# Specification testing with grouped fixed effects 111We are grateful to Elena
Manresa, Chris Muris, Laura Serlenga, Francesco Bartolucci, Andreas Dzemski,
Arturas Juodis, Amrei Stammann, Riccardo Lucchetti, to the audience at the
28th International Panel Data Conference, and to the participants of the 11th
Workshop of Econometrics and Empirical Economics for their helpful comments
and suggestions. We also thank Carolina Castagnetti and Federico Belotti for
generously sharing their codes. Francesco Valentini would like to acknowledge
the financial support by the European Union - Next Generation EU. Project
Code: ECS00000041; Project CUP: C43C22000380007; Project Title: Innovation,
digitalization and sustainability for the diffused economy in Central Italy -
VITALITY.

Claudia Pigini222Marche Polytechnic University (Italy). E-mail:
<EMAIL_ADDRESS>Alessandro Pionati333Marche Polytechnic University
(Italy).Corresponding Author. Address: Department of Economics and Social
Sciences, P.le Martelli 8, 60121 Ancona (Italy) E-mail<EMAIL_ADDRESS>Francesco Valentini 444Marche Polytechnic University (Italy). E-mail:
<EMAIL_ADDRESS>
###### Abstract

We propose a bootstrap generalized Hausman test for the correct specification
of unobserved heterogeneity in both linear and nonlinear fixed-effects panel
data models. We consider as null hypotheses two scenarios in which the
unobserved heterogeneity is either time-invariant or specified as additive
individual and time effects. We contrast the standard fixed-effects estimators
with the recently developed two-way grouped fixed-effects estimator, that is
consistent in the presence of time-varying heterogeneity under minimal
specification and distributional assumptions for the unobserved effects. The
Hausman test exploits the general formulation for the variance of the vector
of contrasts and critical values are computed via parametric percentile
bootstrap, so as to account for the non-centrality of the asymptotic
$\chi^{2}$ distribution arising from the incidental parameters and
approximation biases. Monte Carlo evidence shows that the test has correct
size and good power properties. We provide two empirical applications to
illustrate the proposed test: the first one is based on a linear model for the
determinants of the wage of working women and the second analyzes the trade
extensive margin.

Keywords: Additive effects, Group fixed-effects Hausman test, Parametric
bootstrap, Time-varying heterogeneity

JEL Classification: C12, C23, C25

##  1 Introduction

Correct specification of unobserved heterogeneity is crucial in panel data
modeling. For long, empirical applications have only considered individual
time-constant fixed effects, but the assumption of time-invariant unobserved
heterogeneity is often hardly tenable in practice, especially over a long time
dimension. Therefore the current mainstream approach includes both subject and
time fixed effects, in order to achieve credible identification of the effects
of interest. The simplest and most widely employed setup is the specification
of additive individual and time heterogeneity, namely the two-way fixed-
effects model, that in the linear model is equivalent to the two-way
correlated random effects approach (Wooldridge, , 2021). For nonlinear models
with additive fixed effects, Fernández-Val and Weidner,  (2016) provide
analytical and jackknife bias corrections for the maximum likelihood (ML)
estimator, which is plagued by the incidental parameters problem.

While of simple implementation, the two-way fixed-effects specification fails
to capture the specific impact common factors may have on each subject. There
is now an important stream of literature focused on developing identification
results and estimation strategies for models with interactive time and
individual fixed effects. Contributions have been spurred by the seminal paper
of Bai,  (2009), who provides identification results along with the
asymptotics for the interactive fixed-effects estimator in linear models. More
recently, interactive fixed-effects have been introduced in nonlinear panel
data and network models by Chen et al.,  (2021).

Testing the assumptions on the unobserved heterogeneity specification has also
received considerable attention in the recent econometric literature.
Bartolucci et al.,  (2015) propose a Hausman-type test for the null hypothesis
of time-constant unobserved heterogeneity in generalized linear models, where
conditional ML estimators are compared with first-differences or pairwise
conditional ML estimators. In the context of large stationary panel models,
the factor specification could be tested by comparing additive to interactive
fixed-effects models, on the basis of the Hausman test suggested by Bai,
(2009) and its fixed-$T$ version, derived by Westerlund,  (2019). However, it
has been shown that the Hausman-type test fails to reject the null hypothesis
when individual factor loadings are independent across equations (Castagnetti
et al., 2015b, ). On this basis, Kapetanios et al.,  (2023) use a Hausman-type
test contrasting additive and interactive fixed-effects to detect such
correlation, whereas Castagnetti et al., 2015a overcome the issue by proposing
an alternative max-type test for the null hypothesis of time-invariant
unobserved heterogeneity.

Despite its increasing popularity, the interactive effects approach based on
Bai, ’s procedure comes with some non trivial issues. First, estimation relies
on solving a non convex objective function with possibly multiple minima (Moon
and Weidner, , 2023). Secondly, the reliability of the iterative procedure
crucially depends on the consistency of the parameter estimates chosen as the
starting point for the algorithm (Hsiao, , 2018). Finally, the number of
latent factors should be known _a priori_ to the econometrician, and even in
this case factors are not uniquely determined (Moon and Weidner, , 2015).
These drawbacks might be even more hampering when nonlinear models with
interacted fixed effects are involved (see Chen et al., , 2021). In light of
these considerations, a simpler specification might therefore be preferable,
provided it gives a good enough representation of the structure of the
unobserved heterogeneity.

In this paper we propose a generalized Hausman test for the fixed-effects
specification, in both linear and nonlinear models and where the unobserved
heterogeneity, under the null hypotheses, is either only individual or
additive. The test contrasts fixed-effects ML estimators with the Two-Way
Grouped Fixed Effects (TW-GFE henceforth) approach, recently put forward by
Bonhomme et al., 2022a . Their proposal is based on a first-step data-driven
approximation of the unobserved heterogeneity, which is clustered by the
kmeans algorithm that uses individual and time-series moments to assign
individual and time group memberships. Cluster dummies are then interacted and
enter the model specification as group effects, and the associated parameters
are estimated along with the regression coefficients in the second step. The
resulting second-step estimator is consistent in the presence of unspecified
forms of the time-varying unobserved heterogeneity with minimal assumptions on
the unobserved components, which makes it a perfect candidate to contrast with
the fixed-effects estimators that are consistent only with time-constant or
time-varying additive heterogeneity. Note that, in order to perform the
proposed test, there is no need to estimate the interactive fixed effects
models, as the TW-GFE encompasses this as well as more sophisticated
specifications for the unobserved heterogeneity.555In principle, our strategy
could be used to test for the null hypothesis of a factor structure against
more complex formulations, provided certain regularity conditions on the
functions governing the unobserved heterogeneity are satisfied.

Under specific choices for the number of clusters outlined by Bonhomme et al.,
2022a for the first step, it can be shown that the TW-GFE estimator is
asymptotically normal, so that the Hausman statistic (Hausman, , 1978) has
asymptotic $\chi^{2}$ distribution. However, as it might be difficult to
verify which estimator is more efficient than the other under the null
hypothesis, we rely on the generalized estimator for the variance of the
vector of contrasts proposed by Bartolucci et al.,  (2015). In addition, the
asymptotic $\chi^{2}$ distribution is non-central because of two sources of
asymptotic bias: the incidental parameters problem, that in nonlinear models
plagues both estimators, and the approximation bias, that affects the TW-GFE.
We therefore compute critical values of the test statistic distribution by
means of parametric percentile bootstrap (MacKinnon, , 2006; Horowitz, ,
2019). The main advantage of this procedure lies in the bootstrap
distributions correctly capturing the non-centrality without the need for any
bias correction of either estimator. A related strategy is adopted by Kim and
Sun,  (2016) and Higgins and Jochmans,  (2023) for inferential procedures in
nonlinear fixed-effects models.

We report the results of an extensive Monte Carlo study showing evidence that
the test has correct size and good power in both linear and non linear
specifications. Size properties, however, crucially depend on how effective is
the clustering procedure in approximating the unobserved heterogeneity for the
TW-GFE, that is choosing a sufficiently large number of groups and ensuring
that the moments used for the kmeans clustering are informative about the
latent trains and common factors. Power properties are studied under the
alternative hypothesis of a factor structure. The test exhibits high rejection
rates with the linear model, while has lower power for the probit model when
the ML estimator for additive specification is contrasted with the TW-GFE.
Correct size is also attained with dynamic factors, which represents a
violation of one of the assumptions needed for consistency of the TW-GFE
while, as expected, in scenarios where moments are not informative the test is
not viable. While computationally more intensive than the testing procedures
put forward by Castagnetti et al., 2015a and Bartolucci et al.,  (2015), the
proposed test represents an improvement as the former can only be applied to
linear models in a large-$T$ framework and the latter, while viable for
generalized linear models, lacks power when time effects are independent.

We also provide two empirical applications for the proposed test. The first
concerns a linear model for the determinants of the wage of working women. The
test detects the presence of time-varying unobserved heterogeneity, when the
TW-GFE is compared with the ML estimator with only individual heterogeneity,
and it also shows that the TW-GFE overfits as the additive fixed-effects
specification is enough to capture unobserved common time trends. In the
second application we analyze the trade extensive margin following Helpman et
al.,  (2008). The test does not provide evidence of a more complex structure
for the unobserved heterogeneity, as it fails to reject the null hypothesis of
additive importer and exported fixed-effects.

Literature review This paper relates to the stream of literature hat has
studied fixed-effects panel data models with grouped structures for the
unobserved heterogeneity. Discrete heterogeneity has long been considered
within the random-effects approach (Heckman and Singer, , 1984), especially by
a large body of statistical literature; see, for instance, MacLahlan and Peel,
(2000) on finite-mixture models and Bartolucci et al.,  (2012) on latent
Markov models. On the contrary, the investigation of grouped patterns of
heterogeneity in fixed-effects models is relatively recent in the econometric
literature.

Hahn and Moon,  (2010) study the asymptotic bias arising form the incidental
parameters problem in nonlinear panel data models where unobserved
heterogeneity is assumed to be discrete with a finite number of support
points. Bester and Hansen,  (2016) investigate the asymptotic behavior of the
ML estimator for nonlinear models with grouped effects, under the assumption
that subjects are clustered according to some external known classification.
Models with unknown grouped membership are studied by Su et al.,  (2016), who
propose penalized techniques for the estimation of models where regularization
by classifier-Lasso shrinks individual effects to group coefficients, by Ando
and Bai,  (2016) who consider unobserved group factor structures in linear
models with interactive fixed effects, and finally by Wang et al.,  (2023),
studying group structures combined with structural breaks.

Discrete unobserved heterogeneity can serve as a regularization device that
allows to identify the parameters of interest in panel data models with time-
varying individual effects but not necessarily characterized by a factor
structure. In this vein, Bonhomme and Manresa,  (2015) introduce a GFE
estimator for linear models where the discrete heterogeneity is assumed to
follow time-varying grouped patterns and cluster membership is left
unrestricted. By contrast, the TW-GFE estimator by Bonhomme et al., 2022a is
consistent even with unspecified forms of time-varying unobserved
heterogeneity. While using discretization as an approximation device
introduces an asymptotic bias, the function of the unobserved heterogeneity
they consider encompasses a variety of specifications, such as additive and
interactive effects, under minimal distributional assumptions. This makes the
TW-GFE estimator a simple and potentially very attractive tool for
practitioners.

Outline The rest of the paper is organized as follows: Section 2 briefly
describes the models and estimators; Section 3 reviews the assumptions
required to characterize the asymptotic distribution of the TW-GFE,
illustrates the proposed approach and the asymptotic behavior of the resulting
test statistic, and finally briefly illustrates the alternative testing
procedures; Section 4 presents the results of the simulation study in both
linear and probit cases and discusses violation of relevant assumptions;
Section 5 illustrates the two empirical applications; Finally, Section 6
concludes.

##  2 Models and estimators

Consider a panel data setup where subjects are indexed by $i=1,\ldots,N$ and
time occasions are indexed by $t=1,\ldots,T$. Throughout the paper, we assume
that observations are independent, conditional on the observed covariates and
unobserved heterogeneity, and that the models are static. The traditional
specification of fixed-effects models depicts unobserved heterogeneity as
individual-specific intercepts, so that the conditional distribution of the
response variable $y_{it}$ given an $r$-vector of exogenous covariates
$x_{it}$ is of the type

$y_{it}|x_{it},\theta_{0},\alpha_{i0}\sim
f(y_{it}|x_{it}^{\prime}\theta_{0}+\alpha_{i0}),\quad$ (1)

where $\theta_{0}$ is the vector of parameters of interest, $\alpha_{i0}$
denotes the permanent individual effect, and $f(\cdot)$ is a generic known
density function, as in Chen et al.,  (2021). When (1) is a linear regression
model, consistent OLS estimators of $\theta$ can be trivially obtained on the
basis of standard de-meaning or first-differences transformations, whereas ML
estimators in non-linear models are consistent but exhibit a bias in their
limiting distribution under rectangular array asymptotics (Li et al., , 2003),
unless probability formulations admit sufficient statistics for the individual
intercepts (Andersen, , 1970; Chamberlain, , 1980). Therefore bias reduction
techniques, such as analytical or jackknife corrections, are required (Hahn
and Newey, , 2004). These estimators are usually referred to as the one-way
fixed-effects (OW-FE) estimators.

In order to account for time-varying heterogeneity, the widespread approach is
to include common time effects, that enter the specification in an additive
manner. The model is then of the type

$y_{it}|x_{it},\theta_{0},\alpha_{i0},\zeta_{t0}\sim
f(y_{it}|x_{it}^{\prime}\theta_{0}+\alpha_{i0}+\zeta_{t0}),$ (2)

where $\zeta_{t0}$ represents such time-varying heterogeneity. Similarly to
the case with only individual effects, a consistent estimator of $\theta$ can
be obtained under suitable transformations when a linear regression model is
specified, while bias corrections have to be implemented for ML estimators
(Fernández-Val and Weidner, , 2016). We denote them as two-way fixed-effects
(TW-FE) estimators.

In this paper, we use the TW-GFE estimator to contrast with the OW-FE and TW-
FE estimators so as to perform specification tests and possibly detect more
sophisticated structures for the unobserved heterogeneity. Consider the
following model formulation

$y_{it}|x_{it},\theta_{0},\alpha_{it0}\sim
f(y_{it}|x_{it}^{\prime}\theta_{0}+\alpha_{it0}).$ (3)

According to Bonhomme et al., 2022a , the time-varying unobserved
heterogeneity $\alpha_{it0}$ is characterized by two vectors $\xi_{i0}$ and
$\lambda_{t0}$, and a function $\alpha$$(\cdot)$, satisfying requirements that
will be discussed later in more detail, such that
$\alpha_{it0}=\alpha(\xi_{i0},\lambda_{t0})$. This characterization of
$\alpha_{it0}$ can be easily reconciled with the structures for the unobserved
heterogeneity in models (1) and (2) as follows:

$\alpha_{it0}:\left\\{\begin{array}[]{lc}\alpha_{i0}\equiv\alpha(\xi_{i0})&\mathrm{in\quad\eqref{eq:ti}}\\\\[2.0pt]
\alpha_{i0}+\zeta_{t0}\equiv\alpha(\xi_{i0},\lambda_{t0})&\mathrm{in\quad\eqref{eq:add}}\\\\[2.0pt]
\alpha_{it0}\equiv\alpha(\xi_{i0},\lambda_{t0})&\mathrm{in\quad\eqref{eq:tv}}\\\
\end{array}\right.$

It is also important to the GFE strategy that covariates are affected by the
same source of heterogeneity, so that $x_{it}$ depends on $\mu_{it0}$, where
$\mu_{it0}=\mu(\xi_{i0},\lambda_{t0})$, with $\mu(\cdot)$ satisfying the same
requirements as $\alpha(\cdot)$.

The first-step estimation of the TW-GFE approach deals with the classification
of subjects and time occasions into two different sets of groups. It is worth
to stress that clustering here serves as an approximation tool for the
unobserved heterogeneity, so that there is no number of clusters to be known
_a priori_. As a consequence, groups should not be intended as aggregation
levels coming from external information (e.g. sectors for firms, see also
Papke and Wooldridge, , 2023). Classification relies on performing kmeans
clustering twice, using the vectors of moments
$h_{i}=\frac{1}{T}\sum_{t=1}^{T}h(y_{it},x_{it})$ and
$w_{t}=\frac{1}{N}\sum_{i=1}^{N}w(y_{it},x_{it})$ of fixed dimensions. Both
vectors have to be _informative_ about $\xi_{i0}$ and $\lambda_{t0}$,
respectively, meaning that $\xi_{i0}$ can be uniquely recovered from $h_{i}$
for large $T$ and $\lambda_{t0}$ can be uniquely recovered from $w_{t}$ for
large $N$. The two kmeans clustering procedures return a number of $K$ groups
for the subjects and a different number of $L$ groups for the time occasions,
from which two sets of dummies identifying the related group memberships are
created. In the second step, cluster dummies for the cross-sectional and time
dimensions are then interacted and enter the linear index of the model
specified for the response variable as $KL$ group fixed effects. Estimation is
then carried out by ML.

##  3 Specification tests

We propose a generalized Hausman test for the specification of the unobserved
heterogeneity considering, as null hypotheses, the models portrayed by
Equations (1) and (2). The OW-FE and TW-FE estimators are consistent, with an
asymptotic bias in case of nonlinear models. The TW-GFE estimator is also
consistent but always asymptotically biased. In the presence of more
sophisticated forms of unobserved heterogeneity – different from those in (1)
and (2) – such as a factor structure, only the TW-GFE estimator is consistent.
In the following, we first outline the assumptions for the asymptotic results
needed to derive the test statistic. We then describe the proposed approach
and, finally, briefly recall the existing procedures to test for time-varying
unobserved heterogeneity.

###  3.1 Assumptions and supporting results

To derive our main results we make the following assumptions, that recall
those in Bonhomme et al., 2022a ; Bonhomme et al., 2022b .

######  Assumption 1.

Unobserved Heterogeneity: (i) There exist $\xi_{i0}$ of fixed dimension
$d_{\xi}$ and $\lambda_{t0}$ of fixed dimension $d_{\lambda}$ and two
functions $\alpha(\cdot)$ and $\mu(\cdot)$ that are Lipschitz-continuous in
both arguments, such that $\alpha_{it0}=\alpha(\xi_{i0},\lambda_{t0})$ and
$\mu_{it0}=\mu(\xi_{i0},\lambda_{t0})$; (ii) the supports of $\xi_{i0}$ and
$\lambda_{t0}$ are compact.

######  Assumption 2.

Moment informativeness: There exist moment vectors

$h_{i}=(1/T)\textstyle{\sum_{t}}h(y_{it},x_{it}^{\prime})\quad\text{and}\quad
w_{t}=(1/N)\textstyle{\sum_{i}}w(y_{it},x_{it}^{\prime})$

of fixed dimension, and two unknown Lipschitz-continuous function $\phi$ and
$\psi$, such that

$\underset{T\to\infty}{\mathrm{plim}}h_{i}=\phi(\xi_{i0})\quad\text{and}\quad\underset{N\to\infty}{\mathrm{plim}}w_{t}=\psi(\lambda_{t0}),$

and $\frac{1}{N}\|\sum_{i=1}^{N}h_{i}-\phi(\xi_{i0})\|^{2}=O_{p}(1/T)$,
$\frac{1}{T}\|\sum_{i=t}^{T}w_{t}-\psi(\lambda_{t0})\|^{2}=O_{p}(1/N)$ as
$N,T\to\infty$.

######  Assumption 3.

Asymptotics: as $N,T\rightarrow\infty$, $N/T\rightarrow\rho^{2}$, with
$0<\rho<\infty$.

######  Assumption 4.

Sampling: (i) $(y_{it},x^{\prime}_{it})^{\prime}$, for $i=1,\dots N$ and
$t=1,\dots T$, are i.i.d. given $\xi_{i0}$ and $\lambda_{t0}$; (ii) $\xi_{i0}$
and $\lambda_{t0}$ are also i.i.d.

######  Assumption 5.

Regularity: Let $\ell_{it}(\alpha_{it},\theta)$= ln
$f(y_{it}|x_{it},\alpha_{it},\theta)$ and let
$\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}\ell_{it}(\bar{\alpha}(\theta,\xi_{i0},\lambda_{t0}),\theta)$
be the target log-likelihood (Arellano and Hahn, , 2007) and
$\bar{\alpha}(\theta,\xi,\lambda)=\underset{\alpha}{\operatorname{argmax}}\;\mathbb{E}_{\xi_{i0}=\xi,\lambda_{t0}=\lambda}(\ell_{it}(\alpha,\theta))$:
(i) $\ell_{it}(\theta,\alpha)$ is three time differentiable in
$(\theta,\alpha)$; $\theta_{0}$ is an interior point of the parameter space
$\Theta$; $\Theta$ is compact;
(ii) $\ell_{it}$ is strictly concave as a function of $\alpha$,
$\emph{inf}_{\xi,\lambda,\theta}\mathbb{E}_{\xi_{i0}=\xi,\lambda_{t0}=\lambda}\left(-\frac{\partial^{2}\ell_{it}(\bar{\alpha}(\theta,\xi,\lambda),\theta)}{\partial\alpha\partial\alpha^{\prime}}\right)>0;\\\
\mathbb{E}[\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}\ell_{it}(\bar{\alpha}(\theta,\xi_{i0},\lambda_{t0}),\theta)]$
has a unique maximum at $\theta_{0}$ on $\Theta$, and its second derivative is
negative definite.
(iii) Regularity conditions on boundedness of moments and asymptotic
covariances in Bonhomme et al., 2022b Assumption S2 (iv,v) apply.

Assumption 1 gives the minimal properties of the unobserved heterogeneity in
the Bonhomme et al., 2022a ’s setting. Assumption 2 formalizes moments
informativeness in order to have an effective individual and time clustering.
Assumption 3 depicts rectangular array asymptotics, which is required for the
characterization of the asymptotic normal distribution of the considered
estimators. Assumption 4 outlines the sampling requirements that are more
restrictive than that usually required to characterize the asymptotic
distribution of ML estimators under rectangular-array asymptotics for fixed-
effects models with time heterogeneity. For example, Fernández-Val and
Weidner,  (2016) assume independence over $i$ while relaxing time independence
by allowing for $\alpha$-mixing.666See Fernández-Val and Weidner,  (2016),
Assumption 4.1 (ii). Assumption 4 is instead required for consistency of the
TW-GFE, which effectively rules out the possibility of applying the proposed
test to models with (i) feedback effects and (ii) unobserved heterogeneity
that depends on dynamic factors. The conditions stated in Assumption 5 are
standard requirements for a well-posed maximization problem.

Under Assumptions 1,4, and 5 the OW-FE and TW-FE estimators of $\theta$,
$\hat{\theta}$, for models (1) and (2), respectively, are consistent as
$N,T\to\infty$. Additionally under Assumption 3, $\hat{\theta}$ has the
following asymptotic distribution

$\sqrt{NT}(\hat{\theta}-\theta_{0})\stackrel{{\scriptstyle
d}}{{\rightarrow}}N(B;\,\,I(\theta_{0})^{-1}),$

where $B$ is constant and equal to $\rho C$ for the OW-FE estimator, while it
is equal to $\rho C_{1}+\rho^{-1}C_{2}$ for the TW-FE estimator. Notice that,
in the case of informational orthogonality between the structural and nuisance
parameters, such as in the linear model, $B=0$, whereas characterizations of
these asymptotic biases are given in Hahn and Newey,  (2004) and Fernández-Val
and Weidner,  (2016) for nonlinear models with OW-FE and TW-FE, respectively.
Finally, $I(\theta_{0})$ is the Information matrix of the profile log-
likelihood.

Consistency of the TW-GFE estimator $\tilde{\theta}$ relies on Lemma S1 in
Bonhomme et al., 2022b , based on Assumptions 1,2,4, and 5. The two-step TW-
GFE estimator is then proved to have asymptotic expansion (Corollary S2
_ibidem_)

$\tilde{\theta}=\theta_{0}+J(\theta_{0})^{-1}\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}s_{it}(\theta_{0})+O_{p}\left(\frac{1}{T}+\frac{1}{N}+\frac{KL}{NT}\right)+O_{p}(K^{-\frac{2}{d_{\xi}}}+L^{-\frac{2}{d_{\lambda}}})+o_{p}\left(\frac{1}{\sqrt{NT}}\right),$
(4)

as $N,T,K,L\rightarrow\infty$, such that $KL/(NT)$ tends to zero. In the above
expression, $J(\cdot)$ and $s_{it}(\cdot)$ are the negative expected Hessian
and the score associated with the likelihood function. Three main different
sources of bias can be identified: the $1/T$ and $1/N$ terms depend on the
number of time occasions and subjects used for $h_{i}$ and $w_{t}$ in the
classification step; the $KL/NT$ term reflects the estimation of $KL$ group-
specific parameters using $NT$ observations; the
$K^{-\frac{2}{d_{\xi}}}+L^{-\frac{2}{d_{\lambda}}}$ terms refer to the
approximation bias arising from the discretization of $\xi_{i0}$ and
$\lambda_{t0}$ via kmeans.

The $O_{p}(\cdot)$ terms in the above expansion can be shown to become
$O_{p}(1/T+1/N)$ under suitable choices for the number of groups, $K$ and $L$,
and for $d_{\xi}=d_{\lambda}=1$. The rule suggested by Bonhomme et al., 2022a
and the consequent simplification of the $O_{p}(\cdot)$ terms are summarized
in the following proposition.

######  Proposition 1.

Number of groups and approximation bias:
i) For $d_{\xi}=d_{\lambda}=1$, the number of groups $K$ and $L$ are chosen
according to the following rules

$\hat{K}=\min_{K\geq
1}\\{K:\hat{Q}(K)\leq\gamma\hat{V}_{h_{i}}\\},\qquad\hat{L}=\min_{L\geq
1}\\{L:\hat{Q}(L)\leq\gamma\hat{V}_{w_{t}}\\},$

where $Q(\cdot)$ is the objective function of the kmeans problem, $V_{h}$ and
$V_{w}$ are the variability of the moments $h_{i}$ and $w_{t}$, respectively,
and $\gamma\in(0,1]$ is a user-specified parameter;
ii) Setting $K$ and $L$ proportional or greater than $\sqrt{T}$ and
$\sqrt{N}$, respectively, will ensure that the approximation errors are
$O_{p}(1/T)$ and $O_{p}(1/N)$, so that the $O_{p}(\cdot)$ terms in (4) become
$O_{p}(1/T+1/N)$.

We refer the reader to Bonhomme et al., 2022a for the proofs and derivations
of these results. Smaller values of $\gamma$ yield a larger number of groups:
lowering this value is suggested if moments are weakly informative about
unobserved heterogeneity. In our simulation study we experiment with different
values of $\gamma$.

In order to derive the asymptotic distribution of the TW-GFE estimator, we
need to provide a minimal characterization of the $O_{p}(1/T+1/N)$ term in
said asymptotic expansion.

######  Assumption 6.

Bias of the TW-GFE estimator: The $O_{p}(1/T+1/N)$ term takes the form

$\frac{D_{1}}{T}+\frac{D_{2}}{N}+o_{p}\left(\frac{1}{T}\vee\frac{1}{N}\right),$

where $D_{1}$ and $D_{2}$ are constant.

This assumption extends Corollary 2 in Bonhomme et al., 2022a , according to
which the $O_{p}(1/T)$ term in the asymptotic expansion for the one-way GFE
estimator with only time-constant unobserved heterogeneity is
$E/T+o_{p}(1/T)$, where $E$ is constant. The asymptotic distribution of
$\tilde{\theta}$ can now be characterized by the following theorem, the proof
of which follow from standard arguments of ML estimation.

######  Theorem 1.

Suppose that Assumptions 1-6 and Lemma S1 and Corollary S2 of Bonhomme et al.,
2022b hold, and let $d_{\xi}=d_{\lambda}=1$ then

$\sqrt{NT}(\tilde{\theta}-\theta_{0})\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(D;\,\,J(\theta_{0})^{-1}\right),$

where $D=D_{1}\rho+D_{2}\rho^{-1}$.

###  3.2 Proposed test

In order to test the null hypothesis of correctly specified unobserved
heterogeneity, we rely on a Hausman test based on the difference
$\hat{\delta}=\hat{\theta}-\tilde{\theta}$, namely, by contrasting FE
estimators with the TW-GFE estimator. We examine two cases. In the first one,
the null hypothesis $H_{0}$ is of time-constant unobserved heterogeneity,
under which both the OW-FE and TW-GFE estimators are consistent. In the second
case, under $H_{0}$ the unobserved heterogeneity has an additive structure,
for which both the TW-FE and TW-GFE estimators are consistent. In both
situations, the null hypothesis can therefore be expressed as

$H_{0}:\underset{N,T\rightarrow\infty}{\mathrm{plim}}\hat{\delta}=0$

Instead, only the TW-GFE estimator is consistent for the true parameters under
the alternative hypothesis of a more complex structure of the unobserved
heterogeneity such as, for instance, the factor one. This justifies the use of
an Hausman-type test.

Our test statistic is therefore

$\hat{H}=NT\hat{\delta}^{\prime}\widehat{W}^{-1}\hat{\delta}$ (5)

where $\widehat{W}$ is a consistent estimator of the variance of the contrasts
$\hat{\delta}$, $W$. The asymptotic distribution of $\hat{H}$ is determined by
the distribution of $\hat{\delta}$. Under the assumption of joint normality of
$\hat{\theta},\tilde{\theta}$ and Assumption 3, with $H_{0}$ true we have that

$\sqrt{NT}\,\,\hat{\delta}\stackrel{{\scriptstyle d}}{{\rightarrow}}N(B-D,W).$

As a result, we have that the limiting distribution of $H$ is a $\chi^{2}$
with $r$ degrees of freedom and non-centrality parameter
$\omega=\delta^{\prime}\delta$, with $\delta=B-D$, entailed by the asymptotic
biases in $\hat{\theta}$ and $\tilde{\theta}$, that is

$\hat{H}\stackrel{{\scriptstyle d}}{{\rightarrow}}\chi^{2}_{r,\omega}.$

A test for $H_{0}$ based on $H$ poses two problems. First, there is no
guarantee that using the traditional formulation of the Hausman test (Hausman,
, 1978) will provide a positive definite $\widehat{W}$, as the test always
involves at least one biased estimator. Secondly, quantiles of the limiting
distribution of $\hat{H}$ are unknown.

In order to tackle the first issue, we rely on the generalized formulation for
the variance of $\hat{\delta}$, put forward by Bartolucci et al.,  (2015). Let
us define $\hat{\phi}$ as the complete vector of estimated parameters for FE
models, including $\hat{\theta}$, OW or TW fixed-effects, and also an
estimator of the error term variance in linear models. Similarly we denote
$\tilde{\phi}$ the complete vector of estimated parameters for the TW-GFE. Let
us denote the estimator of the variance of $\hat{\delta}$ as

$\widehat{W}=NT\left[M\widehat{V}(\hat{\phi},\tilde{\phi})M^{\prime}\right],$

where $M$ is a selector matrix of suitable dimension such that it returns the
difference between the blocks in $\widehat{V}(\hat{\phi},\tilde{\phi})$
corresponding to the variance-covariance matrices of $\hat{\theta}$ and
$\tilde{\theta}$. Moreover

$\displaystyle\hat{V}(\hat{\phi},\tilde{\phi})=\begin{pmatrix}H(\hat{\phi})&0\\\
0&H(\tilde{\phi})\\\
\end{pmatrix}^{-1}S\left(\hat{\phi},\tilde{\phi}\right)\begin{pmatrix}H(\hat{\phi})&0\\\
0&H(\tilde{\phi})\\\ \end{pmatrix}^{-1},$

with

$\displaystyle
S\left(\hat{\phi},\tilde{\phi}\right)=\sum_{i=1}^{N}\sum_{t=1}^{T}\begin{pmatrix}g_{it}(\hat{\phi})\\\
s_{it}(\tilde{\phi})\\\ \end{pmatrix}\left(g_{it}(\hat{\phi})^{\prime},\quad
s_{it}(\tilde{\phi})^{\prime}\right),$

where $H(\hat{\phi})$ and $H(\tilde{\phi})$ are the hessian matrices
associated to the complete parameter vector for FE and TW-GFE respectively,
$g_{it}(\hat{\phi})$ and $s_{it}(\tilde{\phi})$ are the scores of the log-
likelihoods associated to the OW-FE (or TW-FE) and TW-GFE approaches,
respectively.

The second issue concerns the limiting distribution of $\hat{H}$. In order
recover the quantiles of the chi-square with unknown centrality parameter, we
rely on parametric percentile bootstrap, which is used in presence of non-
pivotal statistics (Horowitz, , 2019). In practice, after obtaining the ML
estimates $\hat{\phi}$, $\tilde{\phi}$, and computing $\hat{H}$ on the real
data, we generate the bootstrap samples using $\hat{\phi}$ as the true
parameter values. Then the bootstrap statistic $\hat{H}^{\ast}$ is

$\hat{H}^{\ast}=NT\hat{\delta}^{\ast}\left(\widehat{W}^{\ast}\right)^{-1}\hat{\delta}^{\ast}\stackrel{{\scriptstyle
d^{\ast}}}{{\rightarrow}}\chi^{2}_{r,\omega},$

where $\hat{\delta}^{\ast}=\hat{\phi}^{\ast}-\tilde{\phi}^{\ast}$ uses the ML
estimates obtained from the bootstrap sample and $\stackrel{{\scriptstyle
d^{\ast}}}{{\rightarrow}}$ denotes convergence in distribution of the
bootstrap measure. Therefore, percentiles can be obtained as

$q^{\ast}_{1-\alpha}=\mathrm{inf}\left\\{q^{\ast}:\mathrm{Pr}^{\ast}\left(\hat{H}^{\ast}\leq
q^{\ast}\right)\geq(1-\alpha)\right\\},$

where $\mathrm{Pr}^{\ast}$ denotes the probability conditional on the
bootstrap sample. Therefore we reject $H_{0}$ whenever
$\hat{H}>q^{\ast}_{1-\alpha}$. Our approach is related to the contributions by
Higgins and Jochmans,  (2023) and Kim and Sun,  (2016), who exploit parametric
bootstrap for inference in (dynamic) fixed-effects models.777These authors
provide results on the asymptotic validity of the bootstrap bias correction in
OW-FE models. The same result for TW-FE and TW-GFE does not follow from our
assumptions. A formal derivation is beyond the scope of the paper, whose
intent is not to provide a bias correction, nor inference on the model
parameters. Nevertheless, applications of the parametric bootstrap for testing
purposes in two-way fixed-effects are available (Dzemski, , 2019). An
alternative approach based on pre-pivoting and pairs bootstrap is proposed by
Cavaliere et al.,  (2022); in our setting, though, it seems unnecessary to
perform a bias correction only to carry out a pre-test.

###  3.3 Alternative procedures

The performance of the proposed test can be compared with that of two
alternative tools: max-type test put forward by Castagnetti et al., 2015a (CRT
test henceforth) to detect factor structures in a linear framework and the
test for time-invariant unobserved heterogeneity developed by Bartolucci et
al.,  (2015) (BBP test) that applies to both linear and non-linear frameworks.

####  3.3.1 CRT test

Consider a model such as (3) in which $\alpha_{it0}$ has a factor structure,
that is $\alpha(\xi_{i0},\lambda_{t0})=\alpha_{i0}^{\prime}\zeta_{t0}$.

The procedure tests the null hypothesis of no factor structure, defined as
$H_{0}:\zeta_{t}=\zeta$, that is a model with only individual effects. The
max-type test statistics is formulated as

$S=\text{max}_{1\leq t\leq
T}\left[N(\hat{\zeta}_{t}-\hat{\bar{\zeta}})^{\prime}\hat{\Sigma}_{t}^{-1}(\hat{\zeta}_{t}-\hat{\bar{\zeta}})\right],$

where factors are estimated using the common correlated effects approach by
Pesaran,  (2006),888It is worth recalling that the approach by Castagnetti et
al., 2015a can in general be implemented in models with heterogeneous slopes.
$\hat{\bar{\zeta}}$ is the sample mean of $\hat{\zeta}_{t}$ and the
$\hat{\Sigma}_{t}$ is and estimate of the asymptotic factor covariance matrix
(crf Equation 10 in Castagnetti et al., 2015a, ). The test statistic $S$ has
an asymptotic Gumbel distribution. It is worth noting that CRT test requires
large $T$ settings in order attain the correct size in finite samples.

####  3.3.2 BBP test

Differently from CRT test, the BBP test can be employed with generalized
linear models models. As the test here proposed, the BBP is also a generalized
Hausman test contrasting estimators that are consistent only under time-
constant unobserved heterogeneity with estimators that are consistent even in
presence of a time-varying latent variable.

For linear models, the BBP test contrasts the OW-FE estimator with the first
difference estimator. It is worth stressing that the BBP test has power only
when certain conditions are met, namely $T$ must be greater than 3, otherwise
the estimators coincides, and common factors must have a dynamic structure.
For binary choice and Poisson models, the BBP test compares estimators based
on two different formulation of the conditional likelihood: the standard
conditional ML (CML) estimator and the Pairwise CML estimator (PCML). The
former is consistent with time-constant heterogeneity since the probabilities
for the models considered admit sufficient statistics for the incidental
parameters. The PCML approach considers pairs of consecutive time observations
for every individual and the corresponding log-likelihood is conditioned on
the sufficient statistic, thereby allowing for different individual effects in
every couple of periods. The PMCL estimator is therefore consistent in
presence of time varying unobserved heterogeneity. This test also lacks power
in the same cases described above.

##  4 Simulation study

In the following, we describe the design and report the results of an
extensive simulation study where we investigate the test empirical size and
power properties for the linear and probit model.

###  4.1 Linear model

We design a Monte Carlo experiment where observations are generated by a
linear regression model with two exogenous covariates. We consider two
scenarios for the null hypotheses, where the unobserved heterogeneity is
specified as in models (1) and (2).

In particular, in the case of only individual time-constant effects, for
$i=1,\ldots,N$ and $t=1,\ldots,T$, we generate samples according to the
following equations, which we denote as DGP-FE-L:

$\displaystyle y_{it}$ $\displaystyle=$ $\displaystyle
x_{it1}\theta_{1}+x_{it2}\theta_{2}+\alpha_{i}+\varepsilon_{it},$
$\displaystyle x_{itj}$ $\displaystyle=$
$\displaystyle\Gamma_{i}+N(0,1),\quad\mathrm{for}\quad j=1,2,$

where $\alpha_{i}=\varrho\Gamma_{i}+\sqrt{(1-\varrho^{2})}A_{i}$ , with
$A_{i},\Gamma_{i}\sim N(1,3)$, and $\varrho=0.5$. Finally, $\varepsilon_{it}$
is an idiosyncratic standard normal error term. We let the coefficients
$\theta=(\theta_{1},\theta_{2})^{\prime}$ be equal to $(1,1)^{\prime}$. With
this design we explore the size properties of the proposed test comparing the
OW-FE with the TW-GFE estimators. Similarly, when we allow for additive
individual and time effects, samples are generated according to:

$\displaystyle y_{it}$ $\displaystyle=$ $\displaystyle
x_{it1}\theta_{1}+x_{it2}\theta_{2}+\alpha_{i}+\zeta_{t}+\varepsilon_{it},$
$\displaystyle x_{itj}$ $\displaystyle=$
$\displaystyle\Gamma_{i}+\zeta_{t}+N(0,1),\quad\mathrm{for}\quad j=1,2,$

where $\zeta_{t}\sim N(1,1)$. This design is denoted by DGP-AFE-L and it will
be used to contrast the TW-FE with the TW-GFE estimators under the null
hypothesis of additive effects.

In order to investigate the power of the proposed test, the scenario generated
under the alternative hypothesis is a linear panel data model with interactive
fixed effects. Specifically, samples are generated according to a simplified
version of the design outlined by Bai,  (2009), with one latent factor:

$\displaystyle y_{it}$ $\displaystyle=$ $\displaystyle
x_{it1}\theta_{1}+x_{it2}\theta_{2}+\alpha_{i}\zeta_{t}+\varepsilon_{it},$
$\displaystyle x_{it}$ $\displaystyle=$
$\displaystyle\Gamma_{i}\zeta_{t}+N(0,1).$

We denote the above design as DGP-IFE-L.

For each scenario, we consider $N=50,100$, $T=10,20$, and $399$ bootstrap
draws for each of the $1000$ Monte Carlo replications. It is worth recalling
that the performance of the TW-GFE estimator is closely linked to the number
of groups chosen for the first-step kmeans clusterings. Even following the
rule outlined in Proposition 1, this number depends on the variability in the
data, which affects how informative $h_{i},w_{t}$ are about the unobserved
heterogeneity, and the user-defined parameter $\gamma$. We account for the
former by allowing for large variances in the composite error term, and for
the latter by running scenarios where $\gamma=1,0.5,0.25$, resulting in an
increasing number of clusters.

Tables 1, 2, 3 and 4 report the results of the Monte Carlo experiments in four
cases. First we compare the OW-FE and the TW-GFE estimators using DGP-FE-L to
study the empirical size, while the power of the proposed test is investigated
under the alternative process described by DGP-IFE-L. We then turn to the
comparison between the TW-FE and the TW-GFE estimators in the setting where
heterogeneity is specified as additive effects under the null hypothesis in
DGP-AFE-L and that of DGP-IFE-L under the alternative. The tables report the
average of the Hausman test $H$ in (5) across simulations, along with the
empirical size based on the quantile of the central $\chi^{2}$ with two
degrees of freedom ($\chi^{2}_{2}$ p.05) and the bootstrap rejection rate
(Boot p.05). We also present the average bias (Bias), standard deviation (SD)
and ratio between standard error and SD for both elements of $\hat{\theta}$
and $\tilde{\theta}$. For the TW-GFE estimator, we also report the average
selected number of groups according to rule outlined in Proposition 1 (Avg
$\hat{K}$ and Avg $\hat{L}$).

As expected, the empirical rejection rate based on the percentile of the
centered chi-square distribution does not attain the nominal size (5%), as it
fails to account for the noncentrality parameter arising from the
approximation bias in the TW-GFE. It is in fact worth to notice that, even
with the large number of groups obtained with $\gamma=0.05$, the bias of the
TW-GFE is sensitively larger than that of the OW-FE, and TW-FE estimators.
Indeed the bias of the TW-GFE worsens with fewer clusters, as testified by the
results with $\gamma=0.5,1$ and the resulting values of the Hausman test. We
highlight that the $\gamma$ parameter seems to have no effect on the test
value. Nonetheless, the bootstrap distribution is able to mimic the non-
centrality of the chi-square, giving rise to a rejection rate close to the
nominal one, with a improving performance as $T$ increases. The bootstrap test
also presents good rejection rates when the true data generating process has a
factor structure for the unobserved heterogeneity. Increasing the number of
individuals or enriching information embodied in factors may further increase
the power of our test.

Table 1: Size analysis: DGP-FE-L, OW-FE vs TW-GFE

|  | $\gamma=0.25$ | $\gamma=0.5$ | $\gamma=1$
---|---|---|---|---
|  | T=10 | T=20 | T=10 | T=20 | T=10 | T=20
| $N$ |  $50$ $100$ |  $50$ $100$ |  $50$ $100$ |  $50$ $100$ |  $50$ $100$ |  $50$ $100$
Hausman |  |  |  |  |  |  |  |  |  |  |  |  |
H | 4.251 | 5.932 | 5.478 | 7.814 | 6.385 | 10.055 | 8.558 | 12.390 | 9.742 | 16.719 | 13.061 | 21.244
$\chi^{2}_{2}$ p.05 | 0.248 | 0.379 | 0.301 | 0.481 | 0.401 | 0.643 | 0.478 | 0.673 | 0.569 | 0.818 | 0.638 | 0.832
| Boot p.05 | 0.051 | 0.044 | 0.052 | 0.048 | 0.053 | 0.045 | 0.062 | 0.031 | 0.053 | 0.042 | 0.049 | 0.051
$\hat{\theta}_{1}$ |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | 0.000 | -0.001 | -0.000 | 0.000 | 0.000 | -0.001 | -0.000 | 0.000 | 0.000 | -0.001 | -0.000 | 0.000
SD | 0.048 | 0.033 | 0.032 | 0.023 | 0.048 | 0.033 | 0.032 | 0.023 | 0.048 | 0.033 | 0.032 | 0.023
| SE/SD | 0.935 | 0.967 | 0.985 | 0.966 | 0.935 | 0.967 | 0.985 | 0.966 | 0.935 | 0.967 | 0.985 | 0.966
$\hat{\theta}_{2}$ |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | -0.000 | -0.000 | -0.000 | -0.000 | -0.000 | -0.000 | -0.000 | -0.000 | -0.000 | -0.000 | -0.000 | -0.000
SD | 0.048 | 0.034 | 0.033 | 0.023 | 0.048 | 0.034 | 0.033 | 0.023 | 0.048 | 0.034 | 0.033 | 0.023
| SE/SD | 0.933 | 0.921 | 0.960 | 0.958 | 0.933 | 0.921 | 0.960 | 0.958 | 0.933 | 0.921 | 0.960 | 0.958
$\tilde{\theta_{1}}$ |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | -0.025 | -0.024 | -0.015 | -0.014 | -0.044 | -0.046 | -0.028 | -0.026 | -0.077 | -0.080 | -0.049 | -0.050
SD | 0.057 | 0.038 | 0.039 | 0.027 | 0.067 | 0.044 | 0.047 | 0.032 | 0.082 | 0.056 | 0.057 | 0.039
| SE/SD | 0.860 | 0.907 | 0.863 | 0.893 | 0.784 | 0.844 | 0.757 | 0.776 | 0.715 | 0.734 | 0.664 | 0.689
$\tilde{\theta_{2}}$ |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | -0.025 | -0.024 | -0.015 | -0.015 | -0.045 | -0.045 | -0.028 | -0.026 | -0.078 | -0.080 | -0.049 | -0.050
SD | 0.058 | 0.040 | 0.041 | 0.028 | 0.068 | 0.045 | 0.048 | 0.033 | 0.081 | 0.055 | 0.058 | 0.039
SE/SD | 0.847 | 0.870 | 0.818 | 0.852 | 0.768 | 0.828 | 0.732 | 0.751 | 0.719 | 0.754 | 0.657 | 0.696
Avg$K$ | 41.222 | 74.980 | 43.708 | 81.696 | 35.725 | 60.883 | 39.421 | 69.948 | 28.476 | 44.061 | 33.373 | 54.968
|  Avg$L$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000

1000 Monte Carlo (MC) replications. “$H$” is the average of the Hausman test
statistic, across MC replications. “p.05” denotes the rejection rate for a
nominal size of 5%. “$\chi^{2}_{2}$ p.05” is based on the $95^{th}$ percentile
of the central $\chi^{2}_{2}$ distribution. “Boot p.05” is based on the
$95^{th}$ percentile of the empirical distribution determined via 399
bootstrap replications. “Bias” is the mean bias, “SD” and “SE” denote the
standard deviation over the MC replications and the average estimated standard
error, respectively, those are reported for two regressors. “Avg $\hat{K}$”
and “Avg $\hat{L}$” report the average number of groups for individuals and
time occasions obtained in the first step.

Table 2: Power analysis: DGP-IFE-L, OW-FE vs TW-GFE

|  | $\gamma=0.25$
---|---|---
|  | T=10 | T=20
| $N$ |  $50$ $100$ |  $50$ $100$
Hausman |  |  |  |  |
H | 32.330 | 74.524 | 53.350 | 125.246
$\chi^{2}_{2}$ p.05 | 0.908 | 0.994 | 0.961 | 0.998
| Boot p.05 | 0.682 | 0.889 | 0.838 | 0.960
$\hat{\theta}_{1}$ |  |  |  |  |
Bias | 0.260 | 0.250 | 0.261 | 0.255
SD | 0.107 | 0.079 | 0.085 | 0.060
| SE/SD | 0.826 | 0.792 | 0.738 | 0.741
$\hat{\theta}_{2}$ |  |  |  |  |
Bias | 0.252 | 0.257 | 0.257 | 0.256
SD | 0.108 | 0.078 | 0.083 | 0.058
| SE/SD | 0.816 | 0.804 | 0.761 | 0.770
$\tilde{\theta}_{1}$ |  |  |  |  |
Bias | -0.047 | -0.132 | -0.008 | -0.085
SD | 0.184 | 0.157 | 0.142 | 0.119
| SE/SD | 0.498 | 0.420 | 0.404 | 0.347
$\tilde{\theta}_{2}$ |  |  |  |  |
Bias | -0.049 | -0.132 | -0.008 | -0.085
SD | 0.187 | 0.162 | 0.141 | 0.118
| SE/SD | 0.490 | 0.409 | 0.407 | 0.350
|  Avg$K$ | 15.122 | 19.981 | 19.552 | 26.773
|  Avg$L$ | 4.531 | 5.523 | 5.753 | 7.376

1000 Monte Carlo (MC) replications. “$H$” is the average of the Hausman test
statistic, across MC replications. “p.05” denotes the rejection rate for a
nominal size of 5%. “$\chi^{2}_{2}$ p.05” is based on the $95^{th}$ percentile
of the central $\chi^{2}_{2}$ distribution. “Boot p.05” is based on the
$95^{th}$ percentile of the empirical distribution determined via 399
bootstrap replications. “Bias” is the mean bias, “SD” and “SE” denote the
standard deviation over the MC replications and the average estimated standard
error, respectively, those are reported for two regressors. “Avg $\hat{K}$”
and “Avg $\hat{L}$” report the average number of groups for individuals and
time occasions obtained in the first step.

Table 3: Size analysis: DGP-AFE-L, TW-FE vs TW-GFE

|  | $\gamma=0.25$ | $\gamma=0.5$ | $\gamma=1$
---|---|---|---|---
|  | T=10 | T=20 | T=10 | T=20 | T=10 | T=20
| $N$ |  $50$ $100$ |  $50$ $100$ |  $50$ $100$ |  $50$ $100$ |  $50$ $100$ |  $50$ $100$
Hausman |  |  |  |  |  |  |  |  |  |  |  |  |
H | 12.929 | 26.985 | 15.908 | 31.683 | 13.151 | 29.608 | 18.419 | 43.237 | 9.851 | 20.165 | 17.108 | 42.372
$\chi^{2}_{2}$ p.05 | 0.561 | 0.833 | 0.597 | 0.799 | 0.572 | 0.801 | 0.615 | 0.842 | 0.470 | 0.672 | 0.593 | 0.819
| Boot p.05 | 0.041 | 0.048 | 0.045 | 0.047 | 0.045 | 0.040 | 0.046 | 0.041 | 0.039 | 0.041 | 0.049 | 0.039
$\hat{\theta}_{1}$ |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | 0.000 | -0.001 | -0.000 | 0.000 | 0.000 | -0.001 | -0.000 | 0.000 | 0.000 | -0.001 | -0.000 | 0.000
SD | 0.048 | 0.033 | 0.033 | 0.024 | 0.048 | 0.033 | 0.033 | 0.024 | 0.048 | 0.033 | 0.033 | 0.024
| SE/SD | 0.931 | 0.968 | 0.966 | 0.948 | 0.931 | 0.968 | 0.966 | 0.948 | 0.931 | 0.968 | 0.966 | 0.948
$\hat{\theta}_{2}$ |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | -0.000 | -0.000 | 0.000 | -0.000 | -0.000 | -0.000 | 0.000 | -0.000 | -0.000 | -0.000 | 0.000 | -0.000
SD | 0.050 | 0.034 | 0.033 | 0.023 | 0.050 | 0.034 | 0.033 | 0.023 | 0.050 | 0.034 | 0.033 | 0.023
| SE/SD | 0.899 | 0.917 | 0.945 | 0.960 | 0.899 | 0.917 | 0.945 | 0.960 | 0.899 | 0.917 | 0.945 | 0.960
$\tilde{\theta_{1}}$ |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | -0.117 | -0.157 | -0.070 | -0.098 | -0.109 | -0.174 | -0.077 | -0.132 | -0.048 | -0.115 | -0.064 | -0.138
SD | 0.164 | 0.119 | 0.122 | 0.093 | 0.169 | 0.130 | 0.136 | 0.108 | 0.168 | 0.138 | 0.141 | 0.114
| SE/SD | 0.479 | 0.466 | 0.406 | 0.375 | 0.515 | 0.481 | 0.405 | 0.368 | 0.551 | 0.488 | 0.437 | 0.397
$\tilde{\theta_{2}}$ |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | -0.118 | -0.158 | -0.073 | -0.100 | -0.109 | -0.176 | -0.078 | -0.135 | -0.050 | -0.116 | -0.064 | -0.141
SD | 0.157 | 0.122 | 0.124 | 0.095 | 0.175 | 0.132 | 0.137 | 0.110 | 0.169 | 0.144 | 0.144 | 0.115
| SE/SD | 0.503 | 0.452 | 0.398 | 0.368 | 0.494 | 0.472 | 0.403 | 0.362 | 0.548 | 0.467 | 0.428 | 0.394
|  Avg$K$ | 16.112 | 21.192 | 20.809 | 28.940 | 10.102 | 11.862 | 13.815 | 17.245 | 6.067 | 6.605 | 8.482 | 9.512
|  Avg$L$ | 7.482 | 8.122 | 11.874 | 13.602 | 6.630 | 7.415 | 9.810 | 11.665 | 5.730 | 6.575 | 7.817 | 9.618

1000 Monte Carlo (MC) replications. “$H$” is the average of the Hausman test
statistic, across MC replications. “p.05” denotes the rejection rate for a
nominal size of 5%. “$\chi^{2}_{2}$ p.05” is based on the $95^{th}$ percentile
of central a $\chi^{2}_{2}$ distribution. “Boot p.05” is based on the
$95^{th}$ percentile of the empirical distribution determined via 399
bootstrap replications. “Bias” is the mean bias, “SD” and “SE” denote the
standard deviation over the MC replications and the average estimated standard
error, respectively, those are reported for two regressors. “Avg $\hat{K}$”
and “Avg $\hat{L}$” report the average number of groups for individuals and
time occasions obtained in the first step.

Table 4: Power analysis: DGP-IFE-L, TW-FE vs TW-GFE

|  | $\gamma=0.25$
---|---|---
|  | T=10 | T=20
| $N$ |  $50$ $100$ |  $50$ $100$
Hausman |  |  |  |  |
H | 27.748 | 66.009 | 44.446 | 108.099
$\chi^{2}_{2}$ p.05 | 0.873 | 0.987 | 0.938 | 0.996
| Boot p.05 | 0.614 | 0.842 | 0.779 | 0.944
$\hat{\theta}_{1}$ |  |  |  |  |
Bias | 0.237 | 0.228 | 0.239 | 0.233
SD | 0.108 | 0.080 | 0.089 | 0.062
| SE/SD | 0.812 | 0.786 | 0.709 | 0.718
$\hat{\theta}_{2}$ |  |  |  |  |
Bias | 0.230 | 0.236 | 0.235 | 0.234
SD | 0.111 | 0.081 | 0.088 | 0.060
| SE/SD | 0.796 | 0.776 | 0.717 | 0.745
$\tilde{\theta_{1}}$ |  |  |  |  |
Bias | -0.047 | -0.132 | -0.008 | -0.085
SD | 0.184 | 0.157 | 0.142 | 0.119
| SE/SD | 0.498 | 0.420 | 0.404 | 0.347
$\tilde{\theta_{2}}$ |  |  |  |  |
Bias | -0.049 | -0.132 | -0.008 | -0.085
SD | 0.187 | 0.162 | 0.141 | 0.118
| SE/SD | 0.490 | 0.409 | 0.407 | 0.350
|  Avg$K$ | 15.122 | 19.981 | 19.552 | 26.773
|  Avg$L$ | 4.531 | 5.523 | 5.753 | 7.376

1000 Monte Carlo (MC) replications. “$H$” is the average of the Hausman test
statistic, across MC replications. “p.05” denotes the rejection rate for a
nominal size of 5%. “$\chi^{2}_{2}$ p.05” is based on the $95^{th}$ percentile
of central a $\chi^{2}_{2}$ distribution. “Boot p.05” is based on the
$95^{th}$ percentile of the empirical distribution determined via 399
bootstrap replications. “Bias” is the mean bias, “SD” and “SE” denote the
standard deviation over the MC replications and the average estimated standard
error, respectively, those are reported for two regressors. “Avg $\hat{K}$”
and “Avg $\hat{L}$” report the average number of groups for individuals and
time occasions obtained in the first step.

###  4.2 Probit model

We also investigate the small sample properties of our test in a nonlinear
setting, specifically by considering the probit model. For $i=1,\ldots,N$ and
$t=1,\ldots,T$, we generate samples according to the following equations,
which we denote as DGP-FE-P:

$\displaystyle y_{it}$ $\displaystyle=$
$\displaystyle\mathbf{I}\left(x_{it}\theta+\alpha_{i}+\varepsilon_{it}\geq
0\right),$ $\displaystyle x_{it}$ $\displaystyle=$
$\displaystyle\kappa\left[\Gamma_{i}+N(0,1)\right],$

where $\mathbf{I}(\cdot)$ is an indicator function,
$\alpha_{i}=\varrho\Gamma_{i}+\sqrt{(1-\varrho^{2})}A_{i}$,
$A_{i},\Gamma_{i}\sim N(1,1)$, $\varrho=0.5$, and $\varepsilon_{it}$ is an
idiosyncratic standard normal error term. The slope parameter $\theta$ is set
equal to 1. Following Hahn and Newey,  (2004), the variance contributions of
the three elements in the linear index are roughly 0.2, 1, and 1,
respectively, so that we rescale the variance of $x_{it}$ by letting
$\kappa=\sqrt{(1/10)}$. When we allow for additive individual and time
effects, samples are generated according to

$\displaystyle y_{it}$ $\displaystyle=$
$\displaystyle\mathbf{I}\left(x_{it}\theta+\alpha_{i}+\zeta_{t}+\varepsilon_{it}\geq
0\right),$ $\displaystyle x_{it}$ $\displaystyle=$
$\displaystyle\kappa\left[\Gamma_{i}+\zeta_{t}+N(0,1)\right],$

where $\zeta_{t}\sim N(1,1)$ and the sum $\alpha_{i}+\zeta_{t}$ is rescaled to
have unit variance. This design is denoted by DGP-AFE-P. We evaluate the power
of the proposed test in scenarios generated under the alternative hypothesis
of interactive fixed effects with one latent factor, that is

$\displaystyle y_{it}$ $\displaystyle=$
$\displaystyle\mathbf{I}\\{x_{it}\theta+\alpha_{i}\zeta_{t}+\varepsilon_{it}\geq
0\\},$ $\displaystyle x_{it}$ $\displaystyle=$
$\displaystyle\kappa[\Gamma_{i}\zeta_{t}+N(0,1)],$

where again the product $\alpha_{i}\zeta_{t}$ is rescaled to have unit
variance. We refer to this design as DGP-IFE-P. For each experiment, we
consider $N=50,100$, $T=10,20$, and $399$ bootstrap draws for each of the
$1000$ Monte Carlo replications. When dealing with DGP-FE-P, we try different
values of $\gamma=0.25,0.5,1$ in order to evaluate sensitivity of the test to
the number of groups.

Tables 5 and 6 show the simulation results for experiments where we compare
the OW-FE with TW-GFE estimators under DGP-FE-P and DGP-IFE-P, and the TW-FE
and TW-GFE estimators under DGP-AFE-P and DGP-IFE-P, respectively. As
expected, the bootstrap test approaches the correct size under smaller values
of $\gamma$, i.e. when we impose a larger number of groups, while the one
based on the asymptotic critical value fails to account for the noncentrality
parameter of chi-square distribution. The power analysis shows that our test
behaves well when OW-FE is contrasted with TW-GFE under DGP-IFE-P, while the
power decreases drastically when TW-FE is contrasted with TW-GFE under the
same scenario. This is due to the ability of the two-way fixed effects
specification to approximate a model with interacted effects, as the
relatively small bias of TW-FE estimator with respect to TW-GFE clarifies.
Similar evidence for the linear model is reported by Freeman and Weidner,
(2023). It is useful to recall that population moments used to cluster the
unobserved heterogeneity include the average of the binary dependent variable,
which may not provide enough information to detect more complex forms of
unobserved heterogeneity. Also, the TW-GFE is less biased with respect to OW-
FE under both DGP-FE-P and DGP-IFE-P and under all values of $\gamma$ while,
under DGP-AFE-P, TW-GFE exhibits a larger bias than its counterpart due its
approximation issues.

Table 5: Size and power analyses: Probit model, OW-FE vs TW-GFE

|  | DGP-FE-P |  | DGP-IFE-P
---|---|---|---|---
|  | $\gamma=0.25$ |  | $\gamma=0.5$ |  | $\gamma=1$ |  | $\gamma=0.25$
|  | T=10 | T=20 |  | T=10 | T=20 |  | T=10 | T=20 |  | T=10 | T=20
| $N$ |  $50$ $100$ |  $50$ $100$ |  |  $50$ $100$ |  $50$ $100$ |  |  $50$ $100$ |  $50$ $100$ |  |  $50$ $100$ |  $50$ $100$
Hausman |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$H$ | 0.926 | 1.406 | 0.855 | 1.039 |  | 2.327 | 4.615 | 2.436 | 4.949 |  | 4.049 | 8.140 | 4.673 | 9.276 |  | 13.107 | 31.054 | 19.591 | 51.124
$\chi^{2}_{1}$ p.05 | 0.028 | 0.086 | 0.032 | 0.047 |  | 0.210 | 0.552 | 0.226 | 0.606 |  | 0.448 | 0.827 | 0.539 | 0.886 |  | 0.783 | 0.964 | 0.925 | 0.994
| Boot p.05 | 0.040 | 0.027 | 0.050 | 0.039 |  | 0.021 | 0.017 | 0.033 | 0.027 |  | 0.019 | 0.012 | 0.019 | 0.020 |  | 0.731 | 0.950 | 0.835 | 0.988
$\hat{\theta}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | 0.141 | 0.135 | 0.067 | 0.066 |  | 0.141 | 0.135 | 0.067 | 0.066 |  | 0.141 | 0.135 | 0.067 | 0.066 |  | 0.747 | 0.731 | 0.639 | 0.632
SD | 0.282 | 0.193 | 0.174 | 0.122 |  | 0.282 | 0.193 | 0.174 | 0.122 |  | 0.282 | 0.193 | 0.174 | 0.122 |  | 0.281 | 0.224 | 0.175 | 0.140
| SE/SD | 0.907 | 0.934 | 0.971 | 0.975 |  | 0.907 | 0.934 | 0.971 | 0.975 |  | 0.907 | 0.934 | 0.971 | 0.975 |  | 0.707 | 0.627 | 0.741 | 0.652
$\tilde{\theta}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | 0.110 | 0.090 | 0.066 | 0.051 |  | 0.017 | 0.005 | 0.003 | -0.003 |  | -0.066 | -0.080 | -0.049 | -0.053 |  | 0.176 | 0.126 | 0.142 | 0.086
SD | 0.284 | 0.193 | 0.184 | 0.125 |  | 0.253 | 0.174 | 0.166 | 0.117 |  | 0.239 | 0.165 | 0.158 | 0.113 |  | 0.307 | 0.202 | 0.203 | 0.138
| SE/SD | 0.912 | 0.924 | 0.940 | 0.961 |  | 0.946 | 0.962 | 0.984 | 0.977 |  | 0.941 | 0.958 | 0.999 | 0.980 |  | 0.832 | 0.847 | 0.874 | 0.869
|  Avg $\hat{K}$ | 20.445 | 27.279 | 25.870 | 36.731 |  | 13.713 | 16.601 | 18.482 | 23.625 |  | 8.555 | 9.640 | 12.172 | 14.160 |  | 9.619 | 11.456 | 12.635 | 15.373
|  Avg $\hat{L}$ | 1.992 | 1.996 | 2.302 | 2.283 |  | 1.281 | 1.249 | 1.362 | 1.338 |  | 1.010 | 1.011 | 1.001 | 1.000 |  | 6.724 | 7.369 | 10.186 | 11.654

1000 Monte Carlo (MC) replications. “$H$” is the average of the Hausman test
statistic, across MC replications. “p.05” denotes the rejection rate for a
nominal size of 5%. “$\chi^{2}_{1}$ p.05” is based on the $95^{th}$ percentile
of a central $\chi^{2}_{1}$ distribution. “Boot p.05” is based on the
$95^{th}$ percentile of the empirical distribution determined via 399
bootstrap replications. “Bias” is the mean bias, “SD” and “SE” denote the
standard deviation over the MC replications and the average estimated standard
error, respectively. “Avg $\hat{K}$” and “Avg $\hat{L}$” report the average
number of groups for individuals and time occasions obtained in the first
step.

Table 6: Size and power analyses: Probit model, TW-FE vs TW-GFE

|  | DGP-AFE-P |  | DGP-IFE-P
---|---|---|---|---
|  | $\gamma=0.25$ |  | $\gamma=0.25$
|  | T=10 | T=20 |  | T=10 | T=20
| $N$ |  $50$ $100$ |  $50$ $100$ |  |  $50$ $100$ |  $50$ $100$
Hausman |  |  |  |  |  |  |  |  |  |
H | 0.924 | 0.847 | 1.574 | 1.868 |  | 1.882 | 4.352 | 2.370 | 4.433
$\chi^{2}_{2}$ p.05 | 0.031 | 0.030 | 0.102 | 0.144 |  | 0.152 | 0.367 | 0.196 | 0.381
| Boot p.05 | 0.037 | 0.043 | 0.039 | 0.031 |  | 0.093 | 0.221 | 0.158 | 0.337
$\hat{\theta}$ |  |  |  |  |  |  |  |  |  |
Bias | 0.210 | 0.166 | 0.108 | 0.092 |  | 0.297 | 0.275 | 0.210 | 0.194
SD | 0.316 | 0.192 | 0.182 | 0.130 |  | 0.268 | 0.183 | 0.169 | 0.126
| SE/SD | 0.869 | 0.982 | 0.971 | 0.948 |  | 0.851 | 0.867 | 0.883 | 0.828
$\tilde{\theta}$ |  |  |  |  |  |  |  |  |  |
Bias | 0.263 | 0.177 | 0.215 | 0.170 |  | 0.185 | 0.120 | 0.137 | 0.090
SD | 0.364 | 0.206 | 0.223 | 0.149 |  | 0.299 | 0.189 | 0.207 | 0.135
| SE/SD | 0.829 | 0.963 | 0.903 | 0.916 |  | 0.859 | 0.905 | 0.851 | 0.888
|  Avg $\hat{K}$ | 13.188 | 16.010 | 17.141 | 21.847 |  | 9.784 | 11.490 | 12.528 | 15.121
|  Avg $\hat{L}$ | 6.824 | 7.466 | 10.412 | 11.897 |  | 6.736 | 7.289 | 10.114 | 11.616

1000 Monte Carlo (MC) replications. “$H$” is the average of the Hausman test
statistic, across MC replications. “p.05” denotes the rejection rate for a
nominal size of 5%. “$\chi^{2}_{1}$ p.05” is based on the $95^{th}$ percentile
of a central $\chi^{2}_{1}$ distribution. “Boot p.05” is based on the
$95^{th}$ percentile of the empirical distribution determined via 399
bootstrap replications. “Bias” is the mean bias, “SD” and “SE” denote the
standard deviation over the MC replications and the average estimated standard
error, respectively. “Avg $\hat{K}$” and “Avg $\hat{L}$” report the average
number of groups for individuals and time occasions obtained in the first
step.

###  4.3 Comparison with CRT and BBP tests

Table 7 reports the simulation results for the CRT and BBP tests under the
null and alternative hypotheses described by DGP-FE-L and DGP-IFE-L. In
particular, the CRT test can be implemented by considering the null hypothesis
of no factor structure with time-constant individual effect and the
alternative hypothesis of a factor model with 1 latent factor (DGP-IFE-L). The
test fails to attain the correct size in short panels, as also reported by
Castagnetti et al., 2015a . The BBP test has good size properties in both
linear and nonlinear frameworks, while it displays remarkably low power, due
to absence of a dynamic factor structure in DGP-IFE-L, as also discussed in
Section 3.3.2.

Table 7: Size and power analyses: Linear model, CRT and BBP tests |  | DGP-FE-L |  | DGP-IFE-L
---|---|---|---|---
|  | T=10 | T=20 |  | T=10 | T=20
| $N$ | 50 | 100 | 50 | 100 |  | 50 | 100 | 50 | 100
Linear | CRT | 0.994 | 0.993 | 1.000 | 1.000 |  | 0.998 | 0.998 | 1.000 | 1.000
| BBP | 0.077 | 0.065 | 0.065 | 0.048 |  | 0.111 | 0.155 | 0.093 | 0.137
Probit | BBP | 0.067 | 0.055 | 0.071 | 0.06 |  | 0.135 | 0.132 | 0.128 | 0.173

###  4.4 Robustness analyses

In this section we investigate the finite-sample behavior of the proposed test
under departures from some of the assumptions listed in Section 3.1, which are
needed for the consistency of the TW-GFE estimator. In particular, we explore
two scenarios potentially relevant in empirical applications: the first one is
characterized by dynamic factors, representing a violation of Assumption 4;
the second one concerns the lack of informativeness of the population moments
on the unobserved heterogeneity, which is needed for an effective clustering
procedure and is represented by Assumption 2. For both scenarios we consider
linear models and the choice of the number of groups is based on one value of
$\gamma=0.25$.

####  4.4.1 Dynamic factors

We introduce the following generating equations (DGP-ADF)

$\displaystyle y_{it}$ $\displaystyle=$ $\displaystyle
x_{it1}\theta_{1}+x_{it2}\theta_{2}+\alpha_{i}+\zeta_{t}+\varepsilon_{it},$
$\displaystyle x_{itj}$ $\displaystyle=$
$\displaystyle\Gamma_{i}+\zeta_{t}+N(0,1),\quad\mathrm{for}\quad j=1,2,$

where variables are generate as in DGP-AFE-L except for the factor
$\zeta_{t}$, which is now generated as follows

$\zeta_{t}:\left\\{\begin{array}[]{lc}u_{1}&\mathrm{t=1}\\\\[2.0pt]
\rho\zeta_{t-1}+\sqrt{(1-\rho^{2})}u_{t}&\mathrm{t=2\dots T}\\\
\end{array}\right.$

with $u_{t}\sim N(0,1)$ and $\rho=0.8$. Under DGP-ADF time effects therefore
exhibit a dynamic structure, which effectively violates Assumption 4. The size
of the test is then evaluated by contrasting the TW-FE with the TW-GFE. As for
power, we generate samples as in DGP-IFE-L with a dynamic structure for
$\zeta_{t}$ (DGP-IDF).

Simulation results are reported in Table 8. It is worth noticing that
violating the iid assumption for the time heterogeneity does not affect the
finite sample performance of the proposed test. Interestingly, the dynamic
structure seems to bring more information into the clustering procedure, as
testified by the higher number of groups generated in the kmeans
classification step. This provides a better approximation of the unobserved
heterogeneity and, supposedly, a smaller bias. The test also has a
satisfactory power performance. As it happens with DGP-IFE-L, under DGP-IDF,
when TW-GFE is contrasted with OW-FE and TW-FE estimators, the test is able to
detect a time-varying heterogeneity in the first case, while it fails to
capture the interactive structure when an additive specification is assumed.

Table 8: Size and power analyses with a dynamic factor structure: Linear model

|  | DGP-ADF (TW-FE vs TW-GFE) |  | DGP-IDF (OW-FE vs TW-GFE) |  | DGP-IDF (TW-FE vs TW-GFE)
---|---|---|---|---|---|---
|  | T=10 | T=20 |  | T=10 | T=20 |  | T=10 | T=20
| $N$ |  $50$ $100$ |  $50$ $100$ |  |  $50$ $100$ |  $50$ $100$ |  |  $50$ $100$ |  $50$ $100$
Hausman |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$H$ | 3.515 | 5.098 | 5.029 | 5.592 |  | 27.841 | 58.817 | 52.152 | 112.443 |  | 22.640 | 48.781 | 42.039 | 93.978
$\chi^{2}_{2}$ p.05 | 0.171 | 0.287 | 0.265 | 0.272 |  | 0.859 | 0.938 | 0.885 | 0.954 |  | 0.765 | 0.871 | 0.797 | 0.860
| Boot p.05 | 0.036 | 0.050 | 0.051 | 0.046 |  | 0.744 | 0.874 | 0.852 | 0.947 |  | 0.612 | 0.778 | 0.746 | 0.835
$\hat{\theta}_{1}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | -0.001 | 0.001 | 0.001 | -0.001 |  | 0.236 | 0.240 | 0.245 | 0.248 |  | 0.213 | 0.216 | 0.221 | 0.224
SD | 0.048 | 0.034 | 0.033 | 0.023 |  | 0.086 | 0.063 | 0.074 | 0.053 |  | 0.088 | 0.066 | 0.079 | 0.055
| SE/SD | 0.929 | 0.936 | 0.947 | 0.967 |  | 0.785 | 0.758 | 0.719 | 0.708 |  | 0.769 | 0.731 | 0.678 | 0.686
$\hat{\theta}_{2}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | 0.002 | -0.000 | 0.000 | -0.000 |  | 0.239 | 0.235 | 0.248 | 0.249 |  | 0.216 | 0.212 | 0.224 | 0.226
SD | 0.048 | 0.032 | 0.032 | 0.023 |  | 0.090 | 0.064 | 0.078 | 0.054 |  | 0.094 | 0.067 | 0.080 | 0.057
| SE/SD | 0.928 | 0.984 | 0.986 | 0.965 |  | 0.748 | 0.747 | 0.684 | 0.698 |  | 0.725 | 0.723 | 0.666 | 0.667
$\tilde{\theta}_{1}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | -0.016 | -0.030 | 0.000 | -0.015 |  | 0.024 | -0.009 | 0.040 | 0.000 |  | 0.024 | -0.009 | 0.040 | 0.000
SD | 0.080 | 0.060 | 0.057 | 0.042 |  | 0.161 | 0.149 | 0.148 | 0.151 |  | 0.161 | 0.149 | 0.148 | 0.151
| SE/SD | 0.681 | 0.636 | 0.630 | 0.608 |  | 0.459 | 0.357 | 0.371 | 0.262 |  | 0.459 | 0.357 | 0.371 | 0.262
$\tilde{\theta}_{2}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Bias | -0.014 | -0.033 | -0.001 | -0.014 |  | 0.027 | -0.009 | 0.040 | 0.001 |  | 0.027 | -0.009 | 0.040 | 0.001
SD | 0.081 | 0.058 | 0.057 | 0.040 |  | 0.161 | 0.148 | 0.148 | 0.150 |  | 0.161 | 0.148 | 0.148 | 0.150
| SE/SD | 0.670 | 0.665 | 0.639 | 0.639 |  | 0.460 | 0.360 | 0.370 | 0.264 |  | 0.460 | 0.360 | 0.370 | 0.264
|  Avg$\hat{K}$ | 36.301 | 62.856 | 38.599 | 67.827 |  | 11.415 | 15.618 | 9.491 | 12.376 |  | 11.415 | 15.618 | 9.491 | 12.376
|  Avg$\hat{L}$ | 3.668 | 4.501 | 4.858 | 6.318 |  | 4.724 | 5.705 | 6.538 | 8.228 |  | 4.724 | 5.705 | 6.538 | 8.228

DGP-ADF refers to equation 4.4.1. 1000 Monte Carlo (MC) replications. “$H$” is
the average of the Hausman test statistic, across MC replications. “p.05”
denotes the rejection rate for a nominal size of 5%. “$\chi^{2}_{2}$ p.05” is
based on the $95^{th}$ percentile of a central $\chi^{2}_{2}$ distribution.
“Boot p.05” is based on the $95^{th}$ percentile of the empirical distribution
determined via 399 bootstrap replications. “Bias” is the mean bias, “SD” and
“SE” denote the standard deviation over the MC replications and the average
estimated standard error, respectively. “Avg $\hat{K}$” and “Avg $\hat{L}$”
report the average number of groups for individuals and time occasions
obtained in the first step.

####  4.4.2 Non informative moments

In the following scenario we relax Assumption 2 by letting the moments of
$x_{itj}$ and $y_{it}$ include not Lipschitz-continuous functions of the
unobserved heterogeneity. We therefore generate data according to the
following system (DGP-NIM):

$\displaystyle y_{it}$ $\displaystyle=$ $\displaystyle
x_{it1}\theta_{1}+x_{it2}\theta_{2}+\alpha_{i}+\varepsilon_{it},$
$\displaystyle x_{itj}$ $\displaystyle=$
$\displaystyle\Gamma_{i}+N(0,1),\quad\mathrm{for}\quad j=1,2,$
$\displaystyle\alpha_{i}$ $\displaystyle=$
$\displaystyle\rho\Gamma_{i}+\sqrt{1-\rho^{2}}A_{i}$

where $\rho=0.5$, $\Gamma_{i},A_{i}\sim\sqrt{\mathrm{U}(0,1)}$. In this
setting, we expect population moments to poorly discriminate the latent types,
thereby not bringing relevant information in the first step and worsening the
performance of the test.

Table 9 shows the results for this experiment. As expected, the poor
approximation of the unobserved heterogeneity reflects on the distortion of
the bootstrap empirical size. Even though the bias of the TW-GFE estimator
seems limited, the poor performance of the test might spur from a bias in the
variance estimate, due to possibly unaccounted sources of heterogeneity.

Table 9: Size analysis: DGP-NIM, Linear model, OW-FE vs TW-GFE

|  | $\gamma=0.25$
---|---|---
|  | T=10 | T=20
| $N$ |  $50$ $100$ |  $50$ $100$
Hausman |  |  |  |  |
$H$ | 3.239 | 6.753 | 2.921 | 5.136
$\chi^{2}_{2}$ p.05 | 0.108 | 0.551 | 0.085 | 0.352
| Boot p.05 | 0.011 | 0.009 | 0.033 | 0.017
$\hat{\theta}_{1}$ |  |  |  |  |
Bias | 0.000 | 0.001 | 0.001 | -0.001
SD | 0.048 | 0.034 | 0.033 | 0.023
| SE/SD | 0.924 | 0.939 | 0.965 | 0.979
$\hat{\theta}_{2}$ |  |  |  |  |
Bias | 0.002 | 0.000 | 0.000 | -0.001
SD | 0.047 | 0.032 | 0.031 | 0.023
| SE/SD | 0.946 | 0.981 | 1.009 | 0.973
$\tilde{\theta}_{1}$ |  |  |  |  |
Bias | -0.021 | -0.021 | -0.010 | -0.011
SD | 0.050 | 0.034 | 0.034 | 0.023
| SE/SD | 0.925 | 0.957 | 0.962 | 0.980
$\tilde{\theta}_{2}$ |  |  |  |  |
Bias | -0.019 | -0.021 | -0.010 | -0.011
SD | 0.048 | 0.033 | 0.032 | 0.023
SE/SD | 0.950 | 0.981 | 1.013 | 0.984
$Avg\hat{K}$ | 9.862 | 12.273 | 11.635 | 14.806
| $Avg\hat{L}$ | 2.747 | 2.694 | 3.416 | 3.349

1000 Monte Carlo (MC) replications. “$H$” is the average of the Hausman test
statistic, across MC replications. “p.05” denotes the rejection rate for a
nominal size of 5%. “$\chi^{2}_{2}$ p.05” is based on the $95^{th}$ percentile
of a central $\chi^{2}_{2}$ distribution. “Boot p.05” is based on the
$95^{th}$ percentile of the empirical distribution determined via 399
bootstrap replications. “Bias” is the mean bias, “SD” and “SE” denote the
standard deviation over the MC replications and the average estimated standard
error, respectively. “Avg $\hat{K}$” and “Avg $\hat{L}$” report the average
number of groups for individuals and time occasions obtained in the first
step.

##  5 Empirical applications

We evaluate the behavior of our test with real world data in two applications,
one concerning unobserved heterogeneity in wage determinants of young working
women, the other concerning a gravity equation for the extensive margin of
trade.

###  5.1 Wage of young working women

We here estimate a linear panel data model to study the wage determinants of
young working women. The dataset comes from $10$ non-consecutive waves of the
NLSY survey, in which $315$ employed women, not enrolled in school and at a
stage when education was completed, were interviewed. Interviews were carried
out from $1968$ to $1988$, but only ten non-consecutive periods are considered
for each subject, in order to have data without missing values. The variables
taken into account are: _age_ in current year, _marital status_ , _never
married_ , _not_smsa_ , that is not living in a standard statistical
metropolitan area, _capital city_ , 1 for living in a capital city, _south_ ,
1 for living in south of US, _total work experience_ and _tenure_ , both in
years. The dependent variable is _ln wage_ , measured as logarithm of the
ratio between nominal wage and GNP deflator.

We perform the test in two settings: first contrasting TW-GFE with OW-FE and
then TW-GFE with TW-GFE. Table 10 shows estimates for parameters of interest
when OW-FE is considered while Table 11 presents estimates when TW-FE is
considered. We perform the test with 399 bootstrap replications and 6
different values of $\gamma$. The results of our test strongly suggest to opt
for a specification with time-varying unobserved heterogeneity, as estimating
the model with only time-constant individual effects leads to a systematic
rejection of the null hypothesis with a sufficiently large number of groups,
while the pooled model (TW-GFE with one group) leads to substantially the same
results as the OW-FE. In contrast, the test fails to detect structures for the
time-varying unobserved heterogeneity that are more sophisticated than the
additive one.

It is worth noticing that quantiles of centered chi squared distribution are
not reliable at all and inference based on them would be highly misleading.
Moreover, the bootstrap critical values for the OW-FE vs TW-GFE test reveals a
finite-sample distortion affecting the $\chi^{2}$ distribution, which may be
caused by the variance over-estimation in the OW model as a result of the
neglected heterogeneity. Finally, we highlight that minimal values of $\gamma$
are required in this application in order to find a sufficiently large number
of groups, which is necessary in order to make the test, and the TW-GFE,
reliable.

Table 10: Estimation results and test for wage determinants for young working
women: OW-FE vs TW-GFE

| OW-FE | TW-GFE |
---|---|---|---
|  | $\gamma=0.0025$ | $\gamma=0.025$ | $\gamma=0.05$ | $\gamma=0.25$ | $\gamma=0.5$ | $\gamma=1$ |
age | 0.004 | 0.004 | 0.006 | 0.007 | 0.004 | 0.004 | 0.004 |
| (0.002) | (0.003) | (0.002) | (0.002) | (0.002) | (0.002) | (0.002) |
marital status | 0.027 | 0.022 | 0.023 | 0.025 | 0.026 | 0.026 | 0.026 |
| (0.019) | (0.019) | (0.019) | (0.019) | (0.019) | (0.019) | (0.019) |
never married | -0.044 | -0.048 | -0.048 | -0.049 | -0.046 | -0.046 | -0.046 |
| (0.023) | (0.023) | (0.023) | (0.023) | (0.023) | (0.023) | (0.023) |
not_smsa | -0.217 | -0.221 | -0.224 | -0.227 | -0.227 | -0.227 | -0.227 |
| (0.017) | (0.017) | (0.017) | (0.017) | (0.017) | (0.017) | (0.017) |
capital city | -0.036 | -0.032 | -0.044 | -0.042 | -0.046 | -0.046 | -0.046 |
| (0.016) | (0.016) | (0.016) | (0.016) | (0.016) | (0.016) | (0.016) |
south | -0.208 | -0.214 | -0.206 | -0.203 | -0.201 | -0.201 | -0.201 |
| (0.014) | (0.014) | (0.014) | (0.014) | (0.014) | (0.014) | (0.014) |
total experience | 0.018 | 0.037 | 0.025 | 0.024 | 0.018 | 0.018 | 0.018 |
| (0.003) | (0.004) | (0.003) | (0.003) | (0.003) | (0.003) | (0.003) |
tenure | 0.014 | 0.011 | 0.012 | 0.013 | 0.013 | 0.013 | 0.013 |
| (0.002) | (0.002) | (0.002) | (0.002) | (0.002) | (0.002) | (0.002) |
Hausman test value |  | 35.811 | 24.232 | 19.862 | 3.998 | 3.998 | 3.998 |
$\chi^{2}_{8}$ crit. |  | 15.507 | 15.507 | 15.507 | 15.507 | 15.507 | 15.507 |
Boot crit. |  | 16.169 | 9.811 | 9.664 | 9.816 | 10.037 | 9.466 |
K |  | 124.000 | 12.000 | 6.000 | 1.000 | 1.000 | 1.000 |
L |  | 9.000 | 5.000 | 4.000 | 1.000 | 1.000 | 1.000 |

Standard errors in parentheses. OW-FE refers to fixed-effects estimator, while
estimates of TW-GFE are reported for different values of $\gamma$.
“$\chi^{2}_{8}$ crit.” is the $95^{th}$ percentile of the centered chi-squared
distribution. “Boot crit.” is the $95^{th}$ percentile of the empirical
distribution determined via 399 bootstrap replications. K and L are the number
of estimated groups based on individual and time moments respectively.

Table 11: Test and estimation results of wage determinants for young working
women: TW-FE vs TW-GFE

| TW-FE | TW-GFE
---|---|---
|  | $\gamma=0.0025$ | $\gamma=0.025$ | $\gamma=0.05$ | $\gamma=0.25$ | $\gamma=0.5$ | $\gamma=1$
age | 0.010 | 0.006 | 0.007 | 0.006 | 0.004 | 0.004 | 0.004
| (0.002) | (0.003) | (0.002) | (0.002) | (0.002) | (0.002) | (0.002)
marital status | 0.026 | 0.009 | 0.026 | 0.027 | 0.026 | 0.026 | 0.026
| (0.019) | (0.019) | (0.019) | (0.019) | (0.019) | (0.019) | (0.019)
never married | -0.047 | -0.044 | -0.050 | -0.048 | -0.046 | -0.046 | -0.046
| (0.023) | (0.024) | (0.023) | (0.023) | (0.023) | (0.023) | (0.023)
not_smsa | -0.212 | -0.242 | -0.225 | -0.223 | -0.227 | -0.227 | -0.227
| (0.017) | (0.017) | (0.017) | (0.017) | (0.017) | (0.017) | (0.017)
capital city | -0.030 | -0.048 | -0.042 | -0.043 | -0.046 | -0.046 | -0.046
| (0.016) | (0.017) | (0.016) | (0.016) | (0.016) | (0.016) | (0.016)
south | -0.209 | -0.202 | -0.202 | -0.202 | -0.201 | -0.201 | -0.201
| (0.014) | (0.014) | (0.014) | (0.014) | (0.014) | (0.014) | (0.014)
total experience | 0.035 | 0.036 | 0.027 | 0.022 | 0.018 | 0.018 | 0.018
| (0.004) | (0.004) | (0.004) | (0.003) | (0.003) | (0.003) | (0.003)
tenure | 0.011 | 0.011 | 0.012 | 0.013 | 0.013 | 0.013 | 0.013
| (0.002) | (0.002) | (0.002) | (0.002) | (0.002) | (0.002) | (0.002)
Hausman test value |  | 9.966 | 20.754 | 36.889 | 49.873 | 49.873 | 49.873
$\chi^{2}_{r}$ crit. |  | 15.507 | 15.507 | 15.507 | 15.507 | 15.507 | 15.507
Boot crit. |  | 15.887 | 52.448 | 68.533 | 89.087 | 90.186 | 92.863
K |  | 122.000 | 12.000 | 6.000 | 1.000 | 1.000 | 1.000
L |  | 10.000 | 4.000 | 3.000 | 1.000 | 1.000 | 1.000

Standard errors in parentheses. TW-FE refers to fixed-effects estimator, while
estimates of TW-GFE are reported for different values of $\gamma$.
“$\chi^{2}_{8}$ crit.” is the $95^{th}$ percentile of the centered chi-squared
distribution. “Boot crit.” is the $95^{th}$ percentile of the empirical
distribution determined via 399 bootstrap replications. K and L are the number
of estimated groups based on individual and time moments respectively.

###  5.2 The extensive margin of trade

We apply the proposed test to a dataset consisting of a network of countries
to analyze the extensive margin of trade. We use data from Helpman et al.,
(2008) consisting of a cross section of $157$ countries in $1986$999As in Chen
et al.,  (2021), we eliminate Congo because it did not export to any country
in 1986. The two-way fixed-effects probit model is used to describe the
probability that an exchange between countries, where covariates and the
related homophily parameters as well as node-specific heterogeneity are
included. The dependent variable is binary and denotes whether a trade flow
occurs from country $i$ to country $j$. We include a set of geographical
regressors : _Distance_ , which is the logarithm of the geographical distance
(in kilometers) between the capitals of each pair of countries; _Border_ ,
which is a dummy indicating whether the two countries share a border.
Furthermore, an additional set of variables captures the institutional and
cultural similarities of each pair of countries: _Legal_ , _Language_ , and
_Currency_ are binary variables equal to 1 if countries i and j share the same
legal origin, the same language, and the same currency (or they are in the
same currency union), respectively; Religion is measure of cult similarities,
and for details on its construction we refer the reader to Helpman et al.,
(2008). Finally, _Colonial Ties_ is a dummy variable assuming value 1 if
country $i$ colonized country $j$ (or vice-versa) and _FTA_ is a binary
variable capturing if the two countries belong to a common trade agreement.

Table 12 presents the estimates of parameters of interest and outcome of the
proposed test when TW-FE is contrasted with TW-GFE. We consider here one value
of $\gamma=1$, generating $51$ groups for both the importer and exporter
unobserved heterogeneity.

Estimated coefficients for the TW-FE and TW-GFE are different, although those
arising form the TW-FE are coherent with the results provided by the empirical
literature using the same data, while the TW-GFE results are not viable as
they are plagued by both the incidental parameters problem and approximation
error. The proposed test is based on 399 bootstrap replications and fails to
reject the null hypothesis, thus suggesting that the additive specification
for the unobserved heterogeneity is complex enough to capture underlying
traits of trade decisions. This results is different from that of Chen et al.,
(2021), who however consider the intensive margin of trade and find that the
unobserved heterogeneity could be modeled by an interactive specification for
the unobserved heterogeneity.

Table 12: Empirical application for the extensive margin of trade

| TW-FE | TW-GFE
---|---|---
Distance | -0.664 | -0.444
| (0.021) | (0.027)
Border | -0.375 | -0.069
| (0.095) | (0.103)
Legal | 0.094 | -0.153
| (0.029) | (0.062)
Language | 0.297 | 0.323
| (0.038) | (0.051)
Currency | 0.462 | -0.013
| (0.134) | (0.438)
Religion | 0.255 | 0.462
| (0.060) | (0.066)
Colonial ties | 0.343 | 0.351
| (0.285) | (0.307)
FTA | 2.029 | 1.064
| (0.308) | (0.466)
Hausman test value |  | 61.383
$\chi^{2}_{r}$ crit. |  | 15.507
Boot crit. |  | 458.693
K |  | 51.000
L |  | 51.000

No of countries 157, no of dyads 24.806. Standard errors in parentheses. TW-FE
refers to two-way fixed effects estimator. Estimates of TW-GFE are reported
for $\gamma=1$. “$\chi^{2}_{r}$ crit.” is the $95^{th}$ percentile of the
centered chi-squared distribution. “Boot crit” is the $95^{th}$ percentile of
the empirical distribution determined via 399 bootstrap replications. K and L
are the number of estimated groups based on individual and time moments
respectively.

##  6 Final remarks

We propose a specification test for the form of the unobserved heterogeneity
in panel data models. The test is based on the recently proposed grouped
fixed-effects approach and serves to detect departures from the commonly
assumed time-invariant or additive fixed-effects specifications.

The main advantage of our proposal is that it allows practitioners to avoid
the specification and estimation of models with complex forms of time-varying
heterogeneity, which might pose identification and computational problems in
both linear and nonlinear models. By contrast, the TW-GFE approach is a rather
simple non-iterative two-step strategy, involving unsupervised clustering in
the first step and estimation of group effects in the second.

The proposed approach is a generalized Hausman test whose asymptotic
distribution is a non-central chi square because of the bias arising from
incidental parameters, for both the ML estimators that are being contrasted
(at least in the non-linear case), and from the approximation error induced by
discretization of the unobserved heterogeneity, that is performed with the GFE
approach. The use of bootstrap critical values successfully corrects the
empirical size of the test and yields satisfactory power properties. A partial
exception is the case of binary choice models with additive fixed effects, for
which the test exhibits low power. We also show that test maintains the good
finite-sample properties even in presence of violations of the sampling
assumptions, while its use is not advisable when the information in the data
is not able to provide a good enough approximation for the unobserved
heterogeneity.

The empirical applications considered suggest that a specification with time-
varying additive unobserved heterogeneity is needed when evaluating the wage
determinants for young working women, and that a standard gravity equation
with additive importer and exporter fixed-effects is appropriate to model the
trade extensive margin. The proposed test also emerges as a viable alternative
to existing procedures with short panel datasets.

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