# Modelling with Discretized Variables

Felix Chan† László Mátyás‡ and Ágoston Reguly111Corresponding author; e-mail:
<EMAIL_ADDRESS>
All simulation codes are publicly available at
https://github.com/regulyagoston/Split-sampling

###### Abstract

This paper deals with econometric models in which the dependent variable, some
explanatory variables, or both are observed as censored interval data. This
discretization often happens due to confidentiality of ‘sensitive’ variables
like income. Models using these variables cannot point identify regression
parameters as the conditional moments are unknown, which led the literature to
use interval estimates. Here, we propose a discretization method through which
the regression parameters can be point identified while preserving data
confidentiality. We demonstrate the asymptotic properties of the OLS estimator
for the parameters in multivariate linear regressions for cross-sectional
data. The theoretical findings are supported by Monte Carlo experiments and
illustrated with an application to the Australian gender wage gap.

†Curtin University

‡Central European University

∗Corvinus University of Budapest and Georgia Institute of Technology

JEL: C01, C13, C21, C25, C81, C83

Keywords: Discretization, interval censored variable, semiparametric
identification, split sampling, data confidentiality.

##  1 Introduction

Over the last decade, there has been an explosion in the amount of data
collected on individuals, firms, the economy, and society. Governments and the
private sector gather large amounts of information that researchers and
analysts use on a daily basis. Privacy constraints often prevent access to
sensitive data in its original form. One common practice is to discretize
sensitive data and provide observations that refer to intervals rather than
the actual values.222There are other practices to preserve data privacy, such
as data perturbation, data swapping, or cryptography. Duncan et al., (2001)
and Halevi and Rabin, (2006) provide nice overviews on this issue. One of the
most widely used discretized variables is income. Instead of the actual weekly
personal income, one usually works with variables such as: ‘weekly personal
income below $\$100$, between $\$100$ and $\$400$ or above $\$400$.

In this paper, we investigated the effect of such discretization from three
different perspectives, depending on where such variable appears in a model:
i) the discretized variable is an explanatory variable; ii) the discretized
variable is the outcome variable; and iii) discretization occurs on both sides
of the model. We rely on the early results of Manski and Tamer, (2002), which
show that the conditional expectation function and the regression parameters
in such context cannot be point identified in general.333To be precise, Manski
and Tamer, (2002) shows this for i) and ii), but not for iii). We show the
conditions under which the regression parameters can be identified and propose
a method that is suitable for estimating these parameters, even in finite
samples, while preserving the privacy requirements.444When we mention
sensitive data/privacy requirements, we always refer to non-identifiable data
on the variable level, not personal-identity protection.

The key to our approach is the use of multiple discretization schemes. In the
context of income, instead of using only one discretization scheme (call it
(1)): below $\$100$, between $\$100$ and $\$400$ or above $\$400$, we propose
using others by changing the interval boundaries. For example, discretization
scheme (2): below $\$50$, between $\$50$ and $\$350$ or above $\$350$;
discretization scheme (3): below $\$150$, between $\$150$ and $\$450$ or above
$\$450$ etc. By using multiple discretization schemes, which we call split
sampling,555The term split sampling in this paper is not related to the
technique occasionally used in chromatography (Schomburg et al.,, 1977,
Schomburg et al.,, 1981) or methods in machine learning, which splits the
initial sample into folds. we show convergence in distribution as the number
of observations with the number of discretization schemes goes to infinity. We
also derive the speed of convergence for our method, which shows that even a
few discretization schemes are sufficient to properly approximate the
underlying distribution of the discretized variable in finite samples.

Our identification strategy is based on this convergence in distribution, and
we show how to use this result to point identify the parameters of interest
for all three regression setups. The general idea is to use appropriate
expectations conditional on the discretization intervals for the explanatory
variable(s) or for both outcome and explanatory variables. By transforming our
data, we can point identify the parameters of interest from the original
unknown model. We introduce our notations using univariate cases and then
extend them to multivariate settings. We propose different estimators for each
conditional expectation that can be used to replace the observed data. We show
that these estimators are consistent and derive their asymptotic properties.
After replacing the discretized observations with conditional means, we show
that the OLS estimator provides a consistent estimator of the parameter of
interest, and we derive asymptotic properties for all three cases in the
cross-sectional setup. We extend our discussion to nonlinear models and panel
data. We provide Monte Carlo evidence and apply our procedure to wage data
from the Australian Tax Office (ATO) to investigate the gender wage gap in
Australia.

Although our lead example refers to sensitive data considerations, our method
does not stop here but goes beyond all applications where discretization
occurs. Another example is paper or online surveys. These questionnaires use a
fixed number of possible choices for each question to ensure a higher response
rate.666There is extensive research on the response rates with income and
other socioeconomic factors. See e.g., Moore et al., (2000), Schwartz and
Paulin, (2000), Shah et al., (2001), Riphahn and Serfling, (2005) or Takada et
al., (2014). The Australian Household, Income, and Labour Dynamics in
Australia (HILDA) Survey,777https://melbourneinstitute.unimelb.edu.au/hilda is
a good example of this survey, which collects data on personal income as an
alternative for ATO data, to which only a few people have access. To provide a
few more academic examples, Bhat, (1994) shows the effects of age, employment,
and other socioeconomic variables on income, where income is observed through
three different categories: Micklewright and Schnepf, (2010) compares
individual and household income distributions, controlling for age, gender,
and employment. Income was discretized and observed using single-question
surveys. As a final example, let us mention the methodology of multiple price
lists from experimental economics (see, e.g., Andersen et al., 2006, Andersen
et al., 2008, Wakker, 2010 or Yu et al., 2021). This method has been
successfully used to elicit values from a subject (e.g., willingness to pay
for certain objects) in a transparent manner while providing simple incentives
for truthful revelation. The main disadvantage of this method is that it only
elicits interval responses, which makes the estimation of the elasticity
parameter nontrivial. Our method applies in such settings, ensuring a
consistent estimator for the elasticity parameter.

This paper builds on Manski and Tamer, (2002), who show that the parameters of
a regression model cannot be point identified without any further assumptions
when an interval censored variable888Let us note here that there is a
substantial difference between interval and ordinal variables. The interval
variables have known lower and upper boundaries for each interval, and numeric
intervals can be assigned for each observation. In principle, an interval
variable has a (conditional) expectation; however, it cannot be directly
estimated using discretized data. The information provided by this type of
data includes the lower and upper bounds of each category and their
frequencies. An ordinal variable embodies even more severe information loss
relative to the underlying continuous variable. The observed values have only
a relational connection with each other (e.g., they are higher or lower), they
are sorted into classes, and numerical values cannot be assigned to the
observations. Data from these qualitative variables cannot be used to estimate
any conditional moments of the underlying random variables, and can only be
used to obtain the frequencies of each class. In this paper, we only deal with
interval variables. is used as the dependent or explanatory variable. To
circumvent this identification problem, there are a few solutions, all of
which consider the discretization process as given. These estimation methods
depend on where discretization occurs (outcome, explanatory, or both). Here,
we briefly discuss these methods.

Discretized outcome variable: ordered choice models, interval regressions, and
partially identified parameters. Ordered discrete choice models (Greene and
Hensher,, 2010) such as ordered logit and probit models, are the most popular
solutions in the applied literature. These models can handle interval (and
ordinal) variables and aim to predict or classify the dependent variable
instead of (parameter) interpretation. The key to these models is that,
instead of modeling the conditional expectation, they focus on the conditional
probabilities for each interval (e.g., the probability of an observation
falling into a certain class given a set of covariates). Ordinal choice models
use apriori distributional assumptions to create a mapping between the outcome
variable and explanatory variables. This can be a strong assumption,
particularly in the absence of probabilistic justification. These models tend
to have their names based on the assumed distribution.999E.g., ordered logit
uses a (standardized) logistic distribution, whereas ordered probit uses
(standard) normal distribution. Under the assumed distribution, along with
some mild conditions, the parameters are generally point identifiable up to
scaling.101010‘…neither the sign nor the magnitude of the coefficient is
informative […], so the direct interpretation of the coefficients is
fundamentally ambiguous.’ (Greene and Hensher,, 2010, p. 142). Lewbel, (2019)
calls this type of identification normalization in identification, as the
parameters are point identified; however, they are dependent on the
normalization used. To get meaningful interpretations, it is common to
calculate the (average) partial effects on the probabilities using the assumed
distribution. Greene and Hensher, (2010) gives a thorough overview of ordered
choice models estimated via maximum likelihood.

Interval regressions are extensions of ordered choice models because they are
augmented with information on the observed interval boundaries. They aimed to
model conditional expectations using distributional assumptions for point
identification. Abrevaya and Muris, (2020) discusses the use of fixed effects
in such contexts and extends the fully parametric approach to semiparametric
models. The difficulties and drawbacks of this approach are summarized in
Greene and Hensher, (2010, p. 133) or Cameron and Trivedi, (2010).

The third approach comes from the literature on partially identified
parameters (e.g., see Manski and Tamer, 2002, Manski, 2003, Tamer, 2010,
Molinari, 2020, and Wang and Chen, 2022). This approach focuses on interval
data and assigns numerical intervals to each observation. The main advantage
of this method is that it does not require distributional assumptions and
allows valid statistical inferences on the parameters of interest. The
magnitudes and signs of the estimated parameter vector can be interpreted in
the same way as the classical regression coefficients.111111Moment
(in)equality models generalize Manski and Tamer, (2002) for cases where there
are multiple equations and/or inequalities (i.e., Chernozhukov et al., (2007)
or Andrews and Soares, (2010)). Beresteanu and Molinari, (2008) show the
asymptotic properties of such partially identified parameters. Imbens and
Manski, (2004), Chernozhukov et al., (2007), and Kaido et al., (2019), among
others, derive confidence intervals for these set-identified parameters. These
methods are feasible ways to estimate parameter sets for a given conditional
expectation function without any further assumptions. The drawback is that the
estimated parameters are not point identified but rather a set to which the
parameter vector belongs.

Discretized explanatory variable: partially identified parameters, a two-
sample approach, dummy variables, and mid-point regression. Similar to our
previous case, Manski and Tamer, (2002) proposes partial identification and
assigns an interval for the parameters of interest. They use a modified
minimum-distance estimator to estimate these intervals. The two-sample
approach uses a second sample that is independent of the outcome variable but
not of the discretized one. In the spirit of IV estimation, Klevmarken, (1982)
and Ridder and Moffitt, (2007) propose methods that yield consistent and
normally distributed parameter estimates. Pacini, (2019) introduces
semiparametrically efficient versions of this approach. Another commonly used
method is to create a dummy variable for each interval and use it during
estimation. Although it is empirically a legitimate solution when one wants to
‘control’ such variable(s), it does not provide the same conditional
expectations and parameter(s) such as elasticities, which is the main goal of
this study. Finally, the mid-point regression substitutes the interval data
with the intervals’ midpoints. In this paper, we show that this generally
leads to biased parameter estimates.

Discretization on both sides: we are not aware of any research that discusses
this case. In practice, one can typically see a mixture of previous solutions,
such as ordinal choice models with dummy variables or the mid-point approach.

The paper is organized as follows: Section 2 introduces the identification
problem for all three cases and highlights the transformation of the
conditional expectation function to point identify the parameters of interest
under the assumption that the marginal distribution of the discretized
variable is known. Section 3 describes the proposed split-sampling approach.
We discuss the shifting method in detail and show convergence in distribution
to the unknown marginal distribution. In Section 4 we derive the asymptotic
properties of the least square estimator for all possible cases in a
multivariate linear regression setup. We also cover how to extend our method
to nonlinear models and discuss implications with panel data and fixed-effect
type of estimators. Section 5 presents Monte Carlo evidence along with Section
6 where we show an empirical application on the Australian gender wage gap.
Section 7 concludes.

##  2 The Identification Problem

Consider $Z\sim f_{Z}(z;a_{l},a_{u})$ an i.i.d. random variable, where
$f_{Z}(z;a_{l},a_{u})$ denotes the probability density function (pdf) with
support $[a_{l},a_{u}]$, where $a_{l},a_{u}\in\mathbb{R},\,a_{l}<a_{u}$. We
assume that $f_{Z}(\cdot)$ is unknown and continuous. Let $Z_{i}$ be
$i=1,\ldots,N$, realizations of $Z$. Instead of observing $Z_{i}$, we observe
$Z^{*}_{i}$ through a discretization process:

$Z^{*}_{i}=\begin{cases}v_{1}&\quad\text{if }c_{0}\leq
Z_{i}<c_{1}\quad\text{or}\quad Z_{i}\in
C_{1}=\left[c_{0},c_{1}\right)\quad\text{1st interval}\\\ v_{2}&\quad\text{if
}c_{1}\leq Z_{i}<c_{2}\quad\text{or}\quad Z_{i}\in
C_{2}=\left[c_{1},c_{2}\right)\\\ \vdots&\quad\vdots\\\ v_{m}&\quad\text{if
}c_{m-1}\leq Z_{i}<c_{m}\quad\text{or}\quad Z_{i}\in
C_{m}=\left[c_{m-1},c_{m}\right)\\\ \vdots&\quad\vdots\\\ v_{M}&\quad\text{if
}c_{M-1}\leq Z_{i}<c_{M}\quad\text{or}\quad Z_{i}\in
C_{M}=\left[c_{M-1},c_{M}\right)\\\ &\qquad\qquad\qquad\qquad\hfill\text{last
interval},\\\ \end{cases}$ (1)

where, $v_{m}\in C_{m}$, $m=1,\dots,M$ is the assigned value for each
interval. It can be a measure of centrality (e.g., mid-point), or an
arbitrarily assigned value within its interval. $M$ denotes the number of
intervals that are set or known. For simplicity, we use the terms interval and
class interchangeably from now on. Let us highlight two special cases for the
assigned values $v_{m}$. In the first case, we assign the lower bounds
$v_{m}=c_{m-1},\,\,\forall m$, and we note the random variable
$\underline{Z}_{i}^{*}$, while for the second case $\bar{Z}_{i}^{*}$ takes the
upper bounds $v_{m}=c_{m},\,\,\forall m$. By the design of the discretization,
the unobserved values lie between these lower and upper bounds,
$\underline{Z}^{*}_{i}\leq Z_{i}\leq\bar{Z}_{i}^{*}\,,\,\forall i$.121212Note,
when we use underline or bar for a variable, we always refer to discretized
variable, that takes the lower or upper boundaries, regardless if it is a
scalar, a vector or a matrix.

In this paper, we are interested in the regression parameter $\beta$, where
$g(\cdot)$ is a known continuous function,

$\mathbb{E}\left[Y|X\right]=g(X;\beta)\,.$ (2)

For the sake of simplicity, we introduce our notation in scalar terms; $\beta$
stands for a scalar parameter belonging to a subset of a compact finite-
dimensional space $(\mathcal{B})\subset\mathbb{R}$, while $Y,X$ are scalars.
Facing the discretization of a variable can affect the identification of
$\beta$ in three ways. In the first and simplest case, the explanatory
variable $(X^{*})$ is discretized; the second possibility is when the outcome
variable $(Y^{*})$ is discretized; and lastly, where both variables are
discretized $(Y^{*},X^{*})$. In all cases, we highlight the discretized
variable with an asterisk in the superscript. If we are agnostic about which
variable has been discretized, we refer to using $Z$ for the unknown or
$Z^{*}$ for the discretized version. After deriving the univariate cases, we
extend our argument to multivariate scenarios.

Our road map to identify $\beta$ is as follows:

  1. 1.

We show that if the unknown variables are not directly observed, only set
identification is possible, aligned with the results of Manski and Tamer,
(2002). We discuss which assumptions are needed for set identification.

  2. 2.

We show that if the marginal distribution $f_{Z}(\cdot)$ is known, then under
some mild conditions, we can point identify $\beta$, while preserving the
discretization process.131313We do not require the same set of assumptions
that are needed for set-identification.

    1. 2.a)

We acknowledge that we do not know the marginal distribution for $Z$, thus we
identify it via the split-sampling method. Section 3 describes how to
discretize $Z$ to achieve convergence in distribution:
$Z^{*}\overset{d}{\rightarrow}Z$. This is the nonparametric identification
step of $f_{Z}(\cdot)$.

    2. 2.b)

As a second step, based on $f_{Z}(\cdot)$ we identify $\beta$ for each
different discretiation case.

We express the identification of $\beta$ as point identification for
simplicity. However, if we are precise, we carry out semiparametric
identification, as in the first step we identify the distribution function and
in the second case the scalar (or vector) for $\beta$. (See more on
identification in, e.g., Lewbel, 2019.)

###  2.1 Identification with $X^{*}$

The first case focuses on what happens if the discretized variable is on the
right-hand side of the regression, thus we only observe $X^{*}$, the
discretized version of $X$ from Equation 2.

We know $\underline{X}^{*}\leq X\leq\bar{X}^{*}$ and as Manski and Tamer,
(2002) shows141414See Proposition 1 in Manski and Tamer, (2002)., if
$\mathbb{E}\left[Y|X\right]$ exists, weakly monotonic in $X$ and
$\mathbb{E}\left[Y|X\right]=\mathbb{E}\left[Y|X;c_{m}\right]$, thus the
conditional expectation is mean-independent from the boundary points used in
the discretization,

$\mathbb{E}\left[Y|\underline{X}^{*}\right]\leq\mathbb{E}\left[Y|X\right]\leq\mathbb{E}\left[Y|\bar{X}^{*}\right].$

Remark 1: if $M\to\infty\implies\underline{X}^{*}=X=\bar{X}^{*}$, we have the
canonical – non-discretized – case, that allows for point identification.

However, we do not observe $X$ directly, only through a discretization
process, thus point identification is not possible in general, which led the
literature to set-identification.151515Under Equation 2, any
$\mathrm{b}\in\mathcal{B}$ that satisfies $g(\underline{X};\mathrm{b})\leq
g(X;\beta)\leq g(\bar{X};\mathrm{b})$ is said to be observationally equivalent
to $\beta$. See more on the terminology of observational equivalence in
Chesher and Rosen, (2017) and Lewbel, (2019). In this paper, we maintain the
assumptions, $\underline{X}^{*}<\bar{X}^{*}$, $g(\cdot)$ is continuous and if
split sampling can recover the marginal distribution for $X$, then we can show
that $\beta$ is point identified. The argument is simple, we can take the
conditional expectation of Equation 2 with respect to $X\in C_{m}$ for both
sides, and with the law of iterated expectations, we get

$\mathbb{E}\left[Y|X\in
C_{m}\right]=\mathbb{E}\left[g\left(X;\beta\right)|X\in C_{m}\right]\,.$

This expression simplifies further in case $g(\cdot)$ is linear,

$\mathbb{E}\left[Y|X\in C_{m}\right]=g\left(\mathbb{E}\left[X|X\in
C_{m}\right];\beta\right)\,.$ (3)

Equation 3 shows that knowing the marginal distribution of $X$ is enough to
point identify the same $\beta$ parameter, but instead of $Y$ and $X$, one can
use different conditional expectations on both sides. We show the derivations
and discuss the general case, when $g(\cdot)$ is nonlinear in the online
supplement,161616https://regulyagoston.github.io/papers/MDV_online_supplement.pdf
Section A.1.

###  2.2 Identification with $Y^{*}$

In the second case, we want to measure variation in the discretized variable
and infer the elasticity parameter $\beta$. The conditional expectation
function for the unknown $Y$ variable is given by Equation 2, and assumed to
exist.

$\underline{Y}^{*}$ and $\bar{Y}^{*}$ denote the random variables that take
the lower and upper bounds as the choice value in a given interval,
respectively. Furthermore, as these refer to the same random variable, the
unknown conditional probabilities are the same $\Pr\left[\underline{Y}^{*}\in
C_{m}|X\right]=\Pr\left[Y\in C_{m}|X\right]=\Pr\left[\bar{Y}^{*}\in
C_{m}|X\right]\,,\,\forall m$.171717We use $\Pr\left[\cdot\right]$ to
emphasize discrete probabilities and use $f(\cdot)$ for the unknown continuous
pdf. Using the law of iterated expectation, it can be shown that

$\mathbb{E}\left[Y|X\right]=\sum_{m}\left[\int_{c_{m-1}}^{c_{m}}y\,\,f_{Y|X}\left(y|x\right)dy\right]\Pr\left[Y\in
C_{m}|X\right],\\\ $ (4)

where $\mathbb{E}(Y|Y\in
C_{m},x)=\int_{c_{m-1}}^{c_{m}}yf_{Y|X}\left(y|x\right)dy$, and by design
$c_{m-1}\leq\mathbb{E}(Y|Y\in C_{m},X)\leq c_{m}\,,\,\forall m$. Equation 4 is
useful as it shows that we can point identify $\mathbb{E}[Y|X]$, only if we
know $\mathbb{E}(Y|Y\in C_{m},X)$. However, we do not observe $Y$ directly,
only through a discretization process, thus point identification is not
possible in general. This fact guides to set-identification181818The same
result can be found in Manski, (1989) or Manski and Tamer, (2002). and can be
seen after plugging in $\underline{Y}^{*}$ and $\bar{Y}^{*}$ into Equation 4,

$\mathbb{E}\left[\underline{Y}^{*}|X\right]\leq\mathbb{E}\left[Y|X\right]\leq\mathbb{E}\left[\bar{Y}^{*}|X\right].$
(5)

This bound reduces to a point in the limit when $Y$ is measured (or observed)
precisely or, as we argue in this paper if the marginal distribution of $Y$ is
learned.

Remark 2: When the density of the conditional probability,
$f_{Y|X}\left(y|x\right)$ is assumed to be known, $\beta$ is point identified
up to scaling, which leads to the use of ordered choice models or interval
regressions (see, e.g., Greene and Hensher,, 2010 or Cameron and Trivedi,,
2010).

Let us show that one does not need to know the exact values of $Y$ nor impose
(conditional) distributional assumption to identify $\beta$. We maintain the
assumptions that $\underline{Y}^{*}<\bar{Y}^{*}$ and $g(\cdot)$ is a
continuous function and let us assume that the marginal distribution of $Y$
via split sampling is known. First, we take the conditional expectation of
both sides of Equation 5, that is $\mathbb{E}\left[\cdot|X\in D_{l}\right]$,
where $D_{l}$ denotes a set containing $l=1,\ldots,L$ mutually exclusive
partitions of the domain of $X$. After some manipulation, for linear functions
$g(\cdot)$, one can show

$\mathbb{E}\left[Y|X\in D_{l}\right]=g(\mathbb{E}[X|X\in D_{l}];\beta)\,\,.$
(6)

As $\mathbb{E}\left[Y|X\in D_{l}\right]=\sum_{l}\sum_{m}\mathbb{E}[Y|Y\in
C_{m},X\in D_{l}]\Pr[X\in D_{l}]$, we got a measurable quantity on both sides
of the equation, which uses known quantities and the nonparametrically
estimated marginal distribution of $Y$. See the extension to nonlinear
functional form and derivations in the online supplement, Section A.1.

###  2.3 Identification with $Y^{*}$ and $X^{*}$

Finally, let us consider the case where both the outcome and the explanatory
variables are discretized, thus instead of $Y$ and $X$ from Equation 2, we
only observe $Y^{*}$ and $X^{*}$.

Following the same logic as in Equation 4, we can draw inequalities for the
conditional expectation functions with discretized variables. Let us assume
$\mathbb{E}|Y|X|<\infty$, monotonicity: $\mathbb{E}\left[Y|X\right]$ is weakly
monotonic in $X$; and mean-independence:
$\mathbb{E}\left[Y|X\right]=\mathbb{E}\left[Y|X;c_{m_{Y}},c_{m_{X}}\right]$ to
hold, where $c_{m_{X}}$ and $c_{m_{Y}}$ are the discretization interval
boundaries for $X$ and $Y$. Then one can derive lower and upper bounds for the
conditional expectation function:

$\mathbb{E}\left[\underline{Y}^{*}|\underline{X}^{*}\right]\leq\mathbb{E}\left[Y|X\right]\leq\mathbb{E}\left[\bar{Y}^{*}|\bar{X}^{*}\right],$

where $\underline{Y}^{*},\underline{X}^{*}$ are the random variables with the
lower interval boundaries and $\bar{Y}^{*},\bar{X}^{*}$ are the upper parts.

Again, maintaining the assumptions that the
$\underline{Y}\neq\bar{Y},\underline{X}\neq\bar{X}$, $g(\cdot)$ is continuous
and with split sampling, one can recover the underlying marginal distributions
for both $Y$ and $X$, thus we can point identify $\beta$. To show this, let
$C_{m_{Y}},C_{m_{X}}$ the discretization intervals for $Y$ and $X$
respectively. Using the same assumptions and similar argument as in Equation
6, the linear case leads to

$\mathbb{E}[Y|Y\in C_{m_{Y}},X\in C_{m_{X}}]=g\left(\mathbb{E}\left[X|Y\in
C_{m_{Y}},X\in C_{m_{X}}\right];\beta\right)$ (7)

as we add the conditioning of $Y\in C_{m_{Y}}$ to both sides to get measurable
quantities. This implies, that based on marginal distributions of $Y$ and $X$,
we can get the expectation for both $Y$ and $X$, conditional on $Y$ being in
the interval $C_{m_{Y}}$ and $X$ is in the interval $C_{m_{X}}$. Similar
argument applies in the nonlinear case here as well.

###  2.4 Multivariate case

Instead of a univariate regression function, let us deal with the multivariate
case. Let
$\mathbf{Z}=(\mathbf{z}_{1},\dots,\mathbf{z}_{p},\dots,\mathbf{z}_{P})$
distributed as
$\mathbf{Z}\sim\mathbf{f}_{\mathbf{Z}}(\mathbf{z};\mathbf{a}_{l},\mathbf{a}_{u})$,
i.i.d. random variables with $P$ dimensions, where
$\mathbf{f}_{\mathbf{Z}}(\cdot)$ denotes the (joint) pdf with
$\mathbf{z}=(z_{1},\dots,z_{p},\dots,z_{P})$191919Note here $\mathbf{z}$ is
not a row from $\mathbf{Z}$, but represent a vector that is used with the
distribution $\mathbf{f}_{\mathbf{Z}}(\cdot)$., and known support vector
$\left[\mathbf{a}_{l},\mathbf{a}_{u}\right]$, where
$\mathbf{a}_{l},\mathbf{a}_{u}\in\mathbb{R}^{P}$. We assume
$\mathbf{f}_{\mathbf{Z}}(\cdot)$ is unknown and continuous. Instead of
observing $\mathbf{Z}$, we observe $\mathbf{Z}^{*}$ through a (multivariate)
discretization process similar to Equation 1. The difference is instead of
intervals, we use a multi-dimensional grid202020One-by-one discretization does
not work in our case as it will not be able to recover the joint distribution
of $\mathbf{f}_{\mathbf{Z}}(\cdot)$. See more in online supplement, Section
A.2.3. , where each grid is defined as
$\mathbf{C}_{m}=\left[\mathbf{c}_{m-1},\mathbf{c}_{m}\right)$, where
$\mathbf{c}_{m}=(c_{m,1},\dots,c_{m,p},\dots,c_{m,P})$, with $c_{m,p}$ stands
for the $m$’th grid point for variable $p$. Note that
$\mathbf{c}_{0}=\mathbf{a}_{l}$ and $\mathbf{c}_{M}=\mathbf{a}_{u}$, thus the
first boundary vector and last boundary vector contains the boundaries of the
distribution’s support. $\mathbf{v}_{m}\in\mathbf{C}_{m}$ is the assigned
vector for each grid, that can be a measure of centrality (e.g. focus point)
or other point within the grid. In the case of $P=2$, we have a 2D grid that
defines the discretization scheme, and each $\mathbf{C}_{m}$ stands for a
rectangle with $M^{2}$ different partitions.

As the next step, let us define the underlying model that we are interested
in. Let $\mathbf{y}=(Y_{1},\ldots,Y_{N})^{\prime}$ a $(N\times 1)$ vector,
$\mathbf{X}=(\mathbf{x}_{1},\dots,\mathbf{x}_{k},\dots,\mathbf{x}_{K})$ with
dimensions $(N\times K)$ where
$\mathbf{x}_{k}=(\mathbf{x}_{k,1},\ldots,\mathbf{x}_{k,N})^{\prime}$. The
parameter vector of interest is
$\boldsymbol{\beta}=(\beta_{1},\ldots,\beta_{K})^{\prime}$ that is $(K\times
1)$ vector. We allow further conditioning variables that are not discretized
and denote them by
$\mathbf{W}=(\mathbf{w}_{1},\dots,\mathbf{w}_{j},\dots,\mathbf{w}_{J})$ that
is $(N\times J)$ matrix with
$\mathbf{w}_{j}=(\mathbf{w}_{j,1},\ldots,\mathbf{x}_{j,N})^{\prime}$ and
parameter vector
$\boldsymbol{\gamma}=(\gamma_{1},\ldots,\gamma_{J})^{\prime}$.
$\boldsymbol{\beta}$ and $\boldsymbol{\gamma}$ belong to a subset of a compact
finite-dimensional space $(\mathcal{B},\mathcal{G})$. The regression function
that we are interested in takes the following form,

$\mathbb{E}\left[\mathbf{y}|\mathbf{X},\mathbf{W}\right]=g(\mathbf{X},\mathbf{W};\boldsymbol{\beta},\boldsymbol{\gamma})\,.$
(8)

Here, we only highlight additional assumptions or notations that are needed
for our identification of $\boldsymbol{\beta}$ to simplify our argument. The
proofs are all based on the marginal distribution of $\mathbf{y}$ or joint
distribution of $\mathbf{X}$; or both are known via split sampling and the use
of the law of iterated expectations with linearity assumption on $g(\cdot)$ or
continuous mapping theorem. Here we only highlight the linear cases.

  1. 1.

Identification with $\mathbf{X}^{*}$:
$\mathbb{E}\left[\mathbf{y}|\mathbf{X}\in\mathbf{C}_{m},\mathbf{W}\right]=g\left(\mathbb{E}\left[\mathbf{X},\mathbf{W}|\mathbf{X}\in\mathbf{C}_{m}\right];\boldsymbol{\beta},\boldsymbol{\gamma}\right)$
holds under the same assumptions. Expression simplifies further if
$\mathbf{X}$ and $\mathbf{W}$ are separable.

  2. 2.

Identification with $\mathbf{y}^{*}$: no need to distinguish between
$\mathbf{X}$ and $\mathbf{W}$ as both are non-discretized variables. We focus
only on $\mathbf{X}$ and $\boldsymbol{\beta}$. Let $\mathbf{D}_{l}$ denote a
set containing mutually exclusive partitions of the domain for $\mathbf{X}$.
$\mathbb{E}\left[\mathbf{y}|\mathbf{X}\in\mathbf{D}_{l}\right]=g\left(\mathbb{E}\left[\mathbf{X}|\mathbf{X}\in\mathbf{D}_{l}\right];\boldsymbol{\beta}\right)$
holds under the same assumptions.

  3. 3.

Identification with $\mathbf{y}^{*}$ and $\mathbf{X}$: Let
$\mathbf{C}_{m,\mathbf{X}}$ contain the discretization grid for $\mathbf{X}$
and $C_{m,Y}$ for $\mathbf{y}^{*}$. Under the same assumptions,
$\mathbb{E}\left[\mathbf{y}|\mathbf{y}\in
C_{m,Y},\mathbf{X}\in\mathbf{C}_{m,\mathbf{X}},\mathbf{W}\right]=$
$g\left(\mathbb{E}\left[\mathbf{X},\mathbf{W}|\mathbf{y}\in
C_{m,Y},\mathbf{X}\in\mathbf{C}_{m,\mathbf{X}}\right];\boldsymbol{\beta},\boldsymbol{\gamma}\right)$.
Again, if $\mathbf{X}$ and $\mathbf{W}$ are separable, one can simplify
further by applying discretization of $\mathbf{y}^{*}$ with $\mathbf{W}$
separately.

##  3 Split Sampling

The key to our approach is the use of split sampling, which recovers the
marginal distribution of $Z$ even if it is discretized. We achieve convergence
in distribution with the use of multiple discretization schemes with different
thresholds for each interval. As we increase the number of discretization
schemes to infinity along with the number of observations, we reduce the
widths of the intervals to zero, which leads to mapping the required marginal
distributions. This split sampling method can be viewed as a nonparametric
estimator of the marginal distribution. Similarly, as before, we first discuss
the univariate case and then extend our approach to the multivariate case.

We create multiple samples ($s=1,\dots,S$) and discretize them through
different schemes while fixing the number of intervals ($M$) within each split
sample. The reason for fixing $M$ is simple: in the case of data protection,
it allows to keep the values non-recognizable.212121In the case of
questionnaires, fixing small $M$ is also advisable as one needs to account for
restricted human cognitive capacity. After generating the data for each split
sample, we collect the different outcomes of the discretizations for the
‘working sample’. This ‘working sample’ has $b=1,\dots,B$ overall number of
intervals, where $B$ is much larger than $M$. How much depends on the specific
split sampling method used.

We show that we can create a synthetic variable $Z^{\dagger}$ based on the
working sample, whose marginal distribution $f_{Z^{\dagger}}(\cdot)$
asymptotically converges to $f_{Z}(\cdot)$ for the ‘shifting method’. It is
possible to introduce alternative split sampling methods, and we refer the
reader to the online supplement, Section A.4, where we discuss an alternative
of shifting method in detail.

###  3.1 Split samples, working sample, and an illustrative example

Discretization for the $s$-th split sample looks exactly as the problem
introduced in Equation 1, with the only difference being that the interval
boundaries are different. A split sample looks like,222222To simplify the
notation, we use instead of $Z^{*(s)}$ simply $Z^{(s)}$.

$Z^{(s)}_{i}=\begin{cases}v_{1}^{(s)}&\quad\text{if }Z_{i}\in
C_{1}^{(s)}=[c_{0}^{(s)},c_{1}^{(s)}),\\\
&\qquad\qquad\qquad\qquad\hfill\text{1st interval for split sample }s,\\\
v_{2}^{(s)}&\quad\text{if }Z_{i}\in C_{2}^{(s)}=[c_{1}^{(s)},c_{2}^{(s)}),\\\
\vdots&\quad\vdots\\\ v_{m}^{(s)}&\quad\text{if }Z_{i}\in
C_{m}^{(s)}=[c_{m-1}^{(s)},c_{m}^{(s)}),\\\ \vdots&\quad\vdots\\\
v_{M}^{(s)}&\quad\text{if }Z_{i}\in
C_{M}^{(s)}=[c_{M-1}^{(s)},c_{M}^{(s)}],\\\
&\qquad\qquad\qquad\qquad\hfill\text{last interval for split sample }s.\\\
\end{cases}$ (9)

The number of observations across split samples $(N^{(s)})$ can be the same
or, more likely, different. The construction of the working sample’s interval
is simply given by the union of unique interval boundaries within each split
sample.

$\bigcup_{b=0}^{B}c_{b}^{WS}=\bigcup_{s=1}^{S}\bigcup_{m=0}^{M}c_{m}^{(s)}\,.$
(10)

Let us take a very simple illustrative example. Assume that we have two
intervals $M=2$, and two split samples $S=2$. Let $Z$ be a random variable in
$[0,4]$ and define the interval boundaries in the first split sample as
$[0,2)$ and $[2,4]$, while in the second split sample $[0,1)$ and $[1,4]$.
Next, we create the working sample that has $B=3$ intervals, with boundaries
$[0,1)$, $[1,2)$ and $[2,4]$.

Next, let us assume we know the true discrete probabilities for $Z$,

$\Pr(Z=z)=\Pr(Z)=\begin{cases}&0.5,\,\,\text{if }0\leq z<1,\\\
&0.3,\,\,\text{if }1\leq z<2,\\\ &0.2,\,\,\text{if }2\leq z\leq 4.\\\
\end{cases}$

If we only have information based on our separate split samples, we can not
recover the true probabilities of $Z$. However if we use the information from
both split samples, we can deduce the non-observed interval’s probabilities as
well. Figure 1 visualize the example.

024$\Pr\left[Z^{(1)}\in[0,2)\right]=0.8$$\Pr\left[Z^{(1)}\in[2,4]\right]=0.2$014$\Pr\left[Z^{(2)}\in[0,1)\right]=0.5$$\Pr\left[Z^{(2)}\in[1,4]\right]=0.5$0124$\Pr\left[Z^{WS}\in[0,1)\right]=0.5$$\Pr\left[Z^{WS}\in[0,1)\right]=0.3$$\Pr\left[Z^{WS}\in[2,4]\right]=0.2$
Figure 1: The basic idea of split sampling

To generalize our approach to any type of distribution, let us write the
probability for a random observation to be in a given class of a given split
sample,

$\Pr\left(Z\in
C^{(s)}_{m}\right)=\Pr(Z\in\mathcal{S}_{s})\int_{c^{(s)}_{m-1}}^{c^{(s)}_{m}}f_{Z}(z)\mathrm{d}z,$

where $\mathcal{S}_{s}$ denotes the $s$-th split sample. We can express, the
probability of an observation falling into the working sample’s $C^{WS}_{b}$
interval as

$\Pr\left(Z\in
C^{WS}_{b}\right)=\sum_{s=1}^{S}\Pr(Z\in\mathcal{S}_{s})\sum_{m=1}^{M}\Pr\left(Z\in
C^{WS}_{b}\mid Z\in
C^{(s)}_{m}\right)\int_{c^{(s)}_{m-1}}^{c^{(s)}_{m}}f_{Z}(z)\mathrm{d}z\,.$
(11)

In practice, we recommend splitting the sample size into equal parts, thus
$\Pr(Z\in\mathcal{S}_{s})=1/S$ and use uniform probability for $\Pr\left(Z\in
C^{WS}_{b}\mid Z\in C^{(s)}_{m}\right)$. Then Equation 11 simplifies to

$\Pr\left(Z\in
C^{WS}_{b}\right)=\frac{1}{S}\sum_{s=1}^{S}\sum_{\begin{subarray}{c}m\\\
\text{ if }C^{WS}_{b}\in
C^{(s)}_{m}\end{subarray}}\frac{c^{WS}_{b}-c^{WS}_{b-1}}{c^{(s)}_{m}-c^{(s)}_{m-1}}\int_{c^{(s)}_{m-1}}^{c^{(s)}_{m}}f_{Z}(z)\mathrm{d}z\,.$

###  3.2 The shifting method

The shifting method takes the original discretization as given with fixed
interval widths, and shifts the boundaries of the intervals with a certain
value. The interval widths for the different split samples remain the same,
except for the boundary intervals. As one increases the number of split
samples the size of the shift becomes smaller and smaller. This enables to
mapping of the marginal probability distribution at the limit using inductive
logic. First, we introduce the discretization process proposed for the
shifting method and as a second step, we show how to use the realizations of
this discretization to create a synthetic variable that maps the original
variable’s marginal distribution.

Figure 2 shows an illustrative example for split samples with $S=4$ and $M=4$
classes.232323We can interpret this method in two ways: 1) We can consider the
original discretization and take the number of classes as fixed. Then we shift
the boundaries which results in adding an extra class for each split sample.
Thus the original number of intervals is $M$ and we add one for each (shifted)
split sample. 2) We can look at each (shifted) split sample and take $M$ as
given by the shift and consider the first split sample as a special case where
the size of the shift is zero, thus there are only $M-1$ intervals. For the
convenience of mathematical derivations, we will use the second
interpretation. As Figure 2 shows, for the first split sample ($s=1$) there is
one class fewer $(M-1)$ (or if we prefer, we can look at the benchmark as
where we shifted the boundaries with zero).

06060606240.52.54.51.03.05.01.53.55.5split
samples$S=4,M=4$00.511.522.533.544.555.56Working sample$B=(M-1)\times S$
Figure 2: Example for the shifting method

To derive the properties of the shifting method, we define the uniform class
widths for the first split sample as $\frac{a_{u}-a_{l}}{M-1}$. We split these
intervals into $S$ parts, thus we can shift the class boundaries $S$ times.
This defines the size of the shift as

$h=\frac{a_{u}-a_{l}}{S(M-1)}\,.$

This implies that the number of intervals in the working sample is $B=S(M-1)$.
The boundary points for each split sample are given by

$c^{(s)}_{m}=\begin{cases}a_{l}\text{ or }-\infty,&\text{ if }m=0,\\\
a_{l}+(s-1)\frac{a_{u}-a_{l}}{S(M-1)}+(m-1)\frac{a_{u}-a_{l}}{M-1}&\text{ if
}0<m<M,\\\ a_{u}\text{ or }\infty,&\text{ if }m=M.\\\ \end{cases}$

For the working sample we get $c^{WS}_{b}=a_{l}+b\frac{a_{u}-a_{l}}{S(M-1)}$
with the following class widths:

$||C^{(s)}_{m}||=\begin{cases}0,&\text{ if }s=1\text{ and }m=1,\\\
\frac{a_{u}-a_{l}}{M-1},&\text{ if }1<m<M,\\\
(s-1)\frac{a_{u}-a_{l}}{S(M-1)},&\text{ otherwise}.\end{cases}$

Here each interval width is uniformly given by
$||C^{WS}_{b}||=\frac{a_{u}-a_{l}}{S(M-1)}$. Some remarks on these boundary
points:

  * •

The interval widths for the working sample shrink, i.e., $||C^{WS}_{b}||\to 0$
as $S\to\infty$.

  * •

The interval widths $C^{(s)}_{2},C^{(s)}_{3},\dots,C^{(s)}_{M-1}$ for all $s$
are fixed and independent from the number of split samples.

  * •

The boundary classes’ widths, $C^{(s)}_{1},C^{(s)}_{M}$ for
$s\notin\left\\{2,S\right\\}$ are a function of the number of split samples.

  * •

$C^{(2)}_{1}\cong C^{WS}_{1},C^{(S)}_{M}\cong C^{WS}_{B}$ are congruent. In
the case of data protection if $h$ is too small, individuals falling into
these intervals might be identifiable.

  * •

$C^{(1)}_{1}$ has size 0 and does not exist in practice. It is induced by the
formalism we use.

Algorithm 1 summarizes how to create split samples using the shifting method
in practice.

Algorithm 1 The shifting method - creation of split samples

1: For any given $S$ and $M$, set $s=1$ $B=S(M-1)\,\,,\qquad
h=\frac{a_{u}-a_{l}}{B}\,\,,\qquad\Delta=\frac{a_{u}-a_{l}}{M-1}\,.$

2: Set $c^{(s)}_{0}=a_{l}$ and $c^{(s)}_{M}=a_{u}$.

3: If $s=1$, set  $c^{(s)}_{m}=c^{(s)}_{m-1}+\Delta,\qquad m=2,\ldots,M-1$
else  $c^{(s)}_{m}=c^{(s-1)}_{m}+h,\qquad m=1,\ldots,M-1.$ Note: $c^{(1)}_{1}$
does not exist.

4: If $s<S$ then $s:=s+1$ and goto Step 2.

As a second step, we introduce a new synthetic random variable $Z^{\dagger}$
that maps the underlying marginal distribution. Observations from a particular
interval in the split sample $s$ can end up in several candidate intervals
from the working sample. To make it more transparent, let us define the
function $V(s,m):v^{(s)}_{m}\to v^{WS}_{b}$, that maps the discretized value
$v^{(s)}_{m}$ to the working sample’s value $v^{WS}_{b}$,

$V(s,m)=\begin{cases}\\{\emptyset\\},&\text{ if, }s=1\text{ and }m=1,\\\
\bigcup_{b=1}^{s-1}\\{v^{WS}_{b}\\},&\text{ if, }s\neq 1\text{ and }m=1,\\\
\bigcup_{b=s-1+(m-2)(M-1)}^{s-1+(m-1)(M-1)}\\{v^{WS}_{b}\\},&\text{ if,
}1<m<M,\\\ \bigcup_{b=B-S+s-1}^{B}\\{v^{WS}_{b}\\},&\text{ if
}m=M.\end{cases}$ (12)

As we have no prior information on the distribution of $Z$, we uniformly
assign the discretized observations from the split samples to the working
sample during the creation of the synthetic variable $Z^{\dagger}$.

The assignment mechanism can be written as

$Z^{\dagger}|Z^{(s)}\in C^{(s)}_{m}=v\in V(s,m),\text{ with
}\begin{cases}\Pr\left(1\right),&\text{if }s=1\text{ and }m=1,\\\
\Pr\left(1/(s-1)\right),&\text{if }s\neq 1\text{ and }m=1,\\\
\Pr\left(1/S\right),&\text{if }1<m<M,\,\text{ or }\\\
\Pr\left(1/(S-s+1)\right),&\text{if }m=M.\\\ \end{cases}$ (13)

Now, we can write the probability of $Z^{\dagger}\in C^{WS}_{b}$, using
Equation 11 and assuming
${\Pr(Z\in\mathcal{S}_{s})=1/S}$,

$\Pr\left(Z^{\dagger}\in C^{WS}_{b}\right)=\begin{cases}0,&\text{if }s=1\text{
and }m=1,\\\
\frac{1}{S}\sum_{s=2}^{S}\frac{1}{s-1}\int_{C^{(s)}_{1}|C^{WS}_{b}\in
C^{(s)}_{1}}f_{Z}(z)\mathrm{d}z,&\text{if }s\neq 1\text{ and }m=1,\\\
\frac{1}{S^{2}}\sum_{s=1}^{S}\int_{C^{(s)}_{m}|C^{WS}_{b}\in
C^{(s)}_{m}}f_{Z}(z)\mathrm{d}z,&\text{if }1<m<M,\\\
\frac{1}{S}\sum_{s=1}^{S}\frac{1}{S-s+1}\int_{C^{(s)}_{M}|C^{WS}_{b}\in
C^{(s)}_{M}}f_{Z}(z)\mathrm{d}z,&\text{if }m=M\,.\\\ \end{cases}$ (14)

Algorithm 2 describes how to create this synthetic variable $Z^{\dagger}$ that
approximates the underlying distribution of $Z$.

Algorithm 2 The shifting method – creation of synthetic variable
$(Z^{\dagger}_{i})$

1: Set $s:=1,m:=1,Z^{\dagger}_{i}=\emptyset$ .

2: Create the set of observations from the defined split sample class:
$\mathcal{A}_{m}^{(s)}:=\\{Z^{(s)}_{i}\in C^{(s)}_{m}\\}\,\forall i,$ where
$\mathcal{A}_{m}^{(s)}$ has $N^{(s)}_{m}$ number of observations.

3: Create $V(s,m)$, the set of possible working sample choice values,
$V(s,m)=\begin{cases}\\{\emptyset\\},&\text{ if, }s=1\text{ and }m=1,\\\
\bigcup_{b=1}^{s-1}\\{v^{WS}_{b}\\},&\text{ if, }s\neq 1\text{ and }m=1,\\\
\bigcup_{b=s-1+(m-2)(M-1)}^{s-1+(m-1)(M-1)}\\{v^{WS}_{b}\\},&\text{ if,
}1<m<M,\\\ \bigcup_{b=B-S+s-1}^{B}\\{v^{WS}_{b}\\},&\text{ if
}m=M.\end{cases}$

4: Draw $\mathcal{V}_{j}\in V(s,m),j=1,\dots,N^{(s)}_{m}$, with uniform
probabilities given by  $Z^{\dagger}_{i}|Z^{(s)}_{i}\in C^{(s)}_{m}=v\in
V(s,m),\text{ with }\begin{cases}\Pr\left(1\right),&\text{if }s=1\text{ and
}m=1,\\\ \Pr\left(1/(s-1)\right),&\text{if }s\neq 1\text{ and }m=1,\\\
\Pr\left(1/S\right),&\text{if }1<m<M,\,\text{ or }\\\
\Pr\left(1/(S-s+1)\right),&\text{if }m=M.\\\ \end{cases}$ Example: Let
$C^{(2)}_{3}=\left[2.5,4.5\right]$,
$\mathcal{A}^{(s)}_{m}=\\{3.5,3.5,3.5\\},N^{(s)}_{m}=3$,
$V(s,m)=\\{2.75,3.25,3.75,4.25\\}$, the uniform probabilities are $1/4$ for
each choice value. Then we pick values with the defined probability from the
set of $V(s,m)$, $3$ times with repetition, resulting in
$\bigcup_{j=1}^{N^{(s)}_{m}}\mathcal{V}_{j}=\\{2.75,3.25,3.25\\}$

5: Add these new values to $Z^{\dagger}_{i}$,
$Z^{\dagger}_{i}:=\left\\{Z^{\dagger}_{i},\bigcup_{j=1}^{N^{(s)}_{m}}\mathcal{V}_{j}\right\\}$

6: If $s<S$, then $s:=s+1$ and go to Step 2.

7: If $s=S$, then $s:=1$ and set $m=m+1$ and go to Step 2.

Next, we outline the assumptions under which
$Z^{\dagger}\overset{d}{\rightarrow}Z$ as the number of split samples $(S)$
and observations $(N)$ increases.

######  Assumption 1.

Let $Z$ be a continuous random variable with probability density function
$f_{Z}(z;a_{l},a_{u})$ where $a_{l},a_{u},S,N$ and $C^{(s)}_{m}$ are as
defined above then,

  1. 1.a)

$\frac{S}{N}\rightarrow c$ with $c\in(0,1)$ as $N\rightarrow\infty$.

  2. 1.b)

$\int^{b}_{a}f_{Z}(z)dz>0$ for any $(a,b)\subset[a_{l},a_{u}]$.

Assumption 1.a).a) ensures that the number of observations is always higher
than the number of split samples.242424Note, that we can decrease $\mathrm{c}$
to be as close to 0 as we would like to. This is required to identify each
point of $f_{Z}(z;a_{l},a_{u})$ at infinity through the mapping of split
samples to the working set (for each interval of the working set we have
observations). Assumption 1.b).b) imposes a mild assumption on the underlying
distribution: the support of the random variable is not disjoint within the
working samples’ interval, thus
$\int_{c^{WS}_{b-1}}^{c^{WS}_{b}}f_{Z}(z)\mathrm{d}z>0\,,\,\,\forall
c^{WS}_{b}$.

######  Proposition 1.

Under Assumptions 1.a).a), 1.b).b) and $\Pr(Z\in\mathcal{S}_{s})=1/S$,

$\lim_{S\to\infty}\Pr(Z^{\dagger}<a)=\Pr(Z<a)\qquad\forall a\in(a_{l},a_{u})$

or

$\lim_{S\rightarrow\infty}F_{Z^{\dagger}}(a)=F_{Z}(a)\qquad\forall
a\in(a_{l},a_{u}),$

where

$F_{Z^{\dagger}}(a)=\Pr(Z^{\dagger}<a)\text{ and }F_{Z}(a)=\Pr(Z<a).$

See the proof in the online supplement under Section A.2.1.

In addition, we investigate the speed of convergence for
$F_{Z^{\dagger}}(\cdot)\to F_{Z}(\cdot)$ as a function of the number of split
samples $(S)$ to get insight into the choice of $S$ in finite samples. In
general, the rate of convergence is $1/S$ except around the boundaries of the
support.252525Boundaries of the support are defined as: lower bound
$c^{(1)}_{1}+(c^{(S)}_{2}-c^{(1)}_{1})$ and the higher bound:
$c^{(1)}_{M}+(c^{(1)}_{M-1}-c^{(1)}_{M})$ In these regions, the speed of
convergence is $\frac{\log S}{S}$. This result justifies that even with small
values of $S$, one can get a decent mapping of the underlying marginal
distribution of $Z$. See the derivations in online supplement, Section A.2.2.

###  3.3 Further remarks

#### Other split sampling methods

Here, we have discussed one particular split sampling method, the shifting
method. However, it is possible to work out other discretization schemes that
can map the underlying marginal distribution of $Z$. In fact, in the online
supplement, Section A.4, we introduce the magnifying method, which offers an
alternative method to learn $f_{Z}(z;a_{l},a_{u})$. The key to all potential
split sampling methods is to achieve convergence in distribution. Different
discretizations may yield different speeds of convergence and other data
protection properties.262626For example, the magnifying method can not
maintain data protection properties for certain individuals as it achieves
convergence in distribution by narrowing specific parts of the domain to zero
at the limit, revealing the actual value of $Z$. See more details in online
supplement, Section A.4.

#### Multivariate case

One can extend the shifting method to learn the joint distribution of
$\mathbf{Z}$. The shifting method in the multivariate case utilizes the
initial discretization scheme with $\mathbf{C}_{m}$ and shifts the boundary
points of the grids with
$\mathbf{h}=\frac{\mathbf{a}_{u}-\mathbf{a}_{l}}{S(M-1)}$. This results in
$(S(M-1))^{P}$ different grids in the working sample, which is much larger
than the initial $M^{P}$. Changing the grids’ boundary points allows for
mapping the underlying distribution using the same inductive logic to prove
convergence in distribution. For the proof, we need the same assumptions
extended to the multivariate case, $N/S\to
c,\int^{\mathbf{b}}_{\mathbf{a}}\mathbf{f}_{\mathbf{Z}}(\mathbf{z})d\mathbf{z}>0$
for any $(\mathbf{a},\mathbf{b})\subset[\mathbf{a}_{l},\mathbf{a}_{u}]$ and
$\Pr(\mathbf{z}\in\mathcal{S}_{s})=1/S$. See the detailed argument in the
online supplement, Section A.2.3.

#### Infinite support

As a last point, we consider the case when $Z$ has infinite support. If we set
the first and last intervals’ boundary points to infinity, we are facing
censoring. As we show in Section 4.4, one can extend our approach for
censoring by using the continuous mapping theorem. However, this is beyond the
scope of this paper, and we would rather discuss here a simpler solution:
removing those observations that fall into the first or last intervals
$(C^{(s)}_{1},C^{(s)}_{M})$ of the discretized variable. This way, one needs
to truncate not only $Z$, but all other observed variables. The flip side of
this process is that for the truncated distribution, all previous derivations
and results apply.

##  4 Estimation

This section discusses how to use the created $Z^{\dagger}$ variable to
estimate $\beta$ via OLS. In the following, we will assume $g(\cdot)$ is
linear, and later in this section, we will discuss nonlinear cases. As OLS has
well-known properties, we discuss the multivariate setup directly. We rely on
our identification results for each case, depending on where the
discretization happened. The unknown data-generating process (DGP) is

$\mathbf{y}=\mathbf{X}\boldsymbol{\beta}+\mathbf{W}\boldsymbol{\gamma}+\boldsymbol{\varepsilon}\,,\qquad\boldsymbol{\varepsilon}\sim
iid\left(\mathbf{0},\sigma^{2}_{\epsilon}\mathbf{I}\right)\,,$ (15)

where $\mathbf{X}$ and $\mathbf{W}$ are linearly separable and
$\boldsymbol{\varepsilon}$ is homoskedastic. In all three discretization
cases, we propose a two-step procedure:

  1. Step 1.

Introduce OLS estimators for the specific conditional expectations, that we
used to point identify $\boldsymbol{\beta}$ in Section 2. We discuss the
asymptotic properties of these estimators.

  2. Step 2.

Use the estimators from Step 1 to replace variables from Equation 15. We
investigate the asymptotic properties of the OLS estimators for
$\boldsymbol{\beta}$ in the modified regressions, dependent on where
discretization happened.

###  4.1 Explanatory variable

First, let us discuss the case when we observe $\mathbf{X}^{*}$, instead of
$\mathbf{X}$. We rely on the multivariate case of Proposition 1, hence
$\mathbf{X}^{\dagger}\overset{d}{\rightarrow}\mathbf{X}$, where
$\mathbf{X}^{\dagger}=\left(\mathbf{x}^{\dagger}_{1},\dots,\mathbf{x}^{\dagger}_{k},\dots,\mathbf{x}^{\dagger}_{K}\right)$
and
$\mathbf{x}^{\dagger}_{k}=(\mathbf{x}_{k,1},\ldots,\mathbf{x}_{k,N})^{\prime}$.

To identify $\boldsymbol{\beta}$, from the DGP defined by Equation 15, we need
to calculate the sample conditional means for $\mathbf{X}$ given
$\mathbf{X}\in\mathbf{C}^{(s)}_{m}$, thus

$\mathbb{E}\left[\mathbf{y}|\mathbf{X}\in\mathbf{C}^{(s)}_{m},\mathbf{W}\right]=\mathbb{E}\left[\mathbf{X}|\mathbf{X}\in\mathbf{C}^{(s)}_{m}\right]\boldsymbol{\beta}+\mathbf{W}\boldsymbol{\gamma}\,,$
(16)

where
$\mathbf{C}^{(s)}_{m}=\left[\mathbf{c}^{(s)}_{m-1},\mathbf{c}^{(s)}_{m}\right)$,
with
$\mathbf{c}^{(s)}_{m}=\\{c^{(s)}_{m,1},\dots,c^{(s)}_{m,k},\dots,c^{(s)}_{m,K}\\}$
refers to the boundary point vector in split sample $s$ for class $m$ by the
shifting method. Note that these classes are mutually exclusive for given $s$
along $m$ and $k$.272727See more, how the discretization intervals for
multivariate case constructed in the online supplement, Section A.2.3. Also,
note that we do not need to partition $\mathbf{W}$ as it is linearly separable
from $\mathbf{X}$.

Let us define $\kappa(s,\mathbf{m})$ a function that maps the conditional
expectations of $\mathbf{X}$ for given grid $\mathbf{C}^{(s)}_{m}$ 282828Note
that $\mathbf{m}=(m_{1},\dots,m_{K})$ is a vector, defining the grid points in
$\mathbf{C}^{(s)}_{m}$ for each variable $k$. and utilize the results of the
split sampling method,

$\kappa(s,\mathbf{m}):=\mathbb{E}\left[\mathbf{X}|\mathbf{X}\in\mathbf{C}^{(s)}_{m}\right]=\lim_{N,S\to\infty}\mathbb{E}\left[\mathbf{X}^{\dagger}|\mathbf{X}^{\dagger}\in\mathbf{C}^{(s)}_{m}\right]\,.$

Let $\boldsymbol{\kappa}=(\kappa(1,{\bf 1}),\dots,\kappa(S,{\bf
1}),\dots,\kappa(S,\mathbf{m}),\dots,\kappa(S,\mathbf{M}))^{\prime}$ be a
vectorized version of $\kappa(s,\mathbf{m})$ with $(SM^{K}\times 1)$
dimension. We propose an estimator for $\boldsymbol{\kappa}$, using
OLS,292929We can estimate all conditional expectations jointly via OLS.
Otherwise simple conditional means for a given interval $\mathbf{C}^{(s)}_{m}$
would yield the same result, but with this formulation, we can compactly
establish asymptotic properties.

$\hat{\boldsymbol{\kappa}}=\left(\mathbf{1}^{\prime}_{\\{\mathbf{X}^{\dagger}\in\mathbf{C}^{(s)}_{m}\\}}\mathbf{1}_{\\{\mathbf{X}^{\dagger}\in\mathbf{C}^{(s)}_{m}\\}}\right)^{-1}\mathbf{1}^{\prime}_{\\{\mathbf{X}^{\dagger}\in\mathbf{C}^{(s)}_{m}\\}}\mathbf{X}^{\dagger}\,\,,$

where $\mathbf{1}_{\\{\cdot\\}}$ stands for the indicator function and results
in a matrix with dimensions $(N\times SM^{K})$ with the same ordering as
$\boldsymbol{\kappa}$. Let us note, $\hat{\boldsymbol{\kappa}}$ does not
require the actual values of $\mathbf{X}$, only $\mathbf{X}^{\dagger}$ and
$\mathbf{X}^{\dagger}\in\mathbf{C}^{(s)}_{m}$. Under standard OLS assumptions,

$\sqrt{N}\left(\hat{\boldsymbol{\kappa}}-\boldsymbol{\kappa}\right)\mathrel{\overset{\makebox[0.0pt]{\mbox{\scriptsize
a}}}{\sim}}{}\mathcal{N}\left(\mathbf{0},\boldsymbol{\Omega}_{\boldsymbol{\kappa}}\right),$

where $\boldsymbol{\Omega}_{\boldsymbol{\kappa}}$ is the variance-covariance
matrix. See the derivations for the proposed estimator in the online
supplement, Section A.3.1.

#### OLS estimator for $\boldsymbol{\beta}$

As $\mathbf{X}$ is not observed, only its discretized version
$\mathbf{X}^{*}$, let us consider the following modification of Equation 15,
to highlight some consequences of modeling with discretized variables as
explanatory variables.

$\mathbf{y}=\mathbf{X}^{*}\boldsymbol{\beta}+\mathbf{W}\boldsymbol{\gamma}+\left(\mathbf{X}-\mathbf{X}^{*}\right)\boldsymbol{\beta}+\boldsymbol{\varepsilon}_{x}\,.$

The naive OLS estimator for $\boldsymbol{\beta}$ is

$\hat{\boldsymbol{\beta}}^{n}=\left(\mathbf{X}^{*\prime}\mathbf{M}_{\mathbf{W}}\mathbf{X}^{*}\right)^{-1}\mathbf{X}^{*\prime}\mathbf{M}_{\mathbf{W}}\mathbf{y}\,,$

where
$\mathbf{M}_{\mathbf{W}}=\mathbf{I}-\mathbf{W}(\mathbf{W}^{\prime}\mathbf{W})^{-1}\mathbf{W}^{\prime}$
defines the usual residual maker. In general
$\hat{\boldsymbol{\beta}}^{n}\not\to\boldsymbol{\beta}$ as
$\mathbf{X}^{*}\neq\mathbf{X}$, furthermore discretization is contagious thus
$\hat{\boldsymbol{\gamma}}^{n}\not\to\boldsymbol{\gamma}$. This result implies
that the practice of assigning arbitrary values (e.g., mid-point) to the
discretized variable will lead to biases in the estimates even if the
discretized variables are controls. The size of the bias depends on the
underlying distribution of $\mathbf{X}$ and the discretization process. See
further derivation in the online supplement, Section A.5.

To get a consistent estimator, let us follow Equation 16, and define
$\ddot{\mathbf{X}}$, which contains the corresponding conditional expectations
via $\hat{\boldsymbol{\kappa}}$ for discretized values in
$\mathbf{X}^{*}$.303030See more on how to replace the corresponding
conditional expectations in the online supplement, under Section A.3.1,
Algorithm A1.

$\mathbf{y}=\ddot{\mathbf{X}}\boldsymbol{\beta}+\mathbf{W}\boldsymbol{\gamma}+\mathbf{e}\,,$

while $\mathbf{e}=(e_{1},\dots,e_{N})$. The OLS estimator for
$\boldsymbol{\beta}$

$\hat{\boldsymbol{\beta}}=\left(\ddot{\mathbf{X}}^{\prime}\mathbf{M}_{\mathbf{W}}\ddot{\mathbf{X}}\right)^{-1}\ddot{\mathbf{X}}^{\prime}\mathbf{M}_{\mathbf{W}}\mathbf{y}\,,$

Note that $\mathbb{E}[e_{i}]=0$ for all $i$ since $\ddot{\mathbf{X}}$ is a
consistent estimate. Moreover, $\mathbb{E}[e_{i}e_{j}]=0$ for $i\neq j$ due to
$\mathbf{C}^{(s)}_{m}$ are mutually exclusive in $m$ and $k$, and
$\mathbb{E}[e_{i}|\ddot{\mathbf{X}},\mathbf{W}]=0$ since the discretization
does not affect the sampling error. Furthermore, assume
$\mathbb{E}\left[\ddot{\mathbf{X}}\mathbf{e}\right]=\mathbb{E}\left[\mathbf{W}\mathbf{e}\right]=0$.
Then, $\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}=o_{p}(1)$.

The asymptotic distribution of $\hat{\boldsymbol{\beta}}$ can be derived
similarly. As $\boldsymbol{\varepsilon}\sim
iid\left(\mathbf{0},\sigma^{2}_{\epsilon}\mathbf{I}\right)$ and
$\ddot{\mathbf{X}}$ is a consistent estimate of the conditional means,
$\mathbb{E}\left[\mathbf{e}\mathbf{e}^{\prime}|\ddot{\mathbf{X}},\mathbf{W}\right]=\sigma_{e}^{2}\mathbf{I}$.
Under classical assumptions of OLS,
$\sqrt{N}\left(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right)\mathrel{\overset{\makebox[0.0pt]{\mbox{\scriptsize
a}}}{\sim}}{}\mathcal{N}\left(\mathbf{0},\sigma_{e}^{2}\left(\ddot{\mathbf{X}}^{\prime}\mathbf{M}_{\mathbf{W}}\ddot{\mathbf{X}}\right)^{-1}\right)$.

###  4.2 Outcome variable

In the second case, the outcome variable is discretized, thus $\mathbf{y}$ is
not observed, only $\mathbf{y}^{*}$. To get consistent estimates for
$\boldsymbol{\beta}$, let us partition the domain of $\mathbf{X}$’s into
mutually exclusive partitions, denoted by $\mathbf{D}_{l}$. Let us define
$\pi(s,m,\mathbf{l})$ a function that maps the conditional expectations of
$\mathbf{y}$ for a given grid $\mathbf{y}\in C^{(s)}_{m}$ and partition
$\mathbf{X}\in\mathbf{D}_{l}$.

$\pi(s,m,\mathbf{l}):=\mathbb{E}\left[\mathbf{y}|\mathbf{y}\in
C^{(s)}_{m},\mathbf{X}\in\mathbf{D}_{l}\right]=\lim_{N,S\to\infty}\mathbb{E}\left[\mathbf{y}^{\dagger}|\mathbf{y}^{\dagger}\in
C^{(s)}_{m},\mathbf{X}\in\mathbf{D}_{l}\right]\,,$

where $\mathbf{y}^{\dagger}=(Y^{\dagger}_{1},\ldots,Y^{\dagger}_{N})$ and we
utilized the results of the shifting method to show convergence. Let
$\boldsymbol{\pi}=\left(\pi(1,1,{\bf 1}),\dots,\pi(S,1,{\bf
1}),\dots,\pi(S,M,{\bf
1}),\dots,\pi(S,M,\mathbf{l}),\dots,\pi(S,M,\mathbf{L})\right)^{\prime}$ be
the vectorized version of $\pi(s,m,\mathbf{l})$. Estimating
$\boldsymbol{\pi}$, via OLS yields,

$\hat{\boldsymbol{\pi}}=\left(\mathbf{1}^{\prime}_{\\{\mathbf{y}^{\dagger}\in
C^{(s)}_{m},\mathbf{X}\in\mathbf{D}_{l}\\}}\mathbf{1}_{\\{\mathbf{y}^{\dagger}\in
C^{(s)}_{m},\mathbf{X}\in\mathbf{D}_{l}\\}}\right)^{-1}\mathbf{1}^{\prime}_{\\{\mathbf{y}^{\dagger}\in
C^{(s)}_{m},\mathbf{X}\in\mathbf{D}_{l}\\}}\mathbf{y}^{\dagger}\,.$ (17)

Again, vector $\hat{\boldsymbol{\pi}}$ has dimensions of $(SML^{K}\times 1)$,
$\mathbf{1}_{\\{\cdot\\}}$ is the corresponding indicator matrix, and
estimation of $\boldsymbol{\pi}$ does not require the actual values of
$\mathbf{y}$, only $\mathbf{y}^{\dagger}$ and observing
$\mathbf{y}^{\dagger}\in C^{(s)}_{m}$ and $\mathbf{X}\in\mathbf{D}_{l}$. Under
the standard OLS assumption,

$\sqrt{N}\left(\hat{\boldsymbol{\pi}}-\boldsymbol{\pi}\right)\mathrel{\overset{\makebox[0.0pt]{\mbox{\scriptsize
a}}}{\sim}}{}\mathcal{N}\left(\mathbf{0},\boldsymbol{\Omega}_{\boldsymbol{\pi}}\right),$

where $\boldsymbol{\Omega}_{\boldsymbol{\pi}}$ is the corresponding variance-
covariance matrix. See the derivations for the proposed estimator in the
online supplement, Section A.3.2.

#### OLS estimator for $\boldsymbol{\beta}$

To simplify our derivation let us neglect $\mathbf{W}$ from Equation 15. As we
have shown in our identification strategy, the standard regression model can
be rewritten as,

$\mathbb{E}\left[\mathbf{y}|\mathbf{X}\in\mathbf{D}_{l}\right]=\mathbb{E}\left[\mathbf{X}|\mathbf{X}\in\mathbf{D}_{l}\right]\boldsymbol{\beta}\,\,.$
(18)

Let $\tilde{\mathbf{y}}$ be a $N\times 1$ vector, where we replace the
discretized elements of $\mathbf{y}^{*}$ with the consistent estimates of
$\mathbb{E}\left[\mathbf{y}|\mathbf{X}\in\mathbf{D}_{l}\right]$, utilizing
$\hat{\boldsymbol{\pi}}$.313131See more on how to replace the corresponding
conditional expectations in the online supplement, under Section A.3.2,
Algorithm A2.  Let us redefine
$\tilde{\mathbf{X}}=\hat{\mathbb{E}}\left[\mathbf{X}|\mathbf{X}\in\mathbf{D}_{l}\right]$,
as the conditional sample means for $\mathbf{X}$ in partition
$\mathbf{D}_{l}$. Following from equation (18), we get

$\tilde{\mathbf{y}}=\tilde{\mathbf{X}}\boldsymbol{\beta}+\mathbf{u},$

where $\mathbf{u}=\left(u_{1},\ldots,u_{N}\right)^{\prime}$. Note that
$\mathbb{E}[u_{i}]=0$ for all $i$ since $\tilde{\mathbf{y}}$ and
$\tilde{\mathbf{X}}$ are consistent estimates. Moreover,
$\mathbb{E}[u_{i}u_{j}]=0$ for $i\neq j$ due to $\mathbf{D}_{l}$ are mutually
exclusive $\forall l$ and $\mathbb{E}[u_{i}|\tilde{\mathbf{X}}]=0$ since the
partition does not affect the sampling error. Furthermore, assume
$\mathbb{E}\left[\tilde{\mathbf{y}}\right]$ exists and
$\mathbb{E}\left[\tilde{\mathbf{X}}\mathbf{u}\right]=0$. Let
$\hat{\boldsymbol{\beta}}=\left(\tilde{\mathbf{X}}^{\prime}\tilde{\mathbf{X}}\right)^{-1}\tilde{\mathbf{X}}^{\prime}\tilde{\mathbf{y}}$,
then under the usual argument of the classical OLS,
$\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}=o_{p}(1)$.

The asymptotic distribution of $\hat{\boldsymbol{\beta}}$ can be derived in
the same spirit. As $\boldsymbol{\varepsilon}\sim
iid\left(\mathbf{0},\sigma^{2}_{\varepsilon}\mathbf{I}\right)$ and
$\tilde{\mathbf{X}}$ is a consistent estimate of the conditional means,
$\mathbb{E}\left[\mathbf{u}\mathbf{u}^{\prime}|\tilde{\mathbf{X}}\right]=\sigma_{u}^{2}\mathbf{I}$.
Under the classical assumptions of OLS,
$\sqrt{N}\left(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right)\mathrel{\overset{\makebox[0.0pt]{\mbox{\scriptsize
a}}}{\sim}}{}\mathcal{N}\left(\mathbf{0},\sigma_{u}^{2}\left(\tilde{\mathbf{X}}^{\prime}\tilde{\mathbf{X}}\right)^{-1}\right)$.

###  4.3 Discretization on both sides

Our last case is when the discretization happens with both outcome
$\mathbf{y}^{*}$, and with one or more explanatory variables $\mathbf{X}^{*}$.
In this case, we do not need to partition the domain of $\mathbf{X}$, but can
use the discretization grids $\mathbf{C}^{(s)}_{m}$ for the explanatory
variable. Note as $\mathbf{y}$ is discretized, we need to partition
$\mathbf{W}$ with $\mathbf{D}_{l}$, similarly to the case discussed in Section
4.2. To simplify our derivations, let us assume the number of split samples
and the number of classes are the same for both $\mathbf{y}$ and for
$\mathbf{X}$, thus $S=S_{Y}=S_{X}$ and $M=M_{Y}=M_{X}$.323232One can extend
the analysis all results hold. Let us define the following functions that map
the appropriate conditional expectations,

$\displaystyle{\psi}_{Y}(s,\mathbf{m},\mathbf{l}):$
$\displaystyle=\mathbb{E}\left(\mathbf{y}|\mathbf{y}\in
C^{(s)}_{m,Y},\mathbf{X}\in\mathbf{C}^{(s)}_{m,X},\mathbf{W}\in\mathbf{D}_{l}\right)$
$\displaystyle=\lim_{N,S\to\infty}\mathbb{E}\left(\mathbf{y}^{\dagger}|\mathbf{y}^{\dagger}\in
C^{(s)}_{m,Y},\mathbf{X}^{\dagger}\in\mathbf{C}^{(s)}_{m,X},\mathbf{W}\in\mathbf{D}_{l}\right)\,\,,$
$\displaystyle\psi_{X}(s,\mathbf{m},\mathbf{l}):$
$\displaystyle=\mathbb{E}\left(\mathbf{X}|\mathbf{y}\in
C^{(s)}_{m,Y},\mathbf{X}\in\mathbf{C}^{(s)}_{m,X},\mathbf{W}\in\mathbf{D}_{l}\right)$
$\displaystyle=\lim_{N,S\to\infty}\mathbb{E}\left(\mathbf{X}^{\dagger}|\mathbf{y}^{\dagger}\in
C^{(s)}_{m,Y},\mathbf{X}^{\dagger}\in\mathbf{C}^{(s)}_{m,X},\mathbf{W}\in\mathbf{D}_{l}\right)\,,$

where $C^{(s)}_{m,Y}$ is the discretization intervals for $\mathbf{y}$,
$\mathbf{C}^{(s)}_{m,X}$ is the discretization grid for $\mathbf{X}$. Note
that, $s=\\{s_{Y},s_{X}\\}$ and $m=\\{m_{Y},\mathbf{m}_{X}\\}$ in this case,
for $C^{(s)}_{m,Y}$ and for $\mathbf{C}^{(s)}_{m,X}$. We propose conditional
mean estimators, $\hat{\boldsymbol{\psi}}_{Y}$ and
$\hat{\boldsymbol{\psi}}_{X}$ via OLS in the same spirit as above. To simplify
our notation, let
$\mathbf{1}^{\prime}_{\\{\psi\\}}=\mathbf{1}^{\prime}_{\\{\mathbf{y}^{\dagger}\in
C^{(s)}_{m,Y},\,\mathbf{X}^{\dagger}\in\mathbf{C}^{(s)}_{m,X},\mathbf{W}\in\mathbf{D}_{l}\\}}$.
The estimators are,

$\displaystyle\hat{\boldsymbol{\psi}}_{Y}$
$\displaystyle=\left(\mathbf{1}^{\prime}_{\\{\psi\\}}\mathbf{1}_{\\{\psi\\}}\right)^{-1}\mathbf{1}^{\prime}_{\\{\psi\\}}\mathbf{y}^{\dagger}\,,$
(19) $\displaystyle\hat{\boldsymbol{\psi}}_{X}$
$\displaystyle=\left(\mathbf{1}^{\prime}_{\\{\psi\\}}\mathbf{1}_{\\{\psi\\}}\right)^{-1}\mathbf{1}^{\prime}_{\\{\psi\\}}\mathbf{X}^{\dagger}\,.$

Both estimators are consistent and asymptotically normal, see the derivations
in the online supplement, Section A.3.3.

To get a consistent estimator for $\boldsymbol{\beta}$, let us define
$\breve{\mathbf{y}}$ and $\breve{\mathbf{X}}$ that takes the corresponding
values from $\hat{\boldsymbol{\psi}}_{Y}$, and $\hat{\boldsymbol{\psi}}_{X}$
based on the discretized values of $\mathbf{y}^{*}$ and $\mathbf{X}^{*}$. Note
that if there are additional controls $\mathbf{W}$, one needs to replace it
with $\breve{\mathbf{W}}$ that takes the sample conditional means of
$\mathbb{E}\left(\mathbf{W}|\mathbf{y}\in
C^{(s)}_{m,Y},\mathbf{X}\in\mathbf{C}^{(s)}_{m,\mathbf{X}},\mathbf{W}\in\mathbf{D}_{l}\right)$.333333See
more on how to replace the corresponding conditional expectations in the
online supplement, under Section A.3.3, Algorithm A3.  The modified regression
results in

$\breve{\mathbf{y}}=\breve{\mathbf{X}}\boldsymbol{\beta}+\breve{\mathbf{W}}\boldsymbol{\gamma}+\boldsymbol{\nu}\,,$

where $\boldsymbol{\nu}=(\nu_{1},\dots,\nu_{N})$. The OLS estimator for
$\boldsymbol{\beta}$ is,

$\hat{\boldsymbol{\beta}}=\left(\breve{\mathbf{X}}^{\prime}\mathbf{M}_{\breve{\mathbf{W}}}\breve{\mathbf{X}}\right)^{-1}\breve{\mathbf{X}}^{\prime}\mathbf{M}_{\breve{\mathbf{W}}}\breve{\mathbf{y}}\,,$

where $\mathbf{M}_{\breve{\mathbf{W}}}$ is the usual residual maker, using
$\breve{\mathbf{W}}$. Note that $\mathbb{E}[\nu_{i}]=0$ for all $i=1,\dots,N$
since $\breve{\mathbf{y}}$, $\breve{\mathbf{X}}$ and $\breve{\mathbf{W}}$ are
consistent estimates of the corresponding conditional expectations. Moreover,
$\mathbb{E}[\nu_{i}\nu_{j}]=0$ for $i\neq j$ due to $\mathbf{D}_{l}$, and
$\mathbf{C}^{(s)}_{m}$ in $m$ and $k$ are mutually exclusive.
$\mathbb{E}[\nu_{i}|\breve{\mathbf{X}},\breve{\mathbf{W}}]=0$ since the
discretization does not affect the sampling error. Furthermore, assume
$\mathbb{E}\left[\breve{\mathbf{X}}\boldsymbol{\nu}\right]=0$ and
$\mathbb{E}\left[\breve{\mathbf{W}}\boldsymbol{\nu}\right]=0$. Then,
$\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}=o_{p}(1)$. Under classical
assumptions of OLS, and as
$\mathbb{E}\left[\boldsymbol{\nu}\boldsymbol{\nu}^{\prime}|\breve{\mathbf{X}},\breve{\mathbf{W}}\right]=\sigma_{\nu}^{2}\mathbf{I}$,
the asymptotic distribution of $\hat{\beta}$ is given by
$\sqrt{N}\left(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right)\mathrel{\overset{\makebox[0.0pt]{\mbox{\scriptsize
a}}}{\sim}}{}\mathcal{N}\left(\mathbf{0},\sigma_{e}^{2}\left(\breve{\mathbf{X}}^{\prime}\mathbf{M}_{\breve{\mathbf{W}}}\breve{\mathbf{X}}\right)^{-1}\right)$.

###  4.4 Nonlinear models

A possible extension is the use of nonlinear models with one or more variables
discretized. To see how our method can apply to this context, consider

$\mathbf{y}=g(\mathbf{X},\mathbf{W};\boldsymbol{\beta},\boldsymbol{\gamma})+\boldsymbol{\varepsilon},$
(20)

where $g(\cdot)$ denotes a continuous function. Let
$\hat{\boldsymbol{\beta}}(\mathbf{y},\mathbf{X},\mathbf{W})$ denote a
consistent estimator of $\boldsymbol{\beta}$ with
$\rho(\mathbf{X},\mathbf{W})=\sqrt{N}\left[\hat{\boldsymbol{\beta}}(\mathbf{y},\mathbf{X},\mathbf{W})-\boldsymbol{\beta}\right]$
such that
$\rho(\mathbf{y},\mathbf{X},\mathbf{W})\overset{d}{\rightarrow}\mathbf{f}(\bf
0,\boldsymbol{\Omega})$. Under the assumptions made earlier, for all cases,
the nonlinear quantities should converge too.

  * •

regressors are discretized $(\mathbf{X}^{*})$:
$\mathbf{X}^{\dagger}\overset{d}{\rightarrow}\mathbf{X}\implies\rho(\mathbf{y},\mathbf{X}^{\dagger},\mathbf{W})\overset{d}{\rightarrow}\rho(\mathbf{y},\mathbf{X},\mathbf{W})$

  * •

outcome is discretized ($\mathbf{y}^{*}$):
$\mathbf{y}^{\dagger}\overset{d}{\rightarrow}\mathbf{y}\implies\rho(\mathbf{y}^{\dagger},\mathbf{X},\mathbf{W})\overset{d}{\rightarrow}\rho(\mathbf{y},\mathbf{X},\mathbf{W})$

  * •

both variables are discretized ($\mathbf{y}^{*},\mathbf{X}^{*}$):
$(\mathbf{y}^{\dagger},\mathbf{X}^{\dagger})\overset{d}{\rightarrow}(\mathbf{y},\mathbf{X})\implies\rho(\mathbf{y}^{\dagger},\mathbf{X}^{\dagger},\mathbf{W})\overset{d}{\rightarrow}\rho(\mathbf{y},\mathbf{X},\mathbf{W})$,

by the continuous mapping theorem under appropriate regularity conditions. The
technical details of these conditions, however, could be an interesting
subject of future research.

###  4.5 Panel data and perception effect

In the case of panel data, the extension of this methodology is relatively
straightforward. The most important difference is in the identification. If
the interval of an individual does not change over the time periods covered,
the individual effects in the panel and the parameter associated with the
choice variable cannot be identified separately. The within transformation
would wipe out the interval variable as well. When the interval does change
over time, but not much, then we are facing weak identification, i.e., in fact
very little information is available for identification, so the parameter
estimates are going to be highly unreliable. (See more in online supplement,
Section A.5.6.)

To extend our method towards fixed effect type of estimators, let us discuss a
further phenomenon that we call the perception effect. The perception effect
is relevant if the discretization happens through surveys. There is much
evidence in the behavioral literature that the answers to a question may
depend on the way the question is asked (see, e.g., Diamond and Hausman,,
1994, Haisley et al.,, 2008 and Fox and Rottenstreich,, 2003).343434Comments
by Botond Kőszegi on this section are highly appreciated. Note, that this is
present regardless of whether split sampling has been performed or not.
However, with split sampling, there is a way to tackle this issue, much akin
to the approach a similar problem has been dealt with in the panel data
literature. To sketch out the idea in the univariate case, let us define the
perception effect $B_{s}$ for split sample $s$, as

$Z^{**}_{i}=\begin{cases}v_{1}^{(s)}\qquad&\text{if}\quad
c_{0}^{(s)}<Z_{i}+B_{s}<c_{1}^{(s)}\\\ \vdots\qquad&\vdots\\\
v_{m}^{(s)}\qquad&\text{if}\quad
c_{m-1}^{(s)}<Z_{i}+B_{s}<c_{M}^{(s)}\,.\end{cases}$ (21)

Let $Z^{*}_{i}$ be the same quantity, but $B_{s}=0\,,\,\forall s$, thus no
perception effect. $\tilde{Z}^{*}_{i}$ and $\tilde{Z}^{**}_{i}$ denote the
replaced observations in the working sample that derived from $Z^{*}_{i}$ and
$Z^{**}_{i}$, respectively. Following the construction of the working sample,

$\tilde{Z}^{**}_{i}=\tilde{Z}^{*}_{i}+B_{s}\,.$

For example, in the case of discretization happening with the outcome
variable, one can use a similar approach as outlined in Section 4.2, and use
the redefined equation,

$\tilde{Y}^{**}_{i}=\tilde{Y}^{*}_{i}+B_{s}=\beta\tilde{X}_{i}+u_{i}\,.$

Re-write the above in multivariate case,

$\tilde{\mathbf{y}}^{*}+\mathbf{T}\mathbf{B}=\tilde{\mathbf{X}}\boldsymbol{\beta}+\mathbf{u},$

where $\mathbf{B}=\left(B_{1},\ldots,B_{S}\right)^{\prime}$ and $\mathbf{T}$
is a $(N\times S)$ zero-one matrix that extracts the appropriate elements from
$\mathbf{B}$.

Note, the estimator for
$\mathbb{E}\left(\mathbf{y}|\mathbf{X}\in\mathbf{D}_{l}\right)$ needs to be
adjusted for the above to hold if identification of ${\boldsymbol{\beta}}$ is
based on the conditional expectation. The main challenge is to keep track of
the perception effect. This means we need to identify each split sample and
observation when estimating the conditional averages. Specifically,

$N_{l}^{-1}\sum_{\mathbf{X}\in\mathbf{D}_{l},\mathbf{X}\in
s}\tilde{\mathbf{y}}^{**}-\mathbb{E}\left(\mathbf{y}|\mathbf{X}\in\mathbf{D}_{l}\right)+B_{s}=o_{p}(1)\,\,,$

where $N_{l}$ is the number of observations in the partitioned interval
$\mathbf{D}_{l}$.

Now, the estimation of $\boldsymbol{\beta}$ can be done in the spirit of a
fixed effect estimator. Define the usual residual maker,
$\mathbf{M}_{\mathbf{T}}=\mathbf{I}_{N}-\mathbf{T}\left(\mathbf{T}^{\prime}\mathbf{T}\right)^{-1}\mathbf{T}^{\prime}$,
then

$\hat{\boldsymbol{\beta}}=\left(\tilde{\mathbf{X}}^{\prime}\mathbf{M}_{\mathbf{T}}\tilde{\mathbf{X}}\right)^{-1}\tilde{\mathbf{X}}^{\prime}\mathbf{M}_{\mathbf{T}}\tilde{\mathbf{y}}^{**}$
(22)

is a consistent estimator of $\boldsymbol{\beta}$ given the results presented
in this paper and similar arguments for the consistency of the standard fixed
effect estimator in the panel data literature (see, e.g., Matyas, 2024).

While the above discussion focuses on discretization happening with the
outcome variable, similar extensions can be done when discretization happens
with one or more explanatory variables or discretization is present on both
sides of the regression equation. In the online supplement, Section A.6, we
show results for these cases.

Lastly, one can include different fixed effect types of controls such as
interacting class and split sample effects: $B_{sm}$ for $s=1,\ldots,S$ and
$m=1,\ldots,M$. This formulation would take care of all potential perception
effects and allow a Wald-type test to assess the impact of perception effects
on the estimator $\boldsymbol{\beta}$. In the online supplement, Section
A.6.3, we show that this test leads to $\chi^{2}$ statistics, when the null
hypothesis is no perception effects $\left(H_{0}:\,B_{s}=0,\,\,\forall
s\right)$.

##  5 Simulation Evidence

For the simulation experiments, we consider a simple univariate linear
regression model for tractability,

$Y_{i}=X^{\prime}_{i}\beta+\varepsilon_{i}\,,$

where we discretize $Y_{i}$, or $X_{i}$ or both. We set the parameter of
interest $\beta=0.5$ and report the Monte Carlo average bias and its standard
deviation in parenthesis353535avg. bias :=
$\left(MC^{-1}\sum_{mc}(\hat{\beta}_{mc}-\beta)\right)$, std :=
$\sqrt{(MC-1)^{-1}\sum_{mc}(\hat{\beta}_{mc}-\beta)^{2}}$. with $1,000$
repetitions. We only vary $\varepsilon_{i}$ in each repetition and fix
$X_{i}$. We use $N=10,000$ observations for all reported simulation
setup.363636We have experimented with different sample sizes, see our results
in the online supplement, Section A.7. As the size of the bias depends on how
well one can map the conditional expectations, thus from the underlying
distribution, during the simulations we use multiple distributions for
$\varepsilon_{i}$ or $X_{i}$. These distributions have different shapes, which
helps us better understand the effect of dispersion, skewness, kurtosis, and
tail properties on the magnitude of bias. All of the following distributions
are constructed such that the lower and upper boundary points of the support
are -1 and 3.

  * •

Normal: standard normal distribution truncated at $-1$ and $3$.

  * •

Logistic: standard logistic distribution truncated at $-1$ and $3$.

  * •

Log-Normal: standard log-normal distribution truncated at $4$ and subtracted
$1$ (in order to adjust the mean).

  * •

Uniform: uniform distribution between $-1$ and $3$.

  * •

Exponential: exponential distribution with rate parameter $0.5$, truncated at
$4$ and subtracted $1$.

  * •

Weibull: weibull distribution with shape parameter $1.5$ and scale parameter
$1$, truncated at $4$ and subtracted $1$.

To simplify our investigations, we use these distributions to generate
$\varepsilon_{i}$ when the outcome or both the outcome and explanatory
variables are discretized. In these cases $X_{i}\sim\mathcal{N}(0,0.25,-1,1)$,
that is distributed as a truncated normal distribution between -1 and 1 with
mean zero and variance 0.25.373737This choice of variance ensures that even
without truncation, $95\%$ of the probability mass lies between $-1$ and $1$.
When discretization happens with the explanatory variable $X_{i}$, we use the
different distributions to generate $X_{i}$ and use the same truncated normal
distribution for $\varepsilon_{i}\sim\mathcal{N}(0,0.25,-1,1)$. This setup
allows us to control for the domain of $Y_{i}$ in all three cases, which lies
between -2 and 4, thus no bias comes from censoring the tails. For the
discretization of the variable(s) in all cases we use $M=5$ and for split
sampling $S=10$ with equal distances for the intervals between the known
support. We have experimented with different setups; see more in the online
supplement, Section A.7.

To estimate $\beta$, we used the following methods:

  * •

Mid-point regression: A simple linear regression using mid-points for $v_{m}$
and OLS for estimation.

  * •

Shifting method: Shifting method as described in Algorithm 1 and 2.383838We
used mid-values as observations for the working sample’s values
$(v^{WS}_{b})$.

We specify our model to estimate in the following form

$Y_{i}=\alpha+X_{i}^{\prime}\beta+\eta_{i}\,\,,$

where, $\varepsilon_{i}=\alpha+\eta_{i}$ with
$\mathbb{E}(\varepsilon_{i})=\alpha$, to adjust for those distributions, which
does not have zero mean.

### Explanatory variable

In our first simulations, discretization happens with the explanatory variable
$X_{i}$ and the different distributions correspond to the distribution of
$X_{i}$. Table 1 reports the Monte Carlo average bias from the true parameter
along with the Monte Carlo standard deviation in parenthesis.

| Normal | Logistic | Log-Normal | Uniform | Exponential | Weibull
---|---|---|---|---|---|---
Mid-point regression | -0.0252 | -0.0101 | -0.0174 | 0.0002 | 0.0005 | -0.0422
(0.0057) | (0.0046) | (0.0051) | (0.0040) | (0.0102) | (0.0073)
Shifting method $(S=10)$ | -0.0037 | -0.0003 | -0.0022 | 0.0002 | 0.0023 | -0.0015
(0.0060) | (0.0046) | (0.0050) | (0.0038) | (0.0094) | (0.0073)

Table 1: Monte Carlo average bias and standard deviation when the discretized
variable is on the right-hand side

Table 1 shows multiple interesting insights. First, the shifting method
provides a much smaller average bias than the mid-point regression. In most
cases, shifting methods reduce the average bias by the magnitude of five to
ten. A second important insight is when we use a uniform distribution, there
is no bias, as the conditional expectation is the mid-point of the
interval.393939This result is in line with the theoretical results shown in
the online supplement, Section A.5. The results show that the magnitude of the
bias is a function of the curvature of the underlying distribution. E.g., in
the case of the exponential distribution, there is not too much curvature
within the used domain, however for normal/logistic/weibull distributions
there is, thus the mid-point for each interval is further away from the
expected value of the same interval.

### Outcome variable

Now, we discretize the outcome variable $Y_{i}$. Our main concern is the
disturbance term; therefore, we have visited the aforementioned types of
distributions for $\varepsilon_{i}$.404040This creates an asymmetric
distribution for $\varepsilon_{i}$ in several cases, which favors
distribution-independent estimation methods rather than the maximum
likelihood. In the online supplement, Section A.5, we show results with
boundaries where $\varepsilon_{i}$ was truncated symmetrically. The results
and conclusions remain unchanged. To partition the exogenous variable $X_{i}$,
$L$ is set to $50$ with equal distances.

To estimate $\beta$, we have revisited some popular estimation methods that
are applicable, when the outcome variable is discretized:

  * •

Set identification: Estimates the lower and upper boundaries of the parameter
set for $\beta$ using $Y^{*}_{i}$ as interval data. Estimation is based on
Beresteanu and Molinari, (2008).414141We used the published Stata package
(Beresteanu et al.,, 2010) from
https://molinari.economics.cornell.edu/programs.html. This method does not
produce point-estimates for $\beta$, only lower and upper boundaries of the
identified parameter set.

  * •

Ordered probit and logit: Ordered choice models, where $Y^{*}_{i}$ values are
ordinal data, and the model assumes a gaussian or logistic distribution
(Greene and Hensher,, 2010). Note that the estimated maximum likelihood
‘naive’ parameters reported here are not designed to recover $\beta$ and to be
interpreted in the linear regression sense. Therefore, we call the difference
from $\beta$ distortion rather than bias.

  * •

Interval regression: A modification of the ordered choice model, where
$Y^{*}_{i}$ values are interval data and the model assumes gaussian
distribution to model the linear regression model. The maximum likelihood
parameter estimates aim to recover $\beta$ through the distributional
assumption. (See Cameron and Trivedi,, 2010, p. 548-550 or Greene and
Hensher,, 2010, p. 133.)

We have included an intercept wherever possible.424242Ordered choice models’
implementation in Stata removes (restricts to zero) the intercept parameter to
identify $\beta$. We report the Monte Carlo average bias or distortion from
the true parameter along with the standard deviations.

| Normal | Logistic | Log-Normal | Uniform | Exponential | Weibull
---|---|---|---|---|---|---
Set identification† | $\left[-1.1,1.15\right]$ | $\left[-1.09,1.15\right]$ | $\left[-1.09,1.16\right]$ | $\left[-1.07,1.17\right]$ | $\left[-1.06,1.19\right]$ | $\left[-1.09,1.15\right]$
[(0.02),(0.02)] | [(0.03),(0.03)] | [(0.02),(0.02)] | [(0.03),(0.03)] | [(0.03),(0.03)] | [(0.02),(0.02)]
Ordered probit∗ | 0.1971 | 0.0688 | 0.2085 | 0.0158 | 0.0986 | 0.4461
(0.0256) | (0.0253) | (0.0262) | (0.0234) | (0.0241) | (0.0295)
Ordered logit∗ | 0.6509 | 0.3814 | 0.6862 | 0.2379 | 0.4338 | 1.2085
(0.0464) | (0.0455) | (0.0499) | (0.0422) | (0.044) | (0.0546)
Interval regression | 0.0268 | 0.0332 | 0.0371 | 0.0491 | 0.0663 | 0.0397
(0.0198) | (0.0249) | (0.0221) | (0.0271) | (0.0249) | (0.0166)
Mid-point regression | 0.0253 | 0.0322 | 0.0362 | 0.0490 | 0.2077 | 0.0314
(0.0195) | (0.0236) | (0.0216) | (0.0273) | (0.0128) | (0.0157)
Shifting method | -0.0010 | -0.0017 | -0.0010 | -0.0014 | -0.0017 | -0.0003
(0.0211) | (0.0239) | (0.0215) | (0.0271) | (0.0125) | (0.0147)

† Set identification gives the lower and upper boundaries for the valid
parameter set. We report these bounds subtracted from the true parameter;
therefore, it should give a (close) interval around zero.
∗ Distortion from the true $\beta$ is reported. Ordered probit and logit
models’ maximum likelihood parameters do not aim to recover the true $\beta$
parameter; therefore, it is not appropriate to call them biased.

Table 2: Monte Carlo average bias and standard deviation when the discretized
variable is the outcome variable

Table 2 shows that the shifting method consistently provides the smallest
average bias. Set identification gives large intervals for the parameter set
that are unlikely to be useful in practice. Distortions of ordered probit and
logit models are rather large. Interval regression and mid-point regression
perform poorly as the bias is more than ten times larger than for the shifting
method.

### Both variables

In the last simulation, we discretize both $Y_{i}$ and $X_{i}$ variables. We
used different distributions for $\varepsilon_{i}$ and kept $X_{i}$ the same
truncated normal distribution along repetitions. This means that some bias
comes from the discretization of $X_{i}$, but the variation across
distributional results comes from the discretization of $Y_{i}$. For both
discretizations, we used $M=M_{Y}=M_{X}=5$ and $S=S_{Y}=S_{X}=10$.

| Normal | Logistic | Log-Normal | Uniform | Exponential | Weibull
---|---|---|---|---|---|---
Mid-point regression | -0.0853 | -0.0788 | -0.0752 | -0.0635 | 0.0797 | -0.0759
(0.0178) | (0.0213) | (0.0190) | (0.0243) | (0.0116) | (0.0137)
Shifting $(S=10)$ | -0.0027 | 0.0156 | 0.0104 | 0.0156 | 0.0006 | 0.0108
(0.0235) | (0.0269) | (0.0243) | (0.0294) | (0.0132) | (0.0156)

Table 3: Monte Carlo average bias and standard deviation when both variables
are discretized

Table 3 shows the results. Mid-point regression ends up with biased estimates
in all cases, regardless of the used distribution. The shifting method works
decently and reduces the magnitude of bias effectively in all cases.

We have run several other Monte Carlo experiments and the results are similar:
the shifting method outperforms all alternatives. An important result is
consistency for the shifting method, as we increase the number of observations
along with $S$, we get smaller biases. Note that consistency does not hold for
the competing methods. For a detailed discussion of further results see the
online supplement, Section A.7.

##  6 Empirical application: The Australian gender wage gap

We illustrate our approach with a short study of the Australian gender wage
gap from 2017. We obtained wage and socio-economic data from the Australian
Tax Office (ATO). The individuals’ sample files record a $2\%$ sample of the
whole population,434343For details, see ATO’s website: https://www.ato.gov.au/
. including the actual wage values. This enabled us to estimate $\beta$ when
no discretization happens.

In practice, usually, we only observe discretized variables through different
discretization schemes. To mitigate this, we employ equally distanced
discretization methods with $M=3,5$, and $10$. We add the special
discretization scheme based on the Household, Income, and Labour Dynamics in
Australia (HILDA)
Survey.444444https://melbourneinstitute.unimelb.edu.au/hilda. HILDA is an
annual survey, which is well known and widely used without further constraints
in Australia for economic research (see:
https://melbourneinstitute.unimelb.edu.au/hilda/publications) Multiple papers
use HILDA to estimate the gender wage gap (see, e.g., Barón and Cobb-Clark,
2010 or Coelli, 2014). Note that, HILDA uses the discretization process for
total income along with other additional information to calculate hourly
wages. The HILDA Household Questionnaire in 2017 used thirteen different
income categories.454545These categories are the following: negative or zero,
$1-9.999$, $10.000-19.999$, $20.000-29.999$, $30.000-39.999$, $40.000-49.999$,
$50.000-59.999$, $60.000-79.999$, $80.000-99.999$, $100.000-124.999$,
$125.000-149.999$, $150.000-199.999$ and $200.000$ or more. Two extra options
were added: refused and don’t know.

Table 4 shows regression estimates of the gender dummy (1 is female) on log
yearly wage while controlling for age, age square, marital status, occupation,
and regional variables. The ‘Directly observed’ row shows the estimated
parameter when the non-discretized values are used for yearly wages. The row
labeled ‘HILDA’ gives the estimated parameter value when using the HILDA
discretization. The remaining rows show the estimates when the discretization
process is done in equal-sized intervals with $M=3,5,10$. Mid-point regression
does not use split samples, while the shifting method uses $S=10$. In all
cases, the yearly wage was discretized, and then we took the log of the
discretized mid values as this represents the practice.

Discretization |  Estimates for $\hat{\beta}$
---|---
Directly observed | -0.2261
(no discretization) | (0.0023)
HILDA | -0.2242
(0.0023)
Number of intervals | $M=3$ | $M=5$ | $M=10$
Mid-point regression | -0.2747 | -0.2635 | -0.2364
(0.0033) | (0.0031) | (0.0025)
Shifting method ($S=10$) | -0.2214 | -0.2348 | -0.2280
(0.0030) | (0.0027) | (0.0024)
N | 125,995
Table 4: Different estimates for gender wage gap based on different
discretizations of yearly wages

In all cases, the shifting method yields closer estimates than mid-point
regression, whereas the latter statistically produces different results from
the ‘directly observed’ value. HILDA gives a fairly good estimate, which shows
that it is a carefully designed survey and uses further external information.
The shifting method’s advantage compared to HILDA is twofold: i) it is a
general and topic-agnostic tool and thus does not require external information
or further expertise; ii) it does not require the use of surveys, which face
multiple challenges464646For example Fisher, (2019) shows that through
repeated surveys of the same income variable, the reported average income
changes by 8% due to behavioral responses by survey participants. and costs.
For more discussion on the empirical example and for other model setups, see
online supplement, Section A.8.

##  7 Conclusion

This paper deals with econometric models where variables are continuous but
observed through a discretization process that results in interval censored
data. Discretization is becoming increasingly common in practice, particularly
when dealing with sensitive data where discretization protects privacy (e.g.,
income). Modeling with such variables in a regression framework is problematic
as the conditional expectation is unknown and the parameters of interest
cannot be point identified.

Instead of following the partial identification approach, which provides set
intervals for the parameters of interest, we propose new techniques to point
identify the parameters of interest. Our approach hinges on split sampling,
which enables the mapping of the underlying variables’ distributions while
preserving the data privacy requirements. We utilize the mapped (marginal)
distribution and modify the original regression model, which allows the
transformed variables to point identify the original parameters of interest.
Identification depends on where the discretization happens: i) the discretized
variable is an explanatory variable; ii) the discretized variable is the
outcome; and iii) when discretization happens on both sides. The three cases
require different types of transformations of the variables. We examine the
methods of identifying strategies in all instances, initially using univariate
examples and subsequently expanding them to encompass multivariate cases.

During the identification, we assumed the marginal distributions were known,
which is not true in practice. As the next step, we describe an implementation
of split sampling, the shifting method, which systematically shifts the
boundary points of each interval while preserving data confidentiality. We
show convergence in distribution for the underlying unknown variable by the
shifting method, which reduces the size of the shift towards zero by
increasing the number of split samples. This mechanism allows us to
nonparametrically identify the marginal distribution.

For the three modeling purposes, we can use the mapped distribution of the
underlying variable. We derive different consistent estimators based on the
mapped marginal distribution, which leads to consistent parameter estimates
for all cases in a cross-sectional setup. We show the asymptotic properties of
these estimators and discuss extensions to nonlinear models, panel data and
fixed effect type of estimators. We provide Monte Carlo evidence to show that
our methods have superior finite sample properties compared to the ‘usual’
methods. Finally, we investigate our method in practice using the case study
of the Australian gender wage gap. Results show that with the shifting method,
we get close estimates for the gender wage gap parameter even when the number
of intervals is small, only 3 or 5.

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