# Identification and Estimation of Categorical Random Coefficient
Models††thanks: We would like to thank Timothy Armstrong, Hidehiko Ichimura,
Esfandiar Maasoumi, Geert Ridder, Ron Smith and Hayun Song for helpful
comments, and the editor and two anonymous referees for constructive comments
and suggestions.

Zhan Gao  Ph.D. student, Department of Economics, University of Southern
California, 3620 South Vermont Avenue, Los Angeles, CA 90089, USA. Email:
<EMAIL_ADDRESS>M. Hashem Pesaran 1\. Department of Economics, University of
Southern California, 3620 South Vermont Avenue, Los Angeles, CA 90089, USA.
Email<EMAIL_ADDRESS>2\. Trinity College, Cambridge, United Kingdom

###### Abstract

This paper proposes a linear categorical random coefficient model, in which
the random coefficients follow parametric categorical distributions. The
distributional parameters are identified based on a linear recurrence
structure of moments of the random coefficients. A Generalized Method of
Moments estimation procedure is proposed also employed by Peter Schmidt and
his coauthors to address heterogeneity in time effects in panel data models.
Using Monte Carlo simulations, we find that moments of the random coefficients
can be estimated reasonably accurately, but large samples are required for
estimation of the parameters of the underlying categorical distribution. The
utility of the proposed estimator is illustrated by estimating the
distribution of returns to education in the U.S. by gender and educational
levels. We find that rising heterogeneity between educational groups is mainly
due to the increasing returns to education for those with postsecondary
education, whereas within group heterogeneity has been rising mostly in the
case of individuals with high school or less education.

Keywords: Random coefficient models, categorical distribution, return to
education

JEL Code: C01, C21, C13, C46, J30

##  1 Introduction

Random coefficient models have been used extensively in time series, cross-
section and panel regressions. Nicholls and Pagan (1985) consider the
estimation of first and second moments of the random coefficient $\beta_{i}$
and the error term $u_{i}$, in a linear regression model. In the seminal work,
Beran and Hall (1992) establish the conditions of identifying and estimating
the distribution of $\beta_{i}$ and $u_{i}$ non-parametrically. The baseline
linear univariate regression in Beran and Hall (1992) has been extended in
non-parametric framework by Beran (1993); Beran and Millar (1994); Beran,
Feuerverger, and Hall (1996); Hoderlein, Klemelä, and Mammen (2010, 2017) and
Breunig and Hoderlein (2018), to just name a few. Hsiao and Pesaran (2008)
survey random coefficient models in linear panel data models.

In some econometric applications, Hausman (1981); Hausman and Newey (1995);
Foster and Hahn (2000) for examples, the main interest is to estimate the
consumer surplus distribution based on a linear demand system where the
coefficient associated with the price is random. In such settings, the
distribution of the random coefficients is needed when computing the consumer
surplus function, and the non-parametric estimation is more general, flexible
and suitable for the purpose. On the other hand, parametric models may be
favored in applications in which the implied economic meaning of the
distribution of the random coefficients is of interests. Examples include
estimation of the return to education (Lemieux, 2006b, c) and the labor supply
equation (Bick, Blandin, and Rogerson, 2022).

In this paper, we consider a linear regression model with a random coefficient
$\beta_{i}$ that is assumed to follow a categorical distribution, i.e.
$\beta_{i}$ has a discrete support
$\left\\{b_{1},b_{2},\cdots,b_{K}\right\\}$, and $\beta_{i}=b_{k}$ with
probability $\pi_{k}$. The discretization of the support of the random
coefficient $\beta_{i}$ naturally corresponds to the interpretation that each
individual belongs to a certain category, or group, $k$ with probability
$\pi_{k}$. Compared to a non-parametric distribution with continuous support,
assuming a categorical distribution allows us not only to model the
heterogeneous responses across individuals but also to interpret the results
with sharper economic meaning. As we will illustrate in the empirical
application in Section 6, it is hard to clearly interpret the distribution of
returns to education without imposing some form of parametric restrictions.

In addition, with the categorical distribution imposed, the identification and
estimation of the distribution of $\beta_{i}$ do not rely on identically
distributed error terms $u_{i}$ and regressors $\mathbf{w}_{i}$, as shown in
Section 2 and 3. Heterogeneously generated errors can be allowed, which is
important in many empirical applications. To the best of our knowledge, this
is the first identification result in linear random coefficient model without
a strict IID setting.

The identification of the distribution of $\beta_{i}$ is established in this
paper based on the identification of the moments of $\beta_{i}$, which
coincides with the identification condition in Beran and Hall (1992) that the
distribution of $\beta_{i}$ is uniquely determined by its moments, which is
assumed to exist up to an arbitrary order. Since under our setup the
distribution of $\beta_{i}$ is parametrically specified, the moments of
$\beta_{i}$ exist and can be derived explicitly. The parameters of the assumed
categorical distribution can then be uniquely determined by a system of
equations in terms of the moments, as in Theorem 2. The parameters of the
categorical distribution are then estimated consistently by the generalized
method of moments (GMM). The estimation procedure based on moment conditions
shares similar spirits as in Ahn, Lee, and Schmidt (2001, 2013) in which Peter
Schmidt and coauthors study panel data models with interactive effects where
they allow for the time effects to vary across individual units. Comparing to
alternative non-parametric random coefficient models, the standard GMM
estimation is easy to implement, and the identified categorical structure has
a clear economic interpretation.

Using Monte Carlo (MC) simulations, we find that moments of the random
coefficients can be estimated reasonably accurately, but large samples are
required for estimation of the parameters of the underlying categorical
distributions. Our theoretical and MC results also suggest that our method is
suitable when the number heterogeneous coefficients and the number of
categories are small (2 or 3). With the number of categories rising the burden
on identification from the moments to the parameters of the categorical
distribution also rises rapidly. The quality of identification also
deteriorates as we need to rely on higher and higher moments to identify a
larger number of categories, since the information content of the moments tend
to decline with their order.

The proposed method is also illustrated by providing estimates of the
distribution of returns to education in the U.S. by gender and educational
levels, using the May and Outgoing Rotation Group (ORG) supplements of the
Current Population Survey (CPS) data. Comparing the estimates obtained over
the sub-periods 1973-75 and 2001-03, we find that rising between group
heterogeneity is largely due to rising returns to education in the case of
individuals with postsecondary education, whilst within group heterogeneity
has been rising in the case of individuals with high school or less education.

Related Literature: This paper draws mainly upon the literature of random
coefficient models. As already mentioned, the main body of the recent
literature is focused on non-parametric identification and estimation.
Following Beran and Hall (1992), Beran (1993) and Beran and Millar (1994)
extend the model to a linear semi-parametric model with a multivariate setup
and propose a minimum distance estimator for the unknown distribution. Foster
and Hahn (2000) extend the identification results in Beran and Hall (1992) and
apply the minimum distance estimator to a gasoline consumption data to
estimate the consumer surplus function. Beran, Feuerverger, and Hall (1996)
and Hoderlein, Klemelä, and Mammen (2010) propose kernel density estimators
based on the Radon inverse transformation in linear models.

In addition to linear models, Ichimura and Thompson (1998) and Gautier and
Kitamura (2013) incorporate the random coefficients in binary choice models.
Gautier and Hoderlein (2015) and Hoderlein, Holzmann, and Meister (2017)
consider triangular models with random coefficients allowing for causal
inference. Matzkin (2012) and Masten (2018) discuss the identification of
random coefficients in simultaneous equation models. Breunig and Hoderlein
(2018) propose a general specification test in a variety of random coefficient
models. Random coefficients are also widely studied in panel data models, for
example Hsiao and Pesaran (2008) and Arellano and Bonhomme (2012)

The rest of the paper is organized as follows: Section 2 establishes the main
identification results. The GMM estimation procedure is proposed and discussed
in Section 3. An extension to a multivariate setting is considered in Section
4. Small sample properties of the proposed estimator are investigated in
Section 5, using Monte Carlo techniques under different regressor and error
distributions. Section 6 presents and discusses our empirical application to
the return to education. Section 7 provides some concluding remarks and
suggestions for future work. Technical proofs are given in Appendix A.1.

Notations: Largest and smallest eigenvalues of the $p\times p$ matrix
$\mathbf{A}=\left(a_{ij}\right)$ are denoted by
$\lambda_{\max}\left(\mathbf{A}\right)$ and
$\lambda_{\min}\left(\mathbf{A}\right),$ respectively, its spectral norm by
$\left\|\mathbf{A}\right\|=\lambda_{\max}^{1/2}\left(\mathbf{A}^{\prime}\mathbf{A}\right)$,
$\mathbf{A}\succ 0$ means that $\mathbf{A}$ is positive definite,
$\mathrm{vech\left(\mathbf{A}\right)}$ denotes the vectorization of distinct
elements of $\mathbf{A}$, $\mathbf{0}$ denotes zero matrix (or vector). For
$\mathbf{a}\in\mathbb{R}^{p}$, $\mathrm{diag}\left(\mathbf{a}\right)$
represents the diagonal matrix with elements of $a_{1},a_{2},\cdots,a_{p}$.
For random variables (or vectors) $u$ and $v$, $u\perp v$ represents $u$ is
independent of $v$. We use $c$ ($C$) to denote some small (large) positive
constants. For a differentiable real-valued function
$f\left(\mathbf{\theta}\right)$,
$\nabla_{\mathbf{\theta}}f\left(\mathbf{\theta}\right)$ denotes the gradient
vector. Operator $\rightarrow_{p}$ denotes convergence in probability, and
$\rightarrow_{d}$ convergence in distribution. The symbols $O(1)$, and
$O_{p}(1)$ denote asymptotically bounded deterministic and random sequences,
respectively.

##  2 Categorical random coefficient model

We suppose the single cross-section observations,
$\left\\{y_{i},x_{i},\mathbf{z}_{i}\right\\}_{i=1}^{n}$, follow the
categorical random coefficient model

$y_{i}=x_{i}\beta_{i}+\mathbf{z}_{i}^{\prime}\mathbf{\gamma}+u_{i},$ (2.1)

where $y_{i},x_{i}\in\mathbb{R}$, $\mathbf{z}_{i}\in\mathbb{R}^{p_{z}},$ and
$\beta_{i}\in\left\\{b_{1},b_{2},\cdots,b_{K}\right\\}$ admits the following
$K$-categorical distribution,

$\beta_{i}=\begin{cases}b_{1},&\text{w.p. }\pi_{1},\\\ b_{2},&\text{w.p.
}\pi_{2},\\\ \vdots&\vdots\\\ b_{K},&\text{w.p. }\pi_{K},\end{cases}$ (2.2)

w.p. denotes “with probability”, $\pi_{k}\in\left(0,1\right)$,
$\sum_{k=1}^{K}\pi_{k}=1$, $b_{1}<b_{2}<\cdots<b_{K}$,
$\mathbf{\gamma}\in\mathbb{R}^{p_{z}}$ is homogeneous and $\mathbf{z}_{i}$
could include an intercept term as its first element. It is assumed that
$\beta_{i}\perp\mathbf{w}_{i}=\left(x_{i},\mathbf{z}_{i}^{\prime}\right)^{\prime}$,
and the idiosyncratic errors $u_{i}$ are independently distributed with mean
$0$.

###### Remark 1

The model can be extended to allow
$\mathbf{x}_{i},\mathbf{\beta}_{i}\in\mathbb{R}^{p}$, with
$\mathbf{\beta}_{i}$ following a multivariate categorical distribution, though
with more complicated notations. We will consider possible extensions in
Section 4.

###### Remark 2

Since we consider a pure cross-sectional setting, the key assumption that
$\beta_{i}$ and $x_{i}$ are independently distributed cannot be relaxed.
Allowing $\beta_{i}$ to vary with $w_{i}$, without any further restrictions,
is tantamount to assuming $y_{i}$ is a general function of $w_{i}$, in effect
rendering a nonparametric specification.

###### Remark 3

The number of categories $K$ is assumed to be fixed and known. Conditions
$\sum_{k=1}^{K}\pi_{k}=1$, $b_{1}<b_{2}<\cdots<b_{K},$ and
$\pi_{k}\in\left(0,1\right)$ together are sufficient for the existence of $K$
categories. For example, if $b_{k}=b_{k^{\prime}}$, then we can merge
categories $k$ and $k^{\prime}$, and the number of categories reduces to
$K-1$. Similarly, if $\pi_{k}=0$ for some $k$, then category $k$ can be
deleted, and the number of categories is again reduced to $K-1$. Information
criteria can be used to determine $K$, but this will not be pursued in this
paper. Model specification tests could also be considered. See, for examples,
Andrews (2001) and Breunig and Hoderlein (2018).

In the rest of this section, we focus on the model (2.1) and establish the
conditions under which the distribution of $\beta_{i}$ is identified.

###  2.1 Identifying the moments of $\beta_{i}$

###### Assumption 1

  1. (a)

(i) $u_{i}$ is distributed independently of
$\mathbf{w}_{i}=\left(x_{i},\mathbf{z}_{i}^{\prime}\right)^{\prime}$ and
$\beta_{i}$. (ii) $\sup_{i}\mathrm{E}\left(\left|u_{i}^{r}\right|\right)<C$,
$r=1,2,\cdots,2K-1$. (iii) $n^{-1}\sum_{i=1}^{n}u_{i}^{4}=O_{p}(1)$.

  2. (b)

(i) Let
$\mathbf{Q}_{n,ww}=n^{-1}\sum_{i=1}^{n}\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}$,
and $\mathbf{q}_{n,wy}=n^{-1}\sum_{i=1}^{n}\mathbf{w}_{i}y_{i}$. Then
$\left\|\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)\right\|<C<\infty$, and
$\left\|\mathrm{E}\left(\mathbf{q}_{n,wy}\right)\right\|<C<\infty$, and there
exists $n_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$,

$0<c<\lambda_{\min}\left(\mathbf{Q}_{n,ww}\right)<\lambda_{\max}\left(\mathbf{Q}_{n,ww}\right)<C<\infty.$

(ii)
$\sup_{i}\mathrm{E}\left(\left\|\mathbf{w}_{i}\right\|^{r}\right)<C<\infty$,
$r=1,2,\cdots,4K-2$.
(iii) $n^{-1}\sum_{i=1}^{n}\left\|\mathbf{w}_{i}\right\|^{4}=O_{p}(1)$.

  3. (c)

$\left\|\mathbf{Q}_{n,ww}-\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)\right\|=O_{p}\left(n^{-1/2}\right)$,
$\left\|\mathbf{q}_{n,wy}-\mathrm{E}\left(\mathbf{q}_{n,wy}\right)\right\|=O_{p}\left(n^{-1/2}\right)$,
and

$\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)=n^{-1}\sum_{i=1}^{n}\mathrm{E}\left(\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\right)\succ
0.$

  4. (d)

$\left\|\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)-\mathbf{Q}_{ww}\right\|=O\left(n^{-1/2}\right)$,
$\left\|\mathrm{E}\left(\mathbf{q}_{n,wy}\right)-\mathbf{q}_{wy}\right\|=O\left(n^{-1/2}\right)$,
where
$\mathbf{q}_{wy}=\lim\limits_{n\to\infty}\mathrm{E}\left(\mathbf{q}_{n,wy}\right)$,
$\mathbf{Q}_{ww}=\lim\limits_{n\to\infty}\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)$
and $\mathbf{Q}_{ww}\succ 0$.

###### Remark 4

Part (a) of Assumption 1 relaxes the assumption that $u_{i}$ is identically
distributed, and allows for heterogeneously generated errors. For
identification of the distribution of $\beta_{i}$, we require $u_{i}$ to be
distributed independently of $\mathbf{w}_{i}$ and $\beta_{i}$, which rules out
conditional heteroskedasticity. However, estimation and inference involving
$\mathrm{E}\left(\beta_{i}\right)$ and $\mathbf{\gamma}$ can be carried out in
presence of conditionally error heteroskedastic, as shown in Theorem 3. Parts
(c) and (d) of Assumption 1 relax the condition that $\mathbf{w}_{i}$ is
identically distributed across $i$. As we proceed, only $\beta_{i}$, whose
distribution is of interest, is assumed to be IID across $i$, and it is not
required for $\mathbf{w}_{i}$ and $u_{i}$ to be identically distributed over
$i$.

###### Remark 5

The high level conditions in Assumption 1, concerning the convergence in
probability of averages such as
$Q_{n,ww}=n^{-1}\sum_{i=1}^{n}\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}$, can be
verified under weak cross-sectional dependence. Let
$f_{i}=f\left(\mathbf{w}_{i},\beta_{i},u_{i}\right)$ be a generic function of
$\mathbf{w}_{i}$, $\beta_{i}$ and $u_{i}$.111$f_{i}$ is assumed to be a
scalar, and we can apply the analysis element-by-element to a matrix, for
example $\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}$. Assume that
$\sup_{i}\mathrm{E}\left(f_{i}^{2}\right)<C$, and
$\sup_{j}\sum_{i=1}^{n}\left|\mathrm{cov}\left(f_{i},f_{j}\right)\right|<C$,
for some fixed $C<\infty$. Then,

$\mathrm{var}\left(\frac{1}{n}\sum_{i=1}^{n}f_{i}\right)\leq\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\left|\mathrm{cov}\left(f_{i},f_{j}\right)\right|\leq\frac{1}{n}\sup_{j}\sum_{i=1}^{n}\left|\mathrm{cov}\left(f_{i},f_{j}\right)\right|\leq\frac{C}{n}.$

By Chebyshev’s inequality, for any $\varepsilon>0$, we have
$M_{\varepsilon}>\sqrt{C/\varepsilon}$ such that

$\Pr\left(\sqrt{n}\left|\frac{1}{n}\sum_{i=1}^{n}\left[f_{i}-\mathrm{E}\left(f_{i}\right)\right]\right|>M_{\varepsilon}\right)\leq\frac{n\mathrm{var}\left(n^{-1}\sum_{i=1}^{n}f_{i}\right)}{C}\varepsilon\leq\varepsilon,$

i.e.
$n^{-1}\sum_{i=1}^{n}\left[f_{i}-\mathrm{E}\left(f_{i}\right)\right]=O_{p}\left(n^{-1/2}\right)$.

Denote
$\mathbf{\phi}_{i}=\left(\beta_{i},\mathbf{\gamma}^{\prime}\right)^{\prime}$
and
$\mathbf{\phi}=\mathrm{E}\left(\mathbf{\phi}_{i}\right)=\left(\mathrm{E}\left(\beta_{i}\right),\mathbf{\gamma}^{\prime}\right)^{\prime}$.
Consider the moment condition,

$\mathrm{E}\left(\mathbf{w}_{i}y_{i}\right)=\mathrm{E}\left(\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\right)\mathbf{\phi},$
(2.3)

and sum (2.3) over $i$

$\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\mathbf{w}_{i}y_{i}\right)=\left[\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\right)\right]\mathbf{\phi}.$
(2.4)

Let $n\to\infty$, then $\mathbf{\phi}$ is identified by

$\mathbf{\phi}=\mathbf{Q}_{ww}^{-1}\mathbf{q}_{wy},$ (2.5)

under Assumption 1.

###### Assumption 2

Let $\tilde{y}_{i}=y_{i}-\mathbf{z}_{i}^{\prime}\mathbf{\gamma}$.

  1. (a)

$\left|n^{-1}\sum_{i=1}^{n}\mathrm{E}\left(\tilde{y}_{i}^{r}x_{i}^{s}\right)-\rho_{r,s}\right|=O\left(n^{-1/2}\right),$
and $\left|\rho_{r,s}\right|<\infty,$ for $r,s=0,1,\cdots,2K-1$.

  2. (b)

$\left|n^{-1}\sum_{i=1}^{n}\mathrm{E}\left(u_{i}^{r}\right)-\sigma_{r}\right|=O\left(n^{-1/2}\right),$
and $\left|\sigma_{r}\right|<\infty$, for $r=2,3,\cdots,2K-1$.

  3. (c)

$n^{-1}\sum_{i=1}^{n}\left[\mathrm{var}(x_{i}^{r})-\left(\rho_{0,2r}-\rho_{0,r}^{2}\right)\right]=O\left(n^{-1/2}\right)$
where $\rho_{0,2r}-\rho_{0,r}^{2}>0,$ for $r=2,3,\cdots,2K-1$.

###### Remark 6

The above assumption allows for a limited degree of heterogeneity of the
moments. As an example, let $\mathrm{E}\left(u_{i}^{r}\right)=\sigma_{ir}$ and
denote the heterogeneity of the $r^{th}$ moment of $u_{i}$ by
$e_{ir}=\sigma_{ir}-\sigma_{r}$. Then

$\left|n^{-1}\sum_{i=1}^{n}\mathrm{E}\left(u_{i}^{r}\right)-\sigma_{r}\right|\leq
n^{-1}\sum_{i=1}^{n}\left|e_{ir}\right|,$

and condition (b) of Assumption 2 is met if $\
\sum_{i=1}^{n}\left|e_{ir}\right|=O(n^{\alpha_{r}})$ with $\alpha_{r}<1/2$.
$\alpha_{r}$ measures the degree of heterogeneity with $\alpha_{r}=1$
representing the highest degree of heterogeneity. A similar idea is used by
Pesaran and Zhou (2018) in their analysis of poolability in panel data models.

###### Theorem 1

Under Assumptions 1 and 2, $\mathrm{E}\left(\beta_{i}^{r}\right)$ and
$\sigma_{r}$, $r=2,3,\cdots,2K-1$ are identified.

Proof. For $r=2,\cdots,2K-1$,

$\displaystyle\mathrm{E}\left(\tilde{y}_{i}^{r}\right)$
$\displaystyle=\mathrm{E}\left(x_{i}^{r}\right)\mathrm{E}\left(\beta_{i}^{r}\right)+\mathrm{E}\left(u_{i}^{r}\right)+\sum_{q=2}^{r-1}\binom{r}{q}\mathrm{E}\left(x_{i}^{r-q}\right)\mathrm{E}\left(u_{i}^{q}\right)\mathrm{E}\left(\beta_{i}^{r-q}\right),$
(2.6) $\displaystyle\mathrm{E}\left(\tilde{y}_{i}^{r}x_{i}^{r}\right)$
$\displaystyle=\mathrm{E}\left(x_{i}^{2r}\right)\mathrm{E}\left(\beta_{i}^{r}\right)+\mathrm{E}\left(x_{i}^{r}\right)\mathrm{E}\left(u_{i}^{r}\right)+\sum_{q=2}^{r-1}\binom{r}{q}\mathrm{E}\left(x_{i}^{2r-q}\right)\mathrm{E}\left(u_{i}^{q}\right)\mathrm{E}\left(\beta_{i}^{r-q}\right).$
(2.7)

where $\binom{r}{q}=\frac{r!}{q!(r-q)!}$ are binomial coefficients, for non-
negative integers $q\leq r$.

Sum over $i$, then by parts (a) and (b) of Assumption 2,

$\displaystyle\rho_{0,r}\mathrm{E}\left(\beta_{i}^{r}\right)+\sigma_{r}$
$\displaystyle=\rho_{r,0}-\sum_{q=2}^{r-1}\binom{r}{q}\rho_{0,r-q}\sigma_{q}\mathrm{E}\left(\beta_{i}^{r-q}\right),$
(2.8)
$\displaystyle\rho_{0,2r}\mathrm{E}\left(\beta_{i}^{r}\right)+\rho_{0,r}\sigma_{r}$
$\displaystyle=\rho_{r,r}-\sum_{q=2}^{r-1}\binom{r}{q}\rho_{0,2r-q}\sigma_{q}\mathrm{E}\left(\beta_{i}^{r-q}\right).$
(2.9)

Derivation details are relegated to Appendix A.1. By part (c) of Assumption 2,
the matrix $\begin{pmatrix}\rho_{0,r}&1\\\
\rho_{0,2r}&\rho_{0,r}\end{pmatrix}$ is invertible for $r=2,3,\cdots,2K-1$. As
a result, we can sequentially solve (2.8) and (2.9) for
$\mathrm{E}\left(\beta_{i}^{r}\right)$ and $\sigma_{r}$, for
$r=2,3,\cdots,2K-1$.

###  2.2 Identifying the distribution of $\beta_{i}$

Beran and Hall (1992, Theorem 2.1, pp. 1972) prove the identification of the
distribution of the random coefficient, $\beta_{i}$, in a canonical model
without covariates, $z_{i}$, under the condition that the distribution of
$\beta_{i}$ is uniquely determined by its moments. We show the identification
of moments of $\beta_{i}$ holds more generally when $x_{i}$ and $u_{i}$ are
not identically distributed and the distribution of $\beta_{i}$ is identified
if it follows a categorical distribution. Note that under (2.2),

$\mathrm{E}\left(\beta_{i}^{r}\right)=\sum_{k=1}^{K}\pi_{k}b_{k}^{r},\;r=0,1,2,\cdots,2K-1,$
(2.10)

with $\mathrm{E}\left(\beta_{i}^{r}\right)$ identified under Assumption 1. To
identify $\mathbf{\pi}=\left(\pi_{1},\pi_{2},...,\pi_{K}\right)^{\prime}$ and
$\mathbf{b}=\left(b_{1},b_{2},...,b_{K}\right)^{\prime}$, we need to verify
that the system of $2K$ equations in (2.10) has a unique solution if
$b_{1}<b_{2}<\cdots<b_{K}$, and $\pi_{k}\in\left(0,1\right)$. In the proof, we
construct a linear recurrence relation and make use of the corresponding
characteristic polynomial.

###### Theorem 2

Consider the random coefficient regression model (2.1), suppose that
Assumptions 1 and 2 hold. Then
$\mathbf{\theta}=\left(\mathbf{\pi}^{\prime},\mathbf{b}^{\prime}\right)^{\prime}$
is identified subject to $b_{1}<b_{2}<\cdots<b_{K}$ and
$\pi_{k}\in\left(0,1\right)$, for all $k=1,2,\cdots,K$.

Proof. We motivate the key idea of the proof in the special case where $K=2,$
and relegate the proof of the general case to the Appendix A.1. Let
$b_{1}=\beta_{L}$, $b_{2}=\beta_{H}$, $\pi_{1}=\pi$ and $\pi_{2}=1-\pi$. Note
that

$\displaystyle\mathrm{E}\left(\beta_{i}\right)$
$\displaystyle=\pi\beta_{L}+\left(1-\pi\right)\beta_{H},$ (2.11)
$\displaystyle\mathrm{E}\left(\beta_{i}^{2}\right)$
$\displaystyle=\pi\beta_{L}^{2}+\left(1-\pi\right)\beta_{H}^{2},$ (2.12)
$\displaystyle\mathrm{E}\left(\beta_{i}^{3}\right)$
$\displaystyle=\pi\beta_{L}^{3}+\left(1-\pi\right)\beta_{H}^{3},$ (2.13)

and $\mathrm{E}\left(\beta_{i}^{k}\right)$, $k=1,2,3$ are identified.
$\left(\pi,\beta_{L},\beta_{H}\right)$ can be identified if the system of
equations (2.11) to (2.13), has a unique solution. By (2.11),

$\pi=\frac{\beta_{H}-\mathrm{E}\left(\beta_{i}\right)}{\beta_{H}-\beta_{L}},\;\text{and}\;1-\pi=\frac{\mathrm{E}\left(\beta_{i}\right)-\beta_{L}}{\beta_{H}-\beta_{L}}.$
(2.14)

Plug (2.14) into (2.12) and (2.13),

$\displaystyle\mathrm{E}\left(\beta_{i}\right)\left(\beta_{L}+\beta_{H}\right)-\beta_{L}\beta_{H}$
$\displaystyle=\mathrm{E}\left(\beta_{i}^{2}\right),$ (2.15)
$\displaystyle\mathrm{E}\left(\beta_{i}^{2}\right)\left(\beta_{L}+\beta_{H}\right)-\mathrm{E}\left(\beta_{i}\right)\beta_{L}\beta_{H}$
$\displaystyle=\mathrm{E}\left(\beta_{i}^{3}\right).$ (2.16)

Denote $\beta_{L+H}=\beta_{L}+\beta_{H}$ and $\beta_{LH}=\beta_{L}\beta_{H}$,
and write (2.15) and (2.16) in matrix form,

$\mathbf{M}\mathbf{D}\mathbf{b}^{\ast}=\mathbf{m},$ (2.17)

where

$\mathbf{M}=\begin{pmatrix}1&\mathrm{E}\left(\beta_{i}\right)\\\
\mathrm{E}\left(\beta_{i}\right)&\mathrm{E}\left(\beta_{i}^{2}\right)\end{pmatrix},\,\mathbf{D}=\begin{pmatrix}-1&0\\\
0&1\end{pmatrix},\,\mathbf{b}^{\ast}=\begin{pmatrix}\beta_{LH}\\\
\beta_{L+H}\end{pmatrix},\;\text{and}\;\mathbf{m}=\begin{pmatrix}\mathrm{E}\left(\beta_{i}^{2}\right)\\\
\mathrm{E}\left(\beta_{i}^{3}\right)\end{pmatrix}.$

Under the conditions $0<\pi<1$ and $\beta_{H}>\beta_{L}$,

$\det\left(\mathbf{M}\right)=\mathrm{var}\left(\beta_{i}\right)=\mathrm{E}\left(\beta_{i}^{2}\right)-\mathrm{E}\left(\beta_{i}\right)^{2}=\pi\left(1-\pi\right)\left(\beta_{H}-\beta_{L}\right)^{2}>0.$

As a result, we can solve (2.17) for $\beta_{L+H}$ and $\beta_{LH}$ as

$\displaystyle\beta_{L+H}$
$\displaystyle=\frac{\mathrm{E}\left(\beta_{i}^{3}\right)-\mathrm{E}\left(\beta_{i}\right)\mathrm{E}\left(\beta_{i}^{2}\right)}{\mathrm{var}\left(\beta_{i}\right)},$
(2.18) $\displaystyle\beta_{LH}$
$\displaystyle=\frac{\mathrm{E}\left(\beta_{i}\right)\mathrm{E}\left(\beta_{i}^{3}\right)-\mathrm{E}\left(\beta_{i}^{2}\right)^{2}}{\mathrm{var}\left(\beta_{i}\right)}.$
(2.19)

$\beta_{L}$ and $\beta_{H}$ are solutions to the quadratic equation,

$\beta^{2}-\beta_{L+H}\beta+\beta_{LH}=0.$ (2.20)

We can verify that $\Delta=\beta_{L+H}^{2}-4\beta_{LH}>0$ by direct
calculation using (2.18) and (2.19). Simplifying $\Delta$ in terms of
$\mathrm{E}\left(\beta_{i}^{k}\right)$ and then plugging in (2.11), (2.12) and
(2.13),

$\displaystyle\Delta=$
$\displaystyle\frac{\left[\mathrm{E}\left(\beta_{i}^{3}\right)-\mathrm{E}\left(\beta_{i}\right)\mathrm{E}\left(\beta_{i}^{2}\right)\right]^{2}-4\mathrm{var}\left(\beta_{i}\right)\left[\mathrm{E}\left(\beta_{i}\right)\mathrm{E}\left(\beta_{i}^{3}\right)-\mathrm{E}\left(\beta_{i}^{2}\right)^{2}\right]}{\left[\mathrm{var}\left(\beta_{i}\right)\right]^{2}}$
$\displaystyle=\left(\beta_{H}-\beta_{L}\right)^{2}>0.$

Then, we obtain the unique solutions,

$\displaystyle\beta_{L}$
$\displaystyle=\frac{1}{2}\left(\beta_{L+H}-\sqrt{\beta_{L+H}^{2}-4\beta_{LH}}\right),$
(2.21) $\displaystyle\beta_{H}$
$\displaystyle=\frac{1}{2}\left(\beta_{L+H}+\sqrt{\beta_{L+H}^{2}-4\beta_{LH}}\right),$
(2.22)

and $\pi$ can be determined by (2.14) correspondingly.

###### Remark 7

The key identifying assumption in (2) is the assumed existence of the strict
ordinal relation $b_{1}<b_{2}<\cdots<b_{K}$ so that $b_{k}$ and
$b_{k^{\prime}}$ are not symmetric for $k\neq k^{\prime}$, and $0<\pi_{k}<1$
so that the distribution of $\beta_{i}$ does not degenerate. When $K=2$, the
conditions $b_{1}<b_{2}<\cdots<b_{K}$, and $\pi_{k}\in\left(0,1\right)$, are
equivalent to
$\mathrm{var}\left(\beta_{i}\right)=\pi_{1}\left(1-\pi_{1}\right)\left(b_{2}-b_{1}\right)^{2}>0$.
In other words, not surprisingly, the categorical distribution of $\beta_{i}$
are identified only if $\mathrm{var}\left(\beta_{i}\right)>0$.

In practice, a test for $\mathbb{H}_{0}:\mathrm{var}\left(\beta_{i}\right)=0$
is possible, by noting that $\mathrm{var}\left(\beta_{i}\right)=0$ is
equivalent to

$\kappa^{2}=\frac{\mathrm{E}\left(\beta_{i}\right)^{2}}{\mathrm{E}\left(\beta_{i}^{2}\right)}=1,$

where $\kappa^{2}$ is well-defined as long as $\beta_{i}\not\equiv 0$. One
important advantage of basing the test of slope homogeneity on $\kappa^{2}$
rather than on $var(\beta_{i})=0$, is that $\kappa^{2}\,$is scale-invariant.
$\mathrm{E}\left(\beta_{i}\right)$ and $\mathrm{E}\left(\beta_{i}^{2}\right)$
are identified as in Section 2.1, whose consistent estimation does not require
$\mathrm{var}\left(\beta_{i}\right)>0$. Consequently, in principle it is
possible to test slope homogeneity by testing $\mathbb{H}_{0}:\kappa^{2}=1$.
However, the problem becomes much more complicated when there are more than
two categories and/or there are more than one regressor under consideration. A
full treatment of testing slope homogeneity in such general settings is beyond
the scope of the present paper.

###### Remark 8

Note that in the special case of the proof of Theorem 2 where $K=2$,
$\beta_{L+H}=\beta_{L}+\beta_{H}$ and $\beta_{LH}=\beta_{L}\beta_{H}$
corresponds to the $b_{1}^{\ast}$ and $b_{2}^{\ast}$ and (2.17) is the same as
(A.1.6) when $K=2$. The special case illustrates the procedure of
identification: identify $\left(b_{k}^{\ast}\right)_{k=1}^{K}$ by the moments
of $\beta_{i}$, then solve for $\left(b_{k}\right)_{k=1}^{K}$ and finally
identify $\left(\pi_{k}\right)_{k=1}^{K}$.

##  3 Estimation

In this section, we propose a generalized method of moments estimator for the
distributional parameters of $\beta_{i}$. To reduce the complexity of the
moment equations, we first obtain a $\sqrt{n}$-consistent estimator of
$\gamma$ and consider the estimation of the distribution of $\beta_{i}$ by
replacing $\gamma$ by $\hat{\gamma}$.

###  3.1 Estimation of $\gamma$

Let
$\mathbf{\phi}=\left(\mathrm{E}\left(\beta_{i}\right),\mathbf{\gamma}^{\prime}\right)^{\prime}$,
$v_{i}=\beta_{i}-\mathrm{E}\left(\beta_{i}\right)$ and using the notation in
Assumption 1, (2.1) can be written as

$y_{i}=\mathbf{w}_{i}^{\prime}\mathbf{\phi}+\xi_{i},$ (3.1)

where $\xi_{i}=u_{i}+x_{i}v_{i}$. Then $\mathbf{\phi}$ can be estimated
consistently by $\hat{\mathbf{\phi}}=\mathbf{Q}_{n,ww}^{-1}\mathbf{q}_{n,wy}$
where $\mathbf{Q}_{n,ww}$ and $\mathbf{q}_{n,wy}$ are defined in Assumption 1.

###### Assumption 3

$\left\|n^{-1}\sum_{i=1}^{n}\mathrm{E}\left(\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\xi_{i}^{2}\right)-\mathbf{V}_{w\xi}\right\|=O\left(n^{-1/2}\right)$,
$\mathbf{V}_{w\xi}\succ 0,$ and

$\left\|\frac{1}{n}\sum_{i=1}^{n}\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\xi_{i}^{2}-\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\xi_{i}^{2}\right)\right\|=O_{p}\left(n^{-1/2}\right).$
(3.2)

###### Remark 9

As in the case of Assumption 1, the high level condition (3.2) can be shown to
hold under weak cross-sectional dependence, assuming that elements of
$\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\xi_{i}^{2}$ are cross-sectionally
weakly correlated over $i$. See Remark 5.

###### Theorem 3

Under Assumption 1, $\hat{\mathbf{\phi}}$ is a consistent estimator for
$\mathbf{\phi}$. In addition, under Assumptions 1 and 3, as $n\to\infty$,

$\sqrt{n}\left(\hat{\mathbf{\phi}}-\mathbf{\phi}\right)\rightarrow_{d}N\left(\mathbf{0},\mathbf{V}_{\phi}\right),$
(3.3)

where
$\mathbf{V}_{\phi}=\mathbf{Q}_{ww}^{-1}\mathbf{V}_{w\xi}\mathbf{Q}_{ww}^{-1}.$
$\mathbf{V}_{\phi}$ is consistently estimated by

$\hat{\mathbf{V}}_{\phi}=\mathbf{Q}_{n,ww}^{-1}\hat{\mathbf{V}}_{w\xi}\mathbf{Q}_{n,ww}^{-1}\to_{p}\mathbf{V}_{\phi},$

as $n\to\infty$, where
$\hat{\mathbf{V}}_{w\xi}=n^{-1}\sum_{i=1}^{n}\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\hat{\xi}_{i}^{2}$,
and $\hat{\xi}_{i}=y_{i}-\mathbf{w}_{i}^{\prime}\hat{\mathbf{\phi}}$.

The proof of Theorem 3 is provided in Section S.2 in the online supplement.

###  3.2 Estimation of the distribution of $\beta_{i}$

Denote the moments of $\beta_{i}$ on the right-hand side of (2.10) by

$\mathbf{m}_{\beta}=(m_{1},m_{2},...,m_{2K-1})^{\prime}=\left[\mathrm{E}\left(\beta_{i}^{r}\right)\right]_{r=1}^{2K-1}\in\Theta_{m}\subset\left\\{\mathbf{m}_{\beta}\in\mathbb{R}^{2K-1}:m_{r}\geq
0,\text{ }r\text{ is even}\right\\},$

and note that

$\mathbf{m}_{\beta}=\left(\begin{array}[]{c}m_{1}\\\ m_{2}\\\ \vdots\\\
m_{2K-1}\end{array}\right)=\left(\begin{array}[]{cccc}b_{1}&b_{2}&\cdots&b_{K}\\\
b_{1}^{2}&b_{2}^{2}&\cdots&b_{K}^{2}\\\ \vdots&\vdots&\vdots&\vdots\\\
b_{1}^{2K-1}&b_{2}^{2K-1}&\cdots&b_{K}^{2K-1}\end{array}\right)\left(\begin{array}[]{c}\pi_{1}\\\
\pi_{2}\\\ \vdots\\\ \pi_{K}\end{array}\right),$ (3.4)

so in general we can write $\mathbf{m}_{\beta}\triangleq
h\left(\mathbf{\theta}\right),$ where
$\mathbf{\theta}=\left(\mathbf{\pi}^{\prime},\mathbf{b}^{\prime}\right)^{\prime}\in\Theta$,
and $\mathbf{\theta}$ can be uniquely determined in terms of
$\mathbf{m}_{\beta}$ by Theorem 2. To estimate $\mathbf{\theta}$, we consider
moment conditions following a similar procedure as in Section 2, and propose a
generalized method of moments (GMM) estimator.

We consider the following moment conditions

$\mathrm{E}\left(\tilde{y}_{i}^{r}\right)=\sum_{q=0}^{r}\binom{r}{q}\mathrm{E}\left(x_{i}^{r-q}\right)\mathrm{E}\left(u_{i}^{q}\right)m_{r-q},\text{
}$

and

$\mathrm{E}\left(\tilde{y}_{i}^{r}x_{i}^{s_{r}}\right)=\sum_{q=0}^{r}\binom{r}{q}\mathrm{E}\left(x_{i}^{r-q+s_{r}}\right)\mathrm{E}\left(u_{i}^{q}\right)m_{r-q},$
(3.5)

where $\mathrm{E}\left(u_{i}\right)=0$,
$\tilde{y}_{i}=y_{i}-\mathbf{z}_{i}^{\prime}\mathbf{\gamma}$,
$r=1,2,...,2K-1$, and $s_{r}=0,1,\cdots,S-r$, where $S$ is a user-specific
tuning parameter, chosen such that the highest order moments of $x_{i}$
included is at most $S$, where $S>2K-1$. 222For identification, we require the
moments of $x_{i}$ to exist up to order $4K-2$. $S$ can take values between
$2K$ to $4K-2$. In practice, the choice of $S$ affects the trade-off between
bias and efficiency.

Let $\sigma_{0}=1$ and $\sigma_{1}=0$ such that $\sigma_{r}$ is well-defined
for $r=0,1,\cdots,2K-1$. Sum (3.5) over $i$ and rearrange terms,

$\displaystyle 0$
$\displaystyle=\sum_{q=0}^{r}\binom{r}{q}\left[\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(x_{i}^{r-q+s_{r}}\right)\mathrm{E}\left(u_{i}^{q}\right)\right]m_{r-q}-\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\tilde{y}_{i}^{r}x_{i}^{s_{r}}\right)$
$\displaystyle=\sum_{q=0}^{r}\binom{r}{q}\left[\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(x_{i}^{r-q+s_{r}}\right)\right]\sigma_{q}m_{r-q}-\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\tilde{y}_{i}^{r}x_{i}^{s_{r}}\right)+\delta_{n}^{(r,s_{r})},$
(3.6)

where

$\delta_{n}^{(r,s_{r})}=\sum_{q=0}^{r}\binom{r}{q}\left[\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(x_{i}^{r-q+s_{r}}\right)\left[\mathrm{E}\left(u_{i}^{q}\right)-\sigma_{q}\right]\right]m_{r-q}=O\left(n^{-1/2}\right),$

as shown in the proof of Theorem 1.

Taking $n\to\infty$ in (3.6),

$\sum_{q=0}^{r}\binom{r}{q}\rho_{0,r-q+s_{r}}\sigma_{q}m_{r-q}-\rho_{r,s_{r}}=0,$
(3.7)

by Assumption 2. We stack the left-hand side of (3.7) over $r=1,2,...,2K-1$,
and $s_{r}=0,1,\cdots,S-r$ and transform
$\mathbf{m}_{\beta}=h\left(\mathbf{\theta}\right)$ to get
$\mathbf{g}_{0}\left(\mathbf{\theta},\mathbf{\sigma},\mathbf{\gamma}\right)$.

To implement the GMM estimation we replace $\tilde{y}_{i}$, by
$\hat{\tilde{y}}_{i}=y_{i}-\mathbf{z}_{i}^{\prime}\hat{\mathbf{\gamma}}$, and
$\rho_{r,s_{r}}$ by
$n^{-1}\sum_{i=1}^{n}\hat{\tilde{y}}_{i}^{r}x_{i}^{s_{r}}$. Noting that
$\mathbf{m}_{\beta}=h\left(\mathbf{\theta}\right)$, denote the sample version
of the left-hand side of (3.7) by

$\hat{g}_{n}^{(r,s_{r})}\left(\mathbf{\theta},\mathbf{\sigma},\hat{\mathbf{\gamma}}\right)=\frac{1}{n}\sum_{i=1}^{n}\hat{g}_{i}^{(r,s_{r})}\left(\mathbf{\theta},\mathbf{\sigma},\hat{\mathbf{\gamma}}\right),$
(3.8)

where

$\hat{g}_{i}^{\left(r,s_{r}\right)}\left(\mathbf{\theta},\mathbf{\sigma},\hat{\mathbf{\gamma}}\right)=\sum_{q=0}^{r}\binom{r}{q}x_{i}^{r-q+s_{r}}\sigma_{q}\left[h\left(\mathbf{\theta}\right)\right]_{r-q}-\hat{\tilde{y}}_{i}^{r}x_{i}^{s_{r}},$

and
$\mathbf{\sigma}=\left(\sigma_{2},\sigma_{3},\cdots,\sigma_{2K-1}\right)^{\prime}$.
Stack the equations in (3.8), over $r=0,1,...,2K-1$ and $s_{r}=0,1,\cdots,S-r$
($S>2K-1$), in vector notations we have

$\mathbf{\hat{g}}_{n}\left(\mathbf{\theta},\mathbf{\sigma},\hat{\mathbf{\gamma}}\right)=\frac{1}{n}\sum_{i=1}^{n}\mathbf{\hat{g}}_{i}\left(\mathbf{\theta},\mathbf{\sigma},\hat{\mathbf{\gamma}}\right).$
(3.9)

Given $\hat{\mathbf{\gamma}}$, the GMM estimator of
$\left(\mathbf{\theta}^{\prime},\mathbf{\sigma}^{\prime}\right)^{\prime}$ is
now computed as

$\left(\hat{\mathbf{\theta}}^{\prime},\hat{\mathbf{\sigma}}^{\prime}\right)^{\prime}=\arg\min_{\mathbf{\theta}\in\Theta,\mathbf{\sigma}\in\mathcal{S}}\hat{\Phi}_{n}\left(\mathbf{\theta},\mathbf{\sigma},\hat{\mathbf{\gamma}}\right),$

where
$\hat{\Phi}_{n}=\mathbf{\hat{g}}_{n}\left(\mathbf{\theta},\mathbf{\sigma},\hat{\mathbf{\gamma}}\right)^{\prime}\mathbf{A}_{n}\mathbf{\hat{g}}_{n}\left(\mathbf{\theta},\mathbf{\sigma},\hat{\mathbf{\gamma}}\right)$,
and $\mathbf{A}_{n}$ is a positive definite matrix. We follow the GMM
literature using the following choice of $\mathbf{A}_{n}$,

$\mathbf{\hat{A}}_{n}=\left[\frac{1}{n}\sum_{i=1}^{n}\mathbf{\hat{g}}_{i}\left(\tilde{\mathbf{\theta}},\tilde{\mathbf{\sigma}},\hat{\mathbf{\gamma}}\right)\mathbf{\hat{g}}_{i}\left(\tilde{\mathbf{\theta}},\tilde{\mathbf{\sigma}},\hat{\mathbf{\gamma}}\right)^{\prime}-\mathbf{\bar{g}}_{n}\mathbf{\bar{g}}_{n}^{\prime}\right]^{-1},$
(3.10)

where
$\mathbf{\bar{g}}_{n}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{\hat{g}}_{i}\left(\tilde{\mathbf{\theta}},\tilde{\mathbf{\sigma}},\hat{\mathbf{\gamma}}\right)$,
and $\tilde{\mathbf{\theta}}$ and $\tilde{\mathbf{\sigma}}$ are preliminary
estimators.

###### Assumption 4

Denote the true values of $\mathbf{\theta}$, $\mathbf{\sigma}$ and
$\mathbf{\gamma}$ by $\mathbf{\theta}_{0}$, $\mathbf{\sigma}_{0}$ and
$\mathbf{\gamma}_{0}$.

  1. (a)

$\Theta$ and $\mathcal{S}$ are compact.
$\mathbf{\theta}_{0}\in\mathrm{int}\left(\Theta\right)$ and
$\mathbf{\sigma}_{0}\in\mathrm{int}\left(\mathcal{S}\right)$.

  2. (b)

$\mathbf{A}_{n}\rightarrow_{p}\mathbf{A}$ as $n\rightarrow\infty$, where
$\mathbf{A}$ is some positive definite matrix.

  3. (c)

$\frac{1}{n}\sum_{i=1}^{n}\left[\hat{\tilde{y}}_{i}^{r}x_{i}^{s_{r}}-\mathrm{E}\left(\tilde{y}_{i}^{r}x_{i}^{s_{r}}\right)\right]=O_{p}\left(n^{-1/2}\right),$

for $r=0,1,2,\cdots,2K-1$, $s_{r}=0,1,\cdots,S-r,$ and $S>2K-1.$

###### Remark 10

Parts (a) and (b) of Assumption 4 are standard regularity conditions in the
GMM literature. Part (c) together with Assumption 2 are high-level regularity
conditions which allow us to generalize the usual IID assumption and nest the
IID data generation process as a special case. The sample analogue terms in
(c) include
$\hat{\tilde{y}}_{i}=y_{i}-\mathbf{z}_{i}^{\prime}\hat{\mathbf{\gamma}}$,
instead of the infeasible
$\tilde{y}_{i}=y_{i}-\mathbf{z}_{i}^{\prime}\mathbf{\gamma}$. The
$\sqrt{n}$-consistency of $\hat{\mathbf{\gamma}}$ shown in Theorem 3 ensures
that replacing $\tilde{y}_{i}$ by $\hat{\tilde{y}}_{i}$ does not alter the
convergence rate.

###### Theorem 4

Let
$\mathbf{\eta}=\left(\mathbf{\theta}^{\prime},\mathbf{\sigma}^{\prime}\right)^{\prime}$
and
$\mathbf{\eta}_{0}=\left(\mathbf{\theta}_{0}^{\prime},\mathbf{\sigma}_{0}^{\prime}\right)^{\prime}$.
Under Assumptions 1, 2, and 4,
$\hat{\mathbf{\eta}}\rightarrow_{p}\mathbf{\eta}_{0}$ as $n\rightarrow\infty$.

The proof of Theorem 4 is provided in Appendix A.1.

###### Assumption 5

Follow the notations as in Assumption 4 and in addition denote
$\mathbf{G}\left(\mathbf{\theta},\mathbf{\sigma},\mathbf{\gamma}\right)=\nabla_{\left(\mathbf{\theta}^{\prime},\mathbf{\sigma}^{\prime}\right)^{\prime}}\mathbf{g}_{0}\left(\mathbf{\theta},\mathbf{\sigma},\mathbf{\gamma}\right)$,
$\mathbf{G}_{0}=\mathbf{G}\left(\mathbf{\theta}_{0},\mathbf{\sigma}_{0},\mathbf{\gamma}_{0}\right)$,
$\mathbf{G}_{\gamma}\left(\mathbf{\theta},\mathbf{\sigma},\mathbf{\gamma}\right)=\nabla_{\mathbf{\gamma}}\mathbf{g}_{0}\left(\mathbf{\theta},\mathbf{\sigma},\mathbf{\gamma}\right)$,
$\mathbf{G}_{0,\gamma}=\mathbf{G}_{\gamma}\left(\mathbf{\theta}_{0},\mathbf{\sigma}_{0},\mathbf{\gamma}_{0}\right)$.

  1. (a)

$\sqrt{n}\mathbf{\hat{g}}_{n}\left(\mathbf{\theta}_{0},\mathbf{\sigma}_{0},\mathbf{\gamma}_{0}\right)\rightarrow_{d}\mathbf{\zeta}\sim
N\left(0,\mathbf{V}\right)$ as $n\to\infty$.

  2. (b)

$\mathbf{G}_{0}^{\prime}\mathbf{AG}_{0}\succ 0$.

###### Remark 11

In Assumption 5, parts (a) is the high level condition required to ensure the
asymptotic normality of
$\mathbf{\hat{g}}_{n}\left(\mathbf{\theta}_{0},\mathbf{\sigma}_{0},\mathbf{\gamma}_{0}\right)$,
which can be verified by Lindeberg central limit theorem under low-level
regularity conditions. Part (c) of Assumption 5 represents the full-rank
condition on $\mathbf{G}_{0}$, required for identification of
$\mathbf{\theta}_{0}$ and $\mathbf{\sigma}_{0}$.

By Theorem 3, we have
$\sqrt{n}\left(\hat{\mathbf{\gamma}}-\mathbf{\gamma}\right)\rightarrow_{d}\zeta_{\gamma}\sim
N(0,V_{\gamma})$. The following theorem shows the asymptotic normality of the
GMM estimator $\hat{\mathbf{\eta}}$.

###### Theorem 5

Under Assumptions 1, 3, 4 and 5,

$\sqrt{n}\left(\hat{\mathbf{\eta}}-\mathbf{\eta}_{0}\right)\rightarrow_{d}\left(\mathbf{G}_{0}^{\prime}\mathbf{A}\mathbf{G}_{0}\right)^{-1}\mathbf{G}_{0}^{\prime}\mathbf{A}\left(\mathbf{\zeta}+\mathbf{G}_{0,\gamma}\mathbf{\zeta}_{\gamma}\right),$

as $n\rightarrow\infty$.

The proof of Theorem 5 is provided in Appendix A.1.

###### Remark 12

In practice, we estimate the variance of the asymptotic distribution of
$\hat{\eta}$ by

$\hat{\mathbf{V}}_{\eta}=\left(\hat{\mathbf{G}}^{\prime}\hat{\mathbf{A}}_{n}\hat{\mathbf{G}}\right)^{-1}\hat{\mathbf{G}}^{\prime}\hat{\mathbf{A}}_{n}\hat{\mathbf{V}}_{\zeta}\hat{\mathbf{A}}_{n}^{\prime}\hat{\mathbf{G}}\left(\hat{\mathbf{G}}^{\prime}\hat{\mathbf{A}}_{n}\hat{\mathbf{G}}\right)^{-1},$
(3.11)

where
$\hat{\mathbf{G}}=\nabla_{\left(\mathbf{\sigma}^{\prime},\mathbf{\theta}^{\prime}\right)^{\prime}}\hat{\mathbf{g}}_{n}\left(\hat{\mathbf{\theta}},\hat{\mathbf{\sigma}},\hat{\gamma}\right)$,
$\hat{\mathbf{A}}_{n}$ is given by (3.10), and

$\hat{\mathbf{V}}_{\zeta}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{\psi}_{n,i}\mathbf{\psi}_{n,i}^{\prime},$

where

$\mathbf{\psi}_{n,i}=\hat{\mathbf{g}}_{i}\left(\hat{\mathbf{\theta}},\hat{\mathbf{\sigma}},\hat{\gamma}\right)+\nabla_{\mathbf{\gamma}}\hat{\mathbf{g}}_{n}\left(\hat{\mathbf{\theta}},\hat{\mathbf{\sigma}},\hat{\gamma}\right)\mathbf{L}\mathbf{Q}_{n,ww}^{-1}\left(\mathbf{w}_{i}\hat{\xi}_{i}\right),$

and $\mathbf{L}=\begin{pmatrix}\mathbf{0}_{p_{z}\times
1}&\mathbf{I}_{p_{z}}\end{pmatrix}$ is the loading matrix that selects
$\mathbf{\gamma}$ out of $\mathbf{\phi}$.

##  4 Multiple regressors with random coefficients

One important extension of the regression model (2.1) is to allow for multiple
regressors with random coefficients having categorical distribution. With this
in mind consider

$y_{i}=\mathbf{x}_{i}^{\prime}\mathbf{\beta}_{i}+\mathbf{z}_{i}^{\prime}\mathbf{\gamma}+u_{i},$
(4.1)

where the $p\times 1$ vector of random coefficients,
$\mathbf{\beta}_{i}\in\mathbb{R}^{p}$ follows the multivariate
distribution333We assume the number of categories $K$ is homogeneous across
$j=1,2,\cdots,p$. This is for notational simplicity, and can be readily
generalized to allow for $K_{j}\neq K_{j^{\prime}}$ without affecting the main
results.

$\mathrm{Pr}\left(\beta_{i1}=b_{1k_{1}},\beta_{i2}=b_{2k_{2}},\cdots,\beta_{ip}=b_{pk_{p}}\right)=\pi_{k_{1},k_{2},\cdots,k_{p}},$
(4.2)

with $k_{j}\in\left\\{1,2,\cdots,K\right\\}$, $b_{j1}<b_{j2}<\cdots<b_{jK}$,
and

$\sum_{k_{1},k_{2},\cdots,k_{p}\in\left\\{1,2,\cdots,K\right\\}}\pi_{k_{1},k_{2},\cdots,k_{p}}=1.$

As in Section 2, $\mathbf{\gamma}\in\mathbb{R}^{p_{z}}$,
$\mathbf{w}_{i}=\left(\mathbf{x}_{i}^{\prime},\mathbf{z}_{i}^{\prime}\right)^{\prime}$,
$\mathbf{\beta}_{i}\perp\mathbf{w}_{i}$, $u_{i}\perp\mathbf{w}_{i}$, and
$u_{i}$ are independently distributed over $i$ with mean $0.$

Example 1 Consider the simple case with $p=2$ and $K=2$. For $j=1,2$, denote
two categories as $\left\\{L,H\right\\}$. The probabilities of four possible
combinations of realized $\mathbf{\beta}_{i}$ is summarized in Table 1, where
$\pi_{LL}+\pi_{LH}+\pi_{HL}+\pi_{HH}=1$.

Table 1: Distribution of $\mathbf{\beta}_{i}$ with $p=2$ and $K=2$ | $k_{2}=L$ | $k_{2}=H$
---|---|---
$k_{1}=L$ | $\pi_{LL}=\Pr\left(\beta_{i1}=b_{1L},\beta_{i2}=b_{2L}\right)$ | $\pi_{LH}=\Pr\left(\beta_{i1}=b_{1L},\beta_{i2}=b_{2H}\right)$
$k_{1}=H$ | $\pi_{HL}=\Pr\left(\beta_{i1}=b_{1H},\beta_{i2}=b_{2L}\right)$ | $\pi_{HH}=\Pr\left(\beta_{i1}=b_{1H},\beta_{i2}=b_{2H}\right)$

We first identify the moments of $\mathbf{\beta}_{i}$. As in Section 2,
$\mathbf{\phi}=\left(\mathrm{E}\left(\mathbf{\beta}_{i}\right)^{\prime},\mathbf{\gamma}^{\prime}\right)^{\prime}$
is identified by

$\mathbf{\phi}=\mathbf{Q}_{ww}^{-1}\mathbf{q}_{wy},$ (4.3)

under Assumption 1. We now consider the identification of the higher order
moments of $\mathbf{\beta}_{i}$ up to the finite order $2K-1$.

Since $\mathbf{\gamma}$ is identified as in (4.3), we treat it as known and
let $\tilde{y}_{i}^{r}=y_{i}-\mathbf{z}_{i}^{\prime}\mathbf{\gamma}$. For
$r=2,3,\cdots,2K-1$, consider the moment conditions

$\displaystyle\mathrm{E}\left(\tilde{y}_{i}^{r}\right)$
$\displaystyle=\mathrm{E}\left[\left(\mathbf{x}_{i}^{\prime}\mathbf{\beta}_{i}+u_{i}\right)^{r}\right]$
$\displaystyle=\mathrm{E}\left[\left(\mathbf{x}_{i}^{\prime}\mathbf{\beta}_{i}\right)^{r}\right]+\mathrm{E}\left(u_{i}^{r}\right)+\sum_{s=2}^{r-1}\binom{r}{s}\mathrm{E}\left[\left(\mathbf{x}_{i}^{\prime}\mathbf{\beta}_{i}\right)^{r-s}\right]\mathrm{E}\left(u_{i}^{s}\right).$
(4.4)

Note that
$\mathbf{x}_{i}^{\prime}\mathbf{\beta}_{i}=\sum_{j=1}^{p}\beta_{ij}x_{ij}$,
and

$\mathrm{E}\left[\left(\sum_{j=1}^{p}\beta_{ij}x_{ij}\right)^{r}\right]=\sum_{\sum_{j=1}^{p}q_{j}=r}\binom{r}{\mathbf{q}}\mathrm{E}\left(\prod_{j=1}^{p}x_{ij}^{q_{j}}\right)\mathrm{E}\left(\prod_{j=1}^{p}\beta_{ij}^{q_{j}}\right),$

where $\binom{r}{\mathbf{q}}=\frac{r!}{q_{1}!q_{2}!\cdots q_{p}!}$, for non-
negative integers $r$, $q_{1}$, $\cdots$, $q_{p}$ with
$r=\sum_{j=1}^{p}q_{j}$, denotes the multinomial coefficients. We stack
$\prod_{j=1}^{p}x_{ij}^{q_{j}}$ with
$\mathbf{q}\in\left\\{\mathbf{q}\in\left\\{0,1,\cdots
r\right\\}^{p}:\sum_{j=1}^{p}q_{j}=r\right\\}$ in a vector form by denoting
444For $\mathbf{x}\in\mathbb{R}^{p}$, note that
$\mathbf{\tau}_{0}\left(\mathbf{x}\right)=1$,
$\mathbf{\tau}_{1}\left(\mathbf{x}\right)=\mathbf{x}$ and
$\mathbf{\tau}_{2}\left(\mathbf{x}\right)=\mathrm{vech}\left(\mathbf{x}\mathbf{x}^{\prime}\right)$.

$\mathbf{\tau}_{r}\left(\mathbf{x}_{i}\right)=\left[\varphi\left(\mathbf{x}_{i},\mathbf{q}_{1}\right),\varphi\left(\mathbf{x}_{i},\mathbf{q}_{2}\right),\cdots,\varphi\left(\mathbf{x}_{i},\mathbf{q}_{\nu_{r}}\right)\right]^{\prime},$

where
$\varphi\left(\mathbf{x}_{i},\mathbf{q}\right)=\prod_{j=1}^{p}x_{ij}^{q_{j}}$
and $\nu_{r}=\binom{r+p-1}{p-1}$ is the number of distinct monomials of degree
$r$ on the variables $x_{i1},x_{i2},\cdots,x_{ip}$. Similarly,

$\mathbf{\tau}_{r}\left(\mathbf{\beta}_{i}\right)=\left[\varphi\left(\mathbf{\beta}_{i},\mathbf{q}_{1}\right),\varphi\left(\mathbf{\beta}_{i},\mathbf{q}_{2}\right),\cdots,\varphi\left(\mathbf{\beta}_{i},\mathbf{q}_{\nu_{r}}\right)\right]^{\prime},$

where
$\varphi\left(\mathbf{\beta}_{i},\mathbf{q}\right)=\prod_{j=1}^{p}\beta_{ij}^{q_{j}}$.

Example 2 Consider $p=2$ and $r=2$, we have

$\displaystyle\mathbf{\tau}_{2}\left(\mathbf{x}_{i}\right)$
$\displaystyle=\left(x_{i1}^{2},x_{i1}x_{i2},x_{i2}^{2}\right)^{\prime},$
$\displaystyle\mathbf{\tau}_{2}\left(\mathbf{\beta}_{i}\right)$
$\displaystyle=\left(\beta_{i1}^{2},\beta_{i1}\beta_{i2},\beta_{i2}^{2}\right)^{\prime},$

and

$\displaystyle\mathrm{E}\left[\left(x_{i1}\beta_{i1}+x_{i2}\beta_{i2}\right)^{2}\right]$
$\displaystyle=\mathrm{E}\left(x_{i1}^{2}\right)\mathrm{E}\left(\beta_{i1}^{2}\right)+2\mathrm{E}\left(x_{i1}x_{i2}\right)\mathrm{E}\left(\beta_{i1}\beta_{i2}\right)+\mathrm{E}\left(x_{i2}^{2}\right)\mathrm{E}\left(\beta_{i2}^{2}\right)$
$\displaystyle=\left[\mathrm{E}\left(x_{i1}^{2}\right),\mathrm{E}\left(x_{i1}x_{i2}\right),\mathrm{E}\left(x_{i2}^{2}\right)\right]\mathrm{diag}\left[\left(1,2,1\right)^{\prime}\right]\left[\mathrm{E}\left(\beta_{i1}^{2}\right),\mathrm{E}\left(\beta_{i1}\beta_{i2}\right),\mathrm{E}\left(\beta_{i2}^{2}\right)\right]^{\prime}$
$\displaystyle=\mathrm{E}\left[\mathbf{\tau}_{2}\left(\mathbf{x}_{i}\right)\right]^{\prime}\mathbf{\Lambda}_{2}\mathrm{E}\left[\mathbf{\tau}_{2}\left(\mathbf{\beta}_{i}\right)\right],$

where
$\mathbf{\Lambda}_{2}=\mathrm{diag}\left[\left(1,2,1\right)^{\prime}\right]$.

Then the moment condition (4.4) can be written as

$\displaystyle\mathrm{E}\left(\tilde{y}_{i}^{r}\right)$
$\displaystyle=\mathrm{E}\left[\mathbf{\tau}_{r}\left(\mathbf{x}_{i}\right)\right]^{\prime}\mathbf{\Lambda}_{r}\mathrm{E}\left[\mathbf{\tau}_{r}\left(\mathbf{\beta}_{i}\right)\right]+\mathrm{E}\left(u_{i}^{r}\right)$
$\displaystyle\quad\quad\quad+\sum_{s=2}^{r-1}\binom{r}{s}\mathrm{E}\left[\mathbf{\tau}_{r-s}\left(\mathbf{x}_{i}\right)\right]^{\prime}\mathbf{\Lambda}_{r-s}\mathrm{E}\left[\mathbf{\tau}_{r-s}\left(\mathbf{\beta}_{i}\right)\right]\mathrm{E}\left(u_{i}^{s}\right),$
(4.5)

where
$\mathbf{\Lambda}_{r}=\mathrm{diag}\left[\left[\binom{r}{\mathbf{q}}\right]_{\sum_{j=1}^{p}q_{j}=r}\right]$
is the $\nu_{r}\times\nu_{r}$ diagonal matrix of multinomial coefficients. We
further consider the moment conditions

$\displaystyle\mathrm{E}\left(\tilde{y}_{i}^{r}\mathbf{\tau}_{r}\left(\mathbf{x}_{i}\right)\right)$
$\displaystyle=\mathrm{E}\left[\mathbf{\tau}_{r}\left(\mathbf{x}_{i}\right)\mathbf{\tau}_{r}\left(\mathbf{x}_{i}\right)^{\prime}\right]\mathbf{\Lambda}_{r}\mathrm{E}\left[\mathbf{\tau}_{r}\left(\mathbf{\beta}_{i}\right)\right]+\mathrm{E}\left[\mathbf{\tau}_{r}\left(\mathbf{x}_{i}\right)\right]\mathrm{E}\left(u_{i}^{r}\right)$
$\displaystyle\quad\quad\quad+\sum_{s=2}^{r-1}\binom{r}{s}\mathrm{E}\left[\mathbf{\tau}_{r}\left(\mathbf{x}_{i}\right)\mathbf{\tau}_{r-s}\left(\mathbf{x}_{i}\right)^{\prime}\right]\mathbf{\Lambda}_{r-s}\mathrm{E}\left[\mathbf{\tau}_{r-s}\left(\mathbf{\beta}_{i}\right)\right]\mathrm{E}\left(u_{i}^{s}\right),$
(4.6)

$r=2,3,\cdots,2K-1$. (4.5) and (4.6) reduce to (2.6) and (2.7) when $p=1$.

###### Assumption 6

  1. (a)

$\left\|n^{-1}\sum_{i=1}^{n}\mathrm{E}\left(\tilde{y}_{i}^{r}\mathbf{\tau}_{s}\left(\mathbf{x}_{i}\right)\right)-\mathbf{\rho}_{r,s}\right\|=O\left(n^{-1/2}\right),$
and $\left\|\mathbf{\rho}_{r,s}\right\|<\infty$, $r,s=0,1,\cdots,2K-1$.

  2. (b)

$\left\|n^{-1}\sum_{i=1}^{n}\mathrm{E}\left[\mathbf{\tau}_{r}\left(\mathbf{x}_{i}\right)\mathbf{\tau}_{s}\left(\mathbf{x}_{i}\right)^{\prime}\right]-\mathbf{\Xi}_{r,s}\right\|=O\left(n^{-1/2}\right),$
and $\left\|\mathbf{\Xi}_{r,s}\right\|<\infty$, $r,s=0,1,\cdots,2K-1$.

  3. (c)

$\left|n^{-1}\sum_{i=1}^{n}\mathrm{E}\left(u_{i}^{r}\right)-\sigma_{r}\right|=O\left(n^{-1/2}\right),$
and $\left|\sigma_{r}\right|<\infty$ for $r=2,3,\cdots,2K-1$.

  4. (d)

$\left\|n^{-1}\sum_{i=1}^{n}\left[\mathrm{var}\left(\mathbf{\tau}_{r}\left(\mathbf{x_{i}}\right)\right)-\left(\mathbf{\Xi}_{r,r}-\mathbf{\rho}_{0,r}\mathbf{\rho}_{0,r}^{\prime}\right)\right]\right\|=O(n^{-1/2})$,
where $\mathbf{\Xi}_{r,r}-\mathbf{\rho}_{0,r}\mathbf{\rho}_{0,r}^{\prime}\succ
0$ for $r=2,3\cdots,2K-1$.

###### Theorem 6

For any $\mathbf{q}\in\left\\{\mathbf{q}\in\left\\{0,1,\cdots
r\right\\}^{p}:\sum_{j=1}^{p}q_{j}=r\right\\}$ and $r=2,3,\cdots,2K-1$,
$\mathrm{E}\left(\prod_{j=1}^{p}\beta_{ij}^{q_{j}}\right)$ and $\sigma_{r}$
are identified under Assumptions 1 and 6.

Proof. For $r=2,3,\cdots,2K-1$, sum (4.5) and (4.6) over $i,$ go through the
same steps as in the proof of Theorem 1, then by Assumptions 6(a) to (c), we
have (for $n\rightarrow\infty$)

$\displaystyle\mathbf{\rho}_{r,0}^{\prime}\mathbf{\Lambda}_{r}\mathrm{E}\left[\mathbf{\tau}_{r}\left(\mathbf{\beta}_{i}\right)\right]+\sigma_{r}$
$\displaystyle=\mathbf{\rho}_{r,0}-\sum_{s=2}^{r-1}\binom{r}{s}\mathbf{\rho}_{0,r-s}\mathbf{\Lambda}_{r-s}\mathrm{E}\left[\mathbf{\tau}_{r-s}\left(\mathbf{\beta}_{i}\right)\right]\sigma_{s},$
(4.7)
$\displaystyle\mathbf{\Xi}_{r,r}\mathbf{\Lambda}_{r}\mathrm{E}\left[\mathbf{\tau}_{r}\left(\mathbf{\beta}_{i}\right)\right]+\mathbf{\rho}_{0,r}\sigma_{r}$
$\displaystyle=\mathbf{\rho}_{r,r}-\sum_{s=2}^{r-1}\binom{r}{s}\mathbf{\Xi}_{r,r-s}\mathbf{\Lambda}_{r-s}\mathrm{E}\left[\mathbf{\tau}_{r-s}\left(\mathbf{\beta}_{i}\right)\right]\sigma_{s}.$
(4.8)

Note that

$\mathbf{M}_{r}=\begin{pmatrix}\mathbf{\Xi}_{r,r}&\mathbf{\rho}_{0,r}\\\
\mathbf{\rho}_{0,r}^{\prime}&1\end{pmatrix}\begin{pmatrix}\mathbf{\Lambda}_{r}&\mathbf{0}\\\
\mathbf{0}&1\end{pmatrix},$

is invertible since
$\det\left(\mathbf{M}_{r}\right)=\det\left(\mathbf{\Xi}_{r,r}-\mathbf{\rho}_{0,r}\mathbf{\rho}_{0,r}^{\prime}\right)\det\left(\mathbf{\Lambda}_{r}\right)>0,$
for $r=2,3,\cdots,R$, by Assumption 6(d). As a result, we can sequentially
solve (4.7) and (4.8) for
$\mathrm{E}\left[\mathbf{\tau}_{r}\left(\mathbf{\beta}_{i}\right)\right]$ and
$\sigma_{r}$, for $r=2,3,\cdots,2K-1$.

We now move from the moments of $\mathbf{\beta}_{i}$ to the distribution of
$\mathbf{\beta}_{i}$. We first focus on the identification of the marginal
probabilities obtained from (4.2) by averaging out the effects of the other
coefficients except for $\beta_{ij}$, namely we initially focus on
identification of $\lambda_{jk}=\Pr\left(\beta_{ij}=b_{jk}\right)$, for
$k=1,2,\cdots,K,$ and $j=1,2,\cdots,p$.

###### Remark 13

Focusing on the marginal distribution of $\beta_{i}$ is similar to focusing on
estimation of partial derivatives in the context of non-parametric estimation,
where the curse of dimensionality applies. Consider the estimation of
regressing $y_{i}$ on
$\mathbf{x}_{i}=\left(x_{i1},x_{i2},\cdots,x_{ip}\right)^{\prime}$,

$y_{i}=F\left(x_{i1},x_{i2},\cdots.x_{ip}\right)+u_{i}.$

Then if $F\left(x_{1},x_{i2},\cdots,x_{ip}\right)$ is a homogeneous function
(of degree $1/\mu$), then

$y_{i}=\sum_{j=1}^{p}\left(\mu\frac{\partial F\left(\cdot\right)}{\partial
x_{ij}}\right)x_{ij}+u_{i},$

and under certain conditions we can treat $\mu\frac{\partial
F\left(\cdot\right)}{\partial x_{ij}}\equiv\beta_{ij}$.

By Theorem 6, $\mathrm{E}\left(\beta_{ij}^{r}\right)$ is identified for
$r=1,2,\cdots,2K-1$ under Assumptions 1 and 6. By (4.2), we have equations

$\mathrm{E}\left(\beta_{ij}^{r}\right)=\sum_{k=1}^{K}\lambda_{jk}b_{jk}^{r},$
(4.9)

$r=0,1,\cdots,2K-1$, which is of the same form as (2.10) and (3.4). To
identify
$\mathbf{\lambda}_{j}=\left(\lambda_{j1},\lambda_{j2},\cdots,\lambda_{jK}\right)^{\prime}$
and $\mathbf{b}_{j}=\left(b_{j1},b_{j2},\cdots,b_{jK}\right)^{\prime}$, we can
verify the system of $2K$ equations in (4.9) has a unique solution if
$b_{j1}<b_{j2}<\cdots<b_{jK}$ and $\lambda_{jk}\in\left(0,1\right)$. The
following corollary is a direct application of Theorem 2.

###### Corollary 7

Consider the model (4.1) and suppose that Assumptions 1 and 6 hold. Then the
parameters
$\mathbf{\theta}_{j}=\left(\mathbf{\lambda}_{j}^{\prime},\mathbf{b}_{j}^{\prime}\right)^{\prime}$
of the marginal distribution of $\beta_{i}$ with respect to $\beta_{ij}$ is
identified subject to $b_{j1}<b_{j2}<\cdots<b_{jK}$ and
$\lambda_{jk}\in\left(0,1\right)$ for $j=1,2,\cdots,p$.

The problem of identification and estimation of the joint distribution of
$\mathbf{\beta}_{i}$ is subject to the curse of dimensionality. We have
$K^{p}-1$ probability weights, $\pi_{k_{1},k_{2},\cdots,k_{p}}$, to be
identified in addition to the $pK$ categorical coefficients $b_{ij}$ that are
identified by Corollary 7. The number of parameters increases rapidly with
$p$. Even in the simplest case with $K=2$, the total number of unknown
parameters is $2p+2^{p}-1$, which grows exponentially.

Note that the marginal probabilities $\lambda_{jk}$ are related to the joint
distribution by

$\lambda_{jk}=\sum_{k_{1},\cdots,k_{j-1},k_{j+1},\cdots,k_{p}\in\left\\{1,2,\cdots,K\right\\}}\pi_{k_{1},k_{2},\cdots,k_{j-1},k,k_{j+1},\cdots,k_{p}},$
(4.10)

$k=1,2,\cdots,K$ and $j=1,2,\cdots,p$. The number of linearly independent
equations in (4.10) is $pK-(p-1)$.

Example 3 Consider the same setup as in Example 1 with $p=2$ and $K=2$. The
marginal probabilities are obtained by

$\displaystyle\lambda_{1L}=\Pr\left(\beta_{i1}=b_{1L}\right)=\pi_{LL}+\pi_{LH},$
$\displaystyle\quad\lambda_{1H}=\Pr\left(\beta_{i1}=b_{1H}\right)=1-\lambda_{1L}=\pi_{HL}+\pi_{HH},$
$\displaystyle\lambda_{2L}=\Pr\left(\beta_{i2}=b_{2L}\right)=\pi_{LL}+\pi_{HL},$
$\displaystyle\quad\lambda_{2H}=\Pr\left(\beta_{i2}=b_{2H}\right)=1-\lambda_{2L}=\pi_{LH}+\pi_{HH}.$
(4.11)

Note that any equation in (4.11) can be expressed as a linear combination of
other three equations, for example
$\lambda_{2H}=\lambda_{1L}+\lambda_{1H}-\lambda_{2L}$.

The equations corresponding to the cross-moments,
$\mathrm{E}\left(\prod_{j=1}^{p}\beta_{ij}^{q_{j}}\right)$, are

$\mathrm{E}\left(\prod_{j=1}^{p}\beta_{ij}^{q_{j}}\right)=\sum_{k_{1},k_{2},\cdots,k_{p}\in\left\\{1,2,\cdots,K\right\\}}\left(\prod_{j=1}^{p}b_{jk_{j}}^{q_{j}}\right)\pi_{k_{1},k_{2},\cdots,k_{p}},$
(4.12)

for $\mathbf{q}\in\left\\{\mathbf{q}\in\left\\{0,1,\cdots
r-1\right\\}^{p}:\sum_{j=1}^{p}q_{j}=r\right\\}$, $r=2,\cdots,2K-1$. The
linear system (4.12) has

$\sum_{r=1}^{2K-1}\binom{r+p-1}{p-1}-p(2K-1)$

equations. Then the total number of equations in (4.10) and (4.12) that can be
utilized to identify joint probabilities is
$C_{r}=\sum_{r=1}^{2K-1}\binom{r+p-1}{p-1}-pK$, which is smaller than the
number of joint probabilities $K^{p}-1$ for large $p$. When $K=2$,
$C_{r}<K^{p}-1$ for $p\geq 7$.

Identification and estimation of the joint distribution of
$\mathbf{\beta}_{i}$ in the general setting will not be pursued in this paper
due to the curse of dimensionality. Instead, we consider special cases, that
are empirically relevant, in which identification of the joint distribution of
$\mathbf{\beta}_{i}$ can be readily established. We first consider small $p$
and $K$, in particular $p=2$ and $K=2$ as in Example 1.

Example 4 Consider the same setup as in Example 1 with $p=2$ and $K=2$. In
addition to (4.11), consider the cross-moment,

$\mathrm{E}\left(\beta_{i1}\beta_{i2}\right)=b_{1L}b_{2L}\pi_{LL}+b_{1L}b_{2H}\pi_{LH}+b_{1H}b_{2L}\pi_{HL}+b_{1H}b_{2H}\pi_{HH}.$
(4.13)

Writing (4.11) and (4.13) in matrix form, we have

$\mathbf{B}\mathbf{\pi}=\mathbf{\lambda},$

where

$\mathbf{B}=\begin{pmatrix}1&1&0&0\\\ 0&0&1&1\\\ 1&0&1&0\\\
b_{1L}b_{2L}&b_{1L}b_{2H}&b_{1H}b_{2L}&b_{1H}b_{2H}\end{pmatrix},\,\mathbf{\pi}=\begin{pmatrix}\pi_{LL}\\\
\pi_{LH}\\\ \pi_{HL}\\\
\pi_{HH}\end{pmatrix},\,\mathbf{\lambda}=\begin{pmatrix}\lambda_{1L}\\\
\lambda_{1H}\\\ \lambda_{2L}\\\
\mathrm{E}\left(\beta_{i1}\beta_{i2}\right)\end{pmatrix}.$

Note that $\mathrm{E}\left(\beta_{i1}\beta_{i2}\right)$ is identified by
Theorem 6, and $b_{jk_{j}}$ and $\lambda_{jk_{j}}$ are identified by Corollary
7, and matrix $\mathbf{B}$ is invertible given that $b_{1L}<b_{1H}$ and
$b_{2L}<b_{2H}$. (See Appendix A.1). As a result, the joint probabilities,
$\mathbf{\pi},$ are identified.

###### Remark 14

The argument in Example 4 is applicable for identification of the joint
distribution of $\left(\beta_{ij},\beta_{i,j^{\prime}}\right)^{\prime}$ for
$j\neq j^{\prime}$ when $p>2$ and $K=2$.

##  5 Finite sample properties using Monte Carlo experiments

We examine the finite sample performance of the categorical coefficient
estimator proposed in Section 3 by Monte Carlo experiments.

###  5.1 Data generating processes

We generate $y_{i}$ as

$y_{i}=\alpha+x_{i}\beta_{i}+z_{i1}\gamma_{1}+z_{i2}\gamma_{2}+u_{i},\text{
for }i=1,2,...,n,$ (5.1)

with $\beta_{i}$ distributed as in (2.2) with $K=2,$ and the parameters
$\pi,\beta_{L}$ and $\beta_{H}$.555A Monte Carlo experiment with $K=3$ is
relegated to Section S.3.5 in the online supplement.

We draw $\beta_{i}$ for each individual $i$ independently by setting
$\beta_{i}=\beta_{L}$ with probability $\pi$ and $\beta_{i}=\beta_{H}$ with
probability $1-\pi$, through a sequence of independent Bernoulli draws. We
consider two sets of parameters in all DGPs, denoted as high variance and low
variance parametrization, respectively,

$\left(\pi,\beta_{L},\beta_{H},\mathrm{E}\left(\beta_{i}\right),\mathrm{var}\left(\beta_{i}\right)\right)=\begin{cases}\left(0.5,1,2,1.5,0.25\right)&\left(high\,variance\right)\\\
\left(0.3,0.5,1.345,1.0915,0.15\right)&\left(low\,variance\right)\end{cases}.$
(5.2)

$\beta_{H}/\beta_{L}=2$ for the high variance parametrization, and
$\beta_{H}/\beta_{L}=2.69$, for the low variance parametrization, which is
motivated by the estimates in our empirical illustration in Section 6.666The
estimates for $\beta_{H}/\beta_{L}$ in our empirical analysis range from 1.50
to 2.79. The values of E$(\beta_{i})$ and $\mathrm{var}\left(\beta_{i}\right)$
are obtained noting that E$(\beta_{i})=\pi\beta_{L}+(1-\pi)\beta_{H}$, and
$\mathrm{var}\left(\beta_{i}\right)=\pi(1-\pi)(\beta_{H}-\beta_{L})^{2}$. The
remaining parameters are set as $\alpha=0.25$, and
$\mathbf{\gamma}=\left(1,1\right)^{\prime},$ across DGPs.

We generate the regressors and the error terms as follows.

DGP 1 (Baseline) We first generate $\tilde{x}_{i}\sim\text{IID}\chi^{2}(2)$,
and then set $x_{i}=(\tilde{x}_{i}-2)/2$ so that $x_{i}$ has $0$ mean and unit
variance. The additional regressors, $z_{ij}$, for $j=1,2$ with homogeneous
slopes are generated as

$z_{i1}=x_{i}+v_{i1}\text{ and }z_{i2}=z_{i1}+v_{i2},$

with $v_{ij}\sim\text{IID }N\left(0,1\right)$, for $j=1,2$. This ensures that
the regressors are sufficiently correlated. The error term, $u_{i}$, is
generated as $u_{i}=\sigma_{i}\varepsilon_{i}$, where $\sigma_{i}^{2}$ are
generated as $0.5(1+\text{IID}\chi^{2}(1))$, and
$\varepsilon_{i}\sim\text{IID}N(0,1)$. Note that $\varepsilon_{i}$ and
$\sigma_{i}^{2}$ are generated independently, and $E(u_{i}^{2})=1$.

DGP 2 (Categorical $x$) This setup deviates from the baseline DGP, and allows
the distribution of $x_{i}$ to differ across $i$. Accordingly, we generate
$x_{i}=\left(\tilde{x}_{1i}-2\right)/2$ where
$\tilde{x}_{1i}\sim\text{IID}\chi^{2}\left(2\right)$ for $i=1,2,\cdots,\lfloor
n/2\rfloor$, and $x_{i}=\left(\tilde{x}_{2i}-2\right)/4$ where
$\tilde{x}_{2i}\sim\text{IID}\chi^{2}\left(4\right)$, for $i=\lfloor
n/2\rfloor+1,\cdots,n$. The additional regressors, $z_{ij}$, for $j=1,2$ with
homogeneous slopes are generated as

$z_{i1}=x_{i}+v_{i1}\text{ and }z_{i2}=z_{i1}+v_{i2},$

with $v_{ij}\sim\text{IID }N\left(0,1\right)$, for $j=1,2$. The error term
$u_{i}$ is generated the same as in DGP 1.

DGP 3 (Categorical $u$) We generate $x_{i}$ and $\mathbf{z}_{i}$ the same as
in DGP 1, but allow the error term $u_{i}$ to have a heterogeneous
distribution over $i$. For $i=1,2,\cdots,\lfloor n/2\rfloor$, we set
$u_{i}=\sigma_{i}\varepsilon_{i},$ where
$\sigma_{i}^{2}\sim\text{IID}\chi^{2}\left(2\right)$ and
$\varepsilon_{i}\sim\text{IID}N(0,1)$, and for $i=\lfloor
n/2\rfloor+1,\cdots,n$, we set $u_{i}=\left(\tilde{u}_{i}-2\right)/2$, where
$\tilde{u}_{i}\sim\text{IID}\chi^{2}\left(2\right)$.

We investigate the finite sample performance of the estimator proposed in
Section 3 across DGP 1 to 3 with low variance and high variance
scenarios.777We can consider a DGP with conditional heteroskedasticity, in
which we follow the baseline DGP and generate the error term as
$u_{i}=x_{i}\varepsilon_{i}$, where $\varepsilon_{i}\sim N(0,1)$. The least
square estimator for $\mathbf{\phi}$ is valid in this setup in terms of
estimation and inference, whereas the GMM estimator for the distributional
parameters $\mathbf{\theta}$ breaks down, which is to be expected since we can
only identify the first moment of $\beta_{i}$ under conditional
heteroskedasticity. The results are available on request. Details of the
computational algorithm used to carry out the Monte Carlo experiments (and the
empirical results that follow) are given in Section S.5 of the online
supplement. An accompanying R package is available at https://github.com/zhan-
gao/ccrm.

###  5.2 Summary of the MC results

Table 2: Bias, RMSE and size of the least square estimator
$\hat{\mathbf{\phi}}$

DGP | Baseline |  Categorical $x$ |  Categorical $u$
---|---|---|---
Sample size $n$ | Bias | RMSE | Size |  Bias | RMSE | Size | Bias | RMSE | Size
high variance: $\mathrm{var}\left(\beta_{i}\right)=0.25$
$\mathrm{E}\left(\beta_{i}\right)=1.5$ | 100 | -0.0024 | 0.2035 | 0.0966 | -0.0037 | 0.2035 | 0.0858 | -0.0042 | 0.2268 | 0.0920
1,000 | -0.0017 | 0.0669 | 0.0568 | -0.0002 | 0.0657 | 0.0540 | -0.0019 | 0.0738 | 0.0540
2,000 | -0.0008 | 0.0463 | 0.0512 | -0.0015 | 0.0475 | 0.0534 | -0.0010 | 0.0523 | 0.0522
5,000 | -0.0004 | 0.0301 | 0.0540 | -0.0008 | 0.0300 | 0.0546 | -0.0007 | 0.0335 | 0.0560
10,000 | 0.0002 | 0.0214 | 0.0508 | 0.0000 | 0.0212 | 0.0510 | 0.0000 | 0.0229 | 0.0456
100,000 | -0.0001 | 0.0066 |  0.0472 | 0.0000 | 0.0066 | 0.0460 | 0.0000 | 0.0075 | 0.0506
$\gamma_{1}=1$ | 100 | -0.0022 | 0.1571 | 0.0604 | -0.0006 | 0.1598 | 0.0666 | 0.0018 | 0.1912 | 0.0656
1,000 | 0.0004 | 0.0501 | 0.0496 | -0.0005 | 0.0496 | 0.0508 | 0.0000 | 0.0600 | 0.0530
2,000 | 0.0003 | 0.0352 | 0.0530 | -0.0004 | 0.0350 | 0.0544 | 0.0002 | 0.0432 | 0.0602
5,000 | -0.0001 | 0.0222 | 0.0470 | 0.0005 | 0.0225 | 0.0548 | 0.0007 | 0.0267 | 0.0522
10,000 | -0.0004 | 0.0157 | 0.0470 | 0.0002 | 0.0157 | 0.0512 | 0.0000 | 0.0188 | 0.0504
100,000 | -0.0001 | 0.0049 |  0.0494 | 0.0000 | 0.0049 | 0.0468 | 0.0000 | 0.0059 | 0.0500
$\gamma_{2}=1$ | 100 | 0.0011 | 0.1115 | 0.0616 | 0.0016 | 0.1121 | 0.0654 | -0.0002 | 0.1364 | 0.0700
1,000 | -0.0003 | 0.0358 | 0.0558 | 0.0001 | 0.0354 | 0.0550 | 0.0006 | 0.0421 | 0.0508
2,000 | -0.0001 | 0.0253 | 0.0522 | 0.0006 | 0.0246 | 0.0502 | -0.0003 | 0.0302 | 0.0560
5,000 | 0.0000 | 0.0158 | 0.0480 | 0.0000 | 0.0159 | 0.0570 | -0.0003 | 0.0185 | 0.0470
10,000 | 0.0002 | 0.0111 | 0.0494 | -0.0002 | 0.0111 | 0.0530 | -0.0001 | 0.0134 | 0.0522
100,000 | 0.0001 | 0.0035 | 0.0488 | 0.0000 | 0.0034 | 0.0446 | 0.0000 | 0.0042 | 0.0496
low variance: $\mathrm{var}\left(\beta_{i}\right)=0.15$
$\mathrm{E}\left(\beta_{i}\right)=1.0915$ | 100 | -0.0006 | 0.1829 | 0.0810 | -0.0023 | 0.1855 | 0.0766 | -0.0025 | 0.2094 | 0.0828
1,000 | -0.0005 | 0.0597 | 0.0610 | 0.0005 | 0.0590 | 0.0478 | -0.0006 | 0.0670 | 0.0542
2,000 | -0.0002 | 0.0408 | 0.0516 | -0.0007 | 0.0427 | 0.0606 | -0.0004 | 0.0475 | 0.0544
5,000 | -0.0002 | 0.0264 | 0.0530 | -0.0006 | 0.0266 | 0.0480 | -0.0005 | 0.0302 | 0.0538
10,000 | 0.0000 | 0.0189 | 0.0546 | -0.0002 | 0.0188 | 0.0486 | -0.0002 | 0.0208 | 0.0482
100,000 | -0.0001 | 0.0059 |  0.0474 | 0.0000 | 0.0059 | 0.0494 | 0.0000 | 0.0068 | 0.0508
$\gamma_{1}=1$ | 100 | -0.0027 | 0.1521 | 0.0614 | -0.0001 | 0.1538 | 0.0622 | 0.0014 | 0.1847 | 0.0624
1,000 | 0.0001 | 0.0480 | 0.0520 | -0.0007 | 0.0481 | 0.0542 | -0.0003 | 0.0584 | 0.0570
2,000 | 0.0002 | 0.0338 | 0.0514 | -0.0006 | 0.0334 | 0.0512 | 0.0001 | 0.0417 | 0.0572
5,000 | -0.0002 | 0.0213 | 0.0474 | 0.0003 | 0.0216 | 0.0532 | 0.0007 | 0.0257 | 0.0498
10,000 | -0.0003 | 0.0150 | 0.0466 | 0.0002 | 0.0152 | 0.0542 | 0.0001 | 0.0183 | 0.0518
100,000 | -0.0001 | 0.0047 |  0.0482 | 0.0000 | 0.0047 | 0.0474 | 0.0000 | 0.0057 | 0.0500
$\gamma_{2}=1$ | 100 | 0.0011 | 0.1081 | 0.0592 | 0.0013 | 0.1079 | 0.0622 | -0.0002 | 0.1323 | 0.0674
1,000 | -0.0003 | 0.0345 | 0.0594 | 0.0003 | 0.0342 | 0.0556 | 0.0006 | 0.0409 | 0.0500
2,000 | 0.0000 | 0.0243 | 0.0534 | 0.0006 | 0.0235 | 0.0450 | -0.0001 | 0.0292 | 0.0576
5,000 | 0.0001 | 0.0152 | 0.0490 | 0.0001 | 0.0152 | 0.0552 | -0.0002 | 0.0179 | 0.0470
10,000 | 0.0002 | 0.0106 | 0.0454 | -0.0002 | 0.0107 | 0.0528 | -0.0002 | 0.0131 | 0.0526
100,000 | 0.0001 | 0.0033 | 0.0442 | 0.0000 | 0.0033 | 0.0448 | 0.0000 | 0.0040 | 0.0486

Notes: The data generating process is (5.1). high variance and low variance
parametrization are described in (5.2). “Baseline”, “Categorical $x$” and
“Categorical $u$” refer to DGP 1 to 3 as in Section 5.1. Generically, bias,
RMSE and size are calculated by
$R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)$,
$\sqrt{R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)^{2}}$,
and
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{0}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
respectively, for true parameter $\theta_{0}$, its estimate
$\hat{\theta}^{(r)}$, the estimated standard error of $\hat{\theta}^{(r)}$,
$\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical value
$\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.

Figure 1: Empirical power functions for the least square estimator
$\hat{\mathbf{\phi}}$ with the high variance parametrization
($\mathrm{var}\left(\beta_{i}\right)=0.25$)

(a) Baseline

(b) Categorical $x$

(c) Categorical $u$

Notes: The data generating process is (5.1) with high variance parametrization
that is described in (5.2). “Baseline”, “Categorical $x$” and “Categorical
$u$” refer to DGP 1 to 3 as in Section 5.1. Generically, power is calculated
by
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{\delta}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
for $\theta_{\delta}$ in a symmetric neighborhood of the true parameter
$\theta_{0}$, the estimate $\hat{\theta}^{(r)}$, the estimated standard error
of $\hat{\theta}^{(r)}$, $\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical
value $\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.

Figure 2: Empirical power functions for the least square estimator
$\hat{\mathbf{\phi}}$ with the low variance parametrization
($\mathrm{var}\left(\beta_{i}\right)=0.15$)

(a) Baseline

(b) Categorical $x$

(c) Categorical $u$

Notes: The data generating process is (5.1) with low variance parametrization
that is described in (5.2). “Baseline”, “Categorical $x$” and “Categorical
$u$” refer to DGP 1 to 3 as in Section 5.1. Generically, power is calculated
by
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{\delta}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
for $\theta_{\delta}$ in a symmetric neighborhood of the true parameter
$\theta_{0}$, the estimate $\hat{\theta}^{(r)}$, the estimated standard error
of $\hat{\theta}^{(r)}$, $\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical
value $\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.

Table 3: Bias, RMSE and size of the GMM estimator for distributional
parameters of $\beta$

DGP | Baseline |  Categorical $x$ |  Categorical $u$
---|---|---|---
Sample size $n$ | Bias | RMSE | Size |  Bias | RMSE | Size | Bias | RMSE | Size
high variance: $\mathrm{var}\left(\beta_{i}\right)=0.25$
$\pi=0.5$ |  100 | 0.0457 | 0.2291 | 0.1737 | 0.0363 | 0.2410 | 0.2130 | 0.0235 | 0.2361 | 0.2231
1,000 | 0.0018 | 0.1019 | 0.1308 | 0.0033 | 0.1178 | 0.1437 | -0.0270 | 0.1741 | 0.2033
2,000 | 0.0017 | 0.0688 | 0.1084 | 0.0015 | 0.0826 | 0.1199 | -0.0174 | 0.1273 | 0.1545
5,000 | -0.0003 | 0.0416 | 0.0936 | -0.0015 | 0.0495 | 0.0908 | -0.0089 | 0.0810 | 0.1048
10,000 | 0.0002 | 0.0301 | 0.0774 | -0.0006 | 0.0351 | 0.0780 | -0.0052 | 0.0582 | 0.0864
100,000 | -0.0001 | 0.0096 |  0.0550 | 0.0002 | 0.0114 | 0.0576 | -0.0009 | 0.0194 | 0.0582
$\beta_{L}=1$ | 100 | 0.1415 | 0.4749 | 0.2472 | 0.1099 | 0.5110 | 0.2138 | 0.1151 | 0.5961 | 0.1820
1,000 | 0.0207 | 0.1242 | 0.1501 | 0.0200 | 0.1454 | 0.1433 | -0.0256 | 0.2373 | 0.1225
2,000 | 0.0129 | 0.0819 | 0.1344 | 0.0116 | 0.1007 | 0.1355 | -0.0094 | 0.1486 | 0.1094
5,000 | 0.0048 | 0.0512 | 0.1052 | 0.0027 | 0.0607 | 0.1000 | -0.0053 | 0.0897 | 0.0850
10,000 | 0.0031 | 0.0365 | 0.0854 | 0.0021 | 0.0428 | 0.0900 | -0.0020 | 0.0633 | 0.0714
100,000 | 0.0002 | 0.0112 | 0.0534 | 0.0007 | 0.0135 | 0.0584 | -0.0002 | 0.0207 | 0.0574
$\beta_{H}=2$ | 100 | -0.0996 | 0.5609 | 0.2014 | -0.0873 | 0.6154 | 0.1963 | -0.1071 | 0.6996 | 0.1866
1,000 | -0.0193 | 0.1407 | 0.1864 | -0.0128 | 0.1581 | 0.1661 | -0.0319 | 0.2400 | 0.2093
2,000 | -0.0099 | 0.0893 | 0.1486 | -0.0099 | 0.1094 | 0.1467 | -0.0239 | 0.1663 | 0.1673
5,000 | -0.0053 | 0.0519 | 0.1092 | -0.0072 | 0.0622 | 0.1082 | -0.0127 | 0.1019 | 0.1156
10,000 | -0.0020 | 0.0362 | 0.0878 | -0.0033 | 0.0430 | 0.0880 | -0.0080 | 0.0718 | 0.0986
100,000 | -0.0005 | 0.0114 |  0.0530 | -0.0003 | 0.0134 | 0.0548 | -0.0017 | 0.0236 | 0.0646
low variance: $\mathrm{var}\left(\beta_{i}\right)=0.15$
$\pi=0.3$ |  100 | 0.2175 | 0.3084 | 0.2183 | 0.2227 | 0.3187 | 0.2464 | 0.2294 | 0.3157 | 0.2500
1,000 | 0.0170 | 0.1536 | 0.1873 | 0.0307 | 0.1837 | 0.2063 | 0.0511 | 0.2295 | 0.2493
2,000 | 0.0014 | 0.1010 | 0.1426 | 0.0105 | 0.1290 | 0.1601 | 0.0181 | 0.1815 | 0.2102
5,000 | -0.0002 | 0.0590 | 0.1084 | 0.0010 | 0.0737 | 0.1158 | 0.0085 | 0.1232 | 0.1468
10,000 | -0.0001 | 0.0415 | 0.0894 | 0.0005 | 0.0515 | 0.0928 | 0.0067 | 0.0906 | 0.1046
100,000 | -0.0001 | 0.0129 |  0.0594 | 0.0003 | 0.0158 | 0.0536 | 0.0108 | 0.0349 | 0.0776
$\beta_{L}=0.5$ | 100 | 0.3365 | 0.5905 | 0.2426 | 0.3153 | 0.6042 | 0.2432 | 0.3384 | 0.6746 | 0.2005
1,000 | 0.0352 | 0.2334 | 0.1560 | 0.0290 | 0.2813 | 0.1544 | 0.0131 | 0.4141 | 0.1233
2,000 | 0.0175 | 0.1414 | 0.1310 | 0.0131 | 0.1835 | 0.1382 | -0.0157 | 0.2988 | 0.1037
5,000 | 0.0085 | 0.0830 | 0.1082 | 0.0041 | 0.1052 | 0.1118 | -0.0057 | 0.1798 | 0.0928
10,000 | 0.0055 | 0.0577 | 0.0966 | 0.0031 | 0.0730 | 0.0934 | 0.0019 | 0.1231 | 0.0760
100,000 | 0.0005 | 0.0180 | 0.0596 | 0.0011 | 0.0222 | 0.0582 | 0.0130 | 0.0443 | 0.0962
$\beta_{H}=1.345$ | 100 | 0.0023 | 0.4727 | 0.1377 | 0.0238 | 0.5290 | 0.1453 | 0.0185 | 0.6500 | 0.1461
1,000 | -0.0081 | 0.1265 | 0.1737 | 0.0042 | 0.1621 | 0.1655 | 0.0120 | 0.2353 | 0.1738
2,000 | -0.0092 | 0.0828 | 0.1428 | -0.0026 | 0.1045 | 0.1475 | 0.0029 | 0.1607 | 0.1710
5,000 | -0.0048 | 0.0489 | 0.1028 | -0.0041 | 0.0586 | 0.1034 | 0.0006 | 0.0970 | 0.1172
10,000 | -0.0025 | 0.0340 | 0.0808 | -0.0024 | 0.0412 | 0.0942 | 0.0019 | 0.0706 | 0.0958
100,000 | -0.0004 | 0.0105 |  0.0486 | -0.0002 | 0.0125 | 0.0548 | 0.0073 | 0.0262 | 0.0696

Notes: The data generating process is (5.1). high variance and low variance
parametrization are described in (5.2). “Baseline”, “Categorical $x$” and
“Categorical $u$” refer to DGP 1 to 3 as in Section 5.1. Generically, bias,
RMSE and size are calculated by
$R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)$,
$\sqrt{R^{-1}\sum_{r=1}^{R}\left(\hat{\theta}^{(r)}-\theta_{0}\right)^{2}}$,
and
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{0}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
respectively, for true parameter $\theta_{0}$, its estimate
$\hat{\theta}^{(r)}$, the estimated standard error of $\hat{\theta}^{(r)}$,
$\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical value
$\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.

Figure 3: Empirical power functions for the GMM estimator of distributional
parameters of $\beta$ with the high variance
parametrization($\mathrm{var}\left(\beta_{i}\right)=0.25$)

(a) Baseline

(b) Categorical $x$

(c) Categorical $u$

Notes: The data generating process is (5.1) with high variance parametrization
that is described in (5.2). “Baseline”, “Categorical $x$” and “Categorical
$u$” refer to DGP 1 to 3 as in Section 5.1. The model is estimated with $S=4$,
the highest order of moments of $x_{i}$ used in estimation. Generically, power
is calculated by
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{\delta}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
for $\theta_{\delta}$ in a symmetric neighborhood of the true parameter
$\theta_{0}$, the estimate $\hat{\theta}^{(r)}$, the estimated standard error
of $\hat{\theta}^{(r)}$, $\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical
value $\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.

Figure 4: Empirical power functions for the GMM estimator of distributional
parameters of $\beta$ with the low variance parametrization
($\mathrm{var}\left(\beta_{i}\right)=0.15$)

(a) Baseline

(b) Categorical $x$

(c) Categorical $u$

Notes: The data generating process is (5.1) with low variance parametrization
that is described in (5.2). “Baseline”, “Categorical $x$” and “Categorical
$u$” refer to DGP 1 to 3 as in Section 5.1. The model is estimated with $S=4$,
the highest order of moments of $x_{i}$ used in estimation. Generically, power
is calculated by
$R^{-1}\sum_{r=1}^{R}\mathbf{1}\left[\left|\hat{\theta}^{(r)}-\theta_{\delta}\right|/\hat{\sigma}_{\hat{\theta}}^{(r)}>\mathrm{cv}_{0.05}\right]$,
for $\theta_{\delta}$ in a symmetric neighborhood of the true parameter
$\theta_{0}$, the estimate $\hat{\theta}^{(r)}$, the estimated standard error
of $\hat{\theta}^{(r)}$, $\hat{\sigma}_{\hat{\theta}}^{(r)}$, and the critical
value $\mathrm{cv}_{0.05}=\Phi^{-1}\left(0.975\right)$ across $R=5,000$
replications, where $\Phi\left(\cdot\right)$ is the cumulative distribution
function of standard normal distribution.

For each sample size $n=100$, $1,000$, $2,000$, $5,000$, $10,000$ and
$100,000$ we run $5,000$ replications of experiments for DGP 1 (baseline), DGP
2 (categorical $x$) and DGP 3 (categorical $u$) with high variance and low
variance parametrization, as set out in (5.2).

We first investigate the finite sample performance of $\hat{\mathbf{\phi}}$,
as an estimator of
$\mathbf{\phi}=\left(\mathrm{E}\left(\beta_{i}\right),\mathbf{\gamma}^{\prime}\right)^{\prime}$.
Bias, root mean squared errors (RMSE) for estimation of
$\mathrm{E}\left(\beta_{i}\right)$, $\gamma_{1}$ and $\gamma_{2}$, as well as
size of testing of the null values at the 5 per cent nominal value are
reported in Table 2. In addition, we plot the associated empirical power
functions in Figure 1 and 2, for cases of high and low $var(\beta_{i})$. The
results show that $\hat{\mathbf{\phi}}$ has very good small sample properties
with small bias and RMSEs, with size very close to the nominal value of 5 per
cent across all DGPs and parametrization, even when sample size is relatively
small. The power of the test increases steadily as the sample size increases.

Then, we turn to the GMM estimator for the distributional parameters of
$\beta_{i}$ proposed in Section 3.2. The bias, RMSE, and the test size based
on the asymptotic distribution given in Theorem 5, for $\pi$, $\beta_{L}$ and
$\beta_{H}$, are reported in Table 3. The empirical power functions are
reported in Figure 3 and 4. The reported results are based on $S=4$, where $S$
$(>2K-1=3)$ denotes the highest order of moments of $x_{i}$ included in
estimation.888We also tried estimation based on a larger number of moments
(using $S=5$ and $S=6$). In the case of current Monte Carlo results, adding
more moments does not seem to add much to the precision of the estimates and
could be counter-productive when $n$ is not sufficiently large. The results
are available in Section S.3.1 in the online supplement.

The upper panel of this table reports the results of the high variance and the
lower panel for the low variance parametrization, as set out in (5.2). For all
parameters and under all DGPs, the bias and RMSE decline steadily with the
sample size as predicted by Theorem 4, and confirm the robustness of the GMM
estimates to the heterogeneity in the regressor and the error processes. But
for a given sample size, the relative precision of the estimates depends on
the variability of $\beta_{i}$, as characterized by the true value of
$\mathrm{var}(\beta_{i})$. The precision of the estimates with high variance
parametrization is relatively higher than that with low variance
parametrization. This is to be expected since, unlike $\mathrm{E}(\beta_{i}),$
the distributional parameters are only identified if
$\mathrm{var}(\beta_{i})>0$. As shown in (2.18) and (2.19) for the current
case of $K=2$, $\mathrm{var}(\beta_{i})$ is in the denominator when we recover
the distributional parameters from the moments of $\beta_{i}$. When
$\mathrm{var}(\beta_{i})$ is small, estimation errors in the moments of
$\beta_{i}$ can be amplified in the estimation of $\pi$, $\beta_{L}$ and
$\beta_{H}$. On the other hand, the larger the variance the more precisely
$\pi$, $\beta_{H}$ and $\beta_{L}$ can be estimated for a given $n$.999Section
S.3.4 in the online supplement presents parametrization with
$\mathrm{var}\left(\beta_{i}\right)=6.35$ and $18.95$, which further confirms
the pattern that the larger the variance the more precisely $\pi$, $\beta_{H}$
and $\beta_{L}$ can be estimated for a given $n$. The size and power also
depends on the parametrization. With both high variance and low variance
parametrization, we can achieve correct size and reasonable power when $n$ is
quite large ($n=100,000$). We plot the empirical power functions for $n\geq
5,000$ for $\pi$, $\beta_{H}$ and $\beta_{L}$ since the size is far above 5
per cent for smaller values of $n$, and power comparisons are not meaningful
in such cases.

###### Remark 15

Note that GMM estimators of moments of $\beta_{i}$, namely
$\mathbf{m}_{\mathbf{\beta}}$, can be obtained using the moment conditions in
(3.7),and the transformations
$\mathbf{m}_{\mathbf{\beta}}=h\left(\mathbf{\theta}\right)$ in (3.4) are
required only to derive the estimators of $\mathbf{\theta}$, the parameters of
the underlying categorical distribution. The Monte Carlo results in Section
S.3.2 in the online supplement show that $\mathbf{m}_{\mathbf{\beta}}$ can be
accurately estimated with relatively small sample sizes. In the estimation of
both $\mathbf{m}_{\mathbf{\beta}}$ and $\mathbf{\theta}$, the same set of
moment conditions are included, so the estimation of distributional parameters
$\mathbf{\theta}$ essentially relies on the relation
$\mathbf{\theta}=h^{-1}\left(\mathbf{m}_{\mathbf{\beta}}\right)$. Sampling
uncertainties in the estimation of $\mathbf{m}_{\mathbf{\beta}}$, particularly
in higher order moments, are potentially amplified through the inverse
transformation $h^{-1}$ that involves matrix inversion, which causes the
difficulties in estimation and inference of $\mathbf{\theta}$ when sample
sizes are small. This is analogous to the problem of precision matrix
estimation from an estimated covariance matrix. In practice, estimation of the
categorical parameters is recommended for applications where the sample size
is relatively large, otherwise it is advisable to focus on estimates of the
lower order moments of $\beta_{i}$.

##  6 Heterogeneous return to education: An empirical application

Since the pioneering work by Becker (1962, 1964) on the effects of investments
in human capital, estimating returns to education has been one of the focal
points of labor economics research. In his pioneering contribution Mincer
(1974) models the logarithm of earnings as a function of years of education
and years of potential labor market experience (age minus years of education
minus six), which can be written in a generic form:

$\log\text{wage}_{i}=\alpha_{i}+\beta_{i}\text{edu}_{i}+\phi\left(\mathbf{z}_{i}\right)+\varepsilon_{i},$
(6.1)

as in Heckman, Humphries, and Veramendi (2018, Equation (1)), where
$\mathbf{z}_{i}$ includes the labor market experience and other relevant
control variables. The above wage equation, also known as the “Mincer
equation”, has become of the workhorse of the empirical works on estimating
the return to education. In the most widely used specification of the Mincer
equation (6.1),

$\phi\left(\mathbf{z}_{i}\right)=\rho_{1}\text{exper}_{i}+\rho_{2}\text{exper}_{i}^{2}+\tilde{\mathbf{z}}_{i}^{\prime}\tilde{\mathbf{\gamma}},$

where $\tilde{\mathbf{z}}_{i}$ is the vector of control variables other than
potential labor market experience.

Along with the advancement of empirical research on this topic, there has been
a growing awareness of the importance of heterogeneity in individual cognitive
and non-cognitive abilities (Heckman, 2001) and their significance for
explaining the observed heterogeneity in return to education. Accordingly, it
is important to allow the parameters of the wage equation to differ across
individuals. In equation (6.1) we allow $\alpha_{i}$ and $\beta_{i}$ to differ
across individuals, but assume that $\phi\left(\mathbf{z}_{i}\right)$ can be
approximated as non-linear functions of experience and other control variables
with homogeneous coefficients.

Specifically, following Lemieux (2006b, c) we also allow for time variations
in the parameters of the wage equation and consider the following categorical
coefficient model over a given cross-section sample indexed by $t$:101010Some
investigators have suggested including higher powers of the experience
variable in the wage equation. Lemieux (2006a), for example, proposes using a
quartic rather than a quadratic function. As a robustness check we also
provide estimation results with quartic experience specification in Section
S.4 in the online supplement.

$\log\text{wage}_{it}=\alpha_{it}+\beta_{it}\text{edu}_{it}+\rho_{1t}\text{exper}_{it}+\rho_{2t}\text{exper}_{it}^{2}+\tilde{\mathbf{z}}_{it}^{\prime}\tilde{\mathbf{\gamma}}_{t}+\varepsilon_{it},$
(6.2)

where the return to education follows the categorical distribution,

$\beta_{it}=\begin{cases}b_{tL}&\text{w.p. }\pi_{t},\\\ b_{tH}&\text{w.p.
}1-\pi_{t},\end{cases}$

and $\tilde{\mathbf{z}}_{it}$ includes gender, martial status and race.
$\alpha_{it}=\alpha_{t}+\delta_{it}$ where $\delta_{it}$ is mean $0$ random
variable assumed to be distributed independently of $\text{edu}_{it}$ and
$\mathbf{z}_{it}=\left(\text{exper}_{it},\text{exper}_{it}^{2}\text{,
}\tilde{\mathbf{z}}_{t}^{\prime}\right)^{\prime}$. Let
$u_{it}=\varepsilon_{it}+\delta_{it}$, and write (6.2) as

$\log\text{wage}_{it}=\alpha_{t}+\beta_{it}\text{edu}_{it}+\rho_{1t}\text{exper}_{it}+\rho_{2t}\text{exper}_{it}^{2}+\tilde{\mathbf{z}}_{it}^{\prime}\tilde{\mathbf{\gamma}}_{t}+u_{it}.$
(6.3)

The correlation between $\alpha_{it}$ and $\text{edu}_{it}$ in (6.1) is the
source of “ability bias” (Griliches, 1977). Given the pure cross-sectional
nature of our analysis, we do not allow for the endogeneity from “ability
bias” or dynamics. To allow for non-zero correlations between $\alpha_{it}$,
eduit and $\mathbf{z}_{it}$, a panel data approach is required, which has its
own challenges, as education and experience variables tend to very slow moving
(if at all) for many individuals in the panel. Time delays between changes in
education and experience, and the wage outcomes also further complicate the
interpretation of the mean estimates of $\beta_{it}$ which we shall be
reporting. To partially address the possible dynamic spillover effects, we
provide estimates of the distribution of $\beta_{it}$ using cross-sectional
data from two different sample periods, and investigate the extent to which
the distribution of return to education has changed over time, by gender and
the level of educational achievements.111111Time variations in return to
education has also been investigated in the literature as a possible
explanation of increasing wage inequality in the U.S. See, for example, the
papers by Lemieux (2006b, c).

We estimate the categorical distribution of the return to education in (6.3)
using the May and Outgoing Rotation Group (ORG) supplements of the Current
Population Survey (CPS) data, as in Lemieux (2006b, c).121212The data is
retrieved from
https://www.openicpsr.org/openicpsr/project/116216/version/V1/view. We pool
observations from 1973 to 1975 for the first sample period,
$t=\left\\{1973-1975\right\\}$ and observations from 2001 to 2003 for the
second sample period, $t=\left\\{2001-2003\right\\}$. Following Lemieux
(2006b), we consider sub-samples of those with less than 12 years of
education, “high school or less”, and those with more than 12 years of
education, “postsecondary education”, as well as the combined sample. We also
present results by gender. The summary statistics are reported in Table 4. As
to be expected, the mean log wages are higher for those with postsecondary
education (for male and female), with the number of years of schooling and
experience rising by about one year across the two sub-period samples. There
are also important differences across male and female, and the two educational
groupings, which we hope to capture in our estimation.

Table 4: Summary Statistics of the May and Outgoing Rotation Group (ORG)
supplements of the Current Population Survey (CPS) data across two periods,
1973 - 75 and 2001 - 03, by years of education and gender

| 1973 - 75 |  | 2001 - 03
---|---|---|---
| High School | Postsecondary | All |  | High School | Postsecondary | All
| or Less | Education |  | or Less | Education
| Both male and female
log wage | 1.59 | 1.94 | 1.69 |  | 1.47 | 1.88 | 1.71
| (0.50) | (0.53) | (0.53) |  | (0.47) | (0.57) | (0.57)
edu. | 10.64 | 15.21 | 12.02 |  | 11.29 | 14.96 | 13.41
| (2.11) | (1.65) | (2.89) |  | (1.68) | (1.82) | (2.53)
age | 36.74 | 34.90 | 36.18 |  | 37.96 | 39.87 | 39.06
| (13.85) | (11.58) |  (13.23) |  | (12.93) | (11.33) | (12.07)
expr. | 20.10 | 13.69 | 18.17 |  | 20.67 | 18.91 | 19.65
| (14.44) | (11.41) | (13.91) |  | (12.95) | (11.17) | (11.98)
marriage | 0.67 | 0.70 | 0.68 |  | 0.52 | 0.60 | 0.57
| (0.47) | (0.46) | (0.47) |  | (0.50) | (0.49) | (0.50)
nonwhite | 0.11 | 0.08 | 0.10 |  | 0.15 | 0.14 | 0.15
| (0.32) | (0.27) | (0.30) |  | (0.36) | (0.35) | (0.35)
$n$ | 77,899 | 33,733 | 111,632 |  | 216,136 | 295,683 | 511,819
| Male
log wage | 1.76 | 2.07 | 1.86 |  | 1.57 | 2.00 | 1.81
| (0.48) | (0.53) | (0.52) |  | (0.48) | (0.58) | (0.58)
edu. | 10.44 | 15.29 | 12.00 |  | 11.19 | 15.02 | 13.31
| (2.26) | (1.69) | (3.08) |  | (1.82) | (1.84) | (2.64)
age | 36.79 | 35.29 | 36.31 |  | 37.21 | 40.24 | 38.89
| (13.82) | (11.24) | (13.07) |  | (12.70) | (11.30) | (12.04)
expr. | 20.35 | 14.00 | 18.32 |  | 20.02 | 19.22 | 19.58
| (14.49) | (11.06) | (13.81) |  | (12.75) | (11.08) | (11.86)
marriage | 0.73 | 0.76 | 0.74 |  | 0.53 | 0.64 | 0.59
| (0.44) | (0.43) | (0.44) |  | (0.50) | (0.48) | (0.49)
nonwhite | 0.10 | 0.06 | 0.09 |  | 0.14 | 0.13 | 0.13
| (0.30) | (0.24) | (0.29) |  | (0.34) | (0.33) | (0.34)
$n$ | 44,299 | 20,851 | 65,150 |  | 116,129 | 144,138 | 260,267
| Female
log wage | 1.35 | 1.71 | 1.45 |  | 1.77 | 1.36 | 1.61
| (0.41) | (0.47) | (0.46) |  | (0.54) | (0.43) | (0.54)
edu. | 10.89 | 15.08 | 12.05 |  | 14.90 | 11.42 | 13.52
| (1.87) | (1.59) | (2.60) |  | (1.79) | (1.49) | (2.40)
age | 36.67 | 34.27 | 36.01 |  | 38.83 | 39.52 | 39.24
| (13.88) | (12.09) | (13.45) |  | (13.14) | (11.35) | (12.10)
expr. | 19.78 | 13.19 | 17.96 |  | 18.61 | 21.41 | 19.73
| (14.36) | (11.94) | (14.04) |  | (11.24) | (13.13) | (12.11)
marriage | 0.60 | 0.60 | 0.60 |  | 0.56 | 0.51 | 0.54
| (0.49) | (0.49) | (0.49) |  | (0.50) | (0.50) | (0.50)
nonwhite | 0.13 | 0.10 | 0.12 |  | 0.15 | 0.17 | 0.16
| (0.33) | (0.30) | (0.33) |  | (0.36) | (0.38) | (0.37)
$n$ | 33,600 | 12,882 | 46,482 |  | 151,545 | 100,007 | 251,552

Notes: “Postsecondary Education” stands for the sub-sample with years of
education higher than 12 and “High School or Less” stands for sub-sample with
years of education less than or equal to 12). edu. and exper. are in years.
marriage and nonwhite are dummy variables. $n$ is the sample size. We report
mean and standard deviation (in parentheses) of each variable. The data is
from the May and Outgoing Rotation Group (ORG) supplements of the Current
Population Survey (CPS) data retrived from
https://www.openicpsr.org/openicpsr/project/116216/version/V1/view.

Table 5: Estimates of the distribution of the return to education across two
periods, 1973 - 75 and 2001 - 03, by years of education and gender

| High School or Less |  |  Postsecondary Edu. |  | All
---|---|---|---|---|---
| 1973 - 75 | 2001 - 03 |  | 1973 - 75 | 2001 - 03 |  | 1973 - 75 | 2001 - 03
| Both Male and Female
$\pi$ | 0.4843 | 0.5069 |  | 0.4398 | 0.3537 |  | 0.4719 | 0.3463
| (4188.8) | (0.0269) |  | (0.0502) | (0.0091) |  | (0.0485) | (0.0047)
$\beta_{L}$ | 0.0608 | 0.0382 |  | 0.0624 | 0.0866 |  | 0.0558 | 0.0645
| (5.0939) | (0.0014) |  | (0.0035) | (0.0009) |  | (0.0020) | (0.0004)
$\beta_{H}$ | 0.0619 | 0.0920 |  | 0.1103 | 0.1401 |  | 0.0941 | 0.1263
| (4.8132) | (0.0019) |  | (0.0032) | (0.0007) |  | (0.0022) | (0.0004)
$\beta_{H}/\beta_{L}$ | 1.0194 | 2.4102 |  | 1.7680 | 1.6178 |  | 1.6879 | 1.9567
| (6.2938) | (0.0428) |  | (0.0618) | (0.0111) |  | (0.0295) | (0.0080)
$\mathrm{E}\left(\beta_{i}\right)$ | 0.0614 | 0.0647 |  | 0.0893 | 0.1212 |  | 0.0760 | 0.1049
$\mathrm{s.d.}\left(\beta_{i}\right)$ | 0.0006 | 0.0269 |  | 0.0238 | 0.0256 |  | 0.0191 | 0.0294
$n$ | 77,899 | 216,136 |  | 33,733 | 295,683 |  | 111,632 | 511,819
| Male
$\pi$ | n/a | 0.4939 |  | 0.4706 | 0.3201 |  | 0.4802 | 0.3290
| n/a | (0.0399) |  | (0.0707) | (0.0104) |  | (0.0815) | (0.0053)
$\beta_{L}$ | 0.0637 | 0.0404 |  | 0.0534 | 0.0743 |  | 0.0536 | 0.0548
| n/a | (0.0019) |  | (0.0046) | (0.0012) |  | (0.0030) | (0.0005)
$\beta_{H}$ | 0.0637 | 0.0911 |  | 0.0995 | 0.1308 |  | 0.0875 | 0.1192
| n/a | (0.0026) |  | (0.0042) | (0.0009) |  | (0.0031) | (0.0005)
$\beta_{H}/\beta_{L}$ | 1.0000 | 2.2526 |  | 1.8641 | 1.7603 |  | 1.6312 | 2.1772
| n/a | (0.0534) |  | (0.1038) | (0.0209) |  | (0.0459) | (0.0144)
$\mathrm{E}\left(\beta_{i}\right)$ | 0.0637 | 0.0661 |  | 0.0778 | 0.1128 |  | 0.0712 | 0.0980
$\mathrm{s.d.}\left(\beta_{i}\right)$ | 0.0000 | 0.0253 |  | 0.0230 | 0.0264 |  | 0.0169 | 0.0303
$n$ | 44,299 | 116,129 |  | 20,851 | 144,138 |  | 65,150 | 260,267
| Female
$\pi$ | 0.4999 | 0.5166 |  | 0.4526 | 0.3906 |  | 0.4566 | 0.3608
| (0.5047) | (0.0283) |  | (0.0829) | (0.0167) |  | (0.0810) | (0.0086)
$\beta_{L}$ | 0.0441 | 0.0348 |  | 0.0823 | 0.0979 |  | 0.0628 | 0.0751
| (0.0133) | (0.0016) |  | (0.0053) | (0.0013) |  | (0.0033) | (0.0007)
$\beta_{H}$ | 0.0723 | 0.0972 |  | 0.1310 | 0.1473 |  | 0.1028 | 0.1333
| (0.0159) | (0.0025) |  | (0.0055) | (0.0011) |  | (0.0038) | (0.0007)
$\beta_{H}/\beta_{L}$ | 1.6392 | 2.7934 |  | 1.5913 | 1.5048 |  | 1.6357 | 1.7756
| (0.1565) | (0.0700) |  | (0.0539) | (0.0121) |  | (0.0353) | (0.0090)
$\mathrm{E}\left(\beta_{i}\right)$ | 0.0582 | 0.0650 |  | 0.1090 | 0.1280 |  | 0.0845 | 0.1123
$\mathrm{s.d.}\left(\beta_{i}\right)$ | 0.0141 | 0.0312 |  | 0.0242 | 0.0241 |  | 0.0199 | 0.0280
$n$ | 33,600 | 100,007 |  | 12,882 | 151,545 |  | 46,482 | 251,552

Notes: This table reports the estimates of the distribution of $\beta_{i}$
with the quadratic in experience specification (6.2), using $S=4$ order
moments of $\text{edu}_{i}$. “Postsecondary Edu.” stands for the sub-sample
with years of education higher than 12 and “High School or Less” stands for
those with years of education less than or equal to 12.
$\mathrm{s.d.}\left(\beta_{i}\right)$ corresponds to the square root of
estimated $\mathrm{var}\left(\beta_{i}\right)$. $n$ is the sample size. “n/a”
is inserted when the estimates show homogeneity of $\beta_{i}$ and $\pi$ is
not identified and cannot be estimated.

Table 6: Estimates of $\mathbf{\gamma}$ associated with control variables
$\mathbf{z}_{i}$ with specification (6.2) across two periods, 1973 - 75 and
2001 - 03, by years of education and gender, which complements Table 5

| High School or Less |  | Postsecondary Edu. |  | All
---|---|---|---|---|---
| 1973 - 75 | 2001 - 03 |  | 1973 - 75 |  2001 - 03 |  | 1973 - 75 | 2001 - 03
| Both male and female
exper. | 0.0305 | 0.0319 |  | 0.0415 | 0.0354 |  | 0.0310 | 0.0321
| (0.0004) | (0.0002) |  | (0.0008) | (0.0003) |  | (0.0003) | (0.0002)
$\mathtt{exper.}^{2}$ ($\times 10^{2}$) | -0.0490 | -0.0505 |  | -0.0826 | -0.0652 |  | -0.0499 | -0.0537
| (0.0009) | (0.0005) |  | (0.0022) | (0.0007) |  | (0.0008) | (0.0005)
marriage | 0.1120 | 0.0751 |  | 0.0886 | 0.0770 |  | 0.1085 | 0.0818
| (0.0036) | (0.0020) |  | (0.0059) | (0.0020) |  | (0.0031) | (0.0014)
nonwhite | -0.0922 | -0.0775 |  | -0.0424 | -0.0571 |  | -0.0715 | -0.0667
| (0.0047) | (0.0024) |  | (0.0088) | (0.0025) |  | (0.0042) | (0.0018)
gender | 0.4157 | 0.2298 |  | 0.2962 | 0.2023 |  | 0.3892 | 0.2167
| (0.0029) | (0.0017) |  | (0.0050) | (0.0018) |  | (0.0025) | (0.0013)
$n$ | 77,899 | 216,136 |  | 33,733 | 295,683 |  | 111,632 | 511,819
| Male
exper. | 0.0369 | 0.0366 |  | 0.0516 | 0.0405 |  | 0.0389 | 0.0371
| (0.0005) | (0.0003) |  | (0.0011) | (0.0005) |  | (0.0005) | (0.0003)
$\mathtt{exper.}^{2}$ ($\times 10^{2}$) | -0.0589 | -0.0589 |  | -0.1016 | -0.0752 |  | -0.0635 | -0.0629
| (0.0012) | (0.0008) |  | (0.0029) | (0.0011) |  | (0.0010) | (0.0007)
marriage | 0.1940 | 0.1123 |  | 0.1497 | 0.1344 |  | 0.1828 | 0.1316
| (0.0053) | (0.0028) |  | (0.0085) | (0.0031) |  | (0.0045) | (0.0021)
nonwhite | -0.1241 | -0.1165 |  | -0.1172 | -0.1010 |  | -0.1178 | -0.1093
| (0.0065) | (0.0035) |  | (0.0127) | (0.0039) |  | (0.0058) | (0.0027)
$n$ | 44,299 | 116,129 |  | 20,851 | 144,138 |  | 65,150 | 260,267
| Female
exper. | 0.0223 | 0.0265 |  | 0.0271 | 0.0313 |  | 0.0208 | 0.0272
| (0.0006) | (0.0003) |  | (0.0011) | (0.0004) |  | (0.0005) | (0.0003)
$\mathtt{exper.}^{2}$ ($\times 10^{2}$) | -0.0376 | -0.0411 |  | -0.0564 | -0.0576 |  | -0.0338 | -0.0450
| (0.0013) | (0.0008) |  | (0.0030) | (0.0010) |  | (0.0012) | (0.0006)
marriage | 0.0115 | 0.0317 |  | -0.0005 | 0.0262 |  | 0.0118 | 0.0322
| (0.0048) | (0.0028) |  | (0.0079) | (0.0026) |  | (0.0041) | (0.0019)
nonwhite | -0.0581 | -0.0441 |  | 0.0395 | -0.0236 |  | -0.0202 | -0.0315
| (0.0065) | (0.0033) |  | (0.0117) | (0.0033) |  | (0.0058) | (0.0024)
$n$ | 33,600 | 100,007 |  | 12,882 | 151,545 |  | 46,482 | 251,552

Notes: This table reports the estimates of $\mathbf{\ \gamma}$ in (6.2).
“Postsecondary Edu.” stands for the sub-sample with years of education higher
than 12 and “High School or Less” stands for those with years of education
less than or equal to 12. The standard error of estimates of coefficients
associated with control variables are estimated based on Theorem 3 and
reported in parentheses. $n$ is the sample size.

We treat the cross-section observations in the two sample periods,
$t=\left\\{1973-1975\right\\}$ and $\left\\{2001-2003\right\\}$, as repeated
cross-sections, rather than a panel data since the data in these two periods
do not cover the same individuals, and represent random samples from the
population of wage earners in two periods. It should also be noted that sample
sizes $(n_{t})$, although quite large, are much larger during
$\left\\{2001-2003\right\\}$, which could be a factor when we come to compare
estimates from the two sample periods. For example, for both male and female
$n_{73-75}=111,632$ as compared to $n_{01-03}=511,819$, a difference which
becomes more pronounced when we consider the number observations in
postsecondary/female category - which rises from $12,882$ for the first period
to $100,007$ in the second period.

We report estimates of $\pi_{t}$, $\beta_{L,t}$ and $\beta_{H,t}$, as well as
corresponding mean and standard deviations (denoted by
s.d.($\hat{\beta}_{it}$)) of the return to education ($\beta_{it}$) for
$t=\left\\{1973-1975\right\\}$ and $\left\\{2001-2003\right\\}$. For a given
$\pi_{t}$, the ratio $\beta_{H,t}/\beta_{L,t}$ provides a measure of within
group heterogeneity and allows us to augment information on changes in mean
with changes in the distribution of return of education. The estimates for the
distribution of the return to education ($\beta_{it}$) are summarized in Table
5, with the estimation results for control variables (such as experience,
experienced squared, and other individual specific characteristic) reported in
Table 6.

As can be seen from Table 5, estimates of
$\mathrm{s.d.}\left(\beta_{it}\right)$ are strictly positive for all sub-
groups, except for the “high school or less” group during the first sample
period. For this group during the first period the estimate of
$\mathrm{s.d.}\left(\beta_{it}\right)$ for the male sub-sample is zero, $\pi$
is not identified, and we have identical estimates for $\beta_{L}$ and
$\beta_{H}$. For this sub-sample, the associated estimates and their standard
errors are shown as unavailable ($n/a$). In case of the female sub-sample as
well as both male and female sub-samples where the estimates of
s.d.($\hat{\beta}_{it}$) are close to zero and $\pi$ is poorly estimated, only
the mean of the return to education is informative. In the case of the samples
where the estimates of $\mathrm{s.d.}\left(\beta_{it}\right)$ are strictly
positive, the estimate of the ratio $\beta_{H,t}/\beta_{L,t}$ provides a good
measure of within group heterogeneity of return to education. The estimates of
$\beta_{H,t}/\beta_{L,t}$, lie between $1.50$ to $2.79$, with the high
estimate obtained for the females with high school or less education during
$\left\\{{2001-03}\right\\}$, and the low estimate is obtained for females
with postsecondary education during the same period.

As our theory suggests the mean estimates of return to education,
$E\left(\beta_{it}\right)$, are very precisely estimated and inferences
involving them tend to be robust to conditional error heteroskedasticity. The
results in Table 5 show that estimates of $E\left(\beta_{it}\right)$ have
increased over the two sample periods $t=\left\\{1973-75\right\\}$ to
$t=\left\\{2001-03\right\\}$, regardless of gender or educational grouping.
The postsecondary educational group show larger increases in the estimates of
$E\left(\beta_{it}\right)$ as compared to those with high school or less.
Estimates of $\mathrm{E}\left(\beta_{it}\right)$ increases by $36$ per cent
for the postsecondary group while the estimates of mean return to education
rises only by around $5$ per cent in the case of those with high school or
less. This result holds for both genders. Comparing the mean returns across
the two educational groups, we find that mean return to education of
individuals with postsecondary education is $45$ per cent higher than those
with high school or less in the $\\{1973-75\\}$ period, but this gap increases
to $87$ per cent in the second period, $\left\\{2001-03\right\\}$. Similar
patterns are observed in the sub-samples by gender. The estimates suggest
rising between group heterogeneity, which is mainly due to the increasing
returns to education for the postsecondary group.

Turning to within group heterogeneity, we focus on the estimates of
$\beta_{H,t}/\beta_{L,t}$ and first note that over the two periods, within
group heterogeneity has been rising mainly in the case of those with high
school or less, for both male and female. For the combined male and female
samples and the male sub-sample, there is little evidence of within group
heterogeneity for the first period $\left\\{1973-75\right\\}$. However, for
the second period $\left\\{2001-03\right\\}$ we find a sizeable degree of
within group heterogeneity where $\beta_{H,t}/\beta_{L,t}$ is estimated to be
around $2.41$, with $\text{s.d.}\left(\beta_{it}\right)\approx 0.03$. For the
female sub-sample with high school or less, little evidence of heterogeneity
was found for the first period, estimates of $\beta_{H,t}/\beta_{L,t}$
increases to $2.79$ for the second sample period, that corresponds to a
commensurate rise in $\text{s.d.}\left(\beta_{i}\right)$ to $0.032$. The
pattern of within group heterogeneity is very different for those with
postsecondary educational. For this group we in fact observe a slight decline
in the estimates of $\beta_{H,t}/\beta_{L,t}$ by gender and over two sample
periods.

Overall, our between and within estimates of mean return to education are in
line with the evidence of rising wage inequality documented in the literature
(Corak, 2013).

##  7 Conclusion

In this paper we consider random coefficient models for repeated cross-
sections in which the random coefficients follow categorical distributions.
Identification is established using moments of the random coefficients in
terms of the moments of the underlying observations. We propose two-step
generalized method of moments to estimate the parameters of the categorical
distributions. The consistency and asymptotic normality of the GMM estimators
are established without the IID assumption typically assumed in the
literature. Small sample properties of the proposed estimator are investigated
by means of Monte Carlo experiments and shown to be robust to heterogeneously
generated regressors and errors, although relatively large samples are
required to estimate the parameters of the underling categorical
distributions. This is largely due to the highly non-linear mapping between
the parameters of the categorical distribution and the higher order moments of
the coefficients. This problem is likely to become more pronounced with a
larger number of categories and coefficients.

In the empirical application, we apply the model to study the evolution of
returns to education over two sub-periods, also considered in the literature
by Lemieux (2006b). Our estimates show that mean (ex post) returns to
education have risen over the periods from 1973 - 75 to 2001 - 2003 mainly in
the case of individuals with postsecondary education, and this result is
robust by gender. We find evidence of within group heterogeneity in the case
of high school or less educational group as compared to those with
postsecondary education.

In our model specification, the number of categories, $K$, is treated as a
tuning parameter and assumed to be known. An information criterion, as in
Bonhomme and Manresa (2015) and Su, Shi, and Phillips (2016), to determine $K$
could be considered. Further investigation of models with multiple regressors
subject to parameter heterogeneity is also required. These and other related
issues are topics for future research.

Appendix

##  Appendix A.1 Proofs

We include proofs and technical details in this section.

Proof of Theorem 1. Sum (2.6) over $i$ and rearrange terms,

$\left(\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(x_{i}^{r}\right)\right)\mathrm{E}\left(\beta_{i}^{r}\right)+\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(u_{i}^{r}\right)=\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\tilde{y}_{i}^{r}\right)-\sum_{q=2}^{r-1}\binom{r}{q}\left(\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(x_{i}^{r-q}\right)\mathrm{E}\left(u_{i}^{q}\right)\right)\mathrm{E}\left(\beta_{i}^{r-q}\right).$
(A.1.1)

Note that

$\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(x_{i}^{r-q}\right)\mathrm{E}\left(u_{i}^{q}\right)=\left(\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(x_{i}^{r-q}\right)\right)\sigma_{q}+\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(x_{i}^{r-q}\right)\left(\mathrm{E}\left(u_{i}^{q}\right)-\sigma_{q}\right),$

and

$\left|\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(x_{i}^{r-q}\right)\left(\mathrm{E}\left(u_{i}^{q}\right)-\sigma_{q}\right)\right|\leq\sup_{i}\left|\mathrm{E}\left(x_{i}^{r-q}\right)\right|\left|\frac{1}{n}\sum_{i=1}^{n}\left(\mathrm{E}\left(u_{i}^{q}\right)-\sigma_{q}\right)\right|=O(n^{-1/2}),$

by Assumption 1(b) and 2(b), then by taking $n\to\infty$ on both sides of
(A.1.1), we have (2.8). Similar steps for (2.7) give (2.9).

Proof of Theorem 2.

Let $m_{r}=\mathrm{E}\left(\beta_{i}^{r}\right)$, $r=1,2,\cdots,2K-1$, which
are taken as known. We show that

$m_{r}=\sum_{k=1}^{K}\pi_{k}b_{k}^{r},$ (A.1.2)

$r=0,1,2,\cdots,2K-1$, has a unique solution
$\mathbf{\theta}=\left(\mathbf{\pi}^{\prime},\mathbf{b}^{\prime}\right)^{\prime}$,
with $b_{1}<b_{2}<\cdots<b_{K}$ and $\pi_{k}\in\left(0,1\right)$ imposed.

Let

$q\left(\lambda\right)=\prod_{k=1}^{K}\left(\lambda-
b_{k}\right)=\lambda^{K}+\left(-1\right)^{1}b^{\ast}_{1}\lambda^{K-1}+\cdots+\left(-1\right)^{K}b^{\ast}_{K},$
(A.1.3)

be the polynomial with $K$ distinct roots $b_{1}$, $b_{2}$, $\cdots$, $b_{K}$.
Note that for each $k$, $\left(b_{k}^{r}\right)_{r=0}^{2K-1}$ satisfies the
linear homogeneous recurrence relation,

$b_{k}^{K+r}=b^{\ast}_{1}b_{k}^{K+r-1}+\left(-1\right)^{1}b^{\ast}_{2}b_{k}^{K+r-2}+\cdots+\left(-1\right)^{K-1}b^{\ast}_{K}b_{k}^{r},$
(A.1.4)

for $r=0,1,\cdots K-1$, since $q$ is the characteristic polynomial of the
linear recurrence relation (A.1.4) and $b_{k}$ is a root of $q$ (Rosen, 2006,
Chapter 5.2). $\left(m_{r}\right)_{r=0}^{2K-1}$ is a linear combination of
$\left(b_{1}^{r}\right)_{r=0}^{2K-1}$, $\left(b_{2}^{r}\right)_{r=0}^{2K-1}$,
$\cdots$, $\left(b_{K}^{r}\right)_{r=0}^{2K-1}$ by (A.1.2), then
$\left(m_{r}\right)_{r=0}^{2K-1}$ also satisfies the linear recurrence
relation (A.1.4), i.e.,

$m_{K+r}=b^{\ast}_{1}m_{K+r-1}+\left(-1\right)^{1}b^{\ast}_{2}m_{K+r-2}+\cdots+\left(-1\right)^{K-1}b^{\ast}_{K}m_{r},$
(A.1.5)

for $r=0,1,\cdots,K-1$. (A.1.5) is a linear system of $K$ equations in terms
of $\left(b^{\ast}_{k}\right)_{k=1}^{K}$. In matrix form,

$\mathbf{M}\mathbf{D}\mathbf{b}^{\ast}=\mathbf{m},$ (A.1.6)

where

$\mathbf{M}=\begin{pmatrix}1&m_{1}&\cdots&m_{K-1}\\\
m_{1}&m_{2}&\cdots&m_{K}\\\ \vdots&\vdots&\ddots&\vdots\\\
m_{K-1}&m_{K}&\cdots&m_{2K-2}\end{pmatrix},$

$\mathbf{D}=\mathrm{diag}\left(\left(-1\right)^{K-1},\left(-1\right)^{K-2},\cdots,1\right)$,
$\mathbf{b}^{\ast}=\left(b^{\ast}_{K},b^{\ast}_{K-1},\cdots,b^{\ast}_{1}\right)^{\prime}$,
and $\mathbf{m}=\left(m_{K},m_{K+1},\cdots,m_{2K-1}\right)^{\prime}$.

Denote
$\mathbf{\psi}_{k}=\left(1,b_{k},b_{k}^{2}\cdots,b_{k}^{K-1}\right)^{\prime}$
and
$\mathbf{\Psi}=\left(\mathbf{\psi}_{1},\mathbf{\psi}_{2},\cdots,\mathbf{\psi}_{K}\right)$.
Then

$\mathbf{M}_{k}=\begin{pmatrix}1&b_{k}&\cdots&b_{k}^{K-1}\\\
b_{k}&b_{k}^{2}&\cdots&b_{k}^{K}\\\ \vdots&\vdots&\ddots&\vdots\\\
b_{k}^{K-1}&b_{k}^{K}&\cdots&b_{k}^{2K-2}\end{pmatrix}=\mathbf{\psi}_{k}\mathbf{\psi}_{k}^{\prime},$

and
$\mathbf{M}=\sum_{k=1}^{K}\pi_{k}\mathbf{M}_{k}=\mathbf{\Psi}\mathrm{diag}\left(\mathbf{\pi}\right)\mathbf{\Psi}^{\prime}$.
Note that $\mathbf{\Psi}^{\prime}$ is a Vandermonde matrix then
$\det\left(\Psi\right)=\prod_{1\leq k<k^{\prime}\leq
K}\left(b_{k^{\prime}}-b_{k}\right)>0$ since $b_{1}<b_{2}<\cdots<b_{K}$.

$\displaystyle\det\left(\mathbf{MD}\right)$
$\displaystyle=\det\left(\mathbf{\Psi}\mathrm{diag}\left(\mathbf{\pi}\right)\mathbf{\Psi}^{\prime}\right)\det\left(\mathbf{D}\right)$
$\displaystyle=\left(\prod_{1\leq k<k^{\prime}\leq
K}\left(b_{k^{\prime}}-b_{k}\right)\right)^{2}\left(\prod_{k=1}^{K}\pi_{k}\right)\left(\left(-1\right)^{\frac{1}{2}K\left(K-1\right)}\right)\neq
0,$

since $\pi_{k}\in\left(0,1\right)$ for any $k$. Then we can identify
$\left(b^{\ast}_{k}\right)_{k=1}^{K}$ by $\left(m_{r}\right)_{r=0}^{2K-1}$ in
(A.1.6), and hence the characteristic polynomial is determined, and we can
identify $\left(b_{k}\right)_{k=1}^{K}$ by (A.1.3).

Since both $\left(b_{k}\right)_{k=1}^{K}$ and
$\left(m_{r}\right)_{r=1}^{2K-1}$ are identified, the first $K$ equations of
(A.1.2) is

$\mathbf{\Psi}^{\prime}\mathbf{\pi}=\left(1,m_{1},m_{2},\cdots,m_{K-1}\right)^{\prime},$

and $\mathbf{\pi}$ is identified by inverting the Vandermonde matrix
$\mathbf{\Psi}^{\prime}$, which completes the proof.

Proof of Theorem 4. Denote

$\Phi_{0}\left(\mathbf{\theta},\mathbf{\sigma},\mathbf{\gamma}\right)=\mathbf{g}_{0}\left(\mathbf{\theta},\mathbf{\sigma},\mathbf{\gamma}\right)^{\prime}\mathbf{A}\mathbf{g}_{0}\left(\mathbf{\theta},\mathbf{\sigma},\mathbf{\gamma}\right),$

where we stack the left-hand side of (3.7) and transform
$\mathbf{m}_{\beta}=h\left(\mathbf{\theta}\right)$ to get
$\mathbf{g}_{0}\left(\mathbf{\theta},\mathbf{\sigma},\mathbf{\gamma}\right)$.
We suppress and the argument $\hat{\gamma}$ and denote
$\mathbf{\eta}=\left(\mathbf{\theta}^{\prime},\mathbf{\sigma}^{\prime}\right)^{\prime}$
for notation simplicity and proceed by verifying the conditions of Newey and
McFadden (1994, Theorem 2.1). Theorem 2 provides the identification results
which together with the positive definiteness of $\mathbf{A}$ verifies that
$\Phi_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)$ is uniquely minimized to
0 at $\mathbf{\eta}_{0}$. The compactness of the parameter space holds by
Assumption 4(a). Note that
$\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)$ is a polynomial in
$\mathbf{\eta}$, which is continuous in $\mathbf{\eta}$.
$\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)$ is bounded on
$\Theta\times\mathcal{S}$. We proceed by verify the uniform convergence
condition. The additive terms in
$\hat{\mathbf{g}}_{n}\left(\mathbf{\eta},\hat{\mathbf{\gamma}}\right)-\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)$
are of the form $H_{n,1}h^{\left(r,q\right)}\left(\mathbf{\eta}\right)$ or
$H_{n,2}$, where

$\displaystyle\left|H_{n,1}\right|$
$\displaystyle=\left|\frac{1}{n}\sum_{i=1}^{n}x_{i}^{r-q+s_{r}}-\rho_{0,r-q+s_{r}}\right|$
$\displaystyle\leq\left|\frac{1}{n}\sum_{i=1}^{n}x_{i}^{r-q+s_{r}}-\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(x_{i}^{r-q+s_{r}}\right)\right|+\left|\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(x_{i}^{r-q+s_{r}}\right)-\rho_{0,r-q+s_{r}}\right|$
$\displaystyle=O_{p}\left(n^{-1/2}\right),$

$h^{\left(r,q\\}\right)}\left(\mathbf{\eta}\right)$ is a polynomial in
$\mathbf{\eta}$, and

$\displaystyle\left|H_{n,2}\right|$
$\displaystyle=\left|\frac{1}{n}\sum_{i=1}^{n}\hat{\tilde{y}}_{i}^{r}x_{i}^{s_{r}}-\rho_{r,s_{r}}\right|$
$\displaystyle\leq\left|\frac{1}{n}\sum_{i=1}^{n}\hat{\tilde{y}}_{i}^{r}x_{i}^{s_{r}}-\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\tilde{y}_{i}^{r}x_{i}^{s_{r}}\right)\right|+\left|\frac{1}{n}\sum_{i=1}^{n}\mathrm{E}\left(\tilde{y}_{i}^{r}x_{i}^{s_{r}}\right)-\rho_{r,s_{r}}\right|$
$\displaystyle=O_{p}\left(n^{-1/2}\right).$

$H_{n,1}=O_{p}\left(n^{-1/2}\right)$ and $H_{n,2}=O_{p}\left(n^{-1/2}\right)$
are due to Assumption 2(a) and 4(c).

By the compactness of $\Theta\times\mathcal{S}$,
$\sup_{\mathbf{\eta}\in\Theta\times\mathcal{S}}h^{\left(r,q\right)}\left(\mathbf{\eta}\right)<C<\infty$
for some positive constant $C$. By triangle inequality, we have

$\sup_{\mathbf{\eta}\in\Theta\times\mathcal{S}}\left\|\hat{\mathbf{g}}_{n}\left(\mathbf{\eta},\hat{\mathbf{\gamma}}\right)-\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)\right\|\rightarrow_{p}0,$
(A.1.7)

as $n\rightarrow\infty$. Following the proof of Newey and McFadden (1994,
Theorem 2.1),

$\displaystyle\left|\hat{\Phi}_{n}\left(\mathbf{\eta},\hat{\mathbf{\
\gamma}}\right)-\Phi_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)\right|$
$\displaystyle\leq$
$\displaystyle\left|\left[\hat{\mathbf{g}}_{n}\left(\mathbf{\eta},\hat{\mathbf{\gamma}}\right)-\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\
\gamma}\right)\right]^{\prime}\mathbf{A}_{n}\left[\hat{\mathbf{g}}_{n}\left(\mathbf{\eta},\hat{\mathbf{\gamma}}\right)-\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)\right]\right|+\left|\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)^{\prime}\left(\mathbf{A}_{n}+\mathbf{A}_{n}^{\prime}\right)\left[\hat{\mathbf{g}}_{n}\left(\mathbf{\eta},\hat{\mathbf{\gamma}}\right)-\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)\right]\right|$
$\displaystyle\quad\quad+\left|\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\
\gamma}\right)^{\prime}\left(\mathbf{A}_{n}-\mathbf{A}\right)\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)\right|$
$\displaystyle\leq$
$\displaystyle\left\|\hat{\mathbf{g}}_{n}\left(\mathbf{\eta},\hat{\mathbf{\
\gamma}}\right)-\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)\right\|^{2}\left\|\mathbf{A}_{n}\right\|+2\left\|\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)\right\|\left\|\hat{\mathbf{g}}_{n}\left(\mathbf{\eta},\hat{\mathbf{\gamma}}\right)-\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\
\gamma}\right)\right\|\left\|\mathbf{A}_{n}\right\|+\left\|\mathbf{g}_{0}\left(\mathbf{\eta},\mathbf{\gamma}\right)\right\|^{2}\left\|\mathbf{A}_{n}-\mathbf{A}\right\|.$

By (A.1.7) and the boundedness of $\mathbf{g}_{0}$,
$\sup_{\mathbf{\eta}\in\eta}\left|\hat{\Phi}_{n}\left(\mathbf{\
\eta},\hat{\mathbf{\gamma}}\right)-\Phi_{n}\left(\mathbf{\
\eta},\mathbf{\gamma}\right)\right|\rightarrow_{p}0$, which completes the
proof.

Proof of Theorem 5. We denote
$\mathbf{\eta}=\left(\mathbf{\theta}^{\prime},\mathbf{\sigma}^{\prime}\right)^{\prime}$
for notation simplicity. The first-order condition,
$\nabla_{\mathbf{\eta}}\hat{\mathbf{g}}_{n}\left(\hat{\mathbf{\eta}},\hat{\mathbf{\gamma}}\right)\mathbf{A}_{n}\hat{\mathbf{g}}_{n}\left(\hat{\mathbf{\eta}},\hat{\mathbf{\gamma}}\right)=\mathbf{0}$,
holds with probability 1. Denote
$\hat{\mathbf{G}}\left(\mathbf{\eta},\mathbf{\gamma}\right)=\nabla_{\mathbf{\eta}}\hat{\mathbf{g}}_{n}\left(\mathbf{\eta},\mathbf{\gamma}\right)$
and expand
$\hat{\mathbf{g}}_{n}\left(\mathbf{\hat{\eta}},\hat{\mathbf{\gamma}}\right)$
in the first-order condition around $\mathbf{\eta}_{0}$, we have

$\displaystyle\sqrt{n}\left(\hat{\mathbf{\eta}}-\mathbf{\eta}_{0}\right)$
$\displaystyle=-\left[\hat{\mathbf{G}}\left(\hat{\mathbf{\eta}},\hat{\mathbf{\gamma}}\right)^{\prime}\mathbf{A}_{n}\hat{\mathbf{G}}\left(\bar{\mathbf{\eta}},\hat{\mathbf{\gamma}}\right)\right]^{-1}\hat{\mathbf{G}}\left(\hat{\mathbf{\eta}},\hat{\mathbf{\
\gamma}}\right)^{\prime}\mathbf{A}_{n}\left(\sqrt{n}\hat{\mathbf{g}}_{n}\left(\mathbf{\eta}_{0},\hat{\mathbf{\gamma}}\right)\right)$
$\displaystyle=-\left[\hat{\mathbf{G}}\left(\hat{\mathbf{\eta}},\hat{\mathbf{\gamma}}\right)^{\prime}\mathbf{A}_{n}\hat{\mathbf{G}}\left(\bar{\mathbf{\eta}},\hat{\mathbf{\gamma}}\right)\right]^{-1}\hat{\mathbf{G}}\left(\hat{\mathbf{\eta}},\hat{\mathbf{\
\gamma}}\right)^{\prime}\mathbf{A}_{n}\left[\sqrt{n}\hat{\mathbf{g}}_{n}\left(\mathbf{\eta}_{0},\mathbf{\gamma}_{0}\right)+\nabla_{\gamma}\hat{\mathbf{g}}_{n}\left(\mathbf{\eta}_{0},\bar{\mathbf{\gamma}}\right)\sqrt{n}\left(\hat{\mathbf{\gamma}}-\mathbf{\gamma}_{0}\right)\right],$

where $\bar{\mathbf{\eta}}$ and $\bar{\mathbf{\gamma}}$ are between
$\hat{\mathbf{\eta}}$ and $\mathbf{\eta}_{0}$; and $\hat{\mathbf{\gamma}}$ and
$\mathbf{\gamma}_{0}$, respectively. Note that by term-by-term convergence, we
have
$\hat{\mathbf{G}}\left(\hat{\mathbf{\eta}},\hat{\mathbf{\gamma}}\right),\hat{\mathbf{G}}\left(\bar{\mathbf{\eta}},\hat{\mathbf{\gamma}}\right)\rightarrow_{p}\mathbf{G}_{0}$
and
$\nabla_{\mathbf{\gamma}}\hat{\mathbf{g}}_{n}\left(\mathbf{\eta}_{0},\bar{\mathbf{\gamma}}\right)\rightarrow_{p}\nabla_{\mathbf{\gamma}}\mathbf{g}_{0}\left(\mathbf{\eta}_{0,}\mathbf{\gamma}_{0}\right)=\mathbf{G}_{0,\gamma}$.
By Assumption 4(b), $\mathbf{A}_{n}\rightarrow_{p}\mathbf{A}$. By Assumption
5(a) and (b) and Slutsky theorem,

$\sqrt{n}\left(\hat{\mathbf{\eta}}-\mathbf{\eta}_{0}\right)\rightarrow_{d}\left(\mathbf{G}_{0}^{\prime}\mathbf{A}\mathbf{G}_{0}\right)^{-1}\mathbf{G}_{0}^{\prime}\mathbf{A}\left(\mathbf{\zeta}+\mathbf{G}_{0,\gamma}\mathbf{\zeta}_{\gamma}\right),$

which completes the proof.

Further details for Example 4. We need to verify the invertibility of the
matrix

$\mathbf{B}=\begin{pmatrix}1&1&0&0\\\ 0&0&1&1\\\ 1&0&1&0\\\
b_{1L}b_{2L}&b_{1L}b_{2H}&b_{1H}b_{2L}&b_{1H}b_{2H}\end{pmatrix}.$

The span of first three rows of $\mathbf{B}$ is

$\mathcal{S}=\left\\{\left(\alpha_{1}+\alpha_{3},\alpha_{1},\alpha_{2}+\alpha_{3},\alpha_{3}\right)^{\prime}:\alpha_{1},\alpha_{2},\alpha_{3}\in\mathbb{R}\right\\}.$

$\left(b_{1L}b_{2L},b_{1L}b_{2H},b_{1H}b_{2L},b_{1H}b_{2H}\right)^{\prime}\notin\mathcal{S}$
is equivalent to $b_{1H}b_{2H}-b_{1H}b_{2L}\neq b_{1L}b_{2H}-b_{1L}b_{2L}$.
This can be verified by

$\left(b_{1H}b_{2H}-b_{1H}b_{2L}\right)-\left(b_{1L}b_{2H}-b_{1L}b_{2L}\right)=\left(b_{1H}-b_{1L}\right)\left(b_{2H}-b_{2L}\right)>0,$

given that $b_{1L}<b_{1H}$ and $b_{2L}<b_{2H}$ hold.

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Online Supplement to

Identification and Estimation of Categorical Random Coefficient Models

by

Zhan Gao and M. Hashem Pesaran

February 2023

##  Appendix S.1 Introduction

This online supplement is composed of four sections. Section S.2 provides
additional proofs and technical details omitted from the main text. Section
S.3 provides additional simulation results. Section S.4 gives additional
empirical results. Details of the computational algorithm used are described
in Section S.5.

##  Appendix S.2 Proofs

We include omitted proofs and technical details in this section.

Proof of Theorem 3. From (3.1), we have

$\frac{1}{n}\sum_{i=1}^{n}\mathbf{w}_{i}y_{i}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{w}_{i}\mathbf{w}_{i}^{\prime}\mathbf{\phi}+\frac{1}{n}\sum_{i=1}^{n}\mathbf{w}_{i}\xi_{i},$

where
$\mathbf{\phi}=\mathrm{E}\left(\mathbf{\phi}_{i}\right)=\left(\mathrm{E}\left(\beta_{i}\right),\mathbf{\gamma}^{\prime}\right)^{\prime},$
and $\xi_{i}=u_{i}+x_{i}v_{i}$, which can be written equivalently as

$\mathbf{q}_{n,wy}=\mathbf{Q}_{n,ww}\mathbf{\phi}+\frac{1}{n}\sum_{i=1}^{n}\mathbf{w}_{i}\xi_{i}.$

Taking expectations of both sides and rearrange terms, we have

$\mathbf{\phi}=\mathrm{E}\left(\mathbf{Q}_{n,ww}\right)^{-1}\mathrm{E}\left(\mathbf{q}_{n,wy}\right).$

Consider

$\displaystyle\mathbf{\hat{\phi}}-\mathbf{\phi}$