# Structural Breaks in Interactive Effects Panels and the Stock Market
Reaction to COVID–19††thanks: Westerlund would like to thank the Knut and
Alice Wallenberg Foundation for financial support through a Wallenberg Academy
Fellowship.

Yiannis Karavias
University of Birmingham Corresponding author: Department of Economics,
University of Birmingham, Edgbaston, B15 2TT, Birmingham, UK. E-mail address:
<EMAIL_ADDRESS>Paresh Narayan
Monash University Joakim Westerlund
Lund University
and
Deakin University

###### Abstract

Dealing with structural breaks is an important step in most, if not all,
empirical economic research. This is particularly true in panel data comprised
of many cross-sectional units, such as individuals, firms or countries, which
are all affected by major events. The COVID–19 pandemic has affected most
sectors of the global economy, and there is by now plenty of evidence to
support this. The impact on stock markets is, however, still unclear. The fact
that most markets seem to have partly recovered while the pandemic is still
ongoing suggests that the relationship between stock returns and COVID–19 has
been subject to structural change. It is therefore important to know if a
structural break has occurred and, if it has, to infer the date of the break.
In the present paper we take this last observation as a source of motivation
to develop a new break detection toolbox that is applicable to different sized
panels, easy to implement and robust to general forms of unobserved
heterogeneity. The toolbox, which is the first of its kind, includes a test
for structural change, a break date estimator, and a break date confidence
interval. Application to a panel covering 61 countries from January 3 to
September 25, 2020, leads to the detection of a structural break that is dated
to the first week of April. The effect of COVID–19 is negative before the
break and zero thereafter, implying that while markets did react, the reaction
was short-lived. A possible explanation for this is the quantitative easing
programs announced by central banks all over the world in the second half of
March.

JEL Classification: C12; C13; C33.

Keywords: Panel data; Structural change; Change-point; Common correlated
effects; Cross-section dependence; COVID–19.

##  1 Introduction

This paper considers what we believe to be a very common scenario in practice.
We have in mind a researcher that seeks to infer a linear relationship between
a dependent variable and a set of regressors. The data set has a panel
structure, in which there are a large number of cross-sectional units, $N$,
that are observed over $T$ time periods. This scenario is relevant because
while the number of time periods is always limited and cannot be increased
other than by the passage of time, statistical agencies keep publishing time
series data for individuals, firms and countries. Thus, while $N$ is usually
quite large, $T$ need not be. One of the concerns here is therefore that $T$
might not be large enough for many econometric approaches to work properly.
Another concern is the presence of unobserved heterogeneity and the
detrimental effect that this may have if said heterogeneity is correlated with
the regressors. The main worry, however, is that the coefficients of some or
indeed all of the regressors may be subject to structural change, because of
some major events that may have caused the relationship to change over time.
The present paper develops a toolbox that enables the researcher to test for
the presence of a common structural break and, if a break is detected, to also
infer the date of the break. The tools are extremely easy to implement,
accommodate general forms of unobserved heterogeneity and they can be used
under quite relaxed conditions on $T$, provided that $N$ is large.

Accounting for structural change has always been an important issue in
economics and elsewhere. Panel data are particularly susceptible to such
change, because of the large number of time series that they contain. For
example, a credit crunch or debt crisis may affect the returns of many firms
and households, an oil price shock may affect the output of many countries,
and a fad or fashion is likely to influence the livelihood of many
individuals. In our empirical application, we consider stock returns for 61
countries, which all plummeted around March 11, 2020, when COVID–19 was
declared a global pandemic by the World Health Organisation (WHO).
Fortunately, the panel data structure not only makes breaks likely, but it
also makes for relatively easy detection. As is well known, with time series
data consistent estimation of the breakpoint is not possible, but only
consistent estimation of the break fraction. By contrast, in panels
consistency is usually possible. Bai (2010) was among the first to make this
point. He considered a highly stylized model with a breaking constant as the
only regressor. The main finding is that the popular ordinary least squares
(OLS) breakpoint estimator based on minimizing the sum of squared residuals is
consistent as $N\to\infty$ regardless of whether $T$ is fixed or going to
infinity. The accuracy of the procedure is therefore greatly enhanced when
compared to the time series case.

The increased estimation accuracy is one of the advantages of using panel
data. Another major advantage is the ability to deal with unobserved
heterogeneity. Such heterogeneity is important in general, and it is
particularly relevant in the type of noisy panels that we have in mind where
typically the regressors explain only a small fraction of the variation in the
dependent variable (see Capelle-Blanchard and Desroziers, 2020, in the context
of COVID–19 and stock returns). This consideration motivated Horváth and
Hušková (2012), Kim (2014), and Westerlund (2019) to extend the work of Bai
(2010) to the case when the stochastic component of the data admits to a
common factor, or “interactive effects”, representation.111Kim (2011) also
allow for common factors. This study is, however, not very constructive, in
the sense that it only establishes that the factors render the OLS breakpoint
estimator inconsistent. Such representations have been shown to be very
effective at capturing unobserved heterogeneity, and they are therefore very
popular. It is, however, still the same breaking constant-only model that is
being considered, and many models of interest involve more general regressors.

Antoch et al. (2019), Baltagi et al. (2016), Boldea et al. (2020), Hidalgo and
Schafgans (2017), and Li et al. (2016) do for a general linear panel data
regression model what Horváth and Hušková (2012), Kim (2014), and Westerlund
(2019) do for the constant-only model. In particular, while Antoch et al.
(2019), and Hidalgo and Schafgans (2017) propose tests for the presence of a
structural break, Baltagi et al. (2016), Boldea et al. (2020), and Li et al.
(2016) take the existence of a break as given and focus instead on the
breakpoint estimation problem. But while highly complementary in terms of the
methods they propose, the assumptions employed are materially different.

Antoch et al. (2019) only require $N$ to be large. However, they assume
instead that the factor loadings are negligible, which means that strong forms
of cross-section dependence are not permitted.222See Chudik et al. (2011) for
a detailed treatment of the concepts of weak and strong cross-section
dependence. It also means that there is no need to account for the factors,
and that their effect on the breakpoint estimation problem is in this sense
trivial. The weak cross-sectional dependence condition is maintained also in
Hidalgo and Schafgans (2017), who in addition require that $N,\,T\to\infty$
with $N/T^{2}\to 0$, which in practice means that $T>>N$. Baltagi et al.
(2016) do not require negligible loadings and are therefore more general in
this regard. The way they do this is by applying the common correlated effects
(CCE) approach of Pesaran (2006), which enables consistent estimation of (the
space spanned by) the unknown factors. But then Baltagi et al. (2016) require
that both $N$ and $T$ are large, which is again rarely the case in practice.
Moreover, the authors only provide a consistency result and they do not
consider the asymptotic distribution of the estimated breakpoint, which is
necessary for the construction of confidence intervals with correct asymptotic
coverage. The same critique applies to the paper of Li et al. (2016), which
uses the principal components method instead of CCE to estimate factors.
Boldea et al. (2020) also do not consider the asymptotic distribution of their
estimated breakpoint, although in their paper $T$ is fixed. Their approach is
similar to the one of Antoch et al. (2019) in the sense that the estimation is
carried out while ignoring the factors. This simplicity does, however, come at
a cost in terms additional restrictive assumptions. Boldea et al. (2020) do
not go as far as Antoch et al. (2019) and require negligible loadings, but
they do assume that the omitted variables bias caused by the factors is time-
invariant, up to the breakpoint, which limits the type of factors that can be
permitted.

Motivated by the above discussion, the present paper develops tools that
enable researchers to both test for the presence of a common structural break,
and to infer the breakpoint of an existing break. The unobserved heterogeneity
is assumed to have a common factor structure that is handled by using a
version of the CCE approach, which is similar yet clearly distinct from the
one employed by Baltagi et al. (2016). The reason for focusing on CCE as
opposed to the otherwise so popular principal components method is in part
because of the extreme simplicity with which the factors are estimated in CCE,
in part because CCE is valid even if $T$ is fixed (see Westerlund et al.,
2019). Needless to say, this last feature, which is not exploited by Baltagi
et al. (2016), is an important advantage when wanting to entertain the
possibility that $T$ might not be large. The idea, which is laid out along
with our model and assumptions in Section 2, is to use the cross-sectional
averages of the regressors to estimate the unknown common factors, and to
simply augment the regression model with these averages.

We begin by considering the problem of estimating the unknown breakpoint given
that a break has occurred. This is done in Section 3. Most papers in the
literature are based on the OLS breakpoint estimator, and so is this paper.
However, instead of minimizing the OLS residuals, which will generally lead to
inconsistency because of the unattended factors (Kim, 2011), we minimize the
CCE residuals. We focus on the results for the case when the magnitude of the
break is bounded from above and below, although we also allow diverging and
shrinking breaks. Moreover, $T$ can be fixed or tending to infinity. According
to the results, the proposed breakpoint estimator is consistent as
$N\to\infty$ with $T$ fixed or as $N,\,T\to\infty$ with $T/N\to 0$, and the
rate of convergence is given by $1/N$. The asymptotic distribution of the
breakpoint estimator is obtained under the same set of conditions on $N$ and
$T$, and is used to construct confidence intervals for the true breakpoint. As
far as we are aware, this paper is the first to provide the rate of
convergence and asymptotic distribution in the presence of common factors, and
it is the first to establish consistency when $T$ is fixed.

While in Section 3 we assume that a break has occurred, in Section 4 we
instead consider the problem of testing for the presence of a break. While
very common in the literature, CUSUM-based test statistics like the one of
Hidalgo and Schafgans (2017) can suffer from low power in certain directions
(see, for example, Andrews, 1993). In this paper, we therefore follow the
recommendation of Andrews (1993), and consider two Wald-type test statistics.
One is designed to test the null hypothesis of no break against the
alternative of a known breakpoint, while in the other the same null is tested
against the alternative of a break at some unknown date. To the best of our
knowledge, these tests are the first that enable break testing in the presence
of common factors. The asymptotic analysis reveals that, while the consistency
and asymptotic distribution of the breakpoint estimator only require
$N\to\infty$, unless $N,\,T\to\infty$ with $T/N\to 0$, the asymptotic
distribution of the Wald test statistics are generally not free of nuisance
parameters. Hence, in terms of the size of $T$, testing for the presence of a
structural break is more demanding than estimating the breakpoint. This is
what theory tells us. According to the Monte Carlo results reported in the
online appendix, however, the new toolbox tends to perform well even if $T$ is
as small as 10, provided that $N$ is large enough. Hence, even if in theory
the Wald tests require $T$ to be large, in small samples this requirement does
not seem very critical.

Section 5 is concerned with our empirical application to the relationship
between stock returns and COVID–19, which is motivated in part by the many
recent calls for econometric research into the effects of the pandemic (see,
for example, the recent special issue of Journal of Econometrics), in part by
existing empirical research. By the end of February, 2020, COVID–19 had led to
a world-wide drop in demand, which in turn brought down investment and
employment. While stock markets initially reacted to news of the pandemic by
losing substantial value, they quickly regained the vast majority of this
loss. The fact that this rebound took place even though the number of new
cases and deaths were still rising is suggestive of structural change. Most
studies of the stock market reaction to COVID–19 either ignore this
possibility altogether or split their sample into subperiods based on major
events (see, for example, Capelle-Blanchard and Desroziers, 2020, and Ramelli
and Wagner, 2020). This means that the breaks are treated as known, if treated
at all, which is risky, as misplaced breaks are just as problematic as omitted
breaks. In the empirical application of the present paper, we offer a more
general treatment. This is done by applying the new toolbox to a sample
covering 61 countries across 38 weeks, from January 3 to September 25, 2020,
which means that $T$ is relatively small. According to the results, the
COVID–19-stock return relationship has been affected by the presence a
structural break in the first week of April, at about the same time as most
central banks announced that they were going to intervene to save the global
economy from collapse. While before the break stock markets reacted
significantly to news about the pandemic, after the break stock markets became
insensitive to such news. This suggests that central banks play a central role
in shaping stock market behaviour in pandemics.

Section 6 concludes the paper. All proofs are provided in the online appendix.

##  2 Model

In this paper, we consider the following linear panel data model with a
structural break at time $b$:

$y_{i,t}=\beta^{\prime}x_{i,t}+\delta^{\prime}z_{i,t}(b)+e_{i,t},$ (2.1)

where $i=1,...,N$ and $t=1,...,T$ index the cross-sectional units and time
periods, respectively. The $k\times 1$ vector $x_{i,t}$ contains the
regressors and the $r\times 1$ vector $z_{i,t}$ is defined as

$z_{i,t}(b)=R^{\prime}x_{i,t}\mathbb{I}(t>b),$ (2.2)

where $\mathbb{I}(t>b)$ is the indicator function taking the value one when
$t>b$ and zero otherwise, and $R$ is an $k\times r$ selection matrix of zeros
and ones with full column rank $r$ that picks out the elements of $x_{i,t}$
whose coefficients are subject to structural change. For example, if $k>r$ and
$R=(0_{r\times(k-r)}^{\prime},I_{r})^{\prime}$, then (2.1) is a partial
structural change model in which only the $r$ last regressors in $x_{i,t}$
appear in $z_{i,t}(b)$. If, on the other hand, $k=r$, then $R=I_{r}$, and so
the model is one of pure structural change. In the empirical application of
Section 5, $y_{i,t}$ is stock returns for country $i$ in week $t$, and
$x_{i,t}$ is comprised of controls and COVID–19 related variables, where the
coefficients of the COVID–19 variables may be breaking. As we explain in that
section, the model can easily be generalized to include multiple structural
changes. The coefficient vectors $\beta$ and $\delta$ are of dimension
$k\times 1$ and $r\times 1$, respectively.

The error $e_{i,t}$ is assumed to admit to a factor structure, which means
that it is allowed to be correlated across $i$. Specifically,

$e_{i,t}=\gamma_{i}^{\prime}f_{t}+\varepsilon_{i,t},$ (2.3)

where $f_{t}$ and $\gamma_{i}$ are $m\times 1$ vectors of common factors and
factor loadings, respectively, and $\varepsilon_{i,t}$ is an idiosyncratic
error term. In our empirical application, the presence of $f_{t}$ in (2.3) is
just natural because many well-known models in finance, like the capital asset
pricing (CAPM) and Fama–French (FF) three factor models, imply that returns
should have a linear factor structure. In this section and the next, we assume
that all the factors are unknown. In Section 5, we demonstrate how the toolbox
can be implemented when some of the factors are observed, as when CAPM holds
and one has data on (world) market returns.

We want to entertain the possibility that the factors are correlated with the
regressors. We therefore follow Pesaran (2006) and assume that

$x_{i,t}=\Gamma_{i}^{\prime}f_{t}+v_{i,t},$ (2.4)

where $\Gamma_{i}$ is a $m\times k$ factor loading matrix and $v_{i,t}$ is a
$k\times 1$ vector of idiosyncratic errors.

For later use, it is convenient to write the above model in matrix form by
stacking the time series observations for each cross-section. The stacked
version of (2.1) is given by

$y_{i}=X_{i}\beta+Z_{i}(b)\delta+e_{i},$ (2.5)

where $y_{i}=(y_{i,1},...,y_{i,T})^{\prime}$ and
$e_{i}=(e_{i,1},...,e_{i,T})^{\prime}$ are $T\times 1$,
$X_{i}=(x_{i,1}^{\prime},...,x_{i,T}^{\prime})^{\prime}$ is $T\times k$, and
$Z_{i}(b)=(z_{i,1}(b)^{\prime},...,z_{i,T}(b)^{\prime})^{\prime}$ is $T\times
r$. Note that because $z_{i,1}(b)=...=z_{i,b}(b)=0_{r\times 1}$, $Z_{i}(b)$
can be written as $Z_{i}(b)=(0_{r\times 1}^{\prime},...,0_{r\times
1}^{\prime},x_{i,b+1}^{\prime}R,...,x_{i,T}^{\prime}R)^{\prime}=X_{i}(b)R$,
where $X_{i}(b)=(0_{k\times 1}^{\prime},...,0_{k\times
1}^{\prime},x_{i,b+1}^{\prime},...,x_{i,T}^{\prime})^{\prime}$. Also,

$e_{i}=F\gamma_{i}+\varepsilon_{i},$ (2.6)

where $F=(f_{1}^{\prime},...,f_{T}^{\prime})^{\prime}$ and
$\varepsilon_{i}=(\varepsilon_{i,1},...,\varepsilon_{i,T})^{\prime}$ are
$T\times m$ and $T\times 1$, respectively. The stacked version of (2.4) is
given by

$X_{i}=F\Gamma_{i}+V_{i},$ (2.7)

where $V_{i}=(v_{i,1}^{\prime},...,v_{i,T}^{\prime})^{\prime}$ is $T\times k$.

The model assumptions depend to a large extent on whether we are estimating
the breakpoint or if we are testing for its existence. Assumptions 2.1 and 2.2
will, however, be maintained throughout this paper.

######  Assumption 2.1.

  1. (a)

$v_{i,t}$ is a covariance stationary process that is independent across $i$
with absolutely summable autocovariances, $E(v_{i,t})=0_{k\times 1}$,
$E(v_{i,t}v_{i,t}^{\prime})=\Sigma_{v,i}$ and $E(\|v_{i,t}\|^{4})<\infty$,
where $\|A\|=\sqrt{\mathrm{tr}(A^{\prime}A)}$ is the Frobenius norm of any
matrix $A$.

  2. (b)

$\varepsilon_{i,t}$ is a covariance stationary process that is independent
across $i$ with absolutely summable autocovariances, $E(\varepsilon_{i,t})=0$,
$E(\varepsilon_{i,t}^{2})=\sigma_{\varepsilon,i}^{2}>0$ and
$E(\varepsilon_{i,t}^{4})<\infty$.

  3. (c)

$\varepsilon_{i,t}$ and $v_{j,s}$ are independent for all $i$, $j$, $s$ and
$t$.

######  Assumption 2.2.

  1. (a)

$T^{-1}F^{\prime}F$ is positive definite with probability approaching one
(w.p.a.1) for all $T$.

  2. (b)

$E(\|f_{t}\|^{4})<\infty$.

  3. (c)

$f_{t}$ is independent of $\varepsilon_{i,s}$ and $v_{i,s}$ for all $i$, $s$
and $t$.

Assumption 2.1 is standard in the interactive effects literature (see, for
example, Baltagi et al., 2016). The only exception known to us is Baltagi et
al. (2017). They do not allow for cross-section dependence, but they do allow
$x_{i,t}$ and $\varepsilon_{i,t}$ to be unit root non-stationary. We allow for
serial correlation and possibly even unit roots in $f_{t}$ (more later) and
hence in $x_{i,t}$ (and $y_{i,t}$), but not in $v_{i,t}$ and
$\varepsilon_{i,t}$. Bai (1997a) has shown that the existence of both serially
correlated errors and lagged dependent variables leads to inconsistent
estimation of the break date. Assumption 2.1 (c) therefore assumes that
$x_{i,t}$ is strictly exogenous. Without unit roots Assumption 2.1 (a)–(c) are
the same as Assumptions 1–3 in Baltagi et al. (2017). Assumption 2.2 (a) and
(b) are met if $f_{t}$ is stationary and not collinear, which is again a
standard requirement in the literature (see Baltagi et al., 2016, Assumption
8). Stationarity is not necessary, though. Note in particular how stationarity
is not required if $T$ is fixed. In fact, $f_{t}$ does not even have to be
stochastic but can also be deterministic. Assumption 2.2 (c) is an identifying
condition that is not particularly restrictive. It ensures that $f_{t}$ is the
only source of cross-section dependence.

##  3 Breakpoint estimation

Let us denote by $b_{0}$ the true value of $b$. The purpose of this section is
to make inference regarding this parameter.

######  Assumption 3.1.

$b_{0}\in\mathcal{B}=[r,T-r-1]$.

######  Assumption 3.2.

  1. (a)

$\mathrm{rank}(\bar{\Gamma})=m\leq k$ for all $N$, including $N\to\infty$.

  2. (b)

$\|\gamma_{i}\|<\infty$ and $\|\bar{\Gamma}\|<\infty$.

Assumption 3.1 requires only that each regime contains at least as many
observations as the number of free parameters. It is therefore very general.
Baltagi et al. (2016) and Westerlund (2019) allow for common factors in very
much the same way as we do. However, they require that the loadings follow
certain probability laws, and that they are independent of all other random
elements of the model. In this section, we treat the loadings as fixed, which
means that we do not make any assumption regarding their distribution or their
correlation with the other random elements of the model. The main restriction
is that $x_{i,t}$ must load on the same factors as $y_{i,t}$, and that the
number of regressors must be at least as large as the number of factors. This
ensures that the factors can be estimated by applying CCE to $x_{i,t}$, as we
will now explain.

Unlike in Antoch et al. (2019), where the factor loadings are assumed to be
negligible, under our conditions valid inference on $b_{0}$ is not possible
without proper accounting for $f_{t}$. The reason is that the factors make
$x_{i,t}$ correlated with $e_{i,t}$, which means that (2.1) cannot be
estimated consistently using OLS. However, we note that $x_{i,t}$ has a pure
factor model representation, suggesting that the factors can be estimated
using methods designed for such models. In this paper, we follow Baltagi et
al. (2016), and use the CCE approach of Pesaran (2006), which is based on
using the cross-sectional average of the observables to estimate the space
spanned by $f_{t}$. The difference is that we do not include the cross-
sectional average of $y_{i,t}$, which in the current context is uninformative
regarding $f_{t}$. This is shown in the online appendix. Hence, in contrast to
Baltagi et al. (2016), in the current paper we only use $\bar{x}_{t}$, where
$\bar{A}_{t}=N^{-1}\sum_{i=1}^{N}A_{i,t}$ is the cross-sectional average of
any variable $A_{i,t}$. In view of (2.4), this average can be written as

$\bar{x}_{t}=\bar{\Gamma}^{\prime}f_{t}+\bar{v}_{t}.$ (3.1)

Let $A^{+}$ denote the Moore–Penrose inverse of any matrix $A$. If Assumption
3.2 is true, so that $\bar{\Gamma}$ has full row rank, the Moore–Penrose
inverse of $\bar{\Gamma}$ is given by
$\bar{\Gamma}^{+}=\bar{\Gamma}^{\prime}(\bar{\Gamma}\bar{\Gamma}^{\prime})^{-1}$.
Hence, $\bar{\Gamma}\bar{\Gamma}^{+}=I_{m}$, which in turn means that (3.1)
can be solved for $f_{t}$ by left-multiplication by $\bar{\Gamma}^{\prime+}$.
It follows that if Assumption 2.1 is also true, so that
$\|\bar{v}_{t}\|=o_{p}(1)$, then

$\|\bar{\Gamma}^{\prime+}\bar{x}_{t}-f_{t}\|=\|\bar{\Gamma}^{\prime+}\bar{v}_{t}\|\leq\|\bar{\Gamma}^{\prime+}\|\|\bar{v}_{t}\|=o_{p}(1).$
(3.2)

We say that $\bar{x}_{t}$ is “rotationally consistent” for $f_{t}$, because it
is consistent up to an invertible rotation matrix. Hence, by augmenting (2.1)
with $\bar{x}_{t}$, provided that $N$ is large, we can control for $f_{t}$,
and in this way break the correlation between the regressors and the error
term.

Define
$M_{\bar{X}}=I_{T}-\bar{X}(\bar{X}^{\prime}\bar{X})^{-1}\bar{X}^{\prime}$,
where
$\bar{X}=N^{-1}\sum_{i=1}^{N}X_{i}=(\bar{x}_{1},...,\bar{x}_{T})^{\prime}$,
and let $\tilde{A}_{i}=M_{\bar{X}}A_{i}$ for any $T$-rowed matrix $A_{i}$. The
augmented model to be estimated can now be written as

$\tilde{y}_{i}=\tilde{X}_{i}\beta+\tilde{Z}_{i}(b_{0})\delta+\tilde{e}_{i}.$
(3.3)

This model can be stacked also over the cross-section, giving

$\tilde{Y}=\tilde{X}\beta+\tilde{Z}(b_{0})\delta+\tilde{E},$ (3.4)

where $\tilde{Y}=(\tilde{y}_{1}^{\prime},...,\tilde{y}_{N}^{\prime})^{\prime}$
and $\tilde{E}=(\tilde{e}_{1}^{\prime},...,\tilde{e}_{N}^{\prime})^{\prime}$
are $NT\times 1$,
$\tilde{X}=(\tilde{X}_{1}^{\prime},...,\tilde{X}_{N}^{\prime})^{\prime}$ is
$NT\times k$ and
$\tilde{Z}(b_{0})=(\tilde{Z}_{1}(b_{0})^{\prime},...,\tilde{Z}_{N}(b_{0})^{\prime})^{\prime}$
is $NT\times r$. Let us further introduce
$M_{\tilde{X}}=I_{NT}-\tilde{X}(\tilde{X}^{\prime}\tilde{X})^{-1}\tilde{X}^{\prime}$.
With $b_{0}$ known, the CCE estimator of $\delta$, which is identically the
OLS estimator obtained from (3.4), and the associated sum of squared residuals
are given by

$\displaystyle\hat{\delta}(b_{0})$
$\displaystyle=(\tilde{Z}(b_{0})^{\prime}M_{\tilde{X}}\tilde{Z}(b_{0}))^{-1}\tilde{Z}(b_{0})^{\prime}M_{\tilde{X}}\tilde{Y},$
(3.5) $\displaystyle SSR(b_{0})$
$\displaystyle=(\tilde{Y}-\tilde{Z}(b_{0})\hat{\delta}(b_{0}))^{\prime}M_{\tilde{X}}(\tilde{Y}-\tilde{Z}(b_{0})\hat{\delta}(b_{0})).$
(3.6)

Of course, in many scenarios of empirical relevance, $b_{0}$ is not known. The
estimator that we will use in its stead is obtained by minimizing $SSR(b)$
over all possible values of $b$;

$\hat{b}=\arg\min_{b\in\mathcal{B}}SSR(b).$ (3.7)

We begin by showing that $\hat{b}$ is consistent. For this to be possible,
however, in addition to Assumptions 2.1–3.2, we need to ensure that the
inverse appearing in $\hat{\delta}(b)$ is well-behaved. This is where
Assumption 3.3 comes in. It demands that the regressors in $x_{i,t}$ have
enough variation across both $i$ and $t$ after projecting out all variation
that can be explained by $f_{t}$. This rules out cross-section-invariant
regressors in $x_{i,t}$.

######  Assumption 3.3.

  1. (a)

$(NT)^{-1}\tilde{X}^{\prime}\tilde{X}$ is positive definite w.p.a.1 for all
$N$ and $T$.

  2. (b)

$(NT)^{-1}\tilde{Z}(b)^{\prime}M_{\tilde{X}}\tilde{Z}(b)$ is positive definite
w.p.a.1 for all $b\in\mathcal{B}$, $N$ and $T$.

We are now ready to state our first main result.

######  Theorem 3.1.

Suppose that Assumptions 2.1, 2.2 and 3.1–3.3 are met. Then, the following
results hold:

  1. (a)

If $N\to\infty$ and $T$ is fixed, or as $N,\,T\to\infty$ with
$\sqrt{N}\|\delta\|\to\infty$ when $m=k$,

$\displaystyle|b_{0}-\hat{b}|=O_{p}(\|\delta\|^{-2}N^{-1}).$ (3.8)

  2. (b)

If $N,\,T\to\infty$ with $\sqrt{N}\|\delta\|\to\infty$ and
$T/(N\|\delta\|^{2})\to 0$ when $m<k$,

$\displaystyle|b_{0}-\hat{b}|=o_{p}(\|\delta\|^{-2}N^{-1}).$ (3.9)

Theorem 3.1 states that $\hat{b}$ is consistent and that the rate of
convergence is $\|\delta\|^{-2}N^{-1}$ or better. The fact that consistency is
possible even if $T$ is fixed is very useful in practice, because it means
that breaks can be detected very quickly. When $T\to\infty$ it matters whether
$m<k$ or $m=k$. Note in particular how the rate of convergence is faster when
$m<k$ then when $m=k$, and that this is true even if $T/(N\|\delta\|^{2})\to
0$ under $m=k$, so that the conditions for (a) and (b) are the same. The
reason is that when $m<k$, unlike what one would expect based on standard
theory for regressions in stationary variables, the effect of the redundant
cross-section averages contained in $\bar{X}$ are not negligible but impact
the asymptotic theory in very much the same way as unit root regressors do in
a spurious regression. Moreover, the redundant averages are correlated with
the breaking regressors in $Z_{i}(b)$, and this increases the signal coming
from $\tilde{Z}_{i}(b)$. As far as we are aware, this is the first time
redundant regressors have been shown to lead to increased accuracy in
breakpoint estimation.

If we are not interested in the distinction between $m<k$ or $m=k$, the
results contained in Theorem 3.1 can be stated as in Corollary 3.1.

######  Corollary 3.1.

Suppose that conditions of Theorem 3.1 are met, and that
$T/(N\|\delta\|^{2})\to 0$ and $\sqrt{N}\|\delta\|\to\infty$. Then, as
$N\to\infty$ with $T$ fixed, or as $N,\,T\to\infty$,

$\displaystyle|b_{0}-\hat{b}|=O_{p}(\|\delta\|^{-2}N^{-1}).$ (3.10)

######  Remark 3.1.

As already mentioned, Baltagi et al. (2016) consider a model that is very
similar to ours and that is estimated using CCE. They show that $\hat{b}$ is
consistent for $b_{0}$; however, they only consider the case when
$N,\,T\to\infty$, and they do not provide the rate of convergence. Moreover,
the proof that they provide is based on the assumption that
$(\bar{y}_{t},\bar{x}_{t}^{\prime})^{\prime}$ is rotationally consistent for
$(f_{t}^{\prime},f_{t}^{\prime}\mathbb{I}(t>b))^{\prime}$, which we show in
the online appendix not to be correct. Bai (2010) is the only other paper that
we are aware of that proves consistency under both fixed and large $T$;
however, his model is very simple in the sense that it does not contain any
regressors except for a breaking mean. Under stationarity, the model
considered by Baltagi et al. (2017) is very similar to our but without
interactive effects. The rate given in Corollary 3.1 is consistent with the
one given in their Theorem 2.

######  Remark 3.2.

Corollary 3.1 requires that $T/(N\|\delta\|^{2})\to 0$ and
$\sqrt{N}\|\delta\|\to\infty$. The latter condition is similar to Assumption 2
in Bai (2010), and is tantamount to requiring $\delta=N^{-\alpha}\delta_{0}$
with $\alpha<1/2$ and $\|\delta_{0}\|\in(0,\infty)$. Hence, while we allow for
it, we do not require $\|\delta\|\to 0$, which is in contrast to studies such
as Antoch et al. (2019), where the magnitude of the break must be shrinking.
The condition that $T/(N\|\delta\|^{2})\to 0$, which is similar in spirit to
Assumption 2 in Baltagi et al. (2016), restricts the relative rate of
expansion of $N$ and $T$, and is only needed when $T$ is large. For example,
if $\|\delta\|=O(1)$, then we require that $T/N\to 0$, as otherwise the error
coming from the estimation of the factors will tend to accumulate as we sum
over time. We also see that the larger is $\|\delta\|$, the weaker the
condition on $T/N$, as to be expected, because a larger break is easier to
discern. If $T$ is fixed, then $T/(N\|\delta\|^{2})\to 0$ is implied by
$\sqrt{N}\|\delta\|\to\infty$.

As Corollary 3.1 makes clear, provided that $T/(N\|\delta\|^{2})\to 0$ and
$\sqrt{N}\|\delta\|\to\infty$, consistency holds irrespectively of whether
$m=k$ or $m<k$, which is of course very useful in practice, as $m$ is unknown
here. This invariance is reflected also in the asymptotic distribution of the
estimated break date, as our next theorem, Theorem 3.2, makes clear. Before we
take the theorem, however, we need to introduce a few more conditions, which
are given in Assumption 3.4.

######  Assumption 3.4.

  1. (a)

$E(\varepsilon_{i,t}\varepsilon_{i,s})=0$ for all $i$ and $t\neq s$.

  2. (b)

$N^{-1}\sum_{i=1}^{N}\sigma_{\varepsilon,i}^{2}E(x_{i,t}x_{i,t}^{\prime})\to\Phi_{X}$
as $N\to\infty$ for all $t$.

  3. (c)

$N^{-1}\sum_{i=1}^{N}E(x_{i,t}x_{i,t}^{\prime})\to\Omega_{X}$ as $N\to\infty$
for all $t$, where $\Omega_{X}$ is positive definite.

Assumption 3.4 is restrictive, but is similar to the conditions used in the
previous literature (see, for example, Bai, 1997a, 2010). It demands that
$\varepsilon_{i,t}$ is serially uncorrelated and that the large-$N$ moments of
$x_{i,t}$ do not depend on time. While indeed quite strong, because of the
presence of $f_{t}$, the first condition does not rule out serial correlation
in $e_{i,t}$. The second requirement is stronger than necessary, and can be
relaxed to accommodate moments that are constant within break regimes but
potentially varying between regimes, as in, for example, Bai (1997a), and
Perron and Yamamoto (2013).

######  Theorem 3.2.

Suppose that Assumptions 2.1, 2.2 and 3.1–3.4 are met, and that
$T/(N\|\delta\|^{2})\to 0$ and $\sqrt{N}\|\delta\|\to\infty$. Then, as
$N\to\infty$ with $T$ fixed, or as $N,\,T\to\infty$ with $T/N\to 0$,

$\displaystyle\frac{(\delta^{\prime}R^{\prime}\Omega_{X}R\delta)^{2}}{\delta^{\prime}R^{\prime}\Phi_{X}R\delta}N(\hat{b}-b_{0})\to_{d}\arg\max_{v\in[-V,V]\subset\mathbb{R}}\left(-\frac{|v|}{2}+B(v)\right)$
(3.11)

where $B(v)$ is standard two-sided Brownian motion on
$v\in[-V,V]\subset\mathbb{R}$.333The two-sided Brownian motion $B(v)$
satisfies $B(0)=0$, and $B(v)=B_{1}(v)$ for $v>0$ and $B(v)=B_{2}(v)$ for
$v<0$, where $B_{1}(v)$ and $B_{2}(v)$ are two independent standard Brownian
motions.

######  Remark 3.3.

Most papers stop at consistency and do not report the asymptotic distribution
of the estimated breakpoint. There are, however, a few exceptions. The study
of Bai (2010) is one of them. He provides the asymptotic distribution of the
estimated breakpoint for a model with a break-in-mean only, and no other
regressors or error cross-section dependence. Kim (2011) extends this analysis
to a model that allows for a break in both the mean and trend, and where the
errors have a factor structure. Baltagi et al. (2018) allow for more general
regressors. However, they assume instead that the errors are cross-sectionally
independent. All three papers require that $T$ is large in their
distributional analyses. As far as we are aware, the asymptotic distribution
reported in Theorem 3.2 is the first to allow for general regressors and
common factors in panels where only $N$ is required to be large.

Theorem 3.2 can be used to construct confidence intervals for $b_{0}$ with
asymptotically correct coverage. Under Assumption 3.4, consistent estimators
of $\Omega_{X}$ and $\Phi_{X}$ can be constructed in the following obvious
manner:

$\displaystyle\hat{\Omega}_{X}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}X_{i}^{\prime}X_{i},$ (3.12)
$\displaystyle\hat{\Phi}_{X}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\hat{\sigma}_{\varepsilon,i}^{2}X_{i}^{\prime}X_{i},$
(3.13)

where
$\hat{\sigma}_{\varepsilon,i}^{2}=T^{-1}\hat{\varepsilon}_{i}^{\prime}\hat{\varepsilon}_{i}$
with the $T\times 1$ vector $\hat{\varepsilon}_{i}$ being the $i$-th block of
the $NT\times 1$ vector
$\hat{\varepsilon}=(\hat{\varepsilon}_{1}^{\prime},...,\hat{\varepsilon}_{N}^{\prime})^{\prime}=M_{\tilde{X}}(\tilde{Y}-\tilde{Z}(\hat{b})\hat{\delta}(\hat{b}))$.
The probability density function of $\arg\max_{v}(-|v|/2+B(v))$ is known
analytically and is given in Bai (1997a). Let us denote by $c_{\alpha}$ the
$(1-\alpha/2)$-th percentile of this distribution function, and let $\lfloor
x\rfloor$ be the integer part of $x$. In analogy to Bai (1997a), an
asymptotically correctly sized $100(1-\alpha)\%$ confidence interval for
$b_{0}$ can now be constructed as

$\displaystyle\left[\hat{b}-\left\lfloor
c_{\alpha}\cdot\frac{\hat{\delta}(\hat{b})^{\prime}R^{\prime}\hat{\Phi}_{X}R\hat{\delta}(\hat{b})}{N(\hat{\delta}(\hat{b})^{\prime}R^{\prime}\hat{\Omega}_{X}R\hat{\delta}(\hat{b}))^{2}}\right\rfloor-1,\,\hat{b}+\left\lfloor
c_{\alpha}\cdot\frac{\hat{\delta}(\hat{b})^{\prime}R^{\prime}\hat{\Phi}_{X}R\hat{\delta}(\hat{b})}{N(\hat{\delta}(\hat{b})^{\prime}R^{\prime}\hat{\Omega}_{X}R\hat{\delta}(\hat{b}))^{2}}\right\rfloor+1\right].$
(3.14)

##  4 Break testing

Testing for the existence of a structural break is a key first step before
estimating the date of the break. In terms of the parameters of (2.1), the
null hypothesis of no structural change is given by $H_{0}:\delta=0_{r\times
1}$. The alternative hypothesis can be formulated in (at least) two ways. We
begin by considering the alternative that there is a single structural change
($\delta\neq 0_{r\times 1}$) at a given date $b$, which may or may not be
equal to $b_{0}$. This hypothesis, henceforth denoted $H_{1}(b)$, can be
tested using the following Wald test statistic:

$W(b)=NT\hat{\delta}(b)^{\prime}\hat{\Sigma}_{\delta}(b)^{-1}\hat{\delta}(b),$
(4.1)

where $\hat{\Sigma}_{\delta}(b)$ is a consistent estimator of the asymptotic
covariance matrix of $\hat{\delta}(b)$, whose construction will be discussed
later. Interestingly, $W(b)$ will not have the expected asymptotic chi-squared
distribution with $r$ degrees of freedom, henceforth denoted $\chi^{2}(r)$,
under $H_{0}$. The intuition behind this result goes as follows. As already
pointed out, because of the presence of $f_{t}$ in both (2.3) and (2.4),
$\tilde{X}_{i}$ is generally endogenous. The exception is in large-$N$
samples, since here $\bar{X}$ is rotationally consistent for $F$, and in this
sense $\tilde{X}_{i}$ is “asymptotically exogenous”. The problem is that while
the use of $M_{\bar{X}}$ takes care of the factors in $X_{i}$, it does not
take care of those in $Z_{i}(b)$, which are breaking. This is a problem
because it means that while $\tilde{X}_{i}$ is asymptotically exogenous,
$\tilde{Z}_{i}(b)$ is not, which in turn invalidates inference based on
$W(b)$. Because of the consistency of $\hat{b}$, in Section 3 the endogeneity
of $\tilde{Z}_{i}(b)$ was not an issue. Of course, if we knew that there was a
break present, as we did in Section 3, there would be no need to test for it
in the first place. The situation considered here is therefore quite different
and this requires some changes.

The first change we make when compared to Section 3, which is quite natural
given the discussion of the last paragraph, is to replace $\bar{X}$ with
$\bar{H}(b)=(\bar{X},\bar{Z}(b))$ and $\tilde{A}_{i}=M_{\bar{X}}A_{i}$ with
$\tilde{A}_{i}(b)=M_{\bar{H}(b)}A_{i}$. The definitions of $\hat{\delta}(b)$
and $W(b)$ are adapted accordingly. The idea here is that by augmenting
$\bar{X}$ with $\bar{Z}(b)$, we can eliminate the factors in both $X_{i}$ and
$Z_{i}(b)$, which means that the endogeneity issue is gone. For this to
happen, however, we need some additional assumptions. In order to appreciate
this, note that

$\displaystyle\left(\begin{array}[]{c}x_{i,t}\\\ z_{i,t}(b)\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}\Gamma_{i}^{\prime}&0_{k\times m}\\\
0_{r\times
m}&R^{\prime}\Gamma_{i}^{\prime}\end{array}\right)\left(\begin{array}[]{c}f_{t}\\\
f_{t}\mathbb{I}(t>b)\end{array}\right)+\left(\begin{array}[]{c}v_{i,t}\\\
R^{\prime}v_{i,t}\mathbb{I}(t>b)\end{array}\right),$ (4.10)

where $(f_{t}^{\prime},f_{t}^{\prime}\mathbb{I}(t>b))^{\prime}$ are the
factors in $(\bar{x}_{i,t}^{\prime},\bar{z}_{i,t}(b)^{\prime})^{\prime}$.
Hence, provided that
$\mathrm{rank}(\bar{\Gamma})=\mathrm{rank}(\bar{\Gamma}R)=m$, such that the
$(k+r)\times 2m$ matrix

$\displaystyle\left(\begin{array}[]{cc}\bar{\Gamma}^{\prime}&0_{k\times m}\\\
0_{r\times m}&R^{\prime}\bar{\Gamma}^{\prime}\end{array}\right)$ (4.13)

has full column rank $2m\leq k+r$, analogous to the discussion of Section 2,
$(\bar{x}_{t}^{\prime},\bar{z}_{t}(b)^{\prime})^{\prime}$ is rotationally
consistent for $(f_{t}^{\prime},f_{t}^{\prime}\mathbb{I}(t>b))^{\prime}$. We
also need to restrict the type of heterogeneity that can be permitted in
$\gamma_{i}$. The way we do this is by assuming that $\gamma_{i}$ admits to a
random coefficient representation, similarly to, for example, Pesaran (2006)
and Karabiyik et al. (2017). Assumption 4.1 below replaces Assumption 3.2 and
is enough to ensure that the effect of the estimation of $f_{t}$ is
asymptotically eliminated.

######  Assumption 4.1.

  1. (a)

$\mathrm{rank}(\bar{\Gamma})=m\leq k$ and $\mathrm{rank}(\bar{\Gamma}R)=m\leq
r$ for all $N$, including $N\to\infty$.

  2. (b)

$\|\bar{\Gamma}\|<\infty$.

  3. (c)

$\gamma_{i}$ is independent across $i$, and of $\varepsilon_{j,t}$, $v_{j,t}$
and $f_{t}$ for all $i$ and $j$ with $E(\gamma_{i})=\gamma$ and
$E(\|\gamma_{i}\|^{2})<\infty$.

Another difference when compared to Section 3, where $T$ could be allowed to
be fixed, is that here both $N$ and $T$ have to be large. The basic reason for
this is that while before we took the break as given, which meant that we
could make use of the consistency of $\hat{b}$ to construct asymptotically
valid confidence intervals, here we do not know if there is a break present
and so we cannot rely on said consistency. This means that the asymptotic
distribution of $W(b)$ is generally not nuisance parameter free. The main
exception is if $T$ is large.444It is possible to conduct asymptotically valid
inference based on $W(b)$ even if $T$ is fixed. However, this requires placing
very strong assumptions regarding the time-variation of the factors, which is
something that we would like to avoid. But if $T$ is large, we can relax the
serial uncorrelatedness and time invariant moment conditions of Assumption
4.2. We also require that Assumption 3.3 holds when $\bar{H}(b)$ is used in
place of $\bar{X}$.

######  Assumption 4.2.

  1. (a)

$(NT)^{-1}\sum_{i=1}^{N}E(V_{i}^{\prime}\Sigma_{\varepsilon,i}V_{i})\to\Phi_{V}$
as $N,\,T\to\infty$, where
$\Sigma_{\varepsilon,i}=E(\varepsilon_{i}\varepsilon_{i}^{\prime})$.

  2. (b)

$(NT)^{-1}\sum_{i=1}^{N}E(V_{i}^{\prime}V_{i})\to\Omega_{V}$ as
$N,\,T\to\infty$, where $\Omega_{V}$ is positive definite.

######  Assumption 4.3.

  1. (a)

$(NT)^{-1}\tilde{X}(b)^{\prime}\tilde{X}(b)$ is positive definite w.p.a.1 for
all $N$ and $T$.

  2. (b)

$(NT)^{-1}\tilde{Z}(b)^{\prime}M_{\tilde{X}(b)}\tilde{Z}(b)$ is positive
definite w.p.a.1 for all $b\in\mathcal{B}^{\prime}$, $N$ and $T$.

A third and final difference when compared to Section 3 is about the range of
values considered for $b$. In Section 3, we only required that
$b\in\mathcal{B}$, which meant that in the large-$T$ case $b/T$ could take on
any value in $[0,1]$. Here this is not possible, for it is only when $b/T$ is
bounded away from zero and one that $W(b)$ converges in distribution (see
Andrews, 1993, for a discussion). In this section, we therefore assume that
$b\in\mathcal{B}^{\prime}$, where

$\mathcal{B}^{\prime}=\\{b:b=\lfloor\tau
T\rfloor~{}\text{with}~{}\tau\in\mathcal{T}=[\epsilon,1-\epsilon]~{}\text{and}~{}\epsilon>0\\}.$
(4.14)

The main implication of this in practice is that we have to truncate, or
“trim”, the range of values considered for $b$ at both beginning and end. A
very common way to do this is to set $\epsilon=0.15$, so that the first and
last 15% of the observations are discarded (see, for example, Andrews, 1993,
and Bai, 1997a). The condition that $b/T$ should bounded away from zero and
one should hold for all $b$, including $b_{0}$. The following assumption
reflects this.

######  Assumption 4.4.

$b_{0}=\lfloor\tau_{0}T\rfloor$, where
$\tau_{0}\in\mathcal{T}_{0}\subset\mathcal{T}$.

We now have all the conditions we need in order to obtain the asymptotic
distribution of $W(b)$.

######  Theorem 4.1.

Suppose that $H_{0}$ holds, and that Assumptions 2.1, 2.2 and 4.1–4.4 are met.
Then, uniformly in $b\in\mathcal{B}^{\prime}$, as $N,\,T\to\infty$ with
$T/N\to 0$,

$W(b)\to_{d}\frac{[J(\tau)-\tau J(1)]^{\prime}[J(\tau)-\tau
J(1)]}{\tau(1-\tau)},$ (4.15)

where $J(\tau)$ is a $r\times 1$ vector standard Brownian motion on
$\tau\in\mathcal{T}\subset(0,1)$.

Because $J(\tau)$ is a standard Brownian motion, $J(\tau)=_{d}N(0_{k\times
1},\tau(1-\tau)I_{r})$, where $=_{d}$ signifies equality in distribution.
Hence, for a given $b$, $[J(\tau)-\tau J(1)]^{\prime}[J(\tau)-\tau
J(1)]/\tau(1-\tau)=_{d}\chi^{2}(r)$, which in turn implies that

$\displaystyle W(b)\to_{d}\chi^{2}(r),$ (4.16)

as $N,\,T\to\infty$ with $T/N\to 0$. This result holds for all $b$ satisfying
$\tau\in\mathcal{T}\subset(0,1)$, including $b_{0}$.

######  Remark 4.1.

It is important to note that while we do require $T$ to be large, $N$ is still
the most important dimension of the data. In the online appendix, we use Monte
Carlo simulations as a means to evaluate the importance of the large-$T$
requirement. According to the results, the Wald tests perform well even if $T$
is as small as 10. The large-$T$ requirement is therefore not very important
in applied work.

So far we have taken the date of the break as given. If the date of the break
is unknown, as it usually is in practice, then $H_{0}$ can be tested against
the alternative hypothesis of a single structural break at some unknown date
$b\in\mathcal{B}^{\prime}$, which we can formulate as
$H_{1}:\bigcup_{b\in\mathcal{B}^{\prime}}H_{1}(b)$. Many researchers follow
Andrews (1993) and take the supremum of Wald test statistics over all possible
breakpoints, and therefore so shall we. The test statistic that we will be
considering is therefore given by

$SW=\sup_{b\in\mathcal{B}^{\prime}}W(b).$ (4.17)

The asymptotic distribution of this test statistic depends on the distribution
of $W(b)$, and is presented in the following corollary to Theorem 4.1.

######  Corollary 4.1.

Suppose that $H_{0}$ holds, and that the conditions of Theorem 4.1 are met.
Then, as $N,\,T\to\infty$ with $T/N\to 0$,

$SW\to_{d}\sup_{\tau\in\mathcal{T}}\frac{[J(\tau)-\tau
J(1)]^{\prime}[J(\tau)-\tau J(1)]}{\tau(1-\tau)}.$ (4.18)

The limiting distribution in Corollary 4.1 is the supremum of the square of a
standardized tied-down Bessel process of order $r$, which has appeared
previously in Andrews (1993), and Hidalgo and Schafgans (2017), among others.
The critical values only depend on $r$ and $\epsilon$, and can be found in
Table I of Andrews (1993).

The above results rely on the availability of a consistent estimator
$\hat{\Sigma}_{\delta}(b)$ of the asymptotic covariance matrix of
$\hat{\delta}(b)$, which is given by
$\Sigma_{\delta}=\Omega_{V}^{-1}\Psi_{V}\Omega_{V}^{-1}$. A natural approach
in the current large-$T$ setting is to take

$\hat{\Sigma}_{\delta}(b)=\hat{\Omega}_{V}(b)^{-1}\hat{\Psi}_{V}(b)\hat{\Omega}_{V}(b)^{-1},$
(4.19)

where

$\displaystyle\hat{\Omega}_{V}(b)$
$\displaystyle=(NT)^{-1}\tilde{Z}(b)^{\prime}M_{\tilde{X}}\tilde{Z}(b),$
(4.20) $\displaystyle\hat{\Psi}_{V}(b)$
$\displaystyle=\hat{\Psi}_{V,0}(b)+\sum_{j=1}^{T-1}k\left(\frac{j}{S_{T}}\right)(\hat{\Psi}_{V,j}(b)+\hat{\Psi}_{V,j}(b)^{\prime}),$
(4.21) $\displaystyle\hat{\Psi}_{V,j}(b)$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=j+1}^{T}\hat{\varepsilon}_{i,t}(b)\hat{\varepsilon}_{i,t-j}(b)\tilde{z}_{i,t}(b)\tilde{z}_{i,t-j}(b)^{\prime}.$
(4.22)

Here $k(\cdot)$ is a real-valued kernel, $S_{T}$ is the bandwidth parameter,
$\hat{\varepsilon}_{i,t}(b)$ is the $t$-th row of the $T\times 1$ vector
$\hat{\varepsilon}_{i}(b)=(\hat{\varepsilon}_{i,1}(b),...,\hat{\varepsilon}_{i,T}(b))^{\prime}$,
which is in turn the $i$-th block of the $NT\times 1$ vector
$\hat{\varepsilon}(b)=(\hat{\varepsilon}_{1}(b)^{\prime},...,\hat{\varepsilon}_{N}(b)^{\prime})^{\prime}=M_{\tilde{X}(b)}(\tilde{Y}(b)-\tilde{Z}(b)\hat{\delta}(b))$,
and the $1\times r$ vector $\tilde{z}_{i,t}(b)^{\prime}$ is the corresponding
row of the $NT\times r$ matrix $M_{\tilde{X}(b)}\tilde{Z}(b)$. Alternatively,
$\hat{\Sigma}_{\delta}(b)$ may be estimated non-parametrically, as in Pesaran
and Tosetti (2011).

######  Remark 4.2.

In the special case when $\varepsilon_{i,t}$ is independently and identically
distributed across both $i$ and $t$ with variance $\sigma_{\varepsilon}^{2}$,
$\Sigma_{\delta}$ reduces to
$\Sigma_{\delta}=\sigma_{\varepsilon}^{2}\Omega_{V}^{-1}$, which can in turn
be estimated using

$\displaystyle\hat{\Sigma}_{\delta}(b)=\hat{\sigma}_{\varepsilon}^{2}(b)\hat{\Omega}_{V}(b)^{-1},$
(4.23)

where
$\hat{\sigma}_{\varepsilon}^{2}(b)=(NT)^{-1}\hat{\varepsilon}(b)^{\prime}\hat{\varepsilon}(b)$.

Once the presence of a break has been established and its location determined,
it is possible to make inference regarding
$\theta=(\beta^{\prime},\delta^{\prime})^{\prime}$. Let us therefore define
$w_{i,t}(b)=(x_{i,t}^{\prime},z_{i,t}(b)^{\prime})^{\prime}$, such that (2.1)
can be written as

$y_{i,t}=\theta^{\prime}w_{i,t}(b_{0})+e_{i,t}.$ (4.24)

The CCE estimator of $\theta$ is given by
$\hat{\theta}=\hat{\theta}(\hat{b})$, where

$\displaystyle\hat{\theta}(b)=(\tilde{W}(b)^{\prime}\tilde{W}(b))^{-1}\tilde{W}(b)^{\prime}\tilde{Y},$
(4.25)

with
$\tilde{W}(b)=(\tilde{W}_{1}(b)^{\prime},...,\tilde{W}_{N}(b)^{\prime})^{\prime}$
being $NT\times(k+r)$, $\tilde{W}_{i}(b)=M_{\bar{H}(b)}W_{i}$ as before, and
$W_{i}(b)=(w_{i,1}(b)^{\prime},...,w_{i,T}(b)^{\prime})^{\prime}$ being
$T\times(k+r)$. By using the results of Westerlund et al. (2019), we can show
that under the conditions of Section 3 with $\tilde{W}(b)$ in place of
$\tilde{X}$, as $N\to\infty$,

$\displaystyle\sqrt{N}(\hat{\theta}-\theta)|F\to_{d}N(0_{(k+r)\times
1},\Omega_{W}^{-1}\Psi_{W}\Omega_{W}^{-1}),$ (4.26)

where

$\displaystyle\Omega_{W}$
$\displaystyle=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}E(\tilde{W}_{i}(b_{0})^{\prime}\tilde{W}_{i}(b_{0})|F),$
(4.27) $\displaystyle\Phi_{W}$
$\displaystyle=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}E(\tilde{W}_{i}(b_{0})^{\prime}\Sigma_{\varepsilon,i}\tilde{W}_{i}(b_{0})|F).$
(4.28)

Hence, $\sqrt{N}(\hat{\theta}-\theta)$ is asymptotically normal conditionally
on $F$, which means that it supports standard normal and chi-squared
inference. If $T\to\infty$, then the asymptotic distribution of $\hat{\theta}$
is the one given by Theorem 4 of Pesaran (2006).

##  5 Application stock market reaction of COVID–19

###  5.1 Motivation

COVID–19 broke out in China in December 2019. Roughly one year later, WHO
(2021) reports 94 million confirmed cases and over two million deaths. We also
know that because of lockdowns, travel restrictions and social distancing
policies, in 2020 GDP dropped by 4.2% globally and real world trade contracted
by 10.3% (OECD, 2020).555By comparison, the lowest global GDP growth rate
during the 2007–2009 global financial crisis was $-1.7$% in 2009. The economic
impact of the pandemic has therefore been substantial. This is what we know.
There are some signs of recovery in the years to come; however, the global
outlook is extremely uncertain, even in the short term. As an indication of
this, the OECD world GDP projections for 2021 ranges from $-2.75$% to $5$%,
depending on, among other things, the evolution of the pandemic, the actions
taken to contain the spread of the virus and their economic impact, and the
time until effective vaccines can be deployed. Hence, even now, one year after
the outbreak, much is uncertain.

The uncertainty we face today is nothing compared to one year ago. At this
time, little was known about the new virus, but it was clear that it was very
infectious and deadly, as, in contrast to previous infectious disease
outbreaks, most countries begun to announce the number of cases and deaths on
a daily basis. Many were chocked by how quickly these numbers were increasing.
Governments scrambled with emergency actions, such as closing schools and
workplaces, travel bans, or even complete curfews, to try to contain the
spread. However, since their effectiveness was far from clear and they made it
impossible for firms and workers to continue their operations without knowing
if and when they would be compensated, these actions added to the already
existing uncertainty, leading to widespread public fear (see Mamaysky, 2020,
and Phan and Narayan, 2020). This was visibly apparent with news coming in of
supermarkets being stocked out of toilet paper (Aggarwal et al., 2020).

In times of extreme uncertainty, stock markets often respond dramatically to
news about the underlying economic and market conditions (see Mamaysky, 2020).
This is what happened during the global financial crisis of 2007–2009 and it
happened again in the initial stages of the pandemic. In January 2020, the
news reporting was comparable to what it was in the beginning of the SARS
(severe acute respiratory syndrome) and Ebola epidemics. By February, however,
COVID–19 started to dominate newspaper discussions of the economy, and by
March, almost all such discussions were about COVID–19 (Baker et al.,
2020).666Not all news were about the economy and many were just rumors, but
they still attracted considerable attention and were therefore important in
setting the public sentiment at the time. For example, on February 17, a run
on toilet paper in Hong Kong was mentioned for the first time, and became a
highly contagious story. Some people in locked-down China reportedly were
reduced to searching for minnows and ragworms to eat. In Italy, there were
stories of medical workers in overwhelmed hospitals being forced to choose
which patients would receive treatment (Shiller, 2020). Stock markets
responded violently. On March 16, the Chicago Board Option Exchange’s
volatility index, the so-called “VIX”, surged past the prior all-time peak
reached during the global financial crisis more than a decade ago. The second-
worst day ever of the Dow Jones industrial index happened on March 16, and
three of the 15 worst days ever of the US market occurred between March 9 and
16 (Wagner, 2020). Stock markets all over the world reacted similarly.

The unprecedented stock market behaviour in the initial stage of COVID–19 has
attracted considerable attention not only in the news but also in research.
The bulk of the evidence seem to suggest that stock markets have generally
responded negatively, although the channel through which this effect works is
still largely unknown. Ashraf (2020a) uses data for 64 countries and finds
that stock prices have reacted negatively to the pandemic, but only when
measured by the number of confirmed cases, as opposed to the death count. This
is largely in agreement with the results of Erdem (2020). Ashraf (2020b)
employs data for 77 countries. He finds that the COVID-19 effect operates not
only through the number of cases, but also through government actions, such as
social distancing measures, containment and health responses, and economic
support packages. Similar findings have been reported by Aggarwal et al.
(2020), and Capelle-Blancard and Desroziers (2020).777Many studies focus on
single countries. There are also those that focus on the volatility of stock
returns, as opposed to stock returns themselves. These are not reviewed here.

The purpose of the current application is to contribute to the above mentioned
literature. This is done in three ways. First, we account for the rebound of
returns. Faced with near economic collapse, starting with the Federal
Reserve’s decision on March 16 to buy USD 700 billion worth of US Treasury
bonds and mortgage-backed securities, central banks around the world announced
aggressive quantitative easing programs (see Hartley and Rebucci, 2020). These
announcements were followed by an abrupt increase in stock prices. The US
S&P500 stock market index, for example, increased by 29% between March 24 and
April 17, a surge that left the index back where it stood in August of 2019
when the US economy was booming. The fact that this rebound took place while
the pandemic was still ongoing is suggestive of a structural break. Most
studies ignore this. The only exceptions known to us are Capelle-Blancard and
Desroziers (2020), Mamaysky (2020), and Ramelli and Wagner (2020), who divide
their samples into sub-periods based on major events. The breaks are therefore
treated as known, which is risky, as misplaced breaks are just as problematic
as omitted breaks. The second contribution is that we account for general
forms of unobserved heterogeneity. Many studies recognize the problem but
assume that it can be handled using country and period fixed effects. However,
fixed effects do not work in general when pair-wise cross-section covariances
of the regression errors differ across countries, and there is plenty of
evidence to support this (see, for example, Zhang et al., 2020). Capelle-
Blancard and Desroziers (2020) use robust standard errors but they can only
handle weak cross-section dependence. The third contribution is that we
account for the smallness of $T$. As alluded to in the previous paragraph,
with COVID–19 being such a recent phenomenon, studies of it are constrained to
data sets with short time span (see, for example, Salisu and Vo, 2020, for a
discussion). Some “compensate” by using a relatively high frequency, such as
daily data, but not all. To take an extreme example, Aggarwal et al. (2020)
use monthly data from December 2019 to May 2020, which means that $T=6$.
Moreover, even if data are daily, the sub-periods considered are very short.
It is therefore important to use appropriate small-$T$ techniques.

###  5.2 Data

Our dependent variable is stock returns (RET), which we compute as the log
difference of the price index.888We experimented using excess returns.
However, because the results were qualitatively the same, and since the
previous literature focuses almost exclusively on raw returns, here we only
report the results based on using raw returns as the dependent variable. We
utilize four control variables; the US Dollar exchange rate (ER), stock market
volatility (VOL), which we proxy using the Chicago Board Options Exchange’s
CBOE volatility index, world market returns (MRET), as measured by the cross-
country average of RET, and the US three-month Treasury bill rate (TBILL)
(see, for example, Aggarwal et al., 2020, Capelle-Blancard and Desroziers,
2020, Mamaysky, 2020, and Salisu and Vo, 2020, for similar control variable
lineups). ER is motivated by the theoretical work of Dornbusch and Fischer
(1980), which says that exchange rates will influence stock returns because
they capture the value of firms’ future cash flows. VOL can be motivated in
part by its ability to predict returns (see, for example, Bollerslev et al.,
2009, and Bollerslev et al., 2015), in part by the theory of Glasserman et al.
(2020), according to which information shocks can lead to large drops in stock
prices and increases in volatility. TBILL captures both the risk-free interest
rate and the importance of the US in shaping stock markets around the world.
The need to control for MRET is due to CAPM.

We use all available measures of COVID–19 that have sufficient time series
data. A total of six variables meet this criterion. The first two capture the
spread of the virus. They are the number of confirmed cases (CASE) and deaths
(DEATH). The next four variables are indices that capture government response
to COVID–19; a government stringency index (STR), a containment and health
index (CONT), a government economic support index (ECON), and an overall
government response index (RESP). STR records the strictness of government
policies that primarily restrict people’s behaviour, such as school and
workplace closures, stay-at-home requirements, and travel bans. CONT captures
mainly social distancing restrictions, but also health system policies such as
testing policy, contact tracing, short term investment in healthcare and
investments in vaccine. ECON is an index that captures government income
support and debt relief. RESP captures all of the above. All indices are on a
scale of zero to 100 with a larger value indicating greater stringency,
greater commitment to health, greater economic support, and greater overall
government response. All data are obtained from Datastream, except for TBILL,
which is from Federal Reserve Bank St Louis. The data are weekly and cover
$N=61$ countries.999The included countries are Argentina, Australia, Austria,
Belgium, Brazil, Bulgaria, Canada, Chile, China, Croatia, Cyprus, Czech
Republic, Denmark, Egypt, Estonia, Finland, France, Germany, Greece, Hong
Kong, Hungary, Iceland, India, Indonesia, Ireland, Israel, Italy, Jamaica,
Japan, Jordan, Kenya, Kuwait, Latvia, Luxembourg, Malaysia, Mexico, Morocco,
the Netherlands, New Zealand, Norway, Oman, Pakistan, Peru, the Philippines,
Poland, Portugal, Romania, Russia, Singapore, Slovakia, Slovenia, South
Africa, South Korea, Spain, Sri Lanka, Sweden, Switzerland, Thailand, Tunisia,
Turkey, and the United Kingdom. As in many other empirical scenarios, the
number of time periods is limited and cannot be increased other than by the
passage of time. We take the largest sample period available to us, which
covers $T=38$ weeks, from January 3 to September 25, 2020. The smallness of
$T$ in this case means that it is important to use techniques that work even
if $T$ is not large. The Monte Carlo results reported in the online appendix
suggest that the proposed toolbox should work well here.

###  5.3 Implementation

Both theory and empirical observations stress the importance of news (see, for
example, Mamaysky, 2020). We therefore follow the bulk of the existing
literature and express all regressors in innovation form by taking first
differences. Ashraf (2020b) considers the same stringency, containment and
health, and economic support indices as we do. While he includes all three
indices at the same time, we do not. The reason is that STR, CONT and RESP are
highly collinear with correlations that range from 0.949 to 0.978 (see
Capelle-Blancard and Desroziers, 2020, for a similar argument). We therefore
include them one at a time.

VOL, MRET and TBILL do not vary by country but only by week. We therefore want
to treat these are observed common factors, a possibility that we did not
consider in Sections 2–4. As pointed out in Section 2, the type of factors
that can be permitted under Assumption 2.2 is very broad. This suggests that
there is no need to discriminate between known and unknown factors, but that
one can just as well treat them all as unknown to be estimated from the data.
In fact, this is the main rationale for writing (2.1) and (2.4) in terms of
(the unknown) $f_{t}$ only. The main drawback of this fully unknown factor
treatment is that it puts strain on the Assumption 3.2 condition that $m\leq
k$, as $k$ is fixed and additional factors increase $m$ even if they are
known. For this reason, it may be preferable to be able to distinguish between
known and unknown factors. Fortunately, in CCE this is very easy. Let us
therefore assume that there are two sets of factors, $f_{t}$ and $d_{t}$,
where $f_{t}$ is a $m\times 1$ vector of unknown factors, just as before,
while $d_{t}$ is a $n\times 1$ vectors of known common regressors, which in
this section is comprised of a constant (country fixed effects), VOL, MRET and
TBILL. Hence, now the total number of factors is equal to $m+n$, out of which
$n$ are known. The model is the same as before but with (2.1) and (2.4)
replaced by

$\displaystyle y_{i,t}$
$\displaystyle=\alpha^{\prime}_{i}d_{t}+\beta^{\prime}x_{i,t}+\delta^{\prime}z_{i,t}(b)+e_{i,t},$
(5.1) $\displaystyle x_{i,t}$
$\displaystyle=A_{i}^{\prime}d_{t}+\Gamma_{i}^{\prime}f_{t}+v_{i,t},$ (5.2)

where $\alpha_{i}$ and $A_{i}$ are $n\times 1$ and $n\times k$, respectively.
Provided that Assumption 3.2 holds, so that $\mathrm{rank}(\bar{\Gamma})=m\leq
k$, similarly to (3.2), we can show that
$(d_{t}^{\prime},\bar{x}_{t}^{\prime})^{\prime}$ is rotationally consistent
for $f_{t}$. Hence, instead of using $M_{\bar{X}}$ to purge the effect of $F$
in the estimation of $b$, $M_{(D,\bar{X})}$ must be used, where
$D=(d_{1},...,d_{T})^{\prime}$ is $T\times n$. This is, however, the only
change needed in order to accommodate the known regressors in $d_{t}$. Note in
particular how the condition that $m\leq k$ is unaffected by $n$. Similarly,
the only change needed in order to account for the known factors in the break
tests is to replace $M_{\bar{H}(b)}$ with $M_{(D,D(b)\bar{H}(b))}$, where
$D(b)=(d_{1}\mathbb{I}(1>b),...,d_{T}\mathbb{I}(T>b))^{\prime}$.

The above discussion suggests that in terms of the known factor-augmented
version of (2.1) in (5.1), in this section $y_{i,t}$ is RET, $d_{t}$ is a
constant, VOL, MRET and TBILL, and $x_{i,t}$ is ER, CASE, DEATH, ECON, and one
of STR, CONT and RESP. We allow the coefficients of the COVID–19 spread and
response variables to be breaking, but not the coefficient of ER. The date of
the break is treated as unknown not only in the estimation but also on the
testing. We therefore focus on the $SW$ test, which we implement using $15$%
trimming ($\epsilon=0.15$) (as in, for example, Andrews, 1993, and Bai,
1997a). Based on its good performance in the Monte Carlo study reported in the
online appendix, the asymptotic covariance matrix of $\hat{\delta}(b)$ is
computed based on the Bartlett kernel with the bandwidth parameter $S_{T}$ set
equal to $S_{T}=\lfloor T^{1/3}\rfloor$.

As in the present paper, Bai (2010) focuses on the single break case, although
he also discuss the possibility of having multiple breaks. As he remarks, if
the number of breaks is given, the one-at-a-time approach of Bai (1997b) can
be used to estimate the breakpoints, and if the number of breaks is unknown, a
test for existence of a break can be applied to each subsample before
estimating another breakpoint. The same approach can be used also in the
current more general context. The results are discussed in the next section.

###  5.4 Results

Following the convention in the literature (see, for example, Ashraf, 2020a,
2020b, Capelle-Blancard and Desroziers, 2020, and Erdem, 2020), Table 1
reports the mean, standard deviation, minimum and maximum of each variable.
RET has a mean value of $-0.392$ with a standard deviation of $4.44$. The fact
that the mean is negative indicates that the pandemic has affected stock
markets negatively. The mean of CASE and DEATH are positive, as expected
because the pandemic has not settled down yet. The results for the response
variables show that governments have responded to the pandemic. In order to
get a feeling for the validity of the conventional fixed effects assumption,
we computed the CD test of Pesaran (2021), which tests the null hypothesis of
no remaining cross-sectional correlation after controlling for fixed effects.
The null is rejected at all conventional significance levels for all
variables, suggesting that, as expected given the above discussion, fixed
effects are not enough to account for the cross-section correlation. The
results of the unit root tests of Elliott et al. (1996), and Pesaran (2007)
confirm that all the variables are stationary.101010The results of the CD test
motivate the use of the CIPS test of Pesaran (2007), which allows for cross-
section correlation. The use of the test of Elliott et al. (1996) is motivated
by its good power properties.

Table 1: Descriptive statistics.

Variable | Mean | SD | Min | Max | UR | CD
---|---|---|---|---|---|---
RET | $-0.392$ | $4.440$ | $-26.795$ | $17.656$ |  $-5.769$*** |  $161.010$***
CASE | $99.125$ | $1314.487$ | $-20289$ | $28789$ |  $-4.137$*** |  $13.631$***
DEATH | $275.842$ | $985.860$ | $-1507$ | $8101$ |  $-2.098$* |  $63.246$***
STR | $1.348$ | $9.163$ | $-47.2$ | $88.9$ |  $-5.768$*** |  $114.333$***
RESP | $1.543$ | $6.732$ | $-30.3$ | $64.3$ |  $-5.633$*** |  $117.615$***
CONT | $1.501$ | $7.207$ | $-35.4$ | $70.8$ |  $-5.651$*** |  $108.350$***
ECON | $1.796$ | $10.293$ | $-50$ | $100$ |  $-5.263$*** |  $55.289$***
ER | $0.460$ | $41.260$ | $-740$ | $1155$ |  $-5.930$*** |  $4.418$***
VOL | $1.663$ | $20.841$ | $-33.678$ | $85.372$ |  $-4.160$*** | $-$
TBILL | $0.399$ | $0.535$ | $0.100$ | $1.520$ |  $-3.405$** | $-$
MRET | $-0.392$ | $3.515$ | $-14.864$ | $5.479$ |  $-4.312$*** | $-$

  * •

_Notes_ : “Mean”, “SD”, “Min” and “Max” refer to the sample average, the
standard deviation, the minimum value and the maximum value of each variable.
The column labelled “UR” reports some unit root test results. If the variable
exhibits cross-sectional variation we employ the CIPS test of Pesaran (2007),
which allows cross-sectional dependence in the form of a common factor. If the
variable only varies over time, we employ unit root test of Elliott et al.
(1996). Both tests are augmented with four lags to capture any serial
correlation in the regression errors. The column labelled “CD” reports the
results obtained by applying Pesaran’s (2004) test for cross-sectional
correlation. The CD results for VOL, TBILL and MRET are not reported as these
variables do not vary by country. The variables are stock returns (RET), cases
(CASE), deaths (DEATH), government stringency (STR), overall government
response (RESP), government containment and health (CONT), government economic
support (ECON), the US dollar exchange rate (ER), stock market volatility
(VOL), the three-month US Treasury Bill rate (TBILL) and average stock returns
(MRET).

Table 2 contains our main results. It reports the estimated coefficients and
their significance, the $SW$ test values and their significance, the estimated
break date, and the associated 95% break date confidence interval. The table
only contains the results for the COVID–19 variables, which are our main
regressors. The controls are included but we do not report their results.
Three specifications are considered, one for each of STR, CONT and RESP.

The first thing to note is that the $SW$ test is highly significant in all
three specifications and that the break date is estimated to the first week of
April, which starts on March 30 and ends on April 5. This is consistent with
the quantitative easing interventions of major central banks and the sharp
stock market rise that followed. For example, the S&P 500 stock market index
lost 34% of its value between February 19 and March 23, but abruptly regained
the vast majority of this loss, rising 29% between March 24 and April 17.
Stock markets all around the world experienced similar surges (see
International Monetary Fund, IMF, 2020). There may be confounding factors that
may have affected stock returns positively. However, we note that the
quantitative easing announcements were among the largest news items at the
time (Mamaysky, 2020). A full table of announcements can be found in Hartley
and Rebucci (2020). The dates of some notable announcements in March 2020 are
the European Central Bank on the 18th, the Bank of England and the Reserve
Bank of Australia on the 19th, the Reserve Bank of New Zealand on the 23rd,
the Bank of Korea on the 25th, the Federal Reserve and the Bank of Canada on
the 27th. Most of the emerging economies’ central banks made their
announcements in the last 10 days of March. Our estimated break date is
located directly after these announcements. Quantitative easing pushes
interest rates down and this has two possible effects, which both result in an
increase in stock prices (see, for example, Bernanke, 2012). First, by
decreasing the discount rate, quantitative easing increases the present value
of future cash flows. Second, quantitative easing makes relatively safe assets
unattractive, which creates an incentive for investors to rebalance their
portfolios to include more stocks, and this in turn pushes stock prices up. We
therefore speculate that it was the quantitative easing announcements that
caused the break in the stock return-COVID–19 relationship.111111We also note
that our estimated breakpoint does not coincide with the sample splits
considered by Capelle-Blancard and Desroziers (2020), Mamaysky (2020), and
Ramelli and Wagner (2020). As explained earlier in this section, the $SW$ test
was applied not only to the full sample but also to the pre- and post-break
periods. There were, however, no significant breaks in the pre- and post-break
periods, and so we conclude that there is just one break.

Table 2: Main results.

Regressor | Coeff | Spec 1 | Spec 2 | Spec 3
---|---|---|---|---
CASE | $\beta$ |  $-0.001$** |  $-0.002$*** |  $-0.002$***
|  | (0.000) | (0.001) | (0.001)
| $\delta$ | 0.001*** | 0.002*** | 0.002***
|  | (0.000) | (0.001) | (0.001)
DEATH | $\beta$ |  $-0.002$* |  $-0.002$** |  $-0.002$**
|  | (0.001) | (0.001) | (0.001)
| $\delta$ | 0.002* | 0.002** | 0.002**
|  | ( 0.001) | (0.001) | (0.001)
ECON | $\beta$ | 0.056** | 0.070*** | 0.060**
|  | (0.028) | (0.027) | (0.026)
| $\delta$ |  $-0.068$** |  $-0.082$*** |  $-0.073$***
|  | (0.029) | (0.027) | (0.027)
STR | $\beta$ |  $-0.077$*** |  |
|  | ( 0.020) |  |
| $\delta$ | 0.075*** |  |
|  | ( 0.022) |  |
GOV | $\beta$ |  |  $-0.075$*** |
|  |  | (0.028) |
| $\delta$ |  | 0.066** |
|  |  | (0.031) |
CONT | $\beta$ |  |  |  $-0.065$***
|  |  |  | (0.023)
| $\delta$ |  |  | 0.056**
|  |  |  | (0.024)
$r$ |  | 4 | 4 | 4
$n$ |  | 4 | 4 | 4
$SW$ |  | 31.182*** | 24.108*** | 24.096***
$\hat{b}$ |  | Apr 5 | Apr 5 | Apr 5
95% CI |  | [Mar 29,Apr 12] | [Mar 29,Apr 12] | [Mar 29,Apr 12]
Unreported controls: Country fixed effects, VOL, TBILL, MRET and ER

  * •

_Notes_ : The dependent variable is stock returns (RET). The main regressors
are cases (CASE), deaths (DEATH), government stringency (STR), overall
government response (RESP), government containment and health (CONT) and
government economic support (ECON). All specifications include country fixed
effects, the US dollar exchange rate (ER), stock market volatility (VOL), the
three-month US Treasury Bill rate (TBILL) and average stock returns (MRET) as
controls. As estimates of the unobserved factors we use the main regressors of
each specification and ER. All specifications also include country fixed
effects, VOL, TBILL and MRET as observed common factors. $\beta$ and $\delta$
refer to the pre-break slope and the size of the break, respectively. $r$ and
$n$ refer to the number of panel data regressors and observed common factors,
respectively. $SW$ refers to the sup-Wald test for the existence of a
structural break, $\hat{b}$ refers to the estimated breakpoint and “95% CI”
refers to the associated $95$% confidence interval. The reported dates refer
to the last day of the relevant week. The numbers within parentheses are the
standard errors. Finally, *, ** and *** denote statistical significance at the
$10$%, $5$% and $1$% levels, respectively.

Another observation is that all the COVID–19 regressors enter significantly
but only before the break. Specifically, the estimated pre-break coefficients
($\beta$) are all significant, as are the estimated breaks ($\delta$), but
they sum up to zero, and the sum is insignificant in all cases. In other
words, the estimated post-break effects ($\beta+\delta$) are insignificant.
Consider CASE and DEATH. Their pre-break effect is significantly negative,
which is consistent with existing results (see, for example, Ashraf, 2020a,
2020b, Capelle-Blancard and Desroziers, 2020, and Erdem, 2020). Hence, as
expected, stock markets therefore initially responded negatively to the news
of the outbreak of the virus. This negative effect is, however, completely
eliminated by the break, which is estimated to be of the same magnitude but of
opposite sign. The post-break effect of CASE and DEATH is therefore estimated
to zero, suggesting that the central bank interventions have had a substantial
positive effect on stock markets.

Let us now move on to the response regressors, STR, CONT, ECON and RESP. The
estimated pre-break effect of ECON is significantly positive, meaning that
stock markets initially responded positively to news of increased government
support, which is again in accordance with our a priori expectations. After
the break, however, stock markets became insensitive to such news. Similarly,
while initially markets responded negatively to announcements of stricter and
more extensive government restrictions, as measured by STR and RESP, after the
break they did not respond at all. The same is true for CONT, which is
probably due to the fact that while this variable captures both social
distancing restrictions and investments in healthcare, the restrictions are
weighted higher in the construction of the index and they did came first. As
an illustration of the effect of STR, CONT and RESP, in Figure 1 we plot the
cross-sectional averages of these variables against that of RET. We see that
while before the break the co-movement between average RET on the one hand and
STR, CONT and RESP on the other hand is clearly negative, after the break the
co-movement is much weaker. These results are quite different from existing
ones. Capelle-Blancard and Desroziers (2020) find that STR has a positive but
insignificant effect, which becomes significant only in the absence of fixed
effects or other control variables. Ashraf (2020b) reports a significantly
negative effect of STR, a significantly positive effect of CONT and an
insignificant effect of ECON. However, these other studies only allow for
fixed effects and they do not take into consideration our estimated
breakpoint, which could very well explain the observed differences in the
results.

Figure 1: Plotting the cross-sectional averages of RET, STR, CONT and RESP.
_Notes:_ The figure plots the weekly cross-sectional averages of stock returns
(RET), government stringency (STR), government containment and health (CONT)
and overall government response (RESP). The break date estimate and the
associated $95$% confidence interval are taken from Table 2. The reported
dates refer to the last day of the relevant week.

According to the results of Baker et al. (2020), and Mamaysky (2020), in the
early phase of the pandemic (late February to late March) stock market
movements were driven by news about the virus. In fact, markets were
“hypersensitive” and overreacted not only to news themselves but also to other
markets’ reaction to news (Mamaysky, 2020). “Markets started to oscillate
wildly, and people suddenly realized that the virus could affect them
directly. Panic selling in the stock market went hand-in-hand with panic
buying in supermarkets” (Wagner, 2020, page 440). This explains why initially
stock markets reacted significantly to all COVID–19 related news (CASE, DEATH,
STR, CONT, ECON and RESP). The powerful central bank interventions acted as a
wake-up call. They signalled a clear commitment to deal with the pandemic,
thereby bringing some certainty to an otherwise extremely uncertain future.
Stock markets reacted positively and progressed on a path to recovery. This is
noteworthy because the economic conditions have been steadily deteriorating as
a result of closures and social distancing (see IMF, 2020). As Krugman (2020)
puts it, “[t]he relationship between stock performance – largely driven by the
oscillation between greed and fear – and real economic growth has always been
somewhere between loose and nonexistent”.

The explanation for our results given in the previous paragraph is consistent
with (at least) two theories. The first is the so-called “overreaction”
hypothesis of Daniel et al. (1998), and Hong and Stein (1999), which states
that investors overreact to negative shocks, such as those that hit stock
markets in the early phase of the pandemic. As more information becomes
available, however, and the central bank announcement were very informative,
investors correct their behavior, which leads to market recovery. The second
theory is that of Glasserman et al. (2020). It states that information shocks,
such as the outbreak of COVID–19, can lead to large drops in prices and
increases in volatility, which in turn cause prices to become hypersensitive
to newsflow. However, information can also push prices out of
hypersensitivity, and our results show that in the post-break regime returns
are no longer reacting to news of the pandemic.

##  6 Conclusions

The main aim of this paper is to provide a toolbox that meets the basic needs
of researchers interested in a linear panel data model with a possible
structural break. The toolbox allows researchers to test for the presence of a
break, and, if a break is detected, to also estimate the location of the break
and construct a confidence interval for the true breakpoint. The toolbox does
not require that the data are independent, nor that $T$ is large, which means
that it widely is applicable.

The new toolbox in employed to investigate the relationship between stock
market returns and COVID–19 in a sample covering 61 countries across 38 weeks.
Stock markets all over the world plunged in the early phase of the pandemic
but they quickly rebounded, and this rebound took place although the end of
the pandemic is still not in sight. Our analysis shows that while initially
responsive, the effect of COVID-19 stopped dead at the end of March-beginning
of April 2020. We attribute this break to the massive quantitative easing
programs announced by central banks around the world in the second half of
March.

## References

Aggarwal, S., S. Nawn and A. Dugar (2020). What Caused Global Stock Market
Meltdown during the COVID Pandemic–Lockdown Stringency or Investor Panic?
_Finance Research Letters_ 31, 101690.

Andrews, D. W. K. (1993). Tests for Parameter Instability and Structural
Change with Unknown Change Point. _Econometrica_ 61, 821–856.

Antoch, J., J. Hanousek, L. Horváth, M. Hušková and S. Wang (2019). Structural
Breaks in Panel Data: Large Number of Panels and Short Length Time Series.
_Econometric Reviews_ 38, 828–855.

Ashraf, B. N. (2020a). Stock Markets’ Reaction to COVID-19: Cases or
Fatalities? _Research in International Business and Finance_ 54, 101249.

Ashraf, B. N. (2020b). Economic Impact of Government Interventions during the
COVID-19 Pandemic: International Evidence from Financial Markets. _Journal of
Behavioral and Experimental Finance_ 27, 100371.

Bai, J. (1997a). Estimation of a Change Point in Multiple Regression Models.
_Review of Economics and Statistics_ 79, 551–563.

Bai, J. (1997b). Estimating Multiple Breaks One at a Time. _Econometric
Theory_ 13, 315–352.

Bai, J. (2010). Common Breaks in Means and Variances for Panel Data. _Journal
of Econometrics_ 157, 78–92.

Baker S. R., N. Bloom, S. J. Davis, K. J. Kost, M. C. Sammon and T. Viratyosin
(2020). The Unprecedented Stock Market Impact of COVID-19. _Covid Economics_
1, 33–42.

Baltagi, B. H., Q. Feng and C. Kao (2016). Estimation of Heterogeneous Panels
with Structural Breaks. _Journal of Econometrics_ 191, 176–195.

Baltagi, B. H., C. Kao and L. Liu (2017). Estimation and Identification of
Change Points in Panel Models with Nonstationary or Stationary Regressors and
Error Term. _Econometric Reviews_ 36, 85–102.

Bernanke, B. (2012). Monetary Policy Since the Onset of the Crisis. Remarks at
the Federal Reserve Bank of Kansas City Economic Symposium, Jackson Hole,
Wyoming, August 31.

Boldea, O., B. Drepper and Z. Gan (2020). Change Point Estimation in Panel
Data with Time-Varying Individual Effects. Forthcoming in _Journal of Applied
Econometrics_.

Bollerslev, T., G. Tauchen and H. Zhou (2009). Expected Stock Returns and
Variance Risk Premia. _Review of Financial Studies_ 22, 4463–4492.

Bollerslev, T., L. Xu and H. Zhou (2015). Stock Return and Cash Flow
Predictability: The Role of Volatility Risk. _Journal of Econometrics_ 187,
458–471.

Capelle-Blancard, G., and A. Desroziers (2020). The Stock Market is not the
Economy? Insights from the Covid-19 crisis. _Covid Economics_ 28, 29–69.

Daniel, K., D. Hirshleifer and A. Subrahmanyam (1998). Investor Psychology and
Security Market Under- and Overreactions. _Journal of Finance_ 53, 1839–1886.

Dornbusch, R., and S. Fischer (1980). Exchange Rates and the Current Account.
_American Economic Review_ 70, 960–971.

Elliott, G., T. J. Rothenberg, and J. H. Stock (1996). Efficient Tests for an
Autoregressive Unit Root. _Econometrica_ 64, 813–836.

Erdem, O., (2020). Freedom and Stock Market Performance during COVID-19
Outbreak. _Finance Research Letters_ 36, 101671.

Glasserman, P., H. Mamaysky, and Y. Shen (2020). Dynamic Information Regimes
in Financial Markets. Working paper.

Hartley, J., S., and A. Rebucci, (2020). An Event Study of COVID–19 Central
Bank Quantitative Easing in Advanced and Emerging Economies. NBER Working
Paper 27339.

Hidalgo, J., and M. Schafgans (2017). Inference and Testing Breaks in Large
Dynamic Panels with Strong Cross Sectional Dependence. _Journal of
Econometrics_ 196, 259–274.

Hong, H., Stein, J., (1999). A Unified Theory of Underreaction, Momentum
Trading, and Overreaction in Asset Markets. _Journal of Finance_ 54,
2143–2184.

IMF (2020). Special Series on COVID-19: The Disconnect between Financial
Markets and the Real Economy. Washington.

Karabiyik, H., S. Reese and J. Westerlund (2017). On the Role of the Rank
Condition in CCE Estimation of Factor-Augmented Panel Regressions. _Journal of
Econometrics_ 197, 60–64.

Kim, D. (2011). Estimating a Common Deterministic Time Trend Break in Large
Panels with Cross Sectional Dependence. _Journal of Econometrics_ 164,
310–330.

Kim, D. (2014). Common Local Breaks in Time Trends for Large Panels.
_Econometric Journal_ 17, 301–337.

Krugman, P. (2020). Crashing Economy, Rising Stocks: What’s going on? New York
Times, April 30.

Li, D., J. Qian and L. Su (2016). Panel Data Models With Interactive Fixed
Effects and Multiple Structural Break. _Journal of the American Statistical
Association_ 111, 1804–1819.

OECD (2020). Economic Outlook, Issue 2, OECD Publishing, Paris.

Pesaran, M. H. (2006). Estimation and Inference in Large Heterogeneous Panels
with a Multifactor Error Structure. _Econometrica_ 74, 967–1012.

Pesaran, M.H. (2007). A Simple Panel Unit Root Test in the Presence of Cross-
Section Dependence. _Journal of Applied Econometrics_ 22, 265–312.

Pesaran, H. M. (2021). General Diagnostic Tests for Cross-Sectional Dependence
in Panels. Forthcoming in _Empirical Economics_.

Pesaran, M. H., and E. Tosetti (2011). Large Panels with Common Factors and
Spatial Correlation. _Journal of Econometrics_ 161(2), 182–202.

Phan, D. H. B., and P. K. Narayan (2020). Country Responses and the Reaction
of the Stock Market to COVID–19–a Preliminary Exposition. _Emerging Markets
Finance and Trade_ 56, 2138–2150.

Ramelli, S., and A. F. Wagner (2020). Feverish Stock Price Reactions to
COVIS-19. Forthcoming in _Review of Corporate Finance Studies_.

Salisu, A. A., and X. V. Vo (2020). Predicting Stock Returns in the Presence
of COVID-19 Pandemic: The Role of Health News. _International Review of
Financial Analysis_ 71, 101546.

Shiller, R. (2020). Opinion: Robert Shiller Explains the Pandemic Stock Market
and why it’s Decoupled from the Economy. Project Syndicate, July 11, 2020.

Wagner, A. F (2020). What the Stock Market Tells us about the Post-COVID-19
World. _Nature Human Behaviour_ 4, 440.

Westerlund, J. (2019). Common Breaks in Means for Cross–Correlated Fixed-$T$
Panel Data. _Journal of Time Series Analysis_ 40, 248–255.

Westerlund, J., Karabiyik, H., and Narayan, P., (2017). Testing for
Predictability in Panels with General Predictors. _Journal of Applied
Econometrics_ 32, 554–574.

Westerlund, J., Y. Petrova and M. Norkute (2019). CCE in Fixed-$T$ Panels.
_Journal of Applied Econometrics_ 34, 746–761.

WHO (2021). Weekly operational update on COVID–19 - 19 January 2021. Retrieved
from: https://www.who.int/publications/m/item/weekly-operational-update-on-
covid-19—19-january-2021.

Zhang, D., M. Hu, Q. Ji (2020) Financial Markets under the Global Pandemic of
COVID-19. _Finance Research Letters_ 36, 101528.