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Heterogeneous Grouping Structures in Panel Data
\title{Heterogeneous Grouping Structures in Panel Data}
\author[1]{Katerina Chrysikou
\thanks{ {[email removed]}} }
\author[1]{George Kapetanios
\thanks{{[email removed]}. }}
\affil[1]{ King's Business School , King's College London}
\date{{ \today}}
\maketitle
\begin{abstract}
In this paper we examine the existence of heterogeneity within a group, in
panels with latent grouping structure. The assumption of within group
homogeneity is prevalent in this literature, implying that the formation of
groups alleviates cross-sectional heterogeneity, regardless of the prior
knowledge of groups. While the latter hypothesis makes inference powerful, it
can be often restrictive. We allow for models with richer heterogeneity that
can be found both in the cross-section and within a group, without imposing
the simple assumption that all groups must be heterogeneous. We further
contribute to the method proposed by \cite{su2016identifying}, by showing that
the model parameters can be consistently estimated and the groups, while
unknown, can be identifiable in the presence of different types of
heterogeneity. Within the same framework we consider the validity of assuming
both cross-sectional and within group homogeneity,
using testing procedures. Simulations demonstrate good finite-sample
performance of the approach in both classification and estimation, while
empirical applications across several datasets provide evidence of multiple
clusters, as well as reject the hypothesis of within group homogeneity.
\newline\noindent{{Keywords}}: Panel Data Models, Latent
Grouping Structures, Heterogeneity tests. \newline
\noindent{JEL Classifications}: C18, C23, C38, C51, C55.
\end{abstract}
\affil{King's Business School , King's College London}
\section{Introduction}
\label{introduction}
The use and analysis of panel datasets and associated models is a vital
research topic in both econometric theory and applications; these models are
frequently applied in many disciplines across economics and other social
sciences. This type of dataset allows the consideration of both
cross-sectional and time dimensions, and provides a rich source of empirical
information. In its majority, panel data analysis relies heavily on the
assumption of homogeneity across units making inference more powerful.
Nevertheless, such an assumption is often restrictive and frequently rejected;
see, among others, \cite{chudik2015common}, \cite{phillips2007transition},
\cite{pesaran2006estimation}, and \cite{browning2014dynamic}. Ignoring
heterogeneity, across units, can result in biased estimation and misleading
inference; see, among others, Chapter 1 in \cite{baltagi2008econometric},
\cite{moulton1986random}, and \cite{moulton1987diagnostics}. On the other
hand, allowing complete heterogeneity leads to potentially inefficient panel
data estimation and inference. To address this issue, researchers often
consider intermediate ways of exploring panel data such as panel structure models.
Panel structure models rely on the hypothesis that all, or a subset of,
parameters are heterogeneous across groups but homogeneous within a group
where neither the number of groups nor the individuals' group-membership is
known. Examples of these models can be found in \cite{phillips2007transition}
who accommodate the hypothesis of convergence of clubs; in this case different
countries belong to different groups and behave differently with the group
structures being latent rather than assumed to be observable.
The identification of latent structures is not an easy task. It is a
computationally expensive procedure and it can be even infeasible to try all
the possible permutations of units across groups. A part of the literature
considers that units in the dataset are grouped using an external observable
classification; see, e.g., \cite{BESTER2016197}. This approach may suffer, as
suitable external variables can be difficult to find in many empirical
settings. Furthermore, a wrong choice for the external classifier may again
lead to misleading inference. For this reason, it can be argued that it is
more useful for the researcher to use data-driven classification procedures
such as \emph{k-means}, see, e.g., \cite{sarafidis2015partially},
\cite{bonhomme2015grouped}, \cite{lin2012estimation}, and \cite{ando2016panel}
; these studies consider \emph{k-means} in linear panel data models, with an unknown
grouping structure. Although unsupervised classification is an appealing
approach and has been theoretically proven to be asymptotically consistent,
see, e.g., \cite{pollard1981strong}, alternative approaches based on the
\emph{Classifier Lasso} (\textsc{c-lasso}), proposed by
\cite{su2016identifying} (SSP) and extended by \cite{su2018identifying} and
\cite{su2019sieve}, have been recently introduced in the literature. In
particular, SSP propose a novel estimation technique which involves
penalisation and simultaneously classifies the data and estimates the
underlying parameters consistently. \cite{meh2022} provides a framework for
joint estimation and identification of latent grouping structures in panel data
models using a pairwise fusion penalized approach. \cite{mammen2022estimation}
presents a new approach to grouping fixed effects in a linear panel model to
reduce their dimensionality and ensure identifiability, by using unsupervised
non-parametric density based clustering, cluster patterns including their
location and number are not restricted.
A different line of work is that of \cite{ren2022matrix} who propose a group
matrix network autoregression (GMNAR) model, which assumes that the subjects
in the same group share the same set of model parameters. In a similar vein,
\cite{zhu2022simultaneous} study dynamic behaviours of heterogeneous
individuals observed in a network, where the dynamic patterns are
characterised by a network vector autoregression model with a latent grouping
structure, where group-wise network effects and time-invariant fixed-effects
can be incorporated. \cite{wu2022} develop methods to recover sparsity
patterns and grouping structures in large panels with both individual and time effects.
In a low dimensional setting, where $T>>N,p$, \cite{wanga2022panel} generalise
their framework and consider panel model with interactive fixed effects such
that individual heterogeneity is captured by latent grouping structure and time
heterogeneity is captured by an unknown structural break. \cite{liu2022panel}
propose a methodology for identifying and estimating explosive bubbles in
mixed-root panel autoregressions with a latent grouping structure.
Full homogeneity placed within groups, is a very common assumption in the literature, but this can be a potentially restrictive hypothesis. It is conceivable and, as we will argue, often the
case, in practice, that each group contains units with heterogeneous
parameters whose mean is group specific and differs across groups.
To fix ideas let the parameter of interest across $N$ panel units be
$\boldsymbol{\beta}_{i} = (\boldsymbol{\beta}_{1,i}, \boldsymbol{\beta}_{2,i},
\ldots, \boldsymbol{\beta}_{p, i})^{\prime}$, where $p$ is the number of
covariates within each unit $i=1,\ldots,N$. Traditionally, under full homogeneity, $\boldsymbol{\beta
}_{i}=\boldsymbol{\beta}$. This is full homogeneity. A heterogeneous extension
is $\boldsymbol{\beta}_{i}=\boldsymbol{\beta}+\boldsymbol{\eta}_{i}$, where
$\boldsymbol{\eta}_{i}$ is an i.i.d. process and focus is placed on estimation
of and inference for $\boldsymbol{\beta}$. Grouping structures specify that
$\boldsymbol{\beta}_{i}=\boldsymbol{\alpha}_{k}$, whereby unit $i$ belongs to
group $k$, where $k=1,\ldots,K$ and $K$ is usually assumed finite. Focus here is placed
on $\boldsymbol{\alpha}_{k}$. Our setting extends this by specifying that
$\boldsymbol{\beta}_{i}=\boldsymbol{\alpha}_{k}+\boldsymbol{\eta}_{i}.$
Clearly $\boldsymbol{\eta}_{i}=\boldsymbol{0}_{p\times1}$ and
$\boldsymbol{\alpha}_{k}=\boldsymbol{\alpha}$, are both assumptions that are
empirically verifiable and potentially invalid. The case where neither holds
has not been explored in the literature and is the main focus, and contribution, of this paper.
Following the work of SSP, we extend the Classifier-Lasso framework to the
above case and further propose an estimation technique based on the
\emph{k-means} clustering method allowing for within-group heterogeneity.
Our focus is mainly to showcase consistency of the estimation of
$\boldsymbol{\beta}_{i}$ and $\boldsymbol{\alpha}_{k}$ providing both theoretical and small sample evidence, rather than the identification of an exact
grouping structure, which under our form of heterogeneity is obviously not
consistently possible. We suggest the use of a simple unsupervised classifier,
like the \emph{k-means}, to determine the grouping structure and, once
determined, we estimate the unit-specific parameters with penalised methods.
Our work extends SSP and contributes to the literature as follows: first, we
allow for a \emph{less} restrictive model, second we allow for richer
cross-sectional heterogeneity, and finally, we provide a data-driven approach
to classify each cross-section. While the latter is a gain of our modelling
approach, heterogeneity is still restrictive and, in certain cases,
contradicts the basic principles of \emph{k-means} clustering. In
particular, when the level of heterogeneity is high, causing the data to be
more noisy cross-sectionally, the clustering process tends to be misleading
since it violates one of the three basic assumptions of \emph{k-means}; that
is, the variance between individuals must be constant and small. Because
\emph{k-means} minimises the within cluster deviation (in terms of squared
Euclidean distances), the optimisation method converges to local minima when
the variance between individuals (cross-sections) is large.
The latter is inconvenient when the empirical data is noisy or even
correlated, but there are other unsupervised classifiers which deal with these
types of cases; see, e.g., \cite{leisch1999bagged} who re-estimates the data
in different partitions using bootstrapping. In this paper, we do not focus
on such a task since our approach does not cover cases where data exhibit
autocorrelation or endogeneity. Obviously, an alternative to our approach
would be the construction of factors instead of groups (clusters). This would
lower the dimension of the data by choosing $K$ factors which explain
considerably the level of heterogeneity in the cross-sections; see, e.g.,
\cite{su2018identifying} for more details.
A further and important contribution of our work is a set of testing procedures, exploring the validity of $\boldsymbol\eta_{i}=\boldsymbol0$
and $\boldsymbol\alpha_{k}=\boldsymbol\alpha$, using testing procedures in the former case. We use
our methods in a variety of empirical settings to explore the validity of the
restrictions and find that in most cases both are not valid, further
showcasing the utility of our framework.
The rest of the paper is organised as follows. Sections \ref{sec1} and
\ref{FeasibleEstimation} outline our theoretical considerations. In Section \ref{MC} we
present extensive Monte Carlo simulations and discussion, Section
\ref{Empirics} presents several empirical examples in support of our method.
We conclude in Section \ref{Discussion}. Technical proofs, of the main (auxiliary) results, and descriptions of the empirical datasets are relegated to the Appendix.
\subsubsection*{Notation}
\noindent\label{notation} For any vector $\boldsymbol{x}\in\mathbb{R}^{n}, $ we denote
the $\ell_{p}$-, and $\ell_{\infty}$ as $\left\| \boldsymbol{x} \right\|
_{p}=\left( \sum_{i=1}^{n} | x_{i} |^{p}\right) ^{1/p}$, $\lVert
\boldsymbol{x} \rVert_{\infty} = \max_{i=1,\ldots,n}|x_{i}|$. Further, $\boldsymbol{1}\{\cdot\}$ denotes an indicator function. We use "$\to_{P}$" to denote
convergence in probability. For two deterministic sequences $a_{n}$ and
$b_{n}$ we define asymptotic proportionality, "$\asymp$", by writing $a_{n}
\asymp b_{n}$ if there exist constants $0 < a_{1} \leq a_{2}$ such that
$a_{1}b_{n} \leq a_{n} \leq a_{2}b_{n}$ for all $n \geq1$. For any set $A$,
$|A|$ denotes its cardinality, while $A^{c}$ denotes its complement. Define $\boldsymbol{A}$, a $n\times m $ matrix, we denote $\Lambda_{\min}(\boldsymbol{A})$ as the smallest eigenvalue of $\boldsymbol{ {A}}$, and $\Lambda_{\max}(\boldsymbol{A})$ as the largest eigenvalue of $\boldsymbol{ {A}}$.
\section{Theoretical considerations}
\label{sec1}
We consider the following balanced panel model:
\begin{align}
y_{it}^{\ast} & =\mu_{i}^{0}+\boldsymbol{\beta}_{i}^{0\prime}\boldsymbol{x}
_{it}^{\ast}+\varepsilon_{it}^{\ast},\\
\boldsymbol{\beta}_{i}^{0} & =\sum_{k=1}^{K}\boldsymbol{\alpha}_{k}
^{0}\boldsymbol{1}\{i\in\boldsymbol{G}_{k}^{0}\}+\boldsymbol{\eta}_{i},\quad
i=1,\ldots,N,\quad t=1,\ldots T \label{model}
\end{align}
where $\mu_{i}^{0}$ is an individual fixed effect, $\boldsymbol{x}_{it}^{\ast
}=\left( {x}_{1,it}^{\ast},\ldots,{x}_{p,it}^{\ast}\right) ^{\prime}$is a
$(p\times1)$ vector of covariates, with $E(x_{l,it}^{\ast}x_{j,it}^{\ast})=0$,
$\varepsilon_{it}^{\ast}=(\varepsilon_{1t},\ldots,\varepsilon_{Nt}^{\ast})$ is
the idiosyncratic error, $\boldsymbol{\alpha}^{0}=(\boldsymbol{\alpha}_{1}
^{0},\ldots,\boldsymbol{\alpha}_{K}^{0})^{\prime}$ a $K\times p$ matrix of
group specific centres for each covariate $j$, where $K$ denotes the number of
groups considered. $\boldsymbol{G}_{k}^{0}$, $k=1,\ldots,K$, are sets of indices,
denoting groups of units. Note that in our analysis, both $N$ and $T$ can be
large and $N$ can possibly grow faster than $T$. Further, $\boldsymbol{\alpha
}_{j}^{0}\neq\boldsymbol{\alpha}_{k}^{0}$ for any $j\neq k,$ $\boldsymbol{G}
_{k}^{0}\subset\{1,2,\ldots,N\}$, $\cup_{k=1}^{K}\boldsymbol{G}_{k}
^{0}=\{1,2,\ldots,N\}$, and $\boldsymbol{G}_{k}^{0}\cap\boldsymbol{G}_{j}
^{0}=\varnothing$ for any $j\neq k$, $j=1,\ldots,K$. We denote the cardinality
of the set $\boldsymbol{G}_{k}$ as $N_{k}=|\boldsymbol{G}_{k}^{0}|$. The slope
heterogeneity, $\boldsymbol{\eta}_{i}=(\eta_{i,1},\ldots,\eta_{i,p})^{\prime}$
is considered to be unknown and allowed to be group specific, along with the
cluster membership. Intuitively, $K$ corresponds to the number of groups and
units (indexed by $i$) within the same group share a common slope
$\boldsymbol{\alpha}_{k}$ but are allowed to differ by $\boldsymbol{\eta}
_{i},i\in\boldsymbol{G}_{k},k=1,\ldots,K$. Further the grouping membership is
common across different covariates.
Often in the panel data literature, a common characteristic can be proxied by
factors (or interactive effects), e.g., as in \cite{bai2009panel}, because of
their ability to effectively summarise information in large data sets. While
existing work suggests that the grouping of different units in a panel model can
be effective under cross-sectional dependence and heteroscedasticity where
$N<T$, see, e.g., \cite{bai2020standard}, and can even accommodate time
variation, see e.g. \cite{su2019sieve}. In this paper, we allow for a degree
of within-group heterogeneity, implying that grouping does not account for the
entirety of cross-sectional heterogeneity. While we are agnostic about the
cross-sectional and within-group idiosyncratic effect, we do assume an
\emph{a-priori} knowledge of the existence of grouping.
For the linear model in \eqref{model} with ${E}\left( \varepsilon
_{it}^{\ast}\mid\boldsymbol{x}_{it}^{\ast},\mu_{i}^{0}\right) =0$, we have
$\widehat{\mu}_{i}=\bar{y}_{i.}^{\ast}-\boldsymbol{\beta}_{i}^{\prime}
\bar{\boldsymbol{x}}_{i}^{\ast}$, where $\bar{y}_{i.}^{\ast}=\frac{1}{T}
\sum_{t=1}^{T}y_{it}^{\ast},{y}_{it}=y_{it}^{\ast}-\bar{y}_{i}^{\ast}$, and
${\bar{\boldsymbol{x}}}_{i}$, $\boldsymbol{x}_{it}$, $\bar{\varepsilon}_{it}$,
and $\varepsilon_{it}$ are analogously defined. Then \eqref{model} becomes
\begin{align}
y_{it}=\boldsymbol{\beta}_{i}^{0}\boldsymbol{x}_{it}+\varepsilon_{it},\quad
i=1,\ldots,N,\quad t=1,\ldots T\label{mod}
\end{align}
Following \cite{su2016identifying} and motivated by the literature on fused
Lasso e.g. \cite{tibshirani2005sparsity}, we minimise the following penalised
profile likelihood (PPL) to estimate the parameters of interest.
\begin{equation}
Q_{iNT,\lambda}\left( \boldsymbol{{\beta}},\boldsymbol{{\alpha}}\right)
=\frac{1}{NT}\sum_{i=1}^{N}\sum_{i=1}^{T}\frac{1}{2}\left( y_{it}
-\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{x}_{it}\right) ^{2}+\frac
{\lambda}{N}\sum_{i}\prod_{k=1}^{K}\left\Vert \boldsymbol{\beta}
_{i}-\boldsymbol{\alpha}_{k}\right\Vert _{2}, \label{minimisation}
\end{equation}
where $\lambda>0$ is the regularisation parameter and $\boldsymbol{{\beta
=(\boldsymbol{\beta}}}_{1},\ldots,\boldsymbol{{\beta}}_{N}\boldsymbol{{)}
}^{\prime}$. Then, \eqref{minimisation} is minimised for some value of
$(\lambda,K)$ and produces $(\boldsymbol{\widehat{\beta}}
,\boldsymbol{\widehat{\alpha}})^{\prime}$. We generalise
\cite{su2016identifying} (\textsc{c-lasso} hereafter) by allowing for heterogeneity within groups.
We show that both the group and the unit-specific parameter can be
consistently estimated even when the underlying generating process is
\eqref{mod}. We obtain estimates of the (unit)group-specific parameters by
using either the \textsc{c-lasso} iterative procedure or \emph{k-means}
estimation, see e.g. \cite{macqueen1967}, \cite{pollard1981strong}.
The penalty term in \eqref{minimisation} takes a mixed additive-multiplicative
form, introduced in \cite{su2016identifying}. In the traditional Lasso
literature, e.g. \cite{tibshirani1996regression}, the penalty term is
additive, and imposes a degree of sparsity in favour of a parsimonious model
at the cost of a degree of bias, whereas in this setting, parsimony results
from grouping. The multiplicative nature of the penalty term in
\eqref{minimisation} is essential as it produces $N$ additive terms for each
of the $K$ separate penalties, such that $\{i=1\ldots,N,\;k=1,\ldots
,K:\;i\in\boldsymbol{G}_{k},\boldsymbol{\beta}_{i}^{0}\in
\boldsymbol{\mathcal{B}}_{k}\subseteq\boldsymbol{\mathcal{B}}\}$, where, for
each group $k$, $\boldsymbol{\mathcal{B}}_{k}\subseteq\boldsymbol{\mathcal{B}
}$ and $\boldsymbol{{\beta}}^{0}\in\boldsymbol{\mathcal{B}}\subseteq
\mathbb{R}^{N\times p}$. The parameters are assumed to exhibit a certain
grouped pattern, the number of unknown slope parameters in
$\{\boldsymbol{{\beta}}_{i}\}$ is of order $O(K)$, instead of $O(N)$, where,
$K(<N)$ is typically a fixed constant in empirical applications.
Contrary to \textsc{c-lasso}, we do not impose homogeneity within any of the
groups $\boldsymbol{\mathcal{B}}_{k}$, hence the unit-specific
$\boldsymbol{{\beta}}_{i}^{0}$ are not penalised down to a specific
group-centre $\boldsymbol{{\alpha}}^{0}_{k}$, simply because
$\boldsymbol{{\beta}}_{i}^{0}\neq\boldsymbol{{\alpha}}^{0}_{k}$, which is more
indicative of a {fused-Lasso} penalty. The component of summation is essential
in order to extract information from the $N$ cross-sectional units, resulting
in the identification of both $\{ \boldsymbol{{\beta}}_{i}^{0} \}$ and $\{
\boldsymbol{{\alpha}}^{0}_{k} \}$. The grouping, while fixed, is not known a-priori.
We present the necessary assumptions that ensure consistency of the proposed estimator:
\begin{assumption}
\label{AssonX}
\begin{enumerate}[nolistsep]
\item $\left\{ \boldsymbol{x}_{it}\right\} $ is a $p$-dimensional ergodic
sequence of r.v.'s such that $E\left( \boldsymbol{x}_{it}|\mathcal{F}
_{t-1}\right) =0$ a.s., $\forall\;i=1,\ldots,N$, $\sup_{i,t}E\left(
{x}_{j,it}^{2}|\mathcal{F}_{t-1}\right) =\sigma^{2}>0$ $\,$
$\sup_{i}\sup_{t}E\left( |{x}_{j,it}|^{2+\nu}\right) <\infty$, for some $\nu>0$ and $j=1,\ldots, p$, where
$\mathcal{F}_{t-1}=\left\{ {x}_{j, it-1}, \ldots, {x}_{j, i1}\right\}$ is the information set at time $t-1$. \label{Xt}
\item $\boldsymbol{\Sigma}_{\boldsymbol{x}}=\mathbb{E}\left[\boldsymbol{x}_{it} \boldsymbol{x}_{it}^{\prime} \mid \mathcal{F}_{t-1}\right] $, we assume that $ \boldsymbol{\Sigma}_{\boldsymbol{x}}=\operatorname{plim}_{T,N \rightarrow \infty} \frac{1}{NT} \sum_{i=1}^N \sum_{t=1}^T\boldsymbol{x}_{it}\boldsymbol{x}_{it}'$ exists and is positive definite, such that $\Lambda_{\min}(\boldsymbol{\Sigma}_{\boldsymbol{x}})>m, \; m>0 $, where $\widehat{\boldsymbol{\Sigma}}_{\boldsymbol{x}}=\frac{1}{NT} \sum_{i=1}^N \sum_{t=1}^T\boldsymbol{x}_{it}\boldsymbol{x}_{it}'$ is the sample covariance matrix, and $\Lambda_{\min}(\boldsymbol{\Sigma}_{\boldsymbol{x}})$ the smallest eigenvalue of $\boldsymbol{\Sigma}_{\boldsymbol{x}}$.\label{posdef}
\item $\left\{ \varepsilon_{it}\right\} $ is an ergodic sequence of r.v.'s
such that
$E\left(\varepsilon_{i t}\right)=0, E\left(\varepsilon_{i t}^2\right)=\sigma_{\varepsilon_i}^2$ and $E\left(\varepsilon_{i t}^{2+\nu}\right)<\infty$ for $i=1, \ldots, N ; t=1, \ldots, T$ and for some $\nu>0$. Let $\mathcal{F}_{-\varepsilon_{i t}}$ be the $\sigma$-field of all stochastic elements in the panel data model, apart from $\varepsilon_{i t}$. Then, $E\left(\varepsilon_{i t} \mid \mathcal{F}_{-\varepsilon_{i t}}\right)=0$.
\label{et}
\item $\{\varepsilon_{it}\}$ is uncorrelated with past, as well as future
realisations of $\{\boldsymbol{\boldsymbol{x}_{it}}\}$, such that
$E({\varepsilon}_{it}|\boldsymbol{x}_{i1},\ldots,\boldsymbol{x}_{iT})=0$,
$\forall\;i=1,\ldots,N,\;t=1,\ldots,T$ \label{exog}
\end{enumerate}
\end{assumption}
\begin{remark}
In Assumption \ref{AssonX}.\ref{Xt} and \ref{AssonX}.\ref{et} we assume that $\{\boldsymbol{x}_{it} \}$ and $\{ \varepsilon_{it}\}$ have finite $2+\nu$ moments for some $\nu>0.$ In Assumption \ref{AssonX}.\ref{posdef} we ensure positive definiteness of the covariance matrix, guaranteeing $\left\|\frac{1}{NT} \sum_{i=1}^N\sum_{t=1}^T\left[\boldsymbol{x}_{it} \boldsymbol{x}_{it} ^{\prime}-\boldsymbol\Sigma_{\boldsymbol{x}}\right]\right\|_2=o_P(1) .$
Note that in the current framework we do not accommodate panels with cross-sectional
dependence.
However, the latter can be relaxed
at the cost of imposing stronger boundedness conditions on the largest
eigenvalue of the variance covariance matrix of $\{\varepsilon_{it}\}$.
\end{remark}
\begin{assumption}
\label{Assonalpha}
\begin{enumerate}[nolistsep]
\item $E\left(\boldsymbol{\eta}_i\right)=0 \text { and } E\left(\boldsymbol{\eta}_i \boldsymbol{\eta}_i^{\prime}\right)=\boldsymbol{\Sigma}_{\eta \eta, i}$
\item $E(\boldsymbol{\eta}_{i}\boldsymbol{\eta}_{j}')=0$ for ${i\neq j= 1, \ldots, N}$. \label{uu}
\item $E\left( \boldsymbol{x}_{it} \boldsymbol{\eta
}_{i}\right)=0 $.\label{etax}
\end{enumerate}
\end{assumption}
\begin{remark}
Substituting $\boldsymbol\beta_{i}^0$ in \eqref{mod}, we have that $y_{it}={\boldsymbol{\alpha}_k^0}'\boldsymbol{ x}_{it }+\boldsymbol{ x}_{it }\boldsymbol{ \eta}'_{i }+\varepsilon_{it}$. Under Assumption \ref{Assonalpha} we obtain a consistent estimation of ${\boldsymbol{\alpha}_k^0}$.
\end{remark}
\begin{remark}
In Assumption \ref{AssonX}.\ref{exog}, weak exogeneity can be considered in
our framework by adding lagged dependent variables, for example considering
dynamic panel models.
Note that in the case of dynamic panel specification, we may maintain the Assumption \ref{Assonalpha}.\ref{etax}, however we can no longer assume that $E\left(\boldsymbol{\eta}_i y_{i, t-1}\right)=\boldsymbol{0}$, where $y_{i, t-1}$ can be defined through continuous substitutions, such that
$
y_{i, t-1}=\sum_{j=0}^{\infty}\left(\eta_{i 1}\right)^j \boldsymbol{x}_{i, t-j-1}^{\prime}\left(\boldsymbol{\alpha}_k^0+\boldsymbol\eta_{i 2}\right)+\sum_{j=0}^{\infty}\left(\eta_{i 1}\right)^j u_{i, t-j-1},
$
and $\boldsymbol{\eta}_i=\left(\eta_{i 1}, \boldsymbol{\eta}_{i 2}^{\prime}\right)^{\prime}$. It follows that $E\left(\boldsymbol{\eta}_i y_{i, t-1}\right) \neq \boldsymbol{0}$.
The violation of the independence between the covariates and the individual effects, $\boldsymbol{\eta}_i$, implies that $y_{i t}|y_{i, t-1}$, and $\boldsymbol{x}_{i t}$ will result to inconsistent estimates of $\boldsymbol{\alpha}^0_k$, even for sufficiently large $T$ and $N$.
\end{remark}
\begin{assumption}
\label{assKm1} \label{tuneparameters}
\begin{enumerate}[nolistsep]
\item The number of clusters, $K$, is fixed and $N_{k}/N\rightarrow\tau_{k}
\in(0,1)$ for each $k=1,\ldots,K$ as $N\rightarrow\infty$, where $N_{k}= |\boldsymbol{G}^0_k|$ in
cluster $k$.\label{a21}
\item $\lambda\asymp\log{T}^{-\alpha}$, for some $\alpha\in(0,1/2]$ as $(N, T)
\rightarrow\infty$.\label{a22}
\item $T \lambda^{2} /(\ln T)^{6+2 \nu} \rightarrow\infty$ and $\lambda(\ln
T)^{\nu} \rightarrow0$ for some $\nu>0$ as $(N, T) \rightarrow\infty
$.\label{Asskm11}
\item $N^{- 1 / 2} T^{-1}(\ln T)^{9} \rightarrow0$ and $N^{2} T^{1-q / 2}
\rightarrow c \in[0, \infty)$ as $(N, T) \rightarrow\infty$.\label{Asskm12}
\end{enumerate}
\end{assumption}
\begin{remark}
Assumption \ref{assKm1}.\ref{Asskm11} imposes conditions on $\lambda$, all of
which hold if $\lambda\propto T^{-\delta} \quad$ for any $\delta\in(0,1 / 2)$.
Assumption \ref{assKm1}.\ref{Asskm12} is needed to ensure some higher-order
terms vanish asymptotically.
\end{remark}
We further introduce an additional assumption on the second moment of
$\boldsymbol{\eta}_{i}$, which serves as a group-uniqueness condition towards
cluster identifiability.\textsc{c-lasso}
\begin{assumption}
\label{clust}
\begin{enumerate}[nolistsep]
\item $\int\|\boldsymbol{\eta}_{i} \|_{2}^{2} P(d \boldsymbol{\eta}
_{i})<\infty$ \label{3cl} and the probability measure $P$ has a continuous
density $f$ on $\mathbb{R}^{p}$.
\item The group centres are pairwise different, such that $\boldsymbol{\alpha
}^{0}_{k}\neq\boldsymbol{\alpha}^{0}_{j}, $ and groups are pairwise disjoint
such that $\boldsymbol{G}_{j}^{0} \cap\boldsymbol{G}_{k}^{0} = \varnothing$,
for $j\neq k , j,k=1,\ldots, K$.
\end{enumerate}
\end{assumption}
\begin{remark}
Assumption \ref{clust}.\ref{3cl} is an identifiability assumption and
imposes finiteness conditions on $\boldsymbol{ {\eta}}_{i}$ for each
$i=1,\ldots, N$.
\end{remark}
With the following theorems, Theorem \ref{SSP_cons} and \ref{consistencyeta},
we provide asymptotic guarantees for the proposed estimator, the
\textsc{c-lasso} estimator and the feasible \emph{k-means} estimator. The
following theorems establish consistency of the adjusted penalised profile
likelihood estimates of the slope parameter ${\boldsymbol{\beta}}$ and the and
group-specific parameter ${\boldsymbol{\alpha}}$.
We consider two types of penalised estimators, the first is \textsc{c-lasso}. We
obtain the set of estimated parameters $(\boldsymbol{\ddot{ \beta} },
\boldsymbol{\ddot{ \alpha}})$, by minimising the objective in
\eqref{minimisation}, $\arg\min_{\boldsymbol{\beta}\in\mathbb{R}^{N\times p},
\boldsymbol{\alpha}\in\mathbb{R}^{p\times K}} Q_{iNT,\lambda}\left(
\boldsymbol{{\beta}},\boldsymbol{{\alpha}}\right) $ for a choice of tuning
parameter, $\lambda$.
In the following theorem we establish asymptotic consistency of
\textsc{c-lasso} under heterogeneous groups.
\begin{theorem}
\label{SSP_cons} Consider Assumptions \ref{AssonX} and \ref{Assonalpha}, the
linear model in \eqref{mod} and the minimisation of \eqref{minimisation}.
Then, for a tuning parameter $\lambda=o(1)$, we write
\begin{align}
\left\| {\boldsymbol{\ddot{\beta}}}_{i}- \boldsymbol{\beta}^{0}_{i}\right\|
_{2} & =O_{P}(T^{-1/2}+\lambda),\label{s1}\\
\Vert\boldsymbol{\ddot{\alpha}} _{k}-\boldsymbol{\alpha} _{k}^{0}
\Vert_{2} & =O_{P}\left( T^{-1/2} \right) , \label{s2}\\
(\boldsymbol{\ddot{\alpha}}_{1}, \ldots, \boldsymbol{\ddot{\alpha}
}_{K}) - ( \boldsymbol{\alpha}_{1}^{0}, \ldots, \boldsymbol{\alpha}_{K}^{0}) &
= O_{P}(T^{-1/2})\label{ssp2}
\end{align}
for $l\neq k, \; l=1,\ldots, K$, where $({\boldsymbol{\ddot{\beta}}},
\boldsymbol{\ddot{\alpha}})$ minimise the \textsc{c-lasso} objective, in \eqref{minimisation}.
\end{theorem}
Similarly to \textsc{c-lasso}, one can use alternative estimators for
$\boldsymbol{{\alpha}}$. An estimator to be considered is the \emph{k-means}.
Let $Q_{1,NT}={(NT)}^{-1}\sum_{i=1}^{N}\sum_{i=1}^{T}\frac{1}{2}\left(
y_{it}-\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{x}_{it}\right) ^{2}$. The
second methodology involves the minimisation of:
\begin{align}
Q_{2,iNT,\lambda}\left( \boldsymbol{{\beta}}\right) =Q_{1,NT}+\frac{\lambda
}{N}\sum_{i}\prod_{k=1}^{K}\left\Vert \boldsymbol{\beta}_{i}-f^{KM}
(\boldsymbol{\beta})\right\Vert _{2},\label{minimisation2}
\end{align}
where $f^{KM}(\boldsymbol{\beta})=\arg\min_{\boldsymbol{\alpha}}\sum
_{i}\left\Vert \boldsymbol{\beta}_{i}-\boldsymbol{\alpha}_{k}\right\Vert
_{2}^{2}$ is the \emph{k-means} solution. Then ${\boldsymbol{\widehat{\beta}}
}=\arg\min_{\boldsymbol{\beta}\in\mathbb{R}^{N\times p}}Q_{2,iNT,\lambda
}\left( \boldsymbol{{\beta}}\right) $ and $\boldsymbol{\widehat{\alpha}
}=f^{KM}(\boldsymbol{\widehat{\beta}})$. Theorem \ref{consistencyeta} has similar
asymptotic properties to the \textsc{c-lasso}, to Theorem \ref{SSP_cons}, but
with slightly slower convergence rates.
\begin{theorem}
\label{consistencyeta} Consider Assumptions \ref{AssonX}--\ref{clust}, the
linear model in \eqref{mod} and the minimisation in \eqref{minimisation2}.
Then, for a tuning parameter $\lambda=o(1)$, we write
\begin{align}
\left\| {\boldsymbol{\widehat{\beta}}}_{i} - \boldsymbol{\beta}^{0}
_{i}\right\| _{2} & =O_{P}(T^{-1/2}+\lambda),\\
\Vert\boldsymbol{\widehat{\alpha}} _{k}-\boldsymbol{\alpha} _{k}^{0}\Vert_{2}
& =O_{P}\left( T^{-1/2} +N^{-1/2}\right) ,\label{ssp3}\\
(\boldsymbol{\widehat{\alpha}}_{1}, \ldots, \boldsymbol{\widehat{\alpha}}_{K})
- ( \boldsymbol{\alpha}_{1}^{0}, \ldots, \boldsymbol{\alpha}_{K}^{0}) &
=O_{P}\left( T^{-1/2}+N^{-1/2}\right) ,
\end{align}
where by Assumption \ref{tuneparameters} \eqref{Asskm12}, $ \ln{T}^{9}/T = o(N^{1/2} ) $ and $(\boldsymbol{\widehat{\beta}},
\widehat{\boldsymbol{\alpha}})$ are centres obtained through the \emph{k-means
Lasso} iterative procedure, outlined in \eqref{minimisation2}.
\end{theorem}
\begin{remark}
\label{consistentbeta} Theorem \eqref{consistencyeta} establishes point-wise
convergence of the estimated slope parameter $\boldsymbol{\widehat{\beta}}
_{i}$. The asymptotic rate which guarantees consistency for
$\boldsymbol{\widehat{\alpha}}$ can be $O_{P}(T^{-1/2})$ which is the rate
using \textsc{c-lasso} in the case where $N$ is of larger order of magnitude
than $T$, i.e. $T=o(N)$.
The latter, while sub-optimal, is of interest as
clustering is a simple method in terms of
asymptotic results and can deal with empirical datasets with a large cross-section
\end{remark}
\subsection{An alternative estimator for the grouping centres}
\label{alternativeGroup} In this section we consider a feasible way to
identify the cluster structure. We devise a simple method, where we obtain the estimates of the unit-specific parameters with \textsc{ols}
\[
\boldsymbol{\widetilde{\beta}}_{i}=\left( \boldsymbol{x}_{i}^{\prime
}\boldsymbol{x}_{i}\right) ^{-1}\boldsymbol{x}_{i}^{\prime}\boldsymbol{y}
_{i}.
\]
and minimise ${Q}_{N}(\boldsymbol{G}_{k},\boldsymbol{\alpha}
)=\frac{1}{N}\sum_{i=1}^{N}\left( \boldsymbol{\beta}_{i}-\boldsymbol{\alpha
}_{k}\right) ^{2}
$ with respect to $\boldsymbol{\alpha}$, for $i=1,\ldots,N,\;k=1,\ldots,K$.
It becomes clear that $\arg\min_{\boldsymbol{\alpha}} {Q}_{N}(G_{k},
\boldsymbol{\alpha}) $ imposes an infeasible problem as there has to be
knowledge of $\boldsymbol{\beta}_{i}$. Therefore, we consider a unit-by-unit
estimation to obtain $\boldsymbol{\widehat{\beta}}_{i}$, using \eqref{inf}.
Substituting \eqref{mod} in \eqref{inf}, we have that
\begin{align}
\boldsymbol{\widetilde{\beta}}_{i}=\boldsymbol{\beta}_{i}+\left( \boldsymbol{x}
_{i}^{\prime}\boldsymbol{x}_{i}\right) ^{-1}\boldsymbol{x}_{i}^{\prime
}\boldsymbol{\varepsilon}_{i}=\boldsymbol{\beta} _{i}+\boldsymbol{\psi}_{i}=
\boldsymbol{\alpha}_{k}+\boldsymbol{\eta} _{i}+\boldsymbol{\psi} _{i}\label{astt}
\end{align}
where $\boldsymbol{x}_{i}=(x_{i1},\ldots,x_{iT})^{\prime}$,
$\boldsymbol{\varepsilon}_{i}=(\varepsilon_{i1},\ldots,\varepsilon
_{iT})^{\prime}$ and $\boldsymbol{\psi}_{i}=\left( \boldsymbol{x}_{i}
^{\prime}\boldsymbol{x}_{i}\right) ^{-1}\boldsymbol{x}_{i}^{\prime
}\boldsymbol{\varepsilon}_{i}$. Then, a feasible \emph{k-means} loss can be
constructed, $\widehat{Q}_{N}(G_{k}, \boldsymbol{\alpha} )$, where the
unit-by-unit estimates can be clustered instead of $\boldsymbol{\beta}_{i}$,
\begin{align}
\widehat{Q}_{N}(G_{k}, \boldsymbol{\alpha} )=\arg\min_{\boldsymbol{\alpha}
}\frac{1}{N}\sum_{i=1}^{N}\left( \boldsymbol{\widetilde{\beta}}_{i}-
\boldsymbol{\alpha}_{k}\right) ^{2}\label{A}
\end{align}
Note that, minimising \eqref{A}, the grouping structure relies on the
estimation of ${{\boldsymbol{\beta}}}_{i}^{\ast}$. We showcase, with the
following example, that ${Q}_{N}(\cdot)$ can be minimised with respect to some
grouping centre $\boldsymbol{\alpha}= \left(\boldsymbol{\alpha}_{j, 1}, \ldots, \boldsymbol{\alpha}_{j, K} \right)'$, $j=1, \ldots, p$.
In the following theorem we show consistency of the estimated centres obtained by
clustering $\boldsymbol{ \widehat{\beta}}$, in \eqref{inf}, through
\emph{k-means}, we call this method \emph{feasible k-means}
\begin{theorem}
\label{Feasible Kmeans} Consider Assumptions \ref{AssonX}--\ref{Assonalpha},
and \ref{clust} then the following holds
\begin{align}
\Vert\boldsymbol{\widetilde{\alpha}} _{k} -\boldsymbol{\alpha} _{k}^{0}\Vert_{2}
& =O_{p}(N^{-1/2}), \label{aahat}
\end{align}
where $\boldsymbol{\widetilde{\alpha}} _{k} $ is the minimiser of \eqref{A}.
\end{theorem}
Naturally, Theorem \ref{Feasible Kmeans} relies heavily on Assumption
\ref{clust}, where $( \boldsymbol{ {\alpha}}^{0}_{1}, \ldots, \boldsymbol{
{\alpha}}^{0}_{K})^{\prime}$ is a unique minimiser of the population
$Q(\cdot)_{N}$, therefore, $(\boldsymbol{\widetilde{\alpha}} _{1}, \ldots,
\boldsymbol{\widetilde{\alpha}} _{K})^{\prime}$ minimises $\widehat{Q}_{N}
(G_{k}, \boldsymbol{\alpha} )$.
\subsection{Classification under group heterogeneity}\label{Class_consistency}
In this section we discuss the classification consistency of the different
estimating techniques.
\begin{definition}
\label{def} Let estimators for the true regression coefficient of unit $i$ be
given by $\boldsymbol{\widehat{\beta}}_{i}$, $i=1,\ldots,N$, and for the set of
cluster centres by $\boldsymbol{\widehat{\alpha}}_{k}$, $k=1,\ldots,K.$ Consider
$\boldsymbol{G}_{k}^{0}\subset\{1,2,\ldots,N\}$, $\cup_{k=1}^{K}
\boldsymbol{G}_{k}^{0}=\{1,2,\ldots,N\}$, and $\boldsymbol{G}_{k}^{0}
\cap\boldsymbol{G}_{j}^{0}=\varnothing$ for any $j\neq k$, $j=1,\ldots,K$.
Then we define $\boldsymbol{\widehat{G}}_{k}\subseteq\{1,2,\ldots,N\}$ as the
set of units whose associated estimated centre, $\boldsymbol{\widehat{\alpha
}}_{k}$ satisfies $p\lim\boldsymbol{\widehat{\alpha}}_{k}=\boldsymbol{\alpha
}_{k}^{0}$, where a unit $i$ is associated with centre
$\boldsymbol{\widehat{\alpha}}_{k}$ if $\arg\min_{j}\left\Vert
\boldsymbol{\widehat{\beta}}_{i}-\boldsymbol{\widehat{\alpha}}_{j}\right\Vert
=\boldsymbol{\widehat{\alpha}}_{k}$.
\end{definition}
Due to the presence of $\boldsymbol{\eta}_{i}$ in the generating process in
\eqref{model}, the probabilities of events of the form $\mathcal{I}
_{1,i}=\{i\in\boldsymbol{\widehat{G}}_{k}|i\notin\boldsymbol{G}_{k}^{0}\}$,
and $\mathcal{I}_{2i}=\{i\notin\boldsymbol{\widehat{G}}_{k}|i\in
\boldsymbol{G}_{k}^{0}\}$, respectively, become difficult to control. The
literature usually defines classification consistency as the property where
$\Pr(\cup_{i}\mathcal{I}_{1,i})\rightarrow0$ and $\Pr(\cup_{i}\mathcal{I}
_{2,i})\rightarrow0.$ In fact in our case it is possible to show the converse.
In particular, we have the following Lemma
\begin{lemma}
\label{probability} Consider Assumptions \ref{AssonX}--\ref{clust}, the model
in \eqref{mod} and the minimisation in \eqref{minimisation2}. Then,
\begin{align}
\lim_{N,T\rightarrow\infty}\Pr(\cup_{i}\mathcal{I}_{1,i}) &
>0,\;and\label{p1}\\
\lim_{N,T\rightarrow\infty}\Pr(\cup_{i}\mathcal{I}_{2,i}) & >0.\label{p2}
\end{align}
\end{lemma}
\begin{remark}\label{ghat}
In Lemma \ref{probability} we show that classification consistency is not
guaranteed due to the underlying generating process of $\boldsymbol{ {\beta}
}_{i}$. Further we show that the probability of Type I and II errors is larger
than 0 as opposed to approaching 0 as $(N,T)\to\infty$. It is further
important to note that similarly to Lemma \ref{probability}, this result can
be shown using alternative estimators such as the \textsc{c-lasso} and the
\emph{feasible k-means}.
\end{remark}
\section{Heterogeneity Inference}
\label{FeasibleEstimation} The possibility of the presence of both a group
structure as well as within group heterogeneity necessitates an exploration of
data dependent methods for heterogeneity inference. Two issues arise. First
determining whether there is heterogeneity within clusters and the second is, under heterogeneity, how to determine the number of clusters. It is important to have the
right sequencing in this investigation. We propose to start with a test of
the following null hypothesis
\begin{align}
H_{0}: \boldsymbol{\beta}_{i} & =\boldsymbol{\alpha}_{{k}},
\text{ against }\\
H_{1}: \boldsymbol{\beta}_{i}& \neq\boldsymbol{\alpha}_{{k}}\; \forall\; i.\label{hypothesis}
\end{align}
Under $H_{0}$, methods such as \textsc{c-lasso} can be used to determine the
number of clusters and obtain consistent estimates of their centres. These methods
can then be used as discussed in Section \ref{tests} below. If the null is not
rejected one reverts to existing methods and find ways to determine the number
of clusters under heterogeneity. Existing methods associated to \emph{k-means}
cluster analysis can be used and are discussed in Section \ref{gap_sec}. In
the following discussion we keep matters simple by not allowing for more
refined features such as weak heterogeneity (defined as heterogeneity that is
asymptotically negligible) or heterogeneity in a proportion of clusters
(although we discuss cluster specific tests below). All such extensions are
possible but beyond the scope of the current paper.
\subsection{Testing for the existence of heterogeneity}\label{tests}
The main interest in this section is to verify the existence
of different types of heterogeneity, i.e. cross-sectional, as well as within
group. We follow the two-step method described in Section
\ref{alternativeGroup} to extract feasibly both the group and unit-specific
parameter and show that using the unit-by-unit estimate as a preliminary
estimator of $\boldsymbol{\beta}_{i}$ can lead to identifiable groups via \emph{k-means},
while the dimensionality of the cross-sectional parameters is allowed to diverge.
We consider the model in \eqref{mod} and simplify our analysis so that $\boldsymbol{\beta} = ({\beta}_{1}, \ldots, {\beta}_{N})^{\prime}$, and $\boldsymbol{\alpha} = ({\alpha}_{1}, \ldots,{\alpha} _{k})$ are $N\times 1$ and $K\times 1$ vectors respectively.
To investigate the existence of heterogeneity in the cross section, and, consequently, within a group ${G}_k$ we construct test statistics under the null hypothesis in \eqref{hypothesis}. Our methodology proceeds in distinct steps; We first obtain the estimates of $\widetilde{\beta}_{i}$ through \textsc{ols} on
$\boldsymbol{{y}}_{i}|\boldsymbol{x}_{i}$ and uncover the grouping structure using \emph{k-means} on ${\widetilde{\beta}}_{i}$ to obtain ${\widetilde{\alpha}}_k,$ the minimiser of \eqref{A}. We then obtain $\boldsymbol{\widehat{\varepsilon}}_{i}=\boldsymbol{{y}}_{i} -\boldsymbol{x}_{i}{\widetilde{\alpha}}_k $ where $i\in \widehat G_{k}$. Note that here it is sensible to assume that $\widehat G_{k}\to_P G_{k}^0$. The latter is clear under $H_0$, contrary, under $H_1$, where the results of Lemma \ref{probability} verify an alternative statement.
Then under the Assumption that
$E({\varepsilon}_{it}|{x}_{it})=0$, and $E({\eta}_{it}|{x}_{it})=0$, $T^{-1/2}
\sum_{i} T^{-1}\boldsymbol{x}_{i}^{\prime}{\varepsilon}_{it} \sim
N(0,\boldsymbol{\sigma}_{x\varepsilon})$, where $\boldsymbol{\sigma
}_{x\varepsilon} = N^{-1} \sum_{i} T^{-1}\boldsymbol{x}_{i}^{\prime
}{\varepsilon}_{it} {\varepsilon}_{it} ^{\prime}\boldsymbol{x}_{i}$. Recall from \eqref{astt} that
\begin{align}
\widetilde{\beta}_{i} = \beta_{i} + \left( \boldsymbol{x}_{i}^{\prime
}\boldsymbol{x}_{i} \right) ^{-1} \boldsymbol{x}_{i}^{\prime}
\boldsymbol{{\varepsilon}}_{i},\label{inf}
\end{align}
where $\boldsymbol{{\varepsilon}}_{i}= ( \boldsymbol{{\varepsilon}}_{i1},
\ldots, \boldsymbol{{\varepsilon}}_{iT} )^{\prime}$. A feasible solution to
\eqref{inf} would be to replace $\boldsymbol{{\varepsilon}}_{i}$ with
$\boldsymbol{\widehat{\varepsilon}}_{i} .$ One can then construct the cross-sectional
heterogeneity test statistic,
\begin{align}
\widehat{S}^{2}_{N} & = \left( \frac{1}{\sqrt{N}} \sum_{i=1}^{N} \frac{ \widehat{t}_{i}^2(\beta)-1 }{\widehat\sigma_{i}}\right) ^{2}
\label{xiT}, \; \text{and }\;
\widehat{t}_{i}(\beta)={\boldsymbol{\widehat{\varepsilon}}_{i}
^{\prime}\boldsymbol{x}_{i}}{\widehat{\sigma}_{i} \left( \boldsymbol{x}
_{i}^{\prime}\boldsymbol{x}_{i}\right) ^{-1/2}},
\end{align}
where $\widehat{\sigma}_{i} =\frac{1}{T-1}\sum_{i} \boldsymbol{\widehat{\varepsilon}}_{i} \boldsymbol{\widehat{\varepsilon}}_{i}
^{\prime}$ a
consistent estimate of $\sigma_i=E\frac{1}{T}\boldsymbol{{\varepsilon}}_{i} \boldsymbol{{\varepsilon}}_{i}
^{\prime}$.
Further, a poolability-type test would rely on the $R^{2}$ statistic,
where one uses the residuals resulting from unit-by-unit \textsc{ols}, see e.g.
\eqref{inf}, to asses the fit against a pooled \textsc{ols} regression, e.g. $
\widehat{\beta}_{\textsc{ols}} = ( \sum_{i=1}^{N} \sum_{t=1}^{T} \boldsymbol{x}
_{it} ^{\prime}\boldsymbol{x}_{it} ) ^{-1} ( \sum_{i=1}^{N}
\sum_{t=1}^{T} \boldsymbol{x}_{it} ^{\prime}y_{it} ),
$
Further, under standard assumptions of independence, see e.g Assumption
\ref{Assonalpha}, the null hypothesis in \eqref{hypothesis} will be rejected once the
unit-by-unit \textsc{ols} reports larger $R^{2}$ than the corresponding $R^{2}$ of the
pooled \textsc{ols}, which indicates that treating the data as heterogeneous increases
the overall panel fit.
To that end we construct the $R^{2}$ statistic, resulting from the auxiliary
regressions of $\boldsymbol{\widehat{\varepsilon}}_{{i}}|\boldsymbol{x}
_{i}$, such that
\begin{align}
\widehat{R}^{2}_{N} & = \left(\frac{1}{\sqrt{N}}\frac{\sum_{i=1}^N T\widehat{r}_{i}^2(\beta) -1 }{\hat{\sigma}_{i, R}} \right)^2
\label{xiR}, \; \text{and }\; \widehat{r}_{i} =\left( \widehat\gamma(\beta) \boldsymbol{x}_i ' \boldsymbol{x}_i \widehat\gamma(\beta ) (\boldsymbol{\widehat{v}}_{i}(\beta)\boldsymbol{\widehat{v}}_{i}(\beta ) ')\right)^{1/2}
\end{align}
where $
\boldsymbol{\widehat{v}}_{i}\boldsymbol{(}\beta\boldsymbol{)=\widehat{\varepsilon}}_{i}-\widehat{\gamma}\left( \beta\right) \boldsymbol{x}_{i}
$
and
$
\widehat{\gamma}\left( \beta\right) ={\boldsymbol{\widehat{\varepsilon}}_{i} ^{\prime}\boldsymbol{x}_{i}}({\boldsymbol{x}_{i}^{\prime
}\boldsymbol{x}_{i}})^{-1},
$ lastly, ${\sigma_{i, R}}$ can be defined similarly to ${\sigma_{i}}$. Under $H_0$, both $\widehat{R}^{2}_{N} , \; \widehat{S}^{2}_{N}$ are asymptotically $\chi_1^2$ distributed, while consistent under the alternative. The latter is the key finding of the following theorem.
Then, we have the following theorem giving the asymptotic distribution of
$S_{N}$ and $R_{N}$, defined in \eqref{xiT} and \eqref{xiR}.
\begin{theorem}
\label{statisticsTheorem} Consider Assumptions \ref{AssonX}, \ref{Assonalpha}
and Assumption \ref{clust}. Then, under the null
\begin{align}
\widehat{S}^{2}_{N} \overset{d}{\to} \chi_{1}^{2},\quad\widehat{R}^{2}_{N}
\overset{d}{\to} \chi_{1}^{2},\label{Statement1_test}
\end{align}
where $\widehat{S}^{2}_{N}$, and
$\widehat{R}^{2}_{N} $ are defined in \eqref{xiT} and \eqref{xiR} respectively.
\end{theorem}
\begin{remark}
While testing for cross-sectional heterogeneity is an important first step towards uncovering a grouping structure, it is also of interest to investigate the existence of heterogeneity within a group $k$. More specifically, in
\eqref{xiT}--\eqref{xiR} we can substitute $\boldsymbol{\widehat{\varepsilon}}_{{i,{k}}}$ instead of $\boldsymbol{\widehat{\varepsilon}
}_{{i}}$ such that
\begin{align}
\widehat{S}_{\widehat{K}}^{2} & = \left( \frac{1}{\sqrt{\widehat N_{k}}} \sum_{i, \; i\in \widehat G_k}
\frac{ \widehat{t}_{k}^2({\widetilde{\alpha}}_k) -1 }{\widehat{\sigma}_{k}
}\right) ^{2} ,\label{xiT_K}\quad \widehat{R}^{2}_{\widehat{K}} =\frac{1}{\sqrt{\widehat N}_{k}} \sum_{i, \; i\in \widehat G_k}
\frac{ (T\widehat{r}_{k}^{2}-1 ) }{\widehat{\sigma}^{2} _{\widehat{r}_{k}, }}
, \; \text{$ i=1, \ldots, N$,}
\end{align}
where $\widehat{t}_{k}({\widetilde{\alpha}}_k) = {\boldsymbol{\widehat{\varepsilon}}_{i}
^{\prime}\boldsymbol{x}_{i}}{\widehat{\sigma}_{k}({\widetilde{\alpha}})\left( \boldsymbol{x}
_{i}^{\prime}\boldsymbol{x}_{i}\right) ^{-1/2}},$ where ${\widetilde{\alpha}}_{k}$ is the minimiser of \eqref{A}. Further, $\widehat{\sigma}_{k}=\frac{1}{T-1}\sum_{i\in G_{k}}
\boldsymbol{\widehat{\varepsilon}}_{{i}
}\boldsymbol{\widehat{\varepsilon}}_{i}^{\prime},
$ and $\widehat{r}^{2}_{k} = \sum_{ i\in G_{k}}({\widetilde{\alpha}}_{k}\boldsymbol{x}_{i} ^{\prime}\boldsymbol{x}_{i}{\widetilde{\alpha}}_{k})^{\prime}({\boldsymbol{\widehat{\varepsilon}
}_{{i}} \boldsymbol{\widehat{\varepsilon}}_{{i}}^{\prime}
})^{-1} $ and $\widehat{\sigma}_{\widehat{r},k} $ can be defined similarly to $\widehat{\sigma}_{k} $ for $1\leq i\leq N.$ Note that using Lemma \ref{probability} under $H_0 $, $\widehat{G}_k\to_P G_k^{0}$ and $\widehat{N}_k\to_P N_k$ and the test-statitsics follow $\chi_1^2$.
The statements in \eqref{xiT_K} can be shown following a similar analysis to show Theorem \ref{statisticsTheorem}.
\end{remark}
\begin{remark}
Under $H_1$ the test statistics \eqref{xiT}, \eqref{xiR},
as well as their group-specific counterparts, \eqref{xiT_K} are consistent. The statement in \eqref{Statement1_test} can
accommodate a reasonably larger dimensionality in terms of the number of covariates, $p$, as long as the clustering remains the same across covariates.
\end{remark}
\subsection{Determination of the number of groups under heterogeneity}
\label{gap_sec}
Allowing for different types of heterogeneity, the identifiability of groups
becomes a difficult task. First, because in real datasets there is no a-priory
knowledge of the necessity of grouping, the number of groups formed or the
grouping structure. Second, in their majority grouping methods require
spherical groups, this way allowing only for a degree of heterogeneity, or are
density based techniques, which in turn require the knowledge of the empirical
distribution for each group. Further, versatile clustering techniques, such as
\emph{k-means}, appear effective, but in cases of no-grouping at all appear
inappropriate as it forces grouping. We address that issue by estimating
the number of groups using techniques known in the statistical literature. The
most commonly used is the \textit{Gap Statistic}, see e.g. \cite{gap}, and can
be constructed as follows,
\begin{align}
\label{gap_in}\text{Gap}_{N}(k) = E & \left[ \log(W_{k}) \right] -
\log(W_{k}),\quad
W_{k}= \sum_{k=1}^{K} \frac{1}{2N_{k}} D_{k}, & \; D_{k} ={ \sum_{i, j\in
G_{k},}} \Vert \boldsymbol{\beta}_i-\boldsymbol{\beta}_j\Vert _2^2,
\end{align}
for ${i\neq j }= 1, \ldots, N$ and $k=1, \ldots, K$, where $\boldsymbol{d}$ is the square euclidean distance from a centre in group
$k$, $W_{k}$ is the pooled within cluster sum of squares around the cluster
means, $E\left[ \log(W_{k}) \right] $ denotes the expectation under a sample
of size $N$ is drawn from a reference distribution and $N_{k}= |G_{k}|.$ Under
Assumption \ref{clust} and well separated groups the number of clusters
${K}^{\ast }$ maximises $\text{Gap}_{N}(k), $ defined in \eqref{gap_in}, such that
\begin{align}
\text{Gap}_{N}(k) \geq \text{Gap}_{N}(k+1) - {N^{-1/2}\sum_{i=1}^N (\log{W}_{k+1, i}) }\label{gap}
\end{align}
We consider a feasible alternative to maximise \eqref{gap_in} with respect to $\hat{K}$, such that for ${i\neq j },$
\begin{align}
\widehat{W}_{k}= \sum_{k=1}^{K} \frac{1}{2\hat{N}_{k}} { \sum_{i, j\in
G_{k},}} \Vert \boldsymbol{\widehat{\beta}}_i-\boldsymbol{\widehat{\beta}}_j\Vert _2^2,\label{wk}
\end{align}
where $\boldsymbol{\widehat{\beta}}_i$ an estimator of $\boldsymbol{\beta}_i$. Note that $\boldsymbol{\widehat{\beta}}_i$ can be a mazimiser of either $Q_{iNT,\lambda}\left( \boldsymbol{{\beta}}\right)$, $Q_{2,iNT,\lambda}\left( \boldsymbol{{\beta}}\right) $, or it can maximise the unit-wise least squares objective, such that $\arg\min_{\boldsymbol{\beta}_i\in \mathbb{R}^p}Q_{1,NT}, \; Q_{1,NT}={(NT)}^{-1}\sum_{i=1}^{N}\sum_{i=1}^{T}\frac{1}{2}\left(
y_{it}-\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{x}_{it}\right) ^{2}$, denoted \eqref{minimisation}, \eqref{minimisation2} and \eqref{inf} respectively.
Consider Assumptions \ref{AssonX}, \ref{Assonalpha}, and \ref{clust}, then $\log\widehat{W}_{k} \to _P \log W_{k} $ where ${W}_k$ is defined in \eqref{gap_in} and $\widehat{W}_k$ in \eqref{wk}.
The gap statistic is an empirical tool to identify feasibly the number of clusters taking into account the within-cluster dispersion, given various clustering algorithms. In Section \ref{MC} we explore the small sample properties of the method using simulations, and report the frequency of selecting the correct numbers of groups, where it is known. Further, as the findings of \cite{gap} suggest, the gap statistic outperforms other methods in the literature, even in the case of no grouping, i.e. $K=1$.
\section{Simulation study}
\label{MC} In this section we explore the finite sample properties of the
different estimators explored in Section \ref{sec1} and \ref{tests} and make comparisons. A detailed description of the methods considered can be found in
\ref{MC1}. Further, this section is divided in two sub-sections: In Section
\ref{MC1} we explore the small sample properties of the different methods in
terms of mean squared error and miss-classification rate, and in Section \ref{MC2}
we focus on the performance of the tests proposed in Section \ref{tests}.
\subsection{Parameter estimation (misclassification) error }
\label{MC1} To evaluate the finite-sample performance of the classification
and estimation procedure, we consider 1000 simulations of model \eqref{mod},
where the number of groups assumed are $K=2$, $\boldsymbol{G}_{k}$ is a
membership vector for group $k=1,\ldots, K$, the simulated true centres are
$\boldsymbol{\alpha} =(0.5, 2)^{\prime}$ and $\eta_{i}\sim N(0, 0.2)$. Further, $x_{it}\sim_{\text{i.i.d}} N(0,1)$, $\varepsilon_{it}\sim_{\text{i.i.d}} N(0,1)$ and
$(\beta_{1},\ldots, \beta_{N/2})^{\prime}\in\boldsymbol{G}_{1}, \;
(\beta_{N/2+1}, \ldots, \beta_{N})^{\prime}\in\boldsymbol{G}_{2},$ where
$N=[20,50,100,200], \; T=[50,100,200,500]$.
We consider various different methods and evaluate the consistency of the
grouped parameters $\beta_{i}$, and individual group centres, as well as the
quality of grouping under homogeneous and heterogeneous parameters. Namely we
consider the \textsc{c-lasso} of \cite{su2016identifying} reported as SSP, the
\textsc{c-lasso} estimator using \emph{k-means} to obtain the group centres,
reported as Km
and the \textsc{c-lasso} without imposing within class homogeneity, reported as \textsc{H-SSP}.
The penalty parameter for the penalisation throughout methods was
carried out via cross validation scheme, from a parameter grid, $\lambda
\in[0.125, 0.25, 0.5, 1, 2]T^{-1/3} $. We carried out a k-fold cross
validation scheme without re-shuffling, where $k=10$.
To asses the consistency of the individual parameters we report the average
mean squared error (MSE) across units under both the null and the alternative,
e.g. \eqref{hypothesis}. We report these findings in Table \ref{betas}. In
Table \ref{alphas} we report the average MSE of the centres of the first group
to save space, e.g. $1000^{-1} \sum_{i=1}^{1000} (\alpha_{1,i}-
\widehat{\alpha}_{1,i})$, where the estimation of the individual $K$ centres
is carried out using four different methods, descried below.
To quantify the quality of grouping we use a dissimilarity measure, Rand
Index, e.g. \cite{rand1971objective}. The Rand index, ranges between 0 and 1,
with 0 indicating that the two data groupings do not agree on any pair of
points and 1 indicating that the data groupings are exactly identical to some
permutation. More specifically, given the following partitions
$\boldsymbol{G}= \{\boldsymbol{G}_{1}, \ldots, \boldsymbol{G}_{r}
\}$, of $\boldsymbol{\beta} $ into $r$ subsets and
$\boldsymbol{\widehat{G}}=\{ \boldsymbol{\widehat{G}}_{1}, \ldots,
\boldsymbol{\widehat{G}}_{s} \}$, a partition of $\boldsymbol{\beta} $ into
$s$ subsets, where $1<r, s\leq N$, we write the following measure
\[
RI = \frac{a+b}{{\binom{N }{2 }}}\equiv\frac{TP+TN}{TP+TN+FP+FN},
\]
where $N$ is the number of units to be clustered, TP is the number of true
positives, TN is the number of true negatives, FP is the number of false
positives, and FN is the number of false negatives. Further, $a=
|\boldsymbol{B}^{*}|, \; \boldsymbol{B}^{*} = \{ (\beta_{i},\beta_{j}) |
\beta_{i},\beta_{j} \in\boldsymbol{G}, \; \beta_{i},\beta_{j} \in
\boldsymbol{\widehat{G}}\}$ and $b = |\boldsymbol{B}^{*}|, \; \boldsymbol{B}
^{*} = \{ (\beta_{i}, \beta_{j} ) | \beta_{i}\in\boldsymbol{I_{1}},\beta
_{j}\in\boldsymbol{G}_{2}, \;\beta_{i}\in\boldsymbol{\widehat{G}}_{1}
,\beta_{j}\in\boldsymbol{\widehat{G}}_{2} \}$, where the former declares the number
of pairs of elements in $\{\beta_{1}, \ldots, \beta_{N}\}$ that are in the
same subset in $\boldsymbol{G}$ and in the same subset in
$\boldsymbol{\widehat{G}}$, and the latter declares the number of pairs of
elements in $\{\beta_{1}, \ldots, \beta_{N}\}$ that are in different subsets
in $\boldsymbol{G}$ and in different subsets in $\boldsymbol{\widehat{G}}$. We
report these findings in Table \ref{RI} both under homogenous and
heterogeneous groups.
Further, since heterogeneity is allowed within a group, we consider additional
metrics to ensure that the grouping is not forced but indeed latent structures
can be found in the cross section. We then estimate the number of groups that
can be found using the unit-by-unit (UE) estimate, ${\beta}^{\ast}_i$ obtained in \eqref{inf}, reporting the frequency of selecting the
true number of groups, $K_{0}=2$ throughout 1000 replications, through the gap
statistic, e.g. \cite{gap}. We report these findings in Table \ref{gap} both
under homogenous and heterogeneous coefficients.
\begin{table}[ht!]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{ l l c c c c c |c c c c c}
\toprule
& \multicolumn{6}{c|}{$\beta_{i} = \alpha_{{i\in G_k}} $} & \multicolumn{5}{c}{$\beta_{i} = \alpha_{{i\in G_k}} + \eta_{i}$} \\ \toprule
&$T/N$& {20}& {50}& {100}& {200}& {500}& {20}& {50}& {100}& {200}& {500}\\\midrule
{SSP}&50& 0.0020 & 0.0008 & 0.0004 & 0.00021&0.0020&0.1829&0.1885&0.1846&0.1855&0.1710\\
{Km}&&0.0149&0.0148&0.0147&0.0147&0.0146&0.0881&0.0887&0.0885&0.0886&0.0887\\
{\textsc{H-SSP}}&& 0.0116 & 0.0109 &0.0107 &0.0107 &0.0107 &0.0889 & 0.0889 & 0.0885 & 0.0888 & 0.0887 \\
\midrule
{SSP}&100&0.0010 & 0.0004 & 0.0002 & 0.0001 &{0.0001}&0.1439&0.1444&0.1483&0.1442&0.1462\\
{Km}&&0.0072&0.0070&0.0069&0.0069&0.0068&0.0818&0.0813&0.0811&0.0811&0.0808\\
\textsc{H-SSP}&& 0.0072 & 0.0070 &0.0069 &0.0069 & 0.0069&0.0820 & 0.0815 & 0.0811 & 0.0811& 0.0809\\
\midrule
{SSP}&200&0.0005 & 0.0002 & 0.0001 & 0.0001&0.0001&0.1203&0.1252&0.1226&0.1256& 0.1280 \\
{Km}&&0.0033&0.0033&0.0033&0.0032&0.0033&0.0770&0.0776&0.0777&0.0780&0.0778\\
\textsc{H-SSP} && 0.0034 & 0.0033 & 0.0033 & 0.0033 & 0.0033& 0.0771 & 0.0777 & 0.0778 & 0.0781 & 0.0778\\
\midrule
{SSP}&500&0.0002&0.0001&0.0000&0.0000&0.0000& 0.1122&0.1139&0.1146&0.1163& 0.1158\\
{Km}&&0.0013&0.0012&0.0012&0.0012& 0.0012&0.0775&0.0771&0.0768&0.0768&0.0767 \\
\textsc{H-SSP}&& 0.0013 & 0.0012 &0.0012 &0.0012 &0.0012&0.0776 &0.0772 & 0.0768& 0.0768 & 0.0767
\\
\bottomrule
\end{tabular}
}\caption{MSE of $\protect\widehat{\beta}_{i, j} $, $i=1,\ldots, N$ and
$j=1,\ldots, M$ the total number of methods examined (four), throughout 1000
replications of model \eqref{mod} under the null, (left panel) and under the
alternative, e.g. \eqref{hypothesis} (right panel), where $\eta_{i}\sim
N(0,0.2)$.}
\label{betas}
\end{table}
\begin{table}[ht!]
\resizebox{\textwidth}{!}{
\begin{tabular}{@{}cccccccccccc@{}}
\toprule
& & \multicolumn{5}{c}{$\beta_{i}=\alpha_{{i\in G_k}}$} & \multicolumn{5}{c}{$\beta_{i}=\alpha_{{i\in G_k}}+\eta_{i}$} \\ \midrule
& $T/N$ & 20 & 50 & 100 & 200 & \multicolumn{1}{c|}{500} & 20 & 50 & 100 & 200 & 500 \\ \midrule
SSP & 50 & 0.0020 & 0.0008 & 0.0004 & 0.0002 & \multicolumn{1}{c|}{0.0001} & 0.0058 & 0.0022 & 0.0013 & 0.0006 & 0.0002 \\
Km & & 0.0022 & 0.0009 & 0.0004 & 0.0003 & \multicolumn{1}{c|}{0.0001} & 0.0060 & 0.0024 & 0.0014 & 0.0007 & 0.0003 \\
\textsc{H-SSP} && 0.0026 & 0.0011 &0.0005 &0.0003 & \multicolumn{1}{c|}{0.0001} &0.0084 &0.0034 &0.0020 &0.0010 &0.0005\\
\midrule
SSP & 100 & 0.0010 & 0.0004 & 0.0002 & 0.0001 & \multicolumn{1}{c|}{0.0001} & 0.0055 & 0.0021 & 0.0010 & 0.0005 & 0.0002 \\
Km & & 0.0010 & 0.0004 & 0.0002 & 0.0001 & \multicolumn{1}{c|}{0.0001} & 0.0057 & 0.0021 & 0.0011 & 0.0006 & 0.0003 \\
\textsc{H-SSP}&& 0.0013 &0.0005 &0.0003 &0.0001 & \multicolumn{1}{c|}{0.0001 } &0.0076 &0.0030 &0.0015 &0.0008 &0.0004\\
\midrule
SSP & 200 & 0.0005 & 0.0002 & 0.0001 & 0.0001 & \multicolumn{1}{c|}{0.00020} & 0.0044 & 0.0018 & 0.0010 & 0.0004 & 0.0008 \\
Km & & 0.0005 & 0.0002 & 0.0001 & 0.0001 & \multicolumn{1}{c|}{0.00022} & 0.0045 & 0.0018 & 0.0010 & 0.0005 & 0.0008 \\
\textsc{H-SSP} &&0.0006 &0.0003 &0.0001 & 0.0001 & \multicolumn{1}{c|}{0.00022} &0.0067 &0.0027 &0.0014 &0.0007 &0.0003 \\
\midrule
SSP & 500 & 0.0002 & 0.0001 & 0.0000 & 0.0000 & \multicolumn{1}{c|}{0.0000} & 0.0043 & 0.0018 & 0.0009 & 0.0004 & 0.0016 \\
Km & & 0.0002 & 0.0001 & 0.0000 & 0.0000 & \multicolumn{1}{c|}{0.0000} & 0.0043 & 0.0018 & 0.0009 & 0.0004 & 0.0018 \\
\textsc{H-SSP}&&0.0002 &0.0001 &0.0000 &0.0000 & \multicolumn{1}{c|}{0.0000} &0.0062 &0.0025 &0.0013 &0.0006 &0.0003\\
\bottomrule
\end{tabular}
}\caption{MSE of $\protect\widehat{\alpha}_{k, j} $, see note in Table
\ref{betas}. }
\label{alphas}
\end{table}As it is expected, in Table \ref{betas}, the MSE decreases as the
sample size increases, while SSP reports a higher MSE compared to all other
methods as both $N,T$ increase in the cases of heterogeneity (see right panel
of Table \ref{betas}), while it also decreases with an increase of $T$. The
latter is due to the fact that SSP does not allow within group heterogeneity ,
forcing the cross-sectional parameters to be penalised down to their
corresponding group centre, while heterogeneity persists within a group, as
the finding of Table \ref{tab1} indicate. $Km$ reports the smallest MSE across
methods in cases of heterogeneity,
In the cases of homogeneity (left panel of Table
\ref{betas}), SSP is the best performing method, as it is expected, while all
methods report smaller MSEs as $T$ increases.
In Table \ref{alphas} all methods reported appear equivalent, while as $T$
increases the error approaches zero. In Table \ref{RI} we report the average
RI (see \eqref{RI}) throughout 1000 replications. As it is expected, under the
null (see left panel of Table \ref{RI}), all methods perform are equivalently,
i.e. most methods report average accuracy around or equal 100\%. In the case
both when generated under the alternative hypothesis, i.e. heterogeneity across
units, SSP reports the smallest accuracy throughout samples, but as the sample
size $T$ increase, accuracy increases as well. The latter is a common finding
across methods.
Table \ref{gap} reports the average frequency of selecting the correct number
of groups via the gap statistic. Table \ref{gap} compliments the results of
Table \ref{RI}, verifying that (1) clustering the cross sectional units is
necessary, and (2) while clustering is necessary, through Table \ref{RI} we
verify that accuracy of grouping is on average high throughout samples.
Finding (1) can be observed both generating under the null and the alternative
as the average frequency of selecting two groups is close to 1 and increases
with the sample size $T$.
\begin{table}[ht!]
\begin{tabular}
[c]{lcccccc|ccccc}
\toprule & & \multicolumn{5}{c|}{$\beta_{i}=\alpha_{{i\in G_{k}}}$} &
\multicolumn{5}{c}{$\beta_{i}=\alpha_{{i\in G_{k}}}+\eta_{i}$}\\
\midrule & $T/N$ & 20 & 50 & 100 & 200 & 500 & 20 & 50 & 100 & 200 & 500\\
\midrule SSP & 50 & 0.990 & 0.990 & 0.990 & 0.990 & 0.990 & 0.900 & 0.880 &
0.880 & 0.880 & 0.880\\
Km & & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
F-Km & & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
\textsc{H-SSP} & & 1 & 1 & 1 & 0.999 & 0.996 & 0.995 & 0.996 & 0.995 & 0.996 &
0.996\\
\midrule SSP & 100 & 1 & 1 & 1 & 1 & 1 & 0.920 & 0.910 & 0.910 & 0.910 &
0.910\\
Km & & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
F-Km & & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
\textsc{H-SSP} & & 1 & 1 & 1 & 1 & 1 & 0.997 & 0.998 & 0.998 & 0.999 & 0.998\\
\midrule SSP & 200 & 1 & 1 & 1 & 1 & 1 & 0.930 & 0.930 & 0.930 & 0.930 &
0.924\\
Km & & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.999\\
F-Km & & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.999\\
\textsc{H-SSP} & & 1 & 1 & 1 & 1 & 1 & 0.999 & 0.999 & 0.999 & 0.999 & 0.999\\
\midrule SSP & 500 & 1 & 1 & 1 & 1 & 1 & 0.940 & 0.940 & 0.940 & 0.930 &
0.934\\
Km & & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.999\\
F-Km & & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.999\\
\textsc{H-SSP} & & 1 & 1 & 1 & 1 & 1 & 0.999 & 0.999 & 0.999 & 0.999 & 0.999\\
\bottomrule
\end{tabular}
\bigskip\caption{Average Rand index (RI) through 1000 replications of 1000
replications of model \eqref{mod} under the null e.g. \eqref{hypothesis} (left
panel) and under the alternative (right panel), where $\eta_{i}\sim N(0,0.2)$.
}
\label{RI}
\end{table}\begin{table}[h]
\centering
\begin{tabular}
[c]{lccccc|cccccl}
\toprule & \multicolumn{5}{c|}{$\beta_{i} = \alpha_{{i\in G_{k}}} + \eta_{i}$}
& \multicolumn{5}{c}{$\beta_{i} = \alpha_{{i\in G_{k}}} $} & \\
\midrule $T/N$ & {20} & {50} & {100} & {200} & {500} & {20} & {50} & {100} &
{200} & {500} & \\
\midrule {50} & 0.954 & 0.998 & 1 & 1 & 1 & 0.956 & 0.999 & 1 & 1 & 1 & \\
\midrule {100} & 0.961 & 1 & 1 & 1 & 1 & 0.950 & 0.997 & 1 & 1 & 1 & \\
\midrule {200} & 0.963 & 0.998 & 1 & 1 & 1 & 0.953 & 0.998 & 1 & 1 & 1 & \\
\midrule {500} & 0.967 & 0.997 & 1 & 1 & 1 & 0.962 & 0.999 & 1 & 1 & 1 & \\
\bottomrule
\end{tabular}
\caption{Frequency of selecting $K_{0}=2$ groups via the gap statistic,
$\eta_{i}\sim N(0,0.2).$}
\label{gap}
\end{table}
\subsection{Testing for the existence of heterogeneity}
\label{MC2} We simulate the model in \eqref{mod} and take the one-way within
transformation, see \eqref{mod}, with $x_{it}\sim_{\text{i.i.d}} N(0,1)$, $\varepsilon_{it}\sim_{\text{i.i.d}} N(0,1)$, $\boldsymbol{\alpha} =(0.5, 2)^{\prime}$, $\eta_{i}\sim N(0,0.2).
$ We generate under the null hypothesis of
\eqref{hypothesis}, to report the size of the test in \eqref{xiT} on the right
panel of Table \ref{tab1} and further power of the test can be reported by
generating under the alternative, see the left panel of Table \ref{tab1}. The
reported quantities of Table \ref{tab2} can be similarly described for the test
in \eqref{xiR} where we assess the goodness of fit under clustered effects.
\newline\indent In Table \ref{tabkh} we report the average frequency of
selecting $K=2$ groups when the true number of groups is $K_{0}=2$ throughout
1000 replications of model \eqref{mod}, while in Table \ref{tabkh1} we report
the average frequency of selecting $K=1$ groups when the true number of groups
is $K_{0}=1$. Further we report four different measures of identifying the number
of latent groups in a vector of estimates \eqref{inf}, the maximum number of
clusters considered is $K_{\max}=5$. Namely, we consider the gap statistic,
e.g. \cite{gap}, reported as 'gap', the Silhouette statistic e.g. \cite{sil},
reported as 'Sil', the Calinski-Harabasz statistic, e.g. \cite{ch} reported as
'CH' and the Davies-Bouldin, e.g. \cite{db} reported as 'DB'. \newline
\indent We are interested in investigating the existence of heterogeneity and
for that matter the existence of groups in two separate cases; First, we
investigate the existence of heterogeneity in the cross-section, using the
unit-by-unit \textsc{ols} estimator in \eqref{inf} (reported as \emph{CS} in Tables
\ref{tab1}--\ref{tab2}) and second we investigate the existence of
heterogeneity after grouping using the \emph{feasible k-means} estimator
(reported as \emph{WG} in Tables \ref{tab1}--\ref{tab2}) minimising
\eqref{A}. All the tests are carried out at $5\%$ level of significance. In
Table \ref{tab1} there is clear indication that asymptotically, as $T$
increases, the test in \eqref{xiT} rejects the null for both types of
heterogeneity frequently, showing that indeed heterogeneity still persists
both before and more importantly after grouping. Similarly to Table
\ref{tab1}, in Table \ref{tab2} the goodness of fit both before and after
grouping is appears better as $T$ increases, showing that grouping is
necessary even when heterogeneity persists within each group. The latter is an
interesting finding coupled with the findings of Table \ref{tabkh}
--\ref{tabkh1}, because there is an implication that in the cases where the
number of groups $K=2$, asymptotically the gap measure
identifies the correct number of groups 100\% o the times, while when there is
no grouping at all, the rate at which the gap statistic selects the correct
number of groups is 100\% throughout $N,\; T$. Therefore, there is clear
indication of grouping structure in the cross-section, and the findings of
Tables \ref{tab1}--\ref{tab2} show that heterogeneity clearly exists within
the different groups as well.
\begin{table}[ht!]
\centering
\resizebox{0.8\linewidth}{!}{
\begin{tabular}{@{}lcccccc|ccccc@{}}
\toprule
& & \multicolumn{5}{c|}{$\beta_{i} = \alpha_{k} + \eta_{i}$} & \multicolumn{5}{c}{$\beta_{i} = \alpha_{k} $} \\ \midrule
& {$N/T$} & 100 & 200 & 500 & 1000 & 2000 & 100 & 200 & 500 & 1000 & 2000 \\ \midrule
CS &{100} & 1 & 1 & 1 & 1 & 1 & 0.14 & 0.14 & 0.15 & 0.13 & 0.15 \\
WG & {} & 0.86 & 0.87 & 0.90 & 0.91 & 0.99 & 0.15 & 0.13 & 0.16 & 0.13 & 0.15 \\ \midrule
CS & {200} & 1 & 1 & 1 & 1 & 1 & 0.11 & 0.11 & 0.11 & 0.10 & 0.11 \\
WG & {} & 0.91 & 0.92 & 0.92 & 0.93 & 0.98 & 0.10 & 0.11 & 0.11 & 0.10 & 0.12 \\ \midrule
CS & {500} & 1 & 1 & 1 & 1 & 1 & 0.06 & 0.07 & 0.07 & 0.05 & 0.07 \\
WG & {} & 0.95 & 0.94 & 0.93 & 0.95 & 0.96 & 0.05 & 0.07 & 0.07 & 0.07 & 0.08 \\ \midrule
CS & {1000} & 1 & 1 & 1 & 1 & 1 & 0.04 & 0.04 & 0.06 & 0.04 & 0.05 \\
WG & {} & 0.96 & 0.95 & 0.96 & 0.96 & 0.97 & 0.04 & 0.05 & 0.03 & 0.06 & 0.06 \\ \midrule
CS & {2000} & 1 & 1 & 1 & 1 & 1 & 0.02 & 0.02 & 0.05 & 0.04 & 0.04 \\
WG & {} & 0.98 & 0.97 & 0.97 & 0.96 & 0.96 & 0.02 & 0.04 & 0.03 & 0.03 & 0.04 \\ \bottomrule
\end{tabular}}
\caption{Size and power of the statistic throughout 1000
replications, generating under the null (right-hand panel) and alternative (right-hand panel) hypothesis, the number of groups is $K_{0}=2$. CS stands for cross-sectional, and WG stands for within group.} \label{tab1}
\end{table}
\begin{table}[ht!]
\centering
\resizebox{0.80\textwidth}{!}{
\begin{tabular}{@{}lccccccccccc@{}}
\toprule
& & \multicolumn{5}{c}{$\beta_{i} = \alpha_{{i\in G_k}} + \eta_{i}$} & \multicolumn{5}{c}{$\beta_{i} = \alpha_{{i\in G_k}} $} \\ \midrule
& N/T & 100 & 200 & 500 & 1000 & \multicolumn{1}{c|}{2000} & 100 & 200 & 500 & 1000 & 2000 \\ \midrule
CS & 100 & 1 & 1 & 1 & 1 & \multicolumn{1}{c|}{1} & 0.06 & 0.04 & 0.08 & 0.13 & 0.07 \\
WG & & 0.17 & 0.57 & 0.98 & 1 & \multicolumn{1}{c|}{1} & 0.03 & 0.09 & 0.03 & 0.04 & 0.04 \\ \midrule
CS & 200 & 1 & 1 & 1 & 1 & \multicolumn{1}{c|}{1} & 0.08 & 0.1 & 0.12 & 0.06 & 0.04 \\
WG & & 0.13 & 0.37 & 0.95 & 1 & \multicolumn{1}{c|}{1} & 0.06 & 0.11 & 0.05 & 0.08 & 0.07 \\\midrule
CS & 500 & 1 & 1 & 1 & 1 & \multicolumn{1}{c|}{1} & 0.05 & 0.09 & 0.07 & 0.09 & 0.01 \\
WG & & 0.1 & 0.24 & 0.73 & 1 & \multicolumn{1}{c|}{1} & 0.07 & 0.05 & 0.05 & 0.05 & 0.04 \\ \midrule
CS & 1000 & 1 & 1 & 1 & 1 & \multicolumn{1}{c|}{1} & 0.04 & 0.07 & 0.04 & 0.05 & 0.04 \\
WG & & 0.13 & 0.13 & 0.59 & 0.97 & \multicolumn{1}{c|}{1} & 0.08 & 0.04 & 0.06 & 0.09 & 0.04 \\ \midrule
CS & 2000 & 1 & 1 & 1 & 1 & \multicolumn{1}{c|}{1} & 0.06 & 0.07 & 0.04 & 0.03 & 0.06 \\
WG & & 0.11 & 0.08 & 0.46 & 0.8 & \multicolumn{1}{c|}{1} & 0.07 & 0.02 & 0.05 & 0.04 & 0.06 \\ \bottomrule
\end{tabular}
}
\caption{Size and power of the statistic in \eqref{xiR} throughout 1000
replications, generating under the null and alternative hypotheses in
\eqref{hypothesis}, the number of groups is $K_{0}=2$.}
\label{tab2}
\end{table}
\begin{table}[ht!]
\resizebox{\textwidth}{!}{\begin{tabular}{@{}lcccccccccccccccc@{}}
\toprule
\multicolumn{17}{l}{$\beta_i=\alpha_{{i\in G_k}}+\eta_{i}$, $\eta_{i}\sim i.i.d$} \\ \midrule
N/T & 100 & & & \multicolumn{1}{c|}{} & 200 & & & \multicolumn{1}{c|}{} & 500 & & & \multicolumn{1}{c|}{} & 1000 & & & \\ \midrule
& Gap & Sil & CH & \multicolumn{1}{c|}{DB} & Gap & Sil & CH & \multicolumn{1}{c|}{DB} & Gap & Sil & CH & \multicolumn{1}{c|}{DB} & Gap & Sil & CH & DB \\ \midrule
100 & 0.763 & 1&0& \multicolumn{1}{c|}{0.955} & 0.830 & 0.998 & 0.002 & \multicolumn{1}{c|}{0.964} & 0.851 & 0.999&0& \multicolumn{1}{c|}{0.959} & 0.849 & 1 & 0.001 & 0.973 \\
200 & 0.932 & 1&0& \multicolumn{1}{c|}{0.999} & 0.942 & 1 & 0.001 & \multicolumn{1}{c|}{0.999} & 0.955 & 1&0& \multicolumn{1}{c|}{0.999} & 0.974 & 1 & 0.001 & 1 \\
500 & 0.995 & 1&0& \multicolumn{1}{c|}{1} & 0.998 & 1&0& \multicolumn{1}{c|}{1} & 0.999 & 1&0& \multicolumn{1}{c|}{1} & 0.999 & 1&0& 1 \\
1000 & 1 & 1&0& \multicolumn{1}{c|}{1} & 1 & 1&0& \multicolumn{1}{c|}{1} & 1 & 1&0& \multicolumn{1}{c|}{1} & 1 & 1&0& 1 \\ \midrule
\multicolumn{17}{l}{$\beta_{i}= a_{{i\in G_k}}$} \\ \midrule
N/T & 100 & & & \multicolumn{1}{c|}{} & 200 & & & \multicolumn{1}{c|}{} & 500 & & & \multicolumn{1}{c|}{} & 1000 & & & \\ \midrule
\multicolumn{1}{l}{} & Gap & Sil & CH & \multicolumn{1}{c|}{DB} & Gap & Sil & CH & \multicolumn{1}{c|}{DB} & Gap & Sil & CH & \multicolumn{1}{c|}{DB} & Gap & Sil & CH & DB \\ \midrule
100 & 1 & 1 & 0.456 & \multicolumn{1}{c|}{1} & 1 & 1 & 0.490 & \multicolumn{1}{c|}{1} & 1 & 1 & 0.466 & \multicolumn{1}{c|}{1} & 1 & 1 & 0.485 & 1 \\
200 & 1 & 1 & 0.652 & \multicolumn{1}{c|}{1} & 1 & 1 & 0.653 & \multicolumn{1}{c|}{1} & 1 & 1 & 0.700 & \multicolumn{1}{c|}{1} & 1 & 1 & 0.705 & 1 \\
500 & 1 & 1 & 0.905 & \multicolumn{1}{c|}{1} & 1 & 1 & 0.922 & \multicolumn{1}{c|}{1} & 1 & 1 & 0.906 & \multicolumn{1}{c|}{1} & 1 & 1 & 0.914 & 1 \\
1000 & 1 & 1 & 0.984 & \multicolumn{1}{c|}{1} & 1 & 1 & 0.989 & \multicolumn{1}{c|}{1} & 1 & 1 & 0.993 & \multicolumn{1}{c|}{1} & 1 & 1 & 0.990 & 1 \\ \bottomrule
\end{tabular}} \caption{ Frequency of selecting $2$ groups when the true number of clusters
is $K_{0}=2.$ }
\label{tabkh}
\end{table}
\begin{table}[ht!]
\resizebox{\textwidth}{!}{\begin{tabular}{@{}ccccccccccccccccc@{}}
\toprule
\multicolumn{17}{l}{$\beta_i=a_{k}+\eta_{i}$, $\eta_{i}\sim i.i.d$} \\ \midrule
N/T & 100 & & & \multicolumn{1}{c|}{} & 200 & & & \multicolumn{1}{c|}{} & 500 & & & \multicolumn{1}{c|}{} & 1000 & & & \\ \midrule
& Gap & Sil & CH & \multicolumn{1}{c|}{DB} & Gap & Sil & CH & \multicolumn{1}{c|}{DB} & Gap & Sil & CH & \multicolumn{1}{c|}{DB} & Gap & Sil & CH & DB \\ \midrule
100 & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & 0 \\
200 & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & 0 \\
500 & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & 0 \\
1000 & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & 0 \\ \midrule
\multicolumn{17}{l}{$\beta_{i}= a_{k}$} \\ \midrule
N/T & 100 & & & \multicolumn{1}{c|}{} & 200 & & & \multicolumn{1}{c|}{} & 500 & & & \multicolumn{1}{c|}{} & 1000 & & & \\ \midrule
& Gap & Sil & CH & \multicolumn{1}{c|}{DB} & Gap & Sil & CH & \multicolumn{1}{c|}{DB} & Gap & Sil & CH & \multicolumn{1}{c|}{DB} & Gap & Sil & CH & DB \\ \midrule
100 & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & 0 \\
200 & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & 0 \\
500 & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & 0 \\
1000 & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & \multicolumn{1}{c|}{0} & 1 & 0 & 0 & 0 \\ \bottomrule
\end{tabular}}\caption{ Frequency of selecting $1$ group when the true
number of clusters is $K_{0}=1.$ }
\label{tabkh1}
\end{table}
\section{Empirical study}
\label{Empirics} In this section we discuss the results of the empirical
analysis, and provide a detailed discussion of the datasets we used, in the Appendix.
In Table \ref{tab_empirics} we report the statistics in \eqref{xiT} and
\eqref{xiR} respectively, with \emph{p-values} and report statistical significance at different levels using $(*)$. Note that
${***}$ signifies 1\% significant, ${**} $ signifies 5\% significant, and ${*}
$ signifies 10\% significant.
Note that the datasets used in the empirical analysis have different number of
variables, hence the critical value for the $\chi_{d}^{2}$ with which the
statistics \eqref{xiT} and \eqref{xiR} are compared, changes, where $d$ are
the degrees of freedom. Namely, for $d=1: c=3.841$, $d=2: c=5.991$, $d=3:
c=7.815$, $d=4: c=9.488$.
Across datasets we use the feasible \emph{k-means} estimator based
on the unit-wise estimation in \eqref{inf} and estimate the number of clusters
using the gap statistic, see e.g. \cite{gap}. The optimal number of clusters
is selected from the range ${K} \in\{ 1, \ldots, 20 \}$ to maximise the
criterion in \eqref{gap_in}. Further we obtain the solution of \eqref{A} based
on the optimal number of clusters $\widehat{K}$ and use \eqref{xiT} to
evaluate whether there is remainder heterogeneity collectively across groups
and asses the fit of the proposed clustering method using \eqref{xiR}.
In our empirical application we use eleven datasets, in which for the means of clustering we have
standardised both $\boldsymbol{X}$ and $\boldsymbol{y}$, the number of
variables for each dataset is indicated with $p$. Throughout the datasets, we observe
substantial heterogeneity across countries was observed in all the
macroeconomic variables, it is important then to construct groups that allow
for within group heterogeneity and at the same time, keeping the groups
identifiable. It is observable that heterogeneity within groups exists at most datasets,
within groups. The only dataset that indicates near-homogeneous groups is the
dataset of the Italian regions, where the F-statistic reported is
statistically significant at 10\% level of significance. Further, the
\textit{"US MSA's"," Gravity"} and \textit{"Gasoline Demand"} datasets report
one group. This finding for the \textit{"US MSAs"}, at least, can appear odd
because the Metropolitan housing areas are 381, and there are grounds for
grouping, as the housing market corresponds to different incomes, while for
the rest of the two a similar hypothesis can be adopted since income is
different across different groups of countries. In these datasets strong
heterogeneity (to noisy to find a signal, i.e. groups) in the cross-section is
observed, indicative of no distinct grouping. Therefore the gap statistic
fails to produce optimal number of groups since it is based on the hypothesis
that the groups while heterogeneous, are clearly distinct.
\begin{table}[ht!]
\centering
\resizebox{0.65\textwidth}{!}{
\begin{tabular}{@{}lllc@{}}
\toprule
\multicolumn{1}{l}{ \textbf{Production functions}} &${\widehat{S}^2_{N, \widehat{K}}}$ &${\widehat{R}^2_{N, \widehat{K}}}$\\\midrule
OECD, $p=2, \; \widehat{K}=2$ & 6.769$^{*}$ & 36.187$^{***}$ & \\
\midrule
EU, $p=2$, $ \widehat{K}=3$ & 14.954$^{***}$ & 85.019 $^{***}$ & \\
\midrule
ITA, $ p = 1, \; \widehat{K}=4$ & 2.081$^{*}$ & 26.889$^{***}$ & \\
\midrule
US States, $p=2$, $ \widehat{K}=3$ & 17.355$^{**}$ & 27.005$^{***}$ & \\
\midrule
UNIDO, $p=1, \; \widehat{K}=3$ & 4.812$^{**}$ & 15.696$^{***}$ & \\
\midrule
Production R\&D, $p=3$, $ \widehat{K}=2$ & 11.475$^{***}$ & 155.059$^{***}$ & \\
\midrule
Gravity, $p=6$, $ \widehat{K}=1$ & 24.337$^{***}$ & 78.541$^{***}$ & \\
\midrule
Gasoline Demand, $p=2$, $ \widehat{K}=1$ & 20.776$^{***}$ & 59.853 $^{***}$ & \\
\midrule
\multicolumn{3}{l}{ \textbf{Income elasticity}}\\\midrule
US States, $ p=1, \widehat{K}=4$ & 12.927$^{***}$ & 29.087$^{***}$ & \\
US MSAs, $p=1, \; \widehat{K}=1$ & 142.939$^{***}$ & 263.817$^{***}$& \\
Saving's rate,$p=4$, $ \widehat{K}=2$ & 10.835$^{**}$ & 181.976$^{***}$ & \\
\bottomrule
\end{tabular}
}
\caption{Statistics and corresponding significance of \eqref{xiT}, \eqref{xiR} under the null. In cases where the test fails to reject the null, each group is homogeneous.}
\label{tab_empirics}
\end{table}
\subsection{Within-group analysis}
\label{wgA} We test for heterogeneity within each group, using \eqref{xiT_K}, across the eleven different datasets considered in Table
\ref{tab_empirics}. First, we note that across datasets heterogeneity has been
observed within groups. It is interesting to point out that our test reports
different levels of significance across different variables and groups,
implying borderline homogeneity for some groups, if the test reports
\textit{p-values} $\approx10\%$. While our findings do not suggest borderline
homogeneity, the findings of the \textit{"Savings"} dataset are worth
discussing. Specifically, we find that in the first group of ${S}_{t}$ (the
ratio of savings to GDP), the \textit{p-value} is $0.0003<<1\%$, while in the
second group a similar result applies. Regarding the second variable ${ I}
_{t}$ (the CPI-based inflation rate) the \textit{p-value} of the first group
is $0.00026$, and for the second group it is $0.00014<<1\%$, while for
${GDP}_{t}$ (the per capita GDP growth rate) similar results are found. These
findings are indicative of heterogeneity within each group for the different
variables, as the test results appear significant at 1\%, with
\textit{p-values} significantly lower than the nominal rate. For the third
variable, ${R}_{t}$ (the real interest rate), \textit{p-values} for the first
and second group are $0.04$, $0.04<5\%$, which indicate that our tests report
statistically significant results at the $5\%$ level of significance. The
latter indicates that interest rates across countries (economic units) can be
indicative of two groups and further those two groups appear less
heterogeneous than the rest.
We use the same analysis in the remainder of the datasets and find, across these datasets, that all groups rejected the null hypothesis at $1\%$ level of significance.
\section{Discussion}
\label{Discussion} We explore the idea of different types of heterogeneity in
large panel data models where the number of units are allowed to diverge faster
than the sample size. We follow the work of \cite{su2016identifying}, and
use several approaches to identify and estimate latent grouping structures in
panel data, developing panel penalized profile likelihood methods for
classification and estimation. More importantly, we argue that, often,
cross-sectional heterogeneity is not alleviated through grouping, creating a
degree of bias and further identification issues in the model. In this paper,
we explore the asymptotic properties of such a model, and emphasize on the
identification of heterogeneous components within a group. We devise a testing
procedure that identifies heterogeneity, for which the statistic is
asymptotically chi-squared under the hypothesis of homogeneity, while it is
consistent otherwise.
These techniques combined, provide a general approach to grouping and
estimating panel models with unknown groups, heterogeneity within and across
groups, and an unknown number of groups. We use known in the literature
statistics, e.g. \cite{gap}, to identify the number of groups, while the
method is forcing neither grouping, nor within group heterogeneity.
Simulations show that the different approaches considered have good
finite-sample performance and can be implemented in empirical datasets.
Further, the two tests considered have good size and power under the null
hypothesis. We use eleven empirical applications which reveal the advantages
of data-determined identification of latent grouping structures in empirical
panel modeling, while most importantly our tests show that the different
groups identified across datasets are heterogeneous.
This work can be extended in a number of different directions, which we leave for future research.
First, it may be appealing to consider a more general framework that
allows the number $(K)$ of groups to grow with the sample size. The
theoretical framework used in this paper can allow $K$ to grow with $N$ at a
slower rate. Second, an extension to panel models with endogeneity will raise
new statistical and computational challenges. Lastly, and in our view, a more
appealing issue is considering mixed panel models, allowing for more
heterogeneity in the sense of noise, while grouping structures can be
considered as signals. It is then natural to devise different testing
procedures to identify both the heterogeneous components and the signals, such
as LM-type of tests for the former and simple t-tests for the latter. This
framework though welcomes issues such as different degrees of bias to control,
while joint estimation procedures can come with an additional mathematical and
computational challenge, empirically though this framework is more realistic
and general.
\newpage