b+\alpha_{0}|V=v)+P(b+\alpha_{0}<U\leq x_{1}|V=v)$
$\displaystyle=P(U\leq x_{1}|V=v),$

where the third equality holds because, since $\alpha_{0}<a-b$ and
$\alpha<\alpha_{0}$, $b+\alpha_{0}\leq x_{1}+\alpha_{0}-\alpha$ for
$x_{1}\in[a,b]$. Let $G_{U,V}$ and $G_{\tilde{U},V}$ be the joint distribution
of $(U,V)$ and $(\tilde{U},V)$ respectively. Then the above calculation with
(H.32) imply that $(\alpha_{0},G_{U,V})$ and $(\alpha,G_{\tilde{U},V})$
produce the identical pair $(P^{11}(x_{1},x),P^{10}(x_{1},x))$. In addition,
the distribution of $V$ is unchanged so that $P(Y_{2}=1|X=x)$ is identified
from data. Therefore, $(\alpha_{0},G_{U,V})$ and $(\alpha,G_{\tilde{U},V})$
are observationally equivalent.

Similarly, when $\alpha_{0}>b-a$, for any $\alpha>\alpha_{0}$, we can define

$\displaystyle\tilde{U}=U+\alpha-\alpha_{0}$ if $\displaystyle U>a+\alpha_{0}$
$\displaystyle\tilde{U}=U$ if $\displaystyle U\leq a+\alpha_{0}$

Then for any $x_{1}\in[a,b]$,

$\displaystyle P(\tilde{U}\leq x_{1}+\alpha|V=v)$
$\displaystyle=P(\tilde{U}\leq x_{1}+\alpha,U\leq
a+\alpha_{0}|V=v)+P(\tilde{U}\leq x_{1}+\alpha,U>a+\alpha_{0}|V=v)$
$\displaystyle=P(U\leq a+\alpha_{0}|V=v)+P(a+\alpha_{0}<U\leq
x_{1}+\alpha_{0}|V=v)$ $\displaystyle=P(U\leq x_{1}+\alpha_{0}|V=v).$
$\displaystyle P(\tilde{U}\leq x_{1}|V=v)$ $\displaystyle=P(\tilde{U}\leq
x_{1},U\leq a+\alpha_{0}|V=v)+P(\tilde{U}\leq x_{1},U>a+\alpha_{0}|V=v)$
$\displaystyle=P(U\leq x_{1}|V=v),$

where we use the facts that $x_{1}\leq a+\alpha_{0}$ and $x_{1}-a<\alpha$ for
$x_{1}\in[a,b]$. So again, $(\alpha_{0},G_{U,V})$ and
$(\alpha,G_{\tilde{U},V})$ are observationally equivalent.

##  Appendix I Proof of Theorem A.2

The sign of $\alpha_{0}$ is identified by the data. In the following, we focus
on deriving the results when $\alpha_{0}>b-a$. By the proof of Theorem A.1, we
have already shown that all $\alpha>\alpha_{0}$ is in the identified set. Now
we consider $\frac{b-a+\alpha_{0}}{2}\leq\alpha<\alpha_{0}$.

$\displaystyle\tilde{U}=U+\alpha-\alpha_{0}$ if $\displaystyle U>a+\alpha$
$\displaystyle\tilde{U}=U$ if $\displaystyle U\leq a+\alpha$

Then for any $x_{1}\in[a,b]$,

$\displaystyle P(\tilde{U}\leq x_{1}+\alpha|V=v)$
$\displaystyle=P(\tilde{U}\leq x_{1}+\alpha,U\leq
a+\alpha|V=v)+P(\tilde{U}\leq x_{1}+\alpha,U>a+\alpha|V=v)$
$\displaystyle=P(U\leq a+\alpha|V=v)+P(a+\alpha<U\leq x_{1}+\alpha_{0}|V=v)$
$\displaystyle=P(U\leq x_{1}+\alpha_{0}|V=v).$ $\displaystyle P(\tilde{U}\leq
x_{1}|V=v)$ $\displaystyle=P(\tilde{U}\leq x_{1},U\leq
a+\alpha|V=v)+P(\tilde{U}\leq x_{1},U>a+\alpha|V=v)$ $\displaystyle=P(U\leq
x_{1}|V=v)+P(U\leq x_{1}+\alpha_{0}-\alpha,U>a+\alpha|V=v).$
$\displaystyle=P(U\leq x_{1}|V=v).$

Here note that the last equality is because $x_{1}+\alpha_{0}-\alpha\leq
b+\alpha_{0}-\alpha\leq a+\alpha$ if $\alpha\geq\frac{b-a+\alpha_{0}}{2}$.
Denote $\alpha^{(1)}=\frac{b-a+\alpha_{0}}{2}$. Then we have shown that there
exists $U^{(1)}(\alpha)$ which only depends on $\alpha$ such that for any
$x_{1}\in[a,b]$, any $v$ and any $\alpha_{0}>\alpha\geq\alpha^{(1)}$

$\displaystyle P(U^{(1)}(\alpha)\leq x_{1}+\alpha|V=v)=P(U\leq
x_{1}+\alpha_{0}|V=v)$ $\displaystyle P(U^{(1)}(\alpha)\leq x_{1}|V=v)=P(U\leq
x_{1}|V=v).$

In particular, there exists $U^{(1)}(\alpha^{(1)})$ such that

$\displaystyle P(U^{(1)}(\alpha^{(1)})\leq x_{1}+\alpha^{(1)}|V=v)=P(U\leq
x_{1}+\alpha_{0}|V=v)$ $\displaystyle P(U^{(1)}(\alpha^{(1)})\leq
x_{1}|V=v)=P(U\leq x_{1}|V=v).$

Now repeating the above construction but replacing $U$ with $U^{(1)}$ and
$\alpha_{0}$ with $\alpha^{(1)}$, we have for any
$\alpha^{(1)}>\alpha\geq\alpha^{(2)}\equiv\frac{b-a+\alpha^{(1)}}{2}$, there
exists $U^{(2)}(\alpha)$ such that for any $x_{1}\in[a,b]$, any $v$ and any
$\alpha^{(1)}>\alpha\geq\alpha^{(2)}$,

$\displaystyle P(U^{(2)}(\alpha)\leq
x_{1}+\alpha^{(2)}|V=v)=P(U^{(1)}(\alpha^{(1)})\leq
x_{1}+\alpha^{(1)}|V=v)=P(U\leq x_{1}+\alpha_{0}|V=v)$ $\displaystyle
P(U^{(2)}(\alpha)\leq x_{1}|V=v)=P(U^{(1)}(\alpha^{(1)})\leq
x_{1}|V=v)=P(U\leq x_{1}|V=v).$

This concludes that any $\alpha$ such that $\alpha_{0}>\alpha\geq\alpha^{(2)}$
is in the identified set. In general, by repeating the procedure k times, we
have that any $\alpha$ such that

$\alpha_{0}>\alpha\geq\alpha^{(k)}=(1-\frac{1}{2^{k}})(b-a)+\frac{\alpha_{0}}{2^{k}}$

is in the identified set. For any $\alpha>b-a$, there exists some finite $k$
such that $\alpha>(1-\frac{1}{2^{k}})(b-a)+\frac{\alpha_{0}}{2^{k}}$. This
concludes the result that $\alpha>b-a$ is in the identified set.

Finally, since if $\alpha>b-a$,
$\partial_{2}P^{11}(x_{1},x)+\partial_{2}P^{10}(\tilde{x}_{1},x)>0$ for all
pairs of $(x_{1},x)$ and $(\tilde{x}_{1},x)$ while, if $\alpha\leq b-a$, at
least there exists one pair $(x_{1},x)$ and $(\tilde{x}_{1},x)$ such that
$\partial_{2}P^{11}(x_{1},x)+\partial_{2}P^{10}(\tilde{x}_{1},x)\leq 0$. This
implies $\alpha\leq b-a$ is not in the identified set. Therefore, the sharp
identified set when $\alpha_{0}>b-a$ is $(b-a,\infty)$.

When $\alpha_{0}<a-b$, a symmetric argument implies that the identified set is
$(-\infty,a-b)$.

##  Appendix J Proof of Theorem G.1

Our first result for this model illustrates how identification can become more
difficult. In our first result for this model, we show when $-W$ has a bounded
support, say $[a,b]$, then $\alpha_{0}$ is not identified if $\alpha_{0}>b-a$.
To establish this, consider an impostor $\alpha$ such that
$\alpha<\alpha_{0}$. In addition, we consider the case where
$\alpha_{0}-\alpha+b<\alpha_{0}+a$ and $\alpha+b<a+1$. Such $\alpha$ exists
because of the fact that $1-(b-a)>\alpha_{0}>b-a.$ Let
$\Delta=\alpha_{0}-\alpha$ and $(\tilde{W},\tilde{\eta}_{1},\tilde{\eta}_{2})$
be mutually independent such that $\tilde{W}$ is distributed as $W-\Delta$,
$\tilde{\eta}_{2}$ is distributed as $\eta_{2}-\Delta$, and

$F_{\tilde{\eta}_{1}}(e)=\begin{cases}F_{\eta_{1}}(e)&\text{ on }e\leq a,\\\
F_{\eta_{1}}(a)&\text{ on }\eta_{1}\in(a,a+\Delta],\\\
F_{\eta_{1}}(e-\Delta)&\text{ on }e\in(a+\Delta,b+\Delta],\\\
\frac{\alpha_{0}+a-e}{\alpha_{0}+a-b-\Delta}F_{\eta_{1}}(b)+\frac{e-b-\Delta}{\alpha_{0}+a-b-\Delta}F_{\eta_{1}}(\alpha_{0}+a)&\text{
on }e\in(b+\Delta,\alpha_{0}+a],\\\ F_{\eta_{1}}(e)&\text{ on
}e\in(\alpha_{0}+a,\alpha_{0}+b),\\\
F_{\eta_{1}}(\alpha_{0}+b)+\frac{e-\alpha_{0}-b}{a+1+\Delta-\alpha_{0}-b}(F_{\eta_{1}}(a+1)-F_{\eta_{1}}(\alpha_{0}+b))&\text{
on }e\in(\alpha_{0}+b,a+1+\Delta],\\\ F_{\eta_{1}}(e-\Delta)&\text{ on
}e\in(a+\Delta+1,b+\Delta+1],\\\
F_{\eta_{1}}(b+1)+\frac{e-(b+\Delta+1)}{a+\alpha_{0}-b-\Delta}(F_{\eta_{1}}(a+\alpha_{0}+1)-F_{\eta_{1}}(b+1))&\text{
on }e\in(b+\Delta+1,a+\alpha_{0}+1],\\\ F_{\eta_{1}}(e)&\text{ on
}e>a+\alpha_{0}+1.\end{cases}$

Then, because $-\tilde{w}=\Delta-w\in[a+\Delta,b+\Delta]$ and $x_{1}=0,1$,

$\displaystyle P(Y_{1}=1,Y_{2}=0|X=x,X_{1}=x_{1})=$ $\displaystyle\int
F_{\eta_{1}}(x_{1}-w)(1-F_{\eta_{2}}(x-w))f_{W}(w)dw$ $\displaystyle=$
$\displaystyle\int
F_{\tilde{\eta_{1}}}(x_{1}-\tilde{w})(1-F_{\tilde{\eta}_{2}}(x-\tilde{w}))f_{\tilde{w}}(\tilde{w})d\tilde{w}.$

Similarly, because
$\alpha-\tilde{w}=\alpha_{0}-w\in[\alpha_{0}+a,\alpha_{0}+b]$ and for
$e\in(\alpha_{0}+a,\alpha_{0}+b]\cup(1+\alpha_{0}+a,1+\alpha_{0}+b]$,
$F_{\tilde{\eta}_{1}}(e)=F_{\eta_{1}}(e)$, we have

$\displaystyle P(Y_{1}=1,Y_{2}=1|X=x,X_{1}=x_{1})=$ $\displaystyle\int
F_{\eta_{1}}(x_{1}+\alpha_{0}-w)F_{\eta_{2}}(x-w)f_{W}(w)dw$ $\displaystyle=$
$\displaystyle\int
F_{\eta_{1}}(x_{1}+\alpha-(w+\alpha-\alpha_{0}))F_{\eta_{2}}(x-w)f_{W}(w)dw$
$\displaystyle=$ $\displaystyle\int
F_{\tilde{\eta_{1}}}(x_{1}+\alpha-\tilde{w})F_{\tilde{\eta_{2}}}(x-\tilde{w})f_{\tilde{w}}(\tilde{w})d\tilde{w}.$

This implies $\alpha_{0}$ is not identified from the impostor $\alpha$.

##  Appendix K Proof of Theorem G.2

We first show that both $\sigma_{0}$ and the density of $\eta_{2}$ are
identified. Note $X$ has full support. This implies the density of $V$ denoted
as $f_{V}(\cdot)$ is identified via

$\displaystyle f_{V}(v)=\partial_{v}E(Y_{2}|X=v).$

In addition, we have

$\displaystyle f_{V}(\cdot)=f_{\eta_{2}}*\phi_{\sigma_{0}}(\cdot),$

where $*$ denotes the convolution operator. Suppose $f_{\eta_{2}}(\cdot)$ and
$\sigma_{0}$ are not identified so that there exist
$f_{\eta_{2}}^{\prime}(\cdot)$ and $\sigma^{\prime}$ such that

$\displaystyle
f_{V}(\cdot)=f_{\eta_{2}}^{\prime}*\phi_{\sigma^{\prime}}(\cdot).$

Without loss of generality, we assume $\sigma^{\prime}\geq\sigma_{0}$,
otherwise, we can just relabel $f_{\eta_{2}}(\cdot)$ and
$f_{\eta_{2}}^{\prime}(\cdot)$. Then we have

$\displaystyle
f_{\eta_{2}}(\cdot)=f_{\eta_{2}}^{\prime}*\phi_{(\sigma^{{}^{\prime}2}-\sigma_{0}^{2})}.$

By Assumption B5, we have $\sigma^{{}^{\prime}}=\sigma_{0}$, which implies
$f_{\eta_{2}}(\cdot)=f_{\eta_{2}}^{\prime}(\cdot)$.

In the following, we proceed given that $f_{\eta_{2}}(\cdot)$ and $\sigma_{0}$
are known. Recall $F_{\eta_{1}}(\cdot)$ as the CDF of $\eta_{1}$. Then,

$\displaystyle P^{11}(x_{1},x)=$ $\displaystyle
P(Y_{1}=1,Y_{2}=1|X_{1}=x_{1},X=x)=\int
F_{\eta_{1}}(x_{1}+\alpha_{0}-\gamma_{0}w)F_{\eta_{2}}(x-\sigma_{0}w)f_{W}(w)dw$

and

$\displaystyle P^{10}(x_{1},x)=$ $\displaystyle
P(Y_{1}=1,Y_{2}=0|X_{1}=x_{1},X=x)=\int
F_{\eta_{1}}(x_{1}-\gamma_{0}w)(1-F_{\eta_{2}}(x-\sigma_{0}w))f_{W}(w)dw.$

Taking derivatives of $P^{11}(x_{1},x)$ and $P^{10}(x_{1},x)$ w.r.t. $x$, we
have

$\displaystyle\partial_{x}P^{11}(x_{1},x)=\int
F_{\eta_{1}}(x_{1}+\alpha_{0}-\gamma_{0}w)f_{\eta_{2}}(x-\sigma_{0}w)f_{W}(w)dw$
(K.33)

and

$\displaystyle-\partial_{x}P^{10}(x_{1},x)=\int
F_{\eta_{1}}(x_{1}-\gamma_{0}w)f_{\eta_{2}}(x-\sigma_{0}w)f_{W}(w)dw.$ (K.34)

Applying Fourier transform on both sides of (K.33) and (K.34), we have

$\displaystyle\mathcal{F}(\partial_{x}P^{11}(x_{1},\cdot))=\mathcal{F}_{\sigma_{0}}(F_{\eta_{1}}(x_{1}+\alpha_{0}-\gamma_{0}\cdot)f_{W}(\cdot))\mathcal{F}(f_{\eta_{2}}(\cdot))$
(K.35)

and

$\displaystyle\mathcal{F}(-\partial_{x}P^{10}(x_{1},\cdot))=\mathcal{F}_{\sigma_{0}}(F_{\eta_{1}}(x_{1}-\gamma_{0}\cdot)f_{W}(\cdot))\mathcal{F}(f_{\eta_{2}}(\cdot)),$
(K.36)

where for a generic function $g(w)$,

$\displaystyle\mathcal{F}_{\sigma_{0}}(g(\cdot))(t)=\frac{1}{\sqrt{2\pi}}\int\exp(-2\pi
it\sigma_{0}w)g(w)dw.$

Then, by (K.35), we can identify $F_{\eta_{1}}(x_{1}+\alpha_{0}-\cdot)$ by

$\displaystyle
F_{\eta_{1}}(x_{1}+\alpha_{0}-\gamma_{0}\cdot)=\mathcal{F}_{\sigma_{0}}^{-1}\left(\frac{\mathcal{F}(\partial_{x}P^{11}(x_{1},\cdot))}{\mathcal{F}(f_{\eta_{2}}(\cdot))}\right)(\cdot)/f_{W}(\cdot).$

Similarly, we can identify

$\displaystyle
F_{\eta_{1}}(x_{1}-\gamma_{0}\cdot)=\mathcal{F}_{\sigma_{0}}^{-1}\left(\frac{\mathcal{F}(-\partial_{x}P^{10}(x_{1},\cdot))}{\mathcal{F}(f_{\eta_{2}}(\cdot))}\right)(\cdot)/f_{W}(\cdot),$

where for a generic function $g(w)$,

$\displaystyle\mathcal{F}_{\sigma_{0}}^{-1}(g(\cdot))(t)=\frac{\sigma_{0}}{\sqrt{2\pi}}\int\exp(2\pi
it\sigma_{0}w)g(w)dw.$

By finding the two pairs $((x_{1},w),(x_{1}^{\prime},w^{\prime}))$ and
$((\tilde{x}_{1},\tilde{w}),(\tilde{x}_{1}^{\prime},\tilde{w}^{\prime}))$ such
that $w-w^{\prime}\neq\tilde{w}-\tilde{w}^{\prime}$,

$\displaystyle
F_{\eta_{1}}(x_{1}+\alpha_{0}-\gamma_{0}w)=F_{\eta_{1}}(x_{1}^{\prime}-\gamma_{0}w^{\prime}),\quad\text{and}\quad
F_{\eta_{1}}(\tilde{x}_{1}+\alpha_{0}-\gamma_{0}\tilde{w})=F_{\eta_{1}}(\tilde{x}_{1}^{\prime}-\gamma_{0}\tilde{w}^{\prime})$

we can identify both $\alpha_{0}$ and $\gamma_{0}$ as the solution of the
following linear system:

$\displaystyle\alpha_{0}+\gamma_{0}(w^{\prime}-w)=x_{1}^{\prime}-x_{1}$
$\displaystyle\alpha_{0}+\gamma_{0}(\tilde{w}^{\prime}-\tilde{w})=\tilde{x}_{1}^{\prime}-\tilde{x}_{1}.$

Online Supplement to “Informational Content of Factor Structures in
Simultaneous Binary Response Models”: Distribution Theory for Closed Form
Estimator

##  Appendix L Nonparametric Factor Structure

In this section we describe an estimator for the case where we have a
nonparametric factor structure. Recall for this model we had the following
relationship between unobservable variables:

$U=g_{0}(V)+\tilde{\Pi}$ (L.1)

where we assumed that $\tilde{\Pi}\perp V$.

Our goal in this more general setup is to identify and estimate both
$\alpha_{0}$ and $g_{0}$. Our identification is based on the condition that

$x_{1}+\alpha_{0}-g_{0}(x)=\tilde{x}_{1}-g_{0}(\tilde{x}).$

if and only if

$\partial_{2}P^{11}(x_{1},x)/f_{V}(x)+\partial_{2}P^{10}(\tilde{x}_{1},\tilde{x})/f_{V}(\tilde{x})=0.$

Using the same $i,j$ pair notation as before, this gives us, in the
nonparametric case,

$X_{1i}-X_{1j}=\alpha_{0}+(g_{0}(X_{i})-g_{0}(X_{j}))$ (L.2)

Note the above equation has a “semi parametric form”, loosely related to the
model considered in, for example, Robinson (1988). However, we point out
crucial differences between what we have above and the standard semi linear
model. Here we are trying to identify the intercept $\alpha_{0}$ which is
usually not identified in the semi linear model as it cannot be separately
identified from the nonparametric function. However, note above on the right
hand side, we do not just have a nonparametric function of $X_{i},X_{j}$, but
the difference of two identical and additively separable functions
$g_{0}(\cdot)$. In fact it is this differencing of these functions which
enables us to separately identify $\alpha_{0}$. Furthermore, as will now see
when turning to our estimator of $\alpha_{0}$, the structure of the
nonparametric component, specifically additive separability of two identical
functions of $X_{i},X_{j}$ respectively, can easily be incorporated into our
approximation of each of them. From a theoretical perspective separable
functions have the advantage of effectively being a one dimensional problem,
as there are no interaction terms to have to deal with. It is well known that
nonparametric estimation of separable functions do not suffer from the “curse
of dimensionality”. See, for example Newey (1994).

To motivate our estimator of $\alpha_{0}$ in this nonparametric factor
structure model, we consider modifying methods used to estimate the semi
linear model, which is usually expressed as

$y_{i}=x_{i}^{\prime}\beta_{0}+g(z_{i})+\varepsilon_{i}$

where $y_{i}$ denotes the observed dependent variable, $x_{i},z_{i}$ are
observed regressors, $g(\cdot)$ is an unknown nuisance function,
$\varepsilon_{i}$ is an unobserved disturbance term, and $\beta_{0}$ is the
unknown regression coefficient vector which is the parameter of interest.
There is a very extensive literature in both econometrics and statistics on
estimation and inference methods for this model- see for example Powell (1994)
for some references.

One popular way to estimate this model is to use an expansion of basis
functions, for example polynomials or splines to approximate $g(\cdot)$, and
from a random sample of $n$ observations of $(y_{i},x_{i},z_{i})$ regress
$y_{i}$ on $x_{i},b(z_{i})$ where $b(z_{i})$ denotes the set of basis
functions used to approximate $g(\cdot)$. As an illustrative example, assuming
$z_{i}$ were scalar, if one were to use polynomials as basis functions, one
would estimate the approximate model,

$y_{i}=x_{i}^{\prime}\beta_{0}+\gamma_{1}z_{i}++\gamma_{2}z_{i}^{2}+\gamma_{3}z_{i}^{3}+....\gamma_{k_{n}}z_{i}^{k_{n}}+u_{in}$

where $k_{n}$ is a positive integer smaller than the sample size $n$, and
$\gamma_{1},\gamma_{2},...\gamma_{k_{n}}$ are additional unknown parameters.
This has been done by regressing $y_{i}$ on
$x_{i},z_{i},z_{i}^{2},...z_{i}^{k_{n}}$, and our estimated coefficient of
$x_{i}$ would be the estimator of $\beta_{0}$. The validity of this approach
has been shown in, for example, Donald and Newey (1994). Now for our problem
at hand, incorporating a nonparametric factor structure, we propose a kernel
weighted least squares estimator. The weights are as they were before,
assigning great weights to pairs of observations where the sum of derivatives
of ratios of choice probabilities are closer to 0.

The dependent variable is identical to as before, the set of $n$ choose 2
pairs $X_{1i}-X_{1j}$. The regressors now reflect the series approximation of
$g_{0}(X_{i})-g_{0}(X_{j})$:

$g_{0}(X_{i})-g_{0}(X_{j})\approx\gamma_{1}(X_{i}-X_{j})+\gamma_{2}(X_{i}^{2}-X_{j}^{2})+\gamma_{3}(X_{i}^{3}-X_{j}^{3})+...\gamma_{k_{n}}(X_{i}^{k_{n}}-X_{j}^{k_{n}})$

So now our estimator would be to regress $X_{1i}-X_{1j}$ on
$1,(X_{i}-X_{j}),(X_{i}^{2}-X_{j}^{2}),...(X_{i}^{k_{n}}-X_{j}^{k_{n}})$,
using the same weights $\hat{\omega}_{ij}$ so the estimator of $\alpha_{0}$,
denoted by $\hat{\alpha}_{NP}$, would be the coefficient on 1. Specifying the
asymptotic properties of this estimator would require additional regularity
conditions, notably the rate at which the sequence of integers $k_{n}$
increases with the sample size $n$.

We again only outline these regularity conditions here, and only to establish
consistency. Since the estimator and proof strategy is very similar to that
for the closed form estimator in the online supplement to this paper, here we
only state the additional one needed for the nonparametric model in this
section.

Assumption BFC

(Basis function conditions) The basis function approximation of the unknown
factor structure function satisfies the following conditions:

BFC.1

The number of basis functions, $k_{n}$, satisfies $k_{n}\rightarrow\infty$ and
$k_{n}/n\rightarrow 0$.

BFC.2

For every $k_{n}$, the smallest eigenvalue of the matrix

$E[P_{k_{n}}P_{k_{n}}^{\prime}]$

is bounded away from 0 uniformly in $k_{n}$, where

$P_{k_{n}}\equiv(1,X_{i}-X_{j}f,X_{i}^{2}-X_{j}^{2},...X_{i}^{k_{n}}-X_{j}^{k_{n}})^{\prime}$

######  Theorem L.1.

Under Assumptions I,K, H, S, PS, FK, FH, BFC,

$\hat{\alpha}_{NP}\stackrel{{\scriptstyle p}}{{\rightarrow}}\alpha_{0}$ (L.3)

##  Appendix M Distribution Theory for Closed Form Estimator

Many of the basic arguments follow those used in Chen and Khan (2008) and
Chen, Khan, and Tang (2016). Recall what the key identification condition that
motivated the weighted least squares estimator: For pairs of observations
$(x_{1},x)$ and $(\tilde{x}_{1},\tilde{x})$ in $\text{Supp}(X_{1},X)$,

$x_{1}+\alpha_{0}-\gamma_{0}x=\tilde{x}_{1}-\gamma_{0}\tilde{x}.$

if and only if

$\partial_{2}P^{11}(x_{1},x)/f_{V}(x)+\partial_{2}P^{10}(\tilde{x}_{1},\tilde{x})/f_{V}(\tilde{x})=0.$

where recall $\partial_{2}$ denotes the partial derivative with respect to the
second argument. Note that even though the random variable $V$ is unobserved,
the density function $f_{V}(\cdot)$ above can be recovered from the data from
the partial derivative of the choice probability in the treatment equation
with respect to the regressor in the treatment equation. Thus the above
equation involves the sum of two ratios of derivatives of choice
probabilities.

Recall $\theta_{0}\equiv(\alpha_{0},\gamma_{0})$. Our estimator of
$\theta_{0}$ is based on pair of observations from the data set. We will
denote the random variables of interest with capital letters, for example
$X_{i},X_{1i}$, and realizations of them with lower letters, for example
$x_{i},x_{1i}$. To denote distinct random variables in the sample when they
form pairs, we will use the subscripts $i,j$.

Note from above, we can express the equation where the pairs receive positive
weights (those whose derivatives of choice probabilities summed up to 0) as

$x_{1i}-x_{1j}=\alpha_{0}+\theta_{0}(x_{i}-x_{j})$ (M.4)

So this motivates regressing the scalar random variable $x_{1i}-x_{1j}$ on the
two by one random vector ${\bf x}_{ij}\equiv(1,x_{i}-x_{j})$. We can now see
that if sufficient such pairs of observations, where the sum of the ratio of
derivative of probabilities could be found to equal 0, $\theta_{0}$ could be
recovered as the unique solution to the system of equations corresponding to
the pairs, as long as the matrix involving the terms ${\bf x}_{ij}$ satisfied
a full rank condition. Such an approach is infeasible for two reasons. The
first reason is that the probability functions, their derivatives, and hence
the ratio of derivatives are unknown. The second reason is that even if these
functions were known, if the probability functions are not discrete valued,
such “matches” will occur with probability zero.

The first problem can be remedied by replacing the true probability function
values with their nonparametric estimates. In the theory here we used a kernel
estimator with kernel function $K(\cdot)$ and bandwidth $H_{n}$, whose
properties are discussed below. The second problem can be dealt with through
the use of “kernel weights” as has been frequently employed in the
semiparametric literature.

Specifically, assuming that the ratio of derivatives of conditional
probability functions were known, we use the following weighting function for
pairs of observations; to illustrate let $P^{k,l,r}$, $k=0,1,l=0,1$ denotes
the ratio of derivatives of choice probabilities. So, for example,
$P^{1,1,r}=\partial_{2}P^{11}(X_{1},X)/f_{V}(X)$ , where $\partial_{2}$
denotes the partial derivative with respect to the second argument. Let
$p^{1r}_{i},p^{0r}_{j}$ denote the $i^{th},j^{th}$ realizations of
$P^{1,1,r},P^{1,0,r}$ respectively; then

$\omega_{ij}=\frac{1}{h_{n}}k\left(\frac{p^{1r}_{i}+p^{0r}_{j}}{h_{n}}\right)$
(M.5)

In (M.5) $h_{n}$ is a bandwidth sequence, which converges to zero as the
sample sizes increases, ensuring that in the limit, only pairs of observations
with probability functions summing up to an arbitrarily small number receive
positive weight. $k(\cdot)$ is the kernel function, which is symmetric around
0, and assumed to have compact support, integrate to 1, and satisfy certain
smoothness conditions discussed later on.

With the weighting matrix defined, a natural estimate of it,
$\hat{\omega}_{ij}$ follows from replacing the true probability function
values with their nonparametric, e.g. kernel, estimates. This suggests a
weighted least squares estimator of $\theta_{0}\equiv(\alpha_{0},\gamma_{0})$,
regressing $x_{1i}-x_{1j}$ on ${\bf x}_{ij}$, with weights
$\hat{\omega}_{ij}$.

Specifically, we propose the following two stage procedure. The first stage is
the kernel estimator of the ratio of derivatives of probability functions, and
the second stage estimator is defined as:

$\hat{\theta}=(\sum_{i\neq j}\tau_{i}\tau_{j}\hat{\omega}_{ij}{\bf x}_{ij}{\bf
x}_{ij}^{\prime})^{-1}(\sum_{i\neq j}-\tau_{i}\tau_{j}\hat{\omega}_{ij}{\bf
x}_{ij}\Delta x_{1ij})$ (M.6)

where $\Delta x_{1ij}\equiv x_{1i}-x_{1j}$, ${\bf
x}_{ij}\equiv(1,x_{i}-x_{j})$ and $\tau_{i}\equiv\tau(x_{1i},x_{i})$ is a
trimming function to remove observations where regressors take values near the
boundary of its support.

We will outline the asymptotic properties of this estimator. Here we use
similar arguments to this used in Chen and Khan (2008) and keep our notation
as close as possible to that used in that paper. To simplify characterizing
the asymptotic properties of this estimator and the regularity conditions we
impose, we first define the following functions of $P^{k,l,r}$ for
$k=l=1,k=1,l=0$ at their $i^{th}$ and $j^{th}$ realized values, denoted by
$p^{1r}_{i},p^{0r}_{j}$

  1. 1.

$f_{P^{k,l,r}_{0}}=f_{P^{k,l,r}_{0}}(P^{k,l,r}_{0i})$, where
$f_{P^{k,l,r}_{0}}(\cdot)$ denotes the density function of $P^{k,l,r}_{0i}$.

  2. 2.

$\mu_{\tau i}=E\left[\tau_{i}|P^{k,l,r}_{0i}\right]$

  3. 3.

$\mu_{\tau xi}=E\left[\tau_{i}\tilde{X}_{i}|P^{k,l,r}_{0i}\right]$

  4. 4.

$\mu_{\tau
xxi}=E\left[\tau_{i}\tilde{X}_{i}\tilde{X}_{i}^{\prime}|P^{k,l,r}_{0i}\right]$

$\mu_{1}(p^{1r}_{i},p^{0r}_{j})\equiv E[{\bf x}_{ij}{\bf
x}_{ij}^{\prime}|p^{1r}_{i},p^{0r}_{j}]$ where ${\bf x}_{i}$ denotes the
$2\times 1$ vector $(1,x_{i})$, $\mu_{0}(p^{0r}_{j})\equiv E[{\bf
x}_{j}|p^{0r}_{j}]$, where ${\bf x}_{j}$ denotes the $2\times 1$ vector
$(1,x_{j})$, $f_{1}(\cdot)$ denotes the density function of the random
variable $P^{1,1,r}$, $f_{0}(\cdot)$ denotes the density function of the
random variable $P^{1,0,r}$.

Our derivation of the asymptotic properties of this estimator are based on the
following assumptions111For notational convenience here we suppress the
presence of the trimming function.:

Assumption I

(Identification) The $2\times 2$ matrix:

$M_{1}=E\left[\mu_{1}(p^{1r}_{i},-p^{1r}_{i})^{\prime}f_{0}(-p^{1r}_{i})\right]$

has full rank.

Assumption K

(Second stage kernel function) The kernel function $k(\cdot)$ used in the
second stage (to match the sum of ratios of derivatives to 0) is assumed to
have the following properties:

K.1

$k(\cdot)$ is twice continuously differentiable, has compact support and
integrates to 1.

K.2

$k(\cdot)$ is symmetric about 0.

K.3

$k(\cdot)$ is an eighth order kernel:

$\displaystyle\int u^{l}k(u)du$ $\displaystyle=$ $\displaystyle 0\ \ \mbox{for
}l=1,2,3,4,5,6,7$ $\displaystyle\int u^{8}k(u)du$ $\displaystyle\neq$
$\displaystyle 0$

Assumption H

(Second stage bandwidth sequence) The bandwidth sequence $h_{n}$ used in the
second stage is of the form:

$h_{n}=cn^{-\delta}$

where $c$ is some constant and $\delta\in(\frac{1}{16},\frac{1}{12})$.

Assumption S

(Order of Smoothness of Density and Conditional Expectation Functions)

S.1

The functions $P^{k,l,r}$ are eighth order continuously differentiable with
derivatives that are bounded on the support of $\tau_{i}$.

S.2

The functions $f_{P^{k,l,r}_{0}}(\cdot)$ (the density function of the random
variable $P^{k,l,r}$) and $E[{\bf x}_{i}|P^{k,l,r}=\cdot]$, where ${\bf
x}_{i}$ denotes the $2\times 1$ vector $(1,x_{i})$ have order of
differentiability of 8, with eight order partial derivatives that are bounded
on the support of $\tau_{i}$.

The final set of assumptions involve restrictions for the first stage kernel
estimator of the ratio of derivatives. This involves smoothness conditions on
the choice probabilities $P^{k,l,r}_{0i}$, smoothness and moment conditions on
the kernel function, and rate conditions on the first stage bandwidth
sequence.

Assumption PS

(Order of smoothness of probability functions and regressor density functions)
The functions $P^{k,l,r}(\cdot)$ and $f_{X_{1},X}(\cdot.\cdot)$ (the density
function of the random vector $(X_{1},X)$) are continuously differentiable of
order $p_{2}$, where $p_{2}>{5}$.

Assumption FK

(First stage kernel function conditions) $K(\cdot)$, used to estimate the
choice probabilities and their derivatives is an even function, integrating to
1 and is of order $p_{2}$ satisfying $p_{2}>{5}$.

Assumption FH

(Rate condition on first stage bandwidth sequence) The first stage bandwidth
sequence $H_{n}$ is of the form:

$H_{n}=c_{2}n^{-\gamma/k}$

where $c_{2}$ is some constant and $\gamma$ satisfies:

$\gamma\in\left(\frac{2}{p_{2}}\left(\frac{1}{3}+\delta\right),\frac{1}{3}-2\delta\right)$

where $\delta$ is regulated by Assumption H.

######  Theorem M.1.

Let

$\psi_{i}=\psi_{1i}+\psi_{2i}+\psi_{3i}+\psi_{4i}$ (M.7)

where $\psi_{ji}\ \ \ j=1-4$ are each mean 0 random variables defined in
equations M.18,M.22,M.25,M.27, respectively, then under Assumptions
I,K,H,S,PS,FK,FH,

$\sqrt{n}(\hat{\theta}-\theta_{0})\Rightarrow N(0,M_{1}^{-1}V_{1}M_{1}^{-1})$
(M.8)

where

$V_{1}=E[\psi_{i}\psi_{i}^{\prime}]$ (M.9)

Proof: Let ${\bf x}_{ij}\equiv(1,(x_{i}-x_{j})),\Delta x_{1ij}\equiv
x_{1i}-x_{1j}$. Then we can express:

$\hat{\theta}-\theta_{0}=\left(\frac{1}{n(n-1)}\sum_{i\neq j}\hat{w}_{ij}{\bf
x}_{ij}{\bf x}_{ij}^{\prime}\right)^{-1}\frac{1}{n(n-1)}\sum_{i\neq
j}\hat{w}_{ij}{\bf x}_{ij}(\Delta x_{1ij}-{\bf x}_{ij}^{\prime}\theta_{0})$

We will first derive a plim for the denominator term and the a linear
representation for the numerator. For the denominator term here we aim to
establish that the double sum $\frac{1}{n(n-1)}\sum_{i\neq j}\hat{w}_{ij}{\bf
x}_{ij}{\bf x}_{ij}^{\prime}$ converges in probability to the $2\times 2$
matrix $M_{1}$. To do so, note by Assumption $K.1$ we can expand
$\hat{w}_{ij}$ around $w_{ij}$. The remainder term involves the difference
between the nonparametrically estimated derivative functions and the true
derivative functions. By Assumptions $K,H,S$ this remainder term is uniformly
(over the support of the trimming function $\tau(\cdot)$) $o_{p}(1)$\- see
e.g. Henderson, Li, Parmeter, and Yao (2015). It thus suffices to establish
the probability limit of $\frac{1}{n(n-1)}\sum_{i\neq j}w_{ij}{\bf x}_{ij}{\bf
x}_{ij}^{\prime}$. To do so we first wish to determine the functional form of
its expectation. For notational ease here we let $p^{1r}_{i},p^{0r}_{j}$
denote $i^{th}$ and $j^{th}$ realized values of $P^{1,1,r},P^{1,0,r}$
respectively, and $\hat{p}^{1r}_{i},\hat{p}^{0r}_{j}$ denote their
nonparametric estimators. Following the same arguments as in Chen and Khan
(2008), Chen, Khan, and Tang (2016), we can write the expectation of
$w_{ij}{\bf x}_{ij}{\bf x}_{ij}^{\prime}$ as

$\int
k((p^{1r}_{i}+p^{0r}_{j})/h_{n})/h_{n}\mu_{1}(p^{1r}_{i},p^{0r}_{j}))f_{1}(p^{1r}_{i})f_{0}(p^{0r}_{j})dp^{1r}_{i}dp^{0r}_{j}$

where $\mu_{1}(p^{1r}_{i},p^{0r}_{j})\equiv E[{\bf x}_{ij}{\bf
x}_{ij}^{\prime}|p^{1r}_{i},p^{0r}_{j}]$, $f_{1}(\cdot)$ denotes the density
function of the random variable $P^{1,1,r}$, $f_{0}(\cdot)$ denotes the
density function of the random variable $P^{1,0,r}$. Changing variables
$u=(p^{1r}_{i}+p^{0r}_{j})/h_{n}$ and taking limits as $h_{n}\rightarrow 0$,
yields that the above integral is

$\int\mu_{1}(p^{1r}_{i},-p^{1r}_{i})f_{1}(p^{1r}_{i})f_{0}(-p^{1r}_{i})dp^{1r}_{i}=E\left[\mu_{1}(p^{1r}_{i},-p^{1r}_{i})f_{0}(-p^{1r}_{i})\right]$

which is $M_{1}$. We next turn attention to the numerator term. This term is
of the form:

$\frac{1}{n(n-1)}\sum_{i\neq j}\hat{w}_{ij}{\bf x}_{ij}(\Delta x_{1ij}-{\bf
x}_{ij}^{\prime}\theta_{0})$

Again, we expand $\hat{w}_{ij}$ around $w_{ij}$. The lead term in this
expansion is of the form:

$\frac{1}{n(n-1)}\sum_{i\neq j}w_{ij}{\bf x}_{ij}(\Delta x_{1ij}-{\bf
x}_{ij}^{\prime}\theta_{0})$

Note that because $p^{1r}_{i}+p^{0r}_{j}=0\Rightarrow\Delta x_{1ij}={\bf
x}_{ij}^{\prime}\theta_{0})$ from our identification result, it follows from
Assumptions K,H that the lead term is $o_{p}(n^{-1/2})$. The linear term in
the expansion is of the form

$\frac{1}{n(n-1)}\sum_{i\neq
j}w_{ij}^{\prime}((\hat{p}^{1r}_{i}-p^{1r}_{i})+(\hat{p}^{0r}_{j}-p^{0r}_{j})){\bf
x}_{ij}(\Delta x_{1ij}-{\bf x}_{ij}^{\prime}\theta_{0})$ (M.10)

We will first focus on the term

$\frac{1}{n(n-1)}\sum_{i\neq
j}w_{ij}^{\prime}(\hat{p}^{1r}_{i}-p^{1r}_{i}){\bf x}_{ij}(\Delta x_{1ij}-{\bf
x}_{ij}^{\prime}\theta_{0})$ (M.11)

Recall $\hat{p}^{1r}_{i}$ denotes a ratio of non parametrically estimated
terms and $p^{1r}_{i}$ denotes the ratio of derivatives. Denote these
estimated and true ratios as $\hat{f}_{vi}^{-1}\hat{p}^{1}_{i}$ ,
$f_{vi}^{-1}p^{1}_{i}$ respectively. Linearizing this ratio, the first term is