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Change point detection in random coefficient autoregressive models
\title[Change points in RCA models ]{Change point detection in random
coefficient autoregressive models}
\author{Lajos Horv\'ath}
\address{Lajos Horv\'ath, Department of Mathematics, University of Utah,
Salt Lake City, UT 84112--0090 USA}
\author{Lorenzo Trapani}
\address{Lorenzo Trapani, School of Economics, University of Nottingham,
University Park, Nottingham NG7 2RD U.K.}
\subjclass{Primary 62M10; Secondary 62G20}
\keywords{Changepoint problem, weighted CUSUM process, Random Coefficient
AutoRegression, Darling-Erd\H{o}s theorem.}
\begin{abstract}
We propose a family of CUSUM-based statistics to detect the presence of
changepoints in the deterministic part of the autoregressive parameter in a
Random Coefficient AutoRegressive (RCA) sequence. In order to ensure the
ability to detect breaks at sample endpoints, we thoroughly study \textit{
weighted} CUSUM statistics, analysing the asymptotics for virtually all
possible weighing schemes, including the standardised CUSUM process (for
which we derive a Darling-Erd\H{o}s theorem) and even heavier weights
(studying the so-called R\'enyi statistics). Our results are valid
irrespective of whether the sequence is stationary or not, and indeed prior
knowledge of stationarity or lack thereof is not required from a practical
point of view. From a technical point of view, our results require the
development of strong approximations which, in the nonstationary case, are
entirely new. Similarly, we allow for heteroskedasticity of unknown form in
both the error term and in the stochastic part of the autoregressive
coefficient, proposing a family of test statistics which are robust to
heteroskedasticity; again, our tests can be readily applied, with no prior
knowledge as to the presence or type of heteroskedasticity. Simulations show
that our procedures work well in finite samples, under all cases considered
(stationarity versus nonstationarity, homoskedasticity versus various forms
of heteroskedasticity). We complement our theory with applications to
financial, economic and epidemiological time series.
\end{abstract}
\maketitle
\section{Introduction\label{intro}}
In this paper we study the stability of the autoregressive parameter of an
RCA(1) sequence:
\begin{comment}
\begin{align}\label{arcnew1}
y_i=\left\{
\begin{array}{ll}
(\beta_k_1+\epsilon_{i,1})y_{i-1}+\epsilon_{i,2},\quad\mbox{if}\;\;\;1\leq i \leq k_1,
\vspace{.3cm}\\
(\beta_A+\epsilon_{i,1})y_{i-1}+\epsilon_{i,2},\quad\mbox{if}\;\;\;k_1+1\leq i \leq k_2,
\\
\quad \vdots\quad \vdots
\\
(\beta_{R+1}+\epsilon_{i,1})y_{i-1}+\epsilon_{i,2}, \;\;\;k_{R}+1\leq i\leq N,
\end{array}
\right.
\end{align}
where $R$ is the number of changes in the autoregression coefficient. We test for the null hypothesis of is no change versus the alternative of at most one change (AMOC) i.e.
\begin{equation}\label{rcanu}
H_0:\;k_1>N
\end{equation}
tested against
\begin{equation}\label{rcaor1}
H_A:\;1<k_1<k_2<\ldots<k_R<N\;\;\mbox{and}\;\;\beta_1\neq \beta_2\neq \ldots\neq \beta_{R+1}.
\end{equation}
\end{comment}
\begin{equation}
y_{i}=\left\{
\begin{array}{ll}
\left( \beta _{0}+\epsilon _{i,1}\right) y_{i-1}+\epsilon _{i,2},\quad
\mbox{if}\;\;\;1\leq i\leq k^{\ast }, & \\
\left( \beta _{A}+\epsilon _{i,1}\right) y_{i-1}+\epsilon _{i,2},\quad
\mbox{if}\;\;\;k^{\ast }+1\leq i\leq N, &
\end{array}
\right. \label{arcnew1}
\end{equation}
where $y_{0}$ denotes an initial value. We test for the null hypothesis of
no change versus the alternative of at most one change (AMOC) i.e.
\begin{align}
H_{0}& :\;k^{\ast }>N, \label{rcanu} \\
H_{A}& :\;1<k^{\ast }<N\;\;\mbox{and}\;\;\beta _{0}\neq \beta _{A}.
\label{racor1}
\end{align}
The RCA model was firstly studied by \citet{andel} and \citet{nichollsquinn}
. It belongs in the wider class of nonlinear models for time series (see
\citealp{fanyao}), which have been proposed \textquotedblleft as a reaction
against the supremacy of linear ones - a situation inherited from strong,
though often implicit, Gaussian assumptions\textquotedblright\ (
\citealp{akharif2003}). Arguably, (\ref{arcnew1}) is very flexible, allowing
for the autoregressive \textquotedblleft root\textquotedblright\ $\beta
_{0}+\epsilon _{i,1}$ to vary over time, and thus for the possibility of
having stationary and nonstationary regimes. This may be a more appropriate
model than a linear specification (see \citealp{lieberman2012};
\citealp{leybourne1996}); \citet{gky} argue that a time-varying parameter
model like (\ref{arcnew1}) can be viewed as a competitor for a model with an
abrupt break in the autoregressive root. Furthermore, equation (\ref{arcnew1}
) also allows for the possibility of (conditional) heteroskedasticity in $
y_{i}$; \citet{tsay1987} shows that the widely popular ARCH\ model by
\citet{engle1982} can be cast into (\ref{arcnew1}), which therefore can be
viewed as a second-order equivalent. Finally, a major advantage of (\ref
{arcnew1}) compared to standard autoregressive models is that estimators of $
\beta _{0}$ are always asymptotically normal, irrespective of whether $y_{i}$
is stationary or nonstationary, thus avoiding the risk of over-differencing
(see \citealp{leybourne1996}).\newline
Given such generality and flexibility, (\ref{arcnew1}) has been used in many
applied sciences, including biology (\citealp{stenseth}), medicine (
\citealp{fryz}), and physics (\citealp{slkezak2019random}). The RCA\ model
has also been applied successfully in the analysis of economic and financial
data, and we refer to the recent contribution by \citet{regis} for a
comprehensive review.
The inferential theory for (\ref{arcnew1}) has been studied extensively.
\citet{schick1996}, \citet{koul1996} and \citet{praskova2004} study Weighted
Least Squares (WLS)\ estimation of $\beta _{0}$; \citet{berkes2009} and
\citet{aue2011} study Quasi Maximum Likelihood estimation, and
\citet{hillpeng2016} develop an Empirical Likelihood estimator. Several
tests have also been developed, including tests for stationarity (see e.g.
\citealp{zhao2012}; and \citealp{trapanistrict}) and for the randomness of
the autoregressive coefficient (\citealp{akharif2003}; \citealp{nagakura2009}
; and \citealp{HT16}).
In contrast, changepoint detection is still underexplored in the RCA
framework. To the best of our knowledge, the only exceptions are
\citet{lee1998}, \citet{lee2003cusum} and \citet{aue2004strong}; in these
papers, a CUSUM\ test is proposed, but only for the stationary case and
based on the unweighted CUSUM\ process. The latter is well-known to suffer
from low power, being in particular less able to detect changepoints
occurring at the beginning/end of the sample. As a solution, the literature
has proposed weighted versions of the CUSUM\ process on the interval $\left[
0,1\right] $, where more emphasis is given to observations at the sample
endpoints (see \citealp{csorgo1997}). Weighing functions are typically of
the form $\left[ t\left( 1-t\right) \right] ^{\kappa }$ with $0\leq \kappa
<\infty $, for $t\in \left[ 0,1\right] $, with more weight placed on
observations at the endpoints as $\kappa $ increases. In particular, the
case $\kappa =\frac{1}{2}$ corresponds to the standardised CUSUM\ process
also proposed by \citet{andrews1993}, whereas the more heavily weighted case
$\kappa >\frac{1}{2}$ corresponds to a family of test statistics known as
\textquotedblleft R\'{e}nyi statistics\textquotedblright\ (see
\citealp{horvathmiller}). When $\kappa >0$, the asymptotics becomes more
complicated, since the weighted statistics diverge at the endpoints $t=0$
and $t=1$, and one can no longer rely on weak convergence to derive the
limiting distributions. In order to overcome this issue, \citet{andrews1993}
proposes trimming the interval on which the weighted CUSUM\ process is
studied; however, this has the undesirable consequence that tests are unable
to detect breaks when these occurs e.g. at the end of the sample.
\bigskip
\textit{Contribution of this paper}
\bigskip
In this paper, we bridge all the gaps mentioned above by proposing a family
of weighted, untrimmed CUSUM statistics. Our paper makes the following four
contributions.
First, we study virtually all possible weighing schemes, deriving the
asymptotics for all $0\leq \kappa <\infty $. From a practical viewpoint,
this entails that our test statistics are designed to detect breaks even
when these are very close to the sample endpoints. Second, all our results
hold irrespective of whether $y_{i}$ is stationary or not; this robustness
arises from using the WLS estimator, and from the well-known fact that the
RCA\ model does not suffer from the \textquotedblleft knife edge
effect\textquotedblright\ which characterizes linear models (
\citealp{lumsdaine1996consistency}). From a practical point of view, this
entails that the tests can be applied with no modifications required, and no
prior knowledge of the stationarity of $y_{i}$ or lack thereof. This feature
is particularly desirable e.g. in the context of detecting the beginning (or
end) of bubbles (see \citealp{harvey2016}): with our set-up, it is possible
to detect changes from stationary to nonstationary/explosive behaviour (as
e.g. in \citealp{horvath2020sequential}; and \citealp{horvath2021sequential}
) which characterize the emergence of a bubble, but it is also possible -
again with no modifications required - to detect changes from explosive to
non-explosive behaviour, as would be the case at the end of a bubble. Being
able to accommodate both cases is a distinctive advantage of the RCA set-up:
whilst tests for changes \textit{towards} an explosive behaviour have been
developed in the literature (see, \textit{inter alia}, \citealp{phillips2011}
; \citealp{phillips2015testing}; and the review by \citealp{homm2012testing}
), tests to detect changes \textit{from} an explosive behaviour are more
rare, possibly due to the more complicated asymptotics in this case. Third,
we allow for heteroskedasticity in both $\epsilon _{i,1}$ and $\epsilon
_{i,2}$, which is usually not considered in the RCA context; interestingly,
for the case $\kappa \geq \frac{1}{2}$, we recover the same, nuisance free
distribution as in the homoskedastic case (in particular, when $\kappa =
\frac{1}{2}$, we obtain a \textquotedblleft classical\textquotedblright\
Darling-Erd\H{o}s limit theorem). Hence, our modified test statistics can be
used from the outset, with no prior knowledge required as to whether $
\epsilon _{i,1}$, or $\epsilon _{i,2}$, or both, is heteroskedastic. Fourth,
our asymptotics is based on strong approximations for the partial sums of an
RCA\ sequence, which are valid irrespective of the stationarity or lack
thereof of $y_{i}$; the strong approximation for the nonstationary case is
entirely new.
\begin{comment}
As a fifth and
final point, our approach can be applied to the whole (very broad) class of
decomposable Bernoulli shifts (see \citealp{wu05}; \citealp{aue09};
\citealp{berkeshormann}); thus, although our focus is on equation (\ref
{arcnew1}), all our proof and results can be extended to sequences which can
be represented as Bernoulli shifts: these include, \textit{inter alia},
linear processes, GARCH\ sequences, threshold autoregressive models and
other nonlinear models (see \citealp{linliu}, for a comprehensive set of
examples).
Our simulations show that our procedures have very good finite sample
properties, being able to detect breaks both in the middle and at endpoints.
As predicted by the theory, R\'{e}nyi-type statistics (based on using $
\kappa >\frac{1}{2}$) prove very effective at detecting breaks at either end
of the sample.
\end{comment}
\newline
The remainder of the paper is organised as follows. We present our test
statistics in Section \ref{tests}, and study their asymptotics in the
homoskedastic case, as a benchmark, in Section \ref{homosk}. The
heteroskedastic case is studied in Section \ref{heterosk}. In Section \ref
{simulations}, we report a simulation exercise; applications to real data
are in Section \ref{empirics}. Section \ref{conclusions} concludes.
Extensions, technical lemmas and all proofs are relegated to the Supplement.
NOTATION. We use the following notation: \textquotedblleft $\overset{{
\mathcal{D}}}{\rightarrow }$\textquotedblright\ for weak convergence;
\textquotedblleft $\overset{\mathcal{P}}{\rightarrow }$\textquotedblright\
for convergence in probability; \textquotedblleft \textit{a.s.}
\textquotedblright\ for \textquotedblleft almost surely\textquotedblright ;
\textquotedblleft $\overset{{\mathcal{D}}}{=}$\textquotedblright\ for
equality in distribution; $\lfloor \cdot \rfloor $ is the integer value
function. Positive, finite constants are denoted as $c_{0}$, $c_{1}$, ...
and their value may change from line to line. Other notation is introduced
further in the paper.
\section{The test statistics\label{tests}}
Our approach is based on comparing the estimates of $\beta _{0}$ before and
after each point in time $k$, by dividing the data into two subsets at $k$
and estimating the autoregressive parameter in both subsamples. As mentioned
above, we use WLS, with weights $1+y_{i-1}^{2}$. This has the advantages of
\textit{(i)} avoiding restrictions on the moments of the observations, and
\textit{(ii)} ensuring standard normal asymptotics irrespective of whether $
y_{i}$ is stationary or not. The WLS estimators are
\begin{equation}
\widehat{\beta }_{k,1}=\left( \sum_{i=2}^{k}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}
\right) ^{-1}\left( \sum_{i=2}^{k}\frac{y_{i}y_{i-1}}{1+y_{i-1}^{2}}\right)
,\quad 2\leq k\leq N, \label{rcabet}
\end{equation}
and
\begin{equation}
\widehat{\beta }_{k,2}=\left( \sum_{i=k+1}^{N}\frac{y_{i-1}^{2}}{
1+y_{i-1}^{2}}\right) ^{-1}\left( \sum_{i=k+1}^{N}\frac{y_{i}y_{i-1}}{
1+y_{i-1}^{2}}\right) ,\quad 1\leq k\leq N-1. \label{rcabet2}
\end{equation}
Our test statistics will be functionals of the process
\begin{equation}
Q_{N}(t)=\left\{
\begin{array}{ll}
0,\quad \mbox{if}\;\;\;0\leq t<2/(N+1), & \\
\displaystyle N^{1/2}(t(1-t))(\widehat{\beta }_{\lfloor (N+1)t\rfloor ,1}-
\widehat{\beta }_{\lfloor (N+1)t\rfloor ,2}),\quad \mbox{if}
\;\;\;2/(N+1)\leq t<1-2/(N+1), & \\
0,\quad \mbox{if}\;\;\;1-2/(N+1)<t\leq 1. &
\end{array}
\right. \label{qnt}
\end{equation}
A \textquotedblleft natural\textquotedblright\ choice to detect the presence
of a possible change is to use the sup-norm of (\ref{qnt}), viz. $
\sup_{0<t<1}\left\vert Q_{N}(t)\right\vert $, but, as mentioned above, this
choice may have low power in detecting changes which occur early or late in
the sample. In order to enhance the power at sample endpoints, one can use
weight functions:
\begin{equation}
\sup_{0<t<1}\frac{\left\vert Q_{N}(t)\right\vert }{w\left( t\right) }.
\label{sup-w-norm}
\end{equation}
\begin{assumption}
\label{as-wc-1} It holds that: (i) $\inf_{\delta \leq t\leq 1-\delta }w(t)>0$
for all $0<\delta <1/2$; (ii) $w(t)$ is non decreasing in a neighborhood of $
0$; (iii) $w(t)$ is non increasing in a neighborhood of $1$.
\end{assumption}
The functions $w(t)$ satisfying Assumption \ref{as-wc-1} belong in a very
wide class; a possible example is $w(t)=\left( t\left( 1-t\right) \right)
^{\kappa }$ with $\kappa >0$. The existence of the limit of (\ref{sup-w-norm}
) can be determined based on the finiteness of the integral functional (see
\citealp{csorgo1993})
\begin{equation}
I(w,c)=\int_{0}^{1}\frac{1}{t(1-t)}\exp \left( -\frac{cw^{2}(t)}{t(1-t)}
\right) dt. \label{defiwc}
\end{equation}
As we show below, (\ref{defiwc}) entails that $w(t)=\left( t\left(
1-t\right) \right) ^{\kappa }$ with $0<\kappa <\frac{1}{2}$ can be employed
in this context.
In order to further enhance the power of our testing procedures, functions
which place more weight at the sample endpoints can also be used, i.e.
\begin{equation}
\sup_{0<t<1}\frac{\left\vert Q_{N}(t)\right\vert }{\left( t\left( 1-t\right)
\right) ^{\kappa }}, \label{sup-R\'{e}nyi}
\end{equation}
with $\kappa \geq \frac{1}{2}$. As mentioned above, when $\kappa =\frac{1}{2}
$, the corresponding limit theorems will be of the Darling-Erd\H{o}s type (
\citealp{darling1956limit}); when $\kappa >\frac{1}{2}$, the test statistics
defined in (\ref{sup-R\'{e}nyi}) are known as \textquotedblleft R\'{e}nyi
statistics\textquotedblright\ (\citealp{horvathmiller}).
\section{Testing for changepoint under homoskedasticity\label{homosk}}
We begin by assuming that the errors $\{\epsilon _{i,1},\epsilon
_{i,2},-\infty <i<\infty \}$ have constant variance.
\begin{assumption}
\label{rcaas} It holds that:\ (i) $\{\epsilon _{i,1},-\infty <i<\infty \}$
and $\{\epsilon _{i,2},-\infty <i<\infty \}$ are independent sequences; (ii)
$\{\epsilon _{i,1},-\infty <i<\infty \}$ are independent and identically
distributed random variables with $E\epsilon _{i,1}=0$, $0<E\epsilon
_{i,1}^{2}=\sigma _{1}^{2}<\infty $ and $E|\epsilon _{i,1}|^{4}<\infty $;
(iii) $\{\epsilon _{i,2},-\infty <i<\infty \}$ are independent and
identically distributed random variables with $E\epsilon _{i,2}=0$, $
0<E\epsilon _{i,2}^{2}=\sigma _{2}^{2}<\infty $ and $E|\epsilon
_{i,2}|^{4}<\infty $.
\end{assumption}
In (\ref{arcnew1}), the stationarity or lack thereof of $y_{i}$ is
determined by the value of $E\ln \left\vert \beta _{0}+\epsilon
_{0,1}\right\vert $ (see \citealp{aue2006}). In particular, if $-\infty \leq
E\ln \left\vert \beta _{0}+\epsilon _{0,1}\right\vert <0$, then $y_{i}$
converges exponentially fast to a strictly stationary solution for all
initial values $y_{0}$. Conversely, if $E\ln \left\vert \beta _{0}+\epsilon
_{0,1}\right\vert \geq 0$, then $y_{i}$ is nonstationary - specifically, $
\left\vert y_{i}\right\vert $ diverges exponentially fast a.s. when $E\ln
\left\vert \beta _{0}+\epsilon _{0,1}\right\vert >0$, whereas it diverges in
probability, but at a rate slower than exponential, in the boundary case $
E\ln \left\vert \beta _{0}+\epsilon _{0,1}\right\vert =0$ (see
\citealp{HT2016}). \newline
We show that the asymptotic variance of the limiting process depends on
whether $y_{i}$ is stationary or not: we therefore study the two cases
(stationarity versus lack thereof) separately. We show that the variance of
the weak limit of $Q_{N}\left( t\right) $ is
\begin{comment}
\begin{equation}
\eta ^{2}=\left\{
\begin{array}{ll}
\displaystyle\left[ E\left( \frac{\bar{y}_{0}^{2}}{1+\bar{y}_{0}^{2}}\right)
^{2}\sigma _{1}^{2}+E\left( \frac{\bar{y}_{0}}{1+\bar{y}_{0}^{2}}\right)
^{2}\sigma _{2}^{2}\right] \left( E\frac{\bar{y}_{0}^{2}}{1+\bar{y}_{0}^{2}}
\right) ^{-2}, & \\
\vspace{0.3cm}\quad \quad \quad \quad \quad \quad \mbox{if}\;\;\;-\infty
\leq E\ln |\beta _{0}+\epsilon _{0,1}|<0,\vspace{0.3cm} & \\
\sigma _{1}^{2},\quad \mbox{if}\;\;\;E\ln |\beta _{0}+\epsilon _{0,1}|\geq 0.
&
\end{array}
\right. \label{etasq}
\end{equation}
\end{comment}
\begin{equation}
\eta ^{2}=\left\{
\begin{array}{l}
\displaystyle\left[ E\left( \frac{\bar{y}_{0}^{2}}{1+\bar{y}_{0}^{2}}\right)
^{2}\sigma _{1}^{2}+E\left( \frac{\bar{y}_{0}}{1+\bar{y}_{0}^{2}}\right)
^{2}\sigma _{2}^{2}\right] \left( E\frac{\bar{y}_{0}^{2}}{1+\bar{y}_{0}^{2}}
\right) ^{-2}, \\
\quad \quad \quad \quad \quad \quad \quad \mbox{if}\;\;\;-\infty \leq E\ln
|\beta _{0}+\epsilon _{0,1}|<0, \\
\sigma _{1}^{2},\quad \mbox{if}\;\;\;E\ln |\beta _{0}+\epsilon _{0,1}|\geq 0.
\end{array}
\right. \label{etasq}
\end{equation}
We require the following notation. In order to study the case $\kappa =\frac{
1}{2}$, we define
\begin{equation}
a(x)=(2\ln x)^{1/2}\quad \mbox{and}\quad b(x)=2\ln x+\frac{1}{2}\ln \ln x-
\frac{1}{2}\ln \pi . \label{da-1/0}
\end{equation}
Also, in order to study the case $\kappa >\frac{1}{2}$, let
\begin{equation}
r_{N}=\min (r_{1}(N),r_{2}(N)),\quad \lim_{N\rightarrow \infty }\frac{r_{N}}{
r_{1}(N)}=\gamma _{1}\quad \mbox{and}\quad \lim_{N\rightarrow \infty }\frac{
r_{N}}{r_{2}(N)}=\gamma _{2}, \label{fadef1}
\end{equation}
\begin{equation}
\mathfrak{a}(\kappa )=\sup_{1\leq t<\infty }\frac{|W(t)|}{t^{\kappa }}.
\label{fadef3}
\end{equation}
\begin{assumption}
\label{thrcaas11}\ It holds that $r_{1}(N)\rightarrow \infty $, $
r_{1}(N)/N\rightarrow 0$, and $r_{2}(N)\rightarrow \infty $, $
r_{2}(N)/N\rightarrow 0$.
\end{assumption}
We start with the stationary case $-\infty \leq E\ln |\beta _{0}+\epsilon
_{0,1}|<0$. In this case, the solution of (\ref{arcnew1}) under the null
hypothesis is close to $\overline{y}_{i}$, the unique anticipative
stationary solution of
\begin{equation}
\overline{y}_{i}=(\beta _{0}+\epsilon _{i,1})\overline{y}_{i-1}+\epsilon
_{i,2},\quad -\infty <i<\infty . \label{rcasta}
\end{equation}
We need the following (technical) assumption, to rule out the degenerate
case that, under stationarity, the denominator of $\eta ^{2}$ defined in (
\ref{etasq}) is zero with probability $1$.
\begin{assumption}
\label{stationary-nonzero} It holds that $P\{\overline{y}_{0}=0\}<1$.
\end{assumption}
\begin{theorem}
\label{thrca1} We assume that $H_{0}$ of (\ref{rcanu}), Assumptions \ref
{as-wc-1}, \ref{rcaas} and \ref{stationary-nonzero} hold, and $-\infty \leq
E\ln |\beta _{0}+\epsilon _{0,1}|<0$.\newline
(i) If $I(w,c)<\infty $ for some $c>0$, then it holds that
\begin{equation*}
\sup_{0<t<1}\frac{\left\vert Q_{N}(t)\right\vert }{w(t)}\;\overset{{\mathcal{
D}}}{\rightarrow }\;\eta \sup_{0<t<1}\frac{|B(t)|}{w(t)},
\end{equation*}
where $\{B(t),0\leq t\leq 1\}$ is a standard Brownian bridge and $\eta $ is
defined in (\ref{etasq}). \newline
(ii) For all $x$, it holds that
\begin{equation*}
\lim_{N\rightarrow \infty }P\Biggl\{a(\ln N)\frac{1}{\eta }
\max_{1<k<N}\left( \frac{k(N-k)}{N}\right) ^{1/2}|\widehat{\beta }_{k,1}-
\widehat{\beta }_{k,2}|\leq x+b(\ln N)\Biggl\}=\exp (-2e^{-x}).
\end{equation*}
(iii) If Assumption \ref{thrcaas11} is satisfied, then it holds that
\begin{equation*}
\left( \frac{r_{N}}{N}\right) ^{\kappa -1/2}\frac{1}{\eta }
\sup_{r_{1}(N)/N\leq t\leq 1-r_{2}(N)/N}\frac{|Q_{N}(t)|}{\left( t\left(
1-t\right) \right) ^{\kappa }}\;\overset{{\mathcal{D}}}{\rightarrow }\;\max
\left( \gamma _{1}^{\kappa -1/2}\mathfrak{a}_{1}(\kappa ),\gamma
_{2}^{\kappa -1/2}\mathfrak{a}_{2}(\kappa )\right) ,
\end{equation*}
for all $\kappa >1/2$, where and $r_{N}$, $\gamma _{1}$ and $\gamma _{2}$
are defined in (\ref{fadef1}), and $\mathfrak{a}_{1}\left( \kappa \right) $
and $\mathfrak{a}_{2}\left( \kappa \right) $ are independent copies of $
\mathfrak{a}\left( \kappa \right) $ defined in (\ref{fadef3}).
\end{theorem}
We now turn to the nonstationary case. We need an additional technical
condition:
\begin{assumption}
\label{rcaap11}\ It holds that $\epsilon _{0,2}$ has a bounded density.
\end{assumption}
\begin{theorem}
\label{thrca111} We assume that $H_{0}$ of (\ref{rcanu}), Assumptions \ref
{as-wc-1}, \ref{rcaas}, \ref{rcaap11} hold, and $0\leq E\ln |\beta
_{0}+\epsilon _{0,1}|<\infty $.\newline
(i) If $I(w,c)<\infty $ for some $c>0$, then it holds that
\begin{equation*}
\sup_{0<t<1}\frac{\left\vert Q_{N}(t)\right\vert }{w(t)}\;\overset{{\mathcal{
D}}}{\rightarrow }\;\eta \sup_{0<t<1}\frac{|B(t)|}{w(t)},
\end{equation*}
where $\{B(t),0\leq t\leq 1\}$ is a standard Brownian bridge and $\eta $ is
defined in (\ref{etasq}). \newline
(ii) For all $x$
\begin{equation*}
\lim_{N\rightarrow \infty }P\Biggl\{a(\ln N)\frac{1}{\eta }
\max_{1<k<N}\left( \frac{k(N-k)}{N}\right) ^{1/2}|\widehat{\beta }_{k,1}-
\widehat{\beta }_{k,2}|\leq x+b(\ln N)\Biggl\}=\exp (-2e^{-x}),
\end{equation*}
where $a(x)$ and $b(x)$ are defined in (\ref{da-1/0}).
(iii) If Assumption \ref{thrcaas11} is satisfied, then it holds that
\begin{equation*}
\left( \frac{r_{N}}{N}\right) ^{\kappa -1/2}\frac{1}{\eta }
\sup_{r_{1}(N)/N\leq t\leq 1-r_{2}(N)/N}\frac{|Q_{N}(t)|}{\left( t\left(
1-t\right) \right) ^{\kappa }}\;\overset{{\mathcal{D}}}{\rightarrow }\;\max
\left( \gamma _{1}^{\kappa -1/2}\mathfrak{a}_{1}(\kappa ),\gamma
_{2}^{\kappa -1/2}\mathfrak{a}_{2}(\kappa )\right) ,
\end{equation*}
for all $\kappa >1/2$, where and $r_{N}$, $\gamma _{1}$ and $\gamma _{2}$
are defined in (\ref{fadef1}), and $\mathfrak{a}_{1}\left( \kappa \right) $
and $\mathfrak{a}_{2}\left( \kappa \right) $ are independent copies of $
\mathfrak{a}\left( \kappa \right) $ defined in (\ref{fadef3}).
\end{theorem}
Theorems \ref{thrca1} and \ref{thrca111} stipulate that the limiting
distributions of the weighted CUSUM statistics are the same irrespective of
whether $y_{i}$ is stationary, explosive or at the boundary: the impact of
nonstationarity is only on $\eta ^{2}$. Hence, it is important to find an
estimator for $\eta ^{2}$ which is consistent for all cases. Let
\begin{equation*}
\widehat{a}_{N,1}=\frac{1}{N-1}\sum_{i=2}^{N}\frac{(y_{i}-\widehat{\beta }
_{N,1}y_{i-1})^{2}y_{i-1}^{2}}{(1+y_{i-1}^{2})^{2}}\quad \mbox{and}\quad
\widehat{a}_{N,2}=\frac{1}{N-1}\sum_{i=2}^{N}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}
}
\end{equation*}
We use the following estimator for $\eta ^{2}$
\begin{equation}
\widehat{\eta }_{N}^{2}=\frac{\widehat{a}_{N,1}}{\widehat{a}_{N,2}^{2}}.
\label{eta-est}
\end{equation}
\begin{corollary}
\label{rcaco1} The results of Theorems \ref{thrca1}--\ref{thrca111} remain
true if $\eta $ is replaced with $\widehat{\eta }_{N}$.
\end{corollary}
Corollary \ref{rcaco1} states that the feasible versions of our test
statistics, based on $\widehat{\eta }_{N}$, have the same distribution as
the infeasible ones, based on $\eta $. Practically, this means that the test
statistics developed above can be implemented with no prior knowledge as to
whether $y_{i}$ is stationary or not.
\section{Change point detection with heteroskedastic errors\label{heterosk}}
In the previous section we assumed, as is typical in the RCA\ literature,
that the innovations $\{\epsilon _{i,1},\epsilon _{i,2},1\leq i\leq N\}$ are
homoskedastic, which may be an undesirable restriction. The literature on
the changepoint problem has recently considered this issue, but
contributions are still relatively rare: exceptions include \citet{xu2015},
\citet{gorecki2017} \citet{bardsley2017} and \citet{horvath2021} (see also
\citealp{xuphillips2008}, for adaptive estimation in autoregressive models).
Heteroskedasticity is particularly interesting and challenging in the RCA\
case: if the distribution of $\epsilon _{i,1}$ is allowed to change, the
observations might change from stationarity to non stationarity even if $
\beta _{0}$ does not undergo any change; however, inference on the RCA\
model will still be asymptotically normal in light of the properties of the
WLS\ estimator discussed above.
In this section, we extend all the results above allowing for
heteroskedasticity in both $\epsilon _{i,1}$ and $\epsilon _{i,2}$. Our
results are valid also in the baseline case of homoskedasticity, and do not
require any explicit knowledge of the form of heteroskedasticity.
\bigskip
Changes in the distribution of $\{\epsilon _{i,1},\epsilon _{i,2},1\leq
i\leq N\}$ at times $1<m_{1}<\ldots <m_{M}<N$ are allowed through the
following assumption.
\begin{assumption}
\label{rcamult} It holds that $m_{\ell }=\lfloor N\tau _{\ell }\rfloor $,
for $1\leq \ell \leq M$, with $0<\tau _{1}<\tau _{2}<\ldots <\tau _{M}<1$.
\end{assumption}
Henceforth, we will use the notation: $m_{0}=0,m_{M+1}=N,\tau _{0}=0$ and $
\tau _{M+1}=1$.
For each subsequence $\{y_{i},m_{\ell -1}<i\leq m_{\ell }\}$, $1\leq \ell
\leq M+1$, the condition for stationarity can be satisfied; in this case,
the elements of this subsequence can be approximated with stationary
variables $\{\bar{y}_{\ell ,j},-\infty <j<\infty \}$ defined by the
recursion
\begin{equation*}
\bar{y}_{\ell ,j}=(\beta _{0}+\epsilon _{\ell ,j,1})\bar{y}_{\ell
,j-1}+\epsilon _{\ell ,j,2},\quad -\infty <j<\infty ,
\end{equation*}
where $\epsilon _{\ell ,j,1}=\epsilon _{j,1},m_{\ell -1}<j\leq m_{\ell }$,
and $\epsilon _{\ell ,j,1},-\infty <j<\infty ,j\not\in (m_{\ell -1},m_{\ell
-1}+1,\ldots ,m_{\ell }]$ are independent and identically distributed copies
of $\epsilon _{m_{\ell },1}$. The random variables $\epsilon _{\ell ,j,2}$
are defined in the same way.
To allow for changes in the distributions of the errors, we replace
Assumptions \ref{rcaas}-\ref{rcaap11} with
\begin{assumption}
\label{rcaas1} It holds that: (i) $\{\epsilon _{i,1},-\infty <i<\infty \}$
and $\{\epsilon _{i,2},-\infty <i<\infty \}$ are independent sequences of
independent random variables; (ii) $P\{y_{0}=0\}<1$; (iii) for each $1\leq
\ell \leq M+1$, $\{\epsilon _{i,1},\;m_{\ell -1}<i\leq m_{\ell }\}$ are
identically distributed with $E\epsilon _{m_{\ell },1}=0,E\epsilon _{m_{\ell
},1}^{2}=\sigma _{m_{\ell },1}^{2}$ and $E|\epsilon _{m_{\ell
},1}|^{4}<\infty $; (iv) for each $1\leq \ell \leq M+1$, $\{\epsilon
_{i,2},m_{\ell -1}<i\leq m_{\ell }\}$ are identically distributed with $
E\epsilon _{m_{\ell },2}=0,E\epsilon _{m_{\ell },2}^{2}=\sigma _{m_{\ell
},2}^{2}$ and $E|\sigma _{m_{\ell },2}|^{4}<\infty $; (v) if $E\ln |\beta
_{0}+\epsilon _{m_{\ell },1}|<0$, $P\left( \bar{y}_{\ell ,0}=0\right) <1$, $
1\leq \ell \leq M+1$; (vi) if $E\ln |\beta _{0}+\epsilon _{m_{\ell },1}|\geq
0$, then $\epsilon _{m_{\ell },2}$ has a bounded density, $1\leq \ell \leq
M+1$.
\end{assumption}
By Assumption \ref{rcaas1}, the WLS estimator may have different variances
in the various regimes. In order to study the limit theory, consider the
following notation:
\begin{equation}
\eta _{\ell }^{2}=\left\{
\begin{array}{ll}
\begin{array}{l}
\displaystyle E\left( \frac{\bar{y}_{\ell ,0}^{2}}{1+\bar{y}_{\ell ,0}^{2}}
\right) ^{2}\sigma _{\ell ,1}^{2}+E\left( \frac{\bar{y}_{\ell ,0}}{1+\bar{y}
_{\ell ,0}^{2}}\right) ^{2}\sigma _{\ell ,2}^{2}, \\
\quad \quad \quad \quad \quad \quad \quad \mbox{if}\;\;\;-\infty \leq E\ln
|\beta _{0}+\epsilon _{m_{\ell },1}|<0,
\end{array}
& \\
\sigma _{\ell ,1}^{2},\quad \mbox{if}\;\;\;E\ln |\beta _{0}+\epsilon
_{m_{\ell },1}|\geq 0 &
\end{array}
\right. \label{eta-l}
\end{equation}
and
\begin{equation}
a_{\ell }=\left\{
\begin{array}{ll}
\displaystyle E\left( \frac{\bar{y}_{\ell ,0}^{2}}{1+\bar{y}_{\ell ,0}^{2}}
\right) ,\quad \mbox{if}\;\;\;-\infty \leq E\ln |\beta _{0}+\epsilon
_{m_{\ell },1}|<0, & \\
1,\quad \mbox{if}\;\;\;E\ln |\beta _{0}+\epsilon _{m_{\ell },1}|\geq 0, &
\end{array}
\right. \label{a-l}
\end{equation}
$1\leq \ell \leq M+1$. Also, let
\begin{equation}
\bar{\eta}(t)=\sum_{j=1}^{\ell -1}\frac{\eta _{j}^{2}}{a_{j}^{2}}(\tau
_{j}-\tau _{j-1})+\frac{\eta _{\ell }^{2}}{a_{\ell }^{2}}(t-\tau _{\ell
-1}),\quad \tau _{\ell -1}<t\leq \tau _{\ell },1\leq \ell \leq M+1,
\label{eta-bar}
\end{equation}
and define the zero mean Gaussian process $\left\{ \Gamma \left( t\right)
,0\leq t\leq 1\right\} $, with $E\left[ \Gamma \left( t\right) \Gamma \left(
s\right) \right] =\bar{\eta}\left( \min \left( t,s\right) \right) $.
We begin by investigating how the limits in Theorems \ref{thrca1} and \ref
{thrca111} behave under heteroskedasticity.
\begin{theorem}
\label{thrca4} We assume that $H_{0}$ of (\ref{rcanu}), and Assumptions \ref
{as-wc-1}, \ref{rcamult} and \ref{rcaas1} hold.\newline
(i) If $I(w,c)<\infty $ for some $c>0$, then it holds that
\begin{equation*}
\sup_{0<t<1}\frac{\left\vert Q_{N}(t)\right\vert }{w(t)}\;\overset{{\mathcal{
D}}}{\rightarrow }\;\sup_{0<t<1}\frac{\left\vert \Gamma (t)-t\Gamma
(1)\right\vert }{w(t)}.
\end{equation*}
\newline
(ii) For all $x$, it holds that
\begin{equation*}
\lim_{N\rightarrow \infty }P\Biggl\{a(\ln N)\sup_{0<t<1}\frac{\left\vert
Q_{N}(t)\right\vert }{\eta _{0}^{1/2}(t,t)}\leq x+b(\ln N)\Biggl\}=\exp
(-2e^{-x}).
\end{equation*}
\newline
(iii) If Assumption \ref{thrcaas11} is also satisfied, then it holds that,
for all $\kappa >1/2$
\begin{equation*}
\left( \frac{r_{N}}{N}\right) ^{\kappa -1/2}\sup_{r_{1}(N)/N\leq t\leq
1-r_{2}(N)/N}\frac{(t(1-t))^{-\kappa +1/2}}{\eta _{0}^{1/2}(t,t)}|{Q}
_{N}(t)|\;\overset{{\mathcal{D}}}{\rightarrow }\;\max \left( \gamma
_{1}^{\kappa -1/2}\mathfrak{a}_{1}(\kappa ),\gamma _{2}^{\kappa -1/2}
\mathfrak{a}_{2}(\kappa )\right) .
\end{equation*}
\end{theorem}
Theorem \ref{thrca4} is only of theoretical interest, but we point out that
heteroskedasticity impacts only on part \textit{(i)}. In that case, the
limiting distribution of the weighted $Q_{N}(t)$ is given by a Gaussian
process with covariance kernel
\begin{equation}
\eta _{0}(t,s)=E\left[ \left( \Gamma (t)-t\Gamma (1)\right) \left( \Gamma
(s)-s\Gamma (1)\right) \right] =\bar{\eta}(\min (t,s))-t\bar{\eta}(s)-s\bar{
\eta}(t)+st\bar{\eta}(1). \label{rcabarga}
\end{equation}
Parts \textit{(ii)-(iii)} of the theorem are the same as in the case of
homoskedasticity.
\begin{comment}
Based on (\ref{rcabarga}), it may seem harder to use the Darling-Erd\H{o}s
and the R\'{e}nyi-type statistics, since the covariance kernel $\eta
_{0}(t,t)$ is not asymptotically proportional to $t(1-t)$. However,
\end{comment}Upon inspecting the proof, in these cases the asymptotic
distribution is driven only by the observations which are as close to sample
endpoints as $o\left( N\right) $. On these intervals, (\ref{eta-bar})
ensures that the asymptotic variance $\eta _{0}(t,t)$ is proportional to $
t(1-t)$.
Finally, note that, in light of the definitions of $\eta _{0}(t,t)$ and $
\eta _{\ell }^{2}$ and $a_{\ell }$, heteroskedasticity in $\epsilon _{i,2}$
does not play a role in the nonstationary case.
\subsection{Feasible tests under heteroskedasticity\label{feasible}}
By Theorem \ref{thrca4}, the implementation of tests based on $Q_{N}(t)$
requires an estimate of $\eta _{0}\left( t,t\right) $. However, this is
fraught with difficulties, since it requires knowledge of the different
regime dates, $m_{\ell }$. Thus, we consider a modification of $Q_{N}(t)$ to
reflect the possible changes in the variances of the errors.
Let
\begin{equation}
\widehat{\mathfrak{c}}_{N,1}(t)=\frac{1}{N}\sum_{i=2}^{\lfloor (N+1)t\rfloor
}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}\quad \mbox{and}\quad \widehat{\mathfrak{c}
}_{N,2}(t)=\frac{1}{N}\sum_{i=\lfloor (N+1)t\rfloor +1}^{N}\frac{y_{i-1}^{2}
}{1+y_{i-1}^{2}},\;\;\;0\leq t\leq 1; \label{c-hat}
\end{equation}
clearly, $\widehat{\mathfrak{c}}_{N,2}(t)=\widehat{\mathfrak{c}}_{N,1}(1)-
\widehat{\mathfrak{c}}_{N,1}(t)$. We then define the modified test statistic
\begin{equation*}
\overline{Q}_{N}(t)=\left\{
\begin{array}{ll}
0,\quad \mbox{if}\;\;\;0<t<2/(N+1),\vspace{0.3cm} & \\
N^{1/2}\widehat{\mathfrak{c}}_{N,1}(t)\widehat{\mathfrak{c}}_{N,2}(t)\left(
\widehat{\beta }_{\lfloor (N+1)t\rfloor ,1}-\widehat{\beta }_{\lfloor
(N+1)t\rfloor ,2}\right) ,\quad \mbox{if}\;\;\;2/(N+1)\leq t<1-2/(N+1),
\vspace{0.3cm} & \\
0,\quad \mbox{if}\;\;\;1-2/(N+1)\leq t<1. &
\end{array}
\right.
\end{equation*}
Under the null of no change, the same arguments as in the proof of Corollary
\ref{rcaco1} guarantee that $\widehat{\mathfrak{c}}_{N,1}(t)$ and $\widehat{
\mathfrak{c}}_{N,2}(t)$ converge to the functions
\begin{equation}
\mathfrak{c}_{1}(t)=\sum_{j=1}^{\ell -1}(\tau _{\ell }-\tau _{\ell
-1})a_{\ell }+(t-\tau _{\ell -1})a_{\ell },\quad \tau _{\ell -1}<t\leq \tau
_{\ell },1\leq \ell \leq M+1, \label{c1-t}
\end{equation}
and $\mathfrak{c}_{2}(t)=\mathfrak{c}_{1}(1)-\mathfrak{c}_{1}(t)$, for $
0\leq t\leq 1$, where $\tau _{\ell }$ is defined in Assumption \ref{rcamult}
, and $a_{\ell },1\leq \ell \leq M+1$ is defined in (\ref{a-l}). In order to
present our main results, we define the zero mean Gaussian process
\begin{equation}
\Theta (t)=\mathfrak{c}_{2}(t)\Delta (t)-\mathfrak{c}_{1}(t)(\Delta
(1)-\Delta (t)),\quad 0\leq t\leq 1; \label{rcadeTh}
\end{equation}
$\{\Delta (t),0\leq t\leq 1\}$ is also a zero mean Gaussian process with $
E\Delta (t)\Delta (s)=\mathfrak{b}(\min (t,s))$, where
\begin{subequations}
\begin{equation}
{\mathfrak{b}}(t)=\sum_{j=1}^{\ell -1}(\tau _{j}-\tau _{j-1})\eta
_{j}^{2}+(t-\tau _{\ell -1})\eta _{\ell }^{2},\quad \tau _{\ell -1}<t\leq
\tau _{\ell },1\leq \ell \leq M+1. \label{b-t}
\end{equation}
Let ${\mathfrak{g}}(t,s)=E\left( \Theta (t)\Theta (s)\right) $; elementary
calculations yield
\end{subequations}
\begin{equation}
{\mathfrak{g}}(t,s)=\mathfrak{c}_{1}(1)\mathfrak{c}_{1}(1)\mathfrak{b}\left(
\min (t,s)\right) -\mathfrak{c}_{1}(1)\mathfrak{c}_{1}(t)\mathfrak{b}(s)-
\mathfrak{c}_{1}(1)\mathfrak{c}_{1}(s)\mathfrak{b}(t)+\mathfrak{c}_{1}(t)
\mathfrak{c}_{1}(s)\mathfrak{b}(1). \label{rcadefg}
\end{equation}
\begin{theorem}
\label{thrca5}We assume that $H_{0}$ of (\ref{rcanu}), and Assumptions \ref
{as-wc-1}, \ref{rcamult} and \ref{rcaas1} hold.\newline
(i) If $I(w,c)<\infty $ for some $c>0$, then
\begin{equation*}
\sup_{0<t<1}\frac{\left\vert \overline{Q}_{N}(t)\right\vert }{w(t)}\;\overset
{{\mathcal{D}}}{\rightarrow }\;\sup_{0<t<1}\frac{\left\vert \Theta
(t)\right\vert }{w(t)},
\end{equation*}
where $\{\Theta (t),0\leq t\leq 1\}$ is the Gaussian process defined in
\eqref{rcadeTh}.\newline
(ii) For all $x$
\begin{equation*}
\lim_{N\rightarrow \infty }P\Biggl\{a(\ln N)\sup_{0<t<1}\frac{\left\vert
\overline{Q}_{N}(t)\right\vert }{\mathfrak{g}^{1/2}\left( t,t\right) }\leq
x+b(\ln N)\Biggl\}=\exp (-2e^{-x}),
\end{equation*}
where $a(x)$, $b(x)$ are defined in (\ref{da-1/0}) and $\mathfrak{g}(t,s)$
is given in (\ref{rcadefg}).\newline
(iii) If Assumption \ref{thrcaas11} also holds and $\kappa >1/2$, then
\begin{equation*}
\left( \frac{r_{N}}{N}\right) ^{\kappa -1/2}\sup_{t_{1}<t<t_{2}}\frac{
(t(1-t))^{-\kappa +1/2}}{\mathfrak{g}^{1/2}\left( t,t\right) }\left\vert
\overline{Q}_{N}(t)\right\vert \;\overset{{\mathcal{D}}}{\rightarrow }\;\max
\left( \gamma _{1}^{\kappa -1/2}\mathfrak{a}_{1}(\kappa ),\gamma
_{2}^{\kappa -1/2}\mathfrak{a}_{2}(\kappa )\right) ,
\end{equation*}
where $t_{1}=r_{N}/N$, $t_{2}=1-t_{1}$, and $r_{N}$, $\gamma _{1}$, $\gamma
_{2}$ are defined in (\ref{fadef1}), $\mathfrak{a}_{1}\left( \kappa \right) $
and $\mathfrak{a}_{2}\left( \kappa \right) $ are independent copies of $
\mathfrak{a}\left( \kappa \right) $ defined in (\ref{fadef3}), and $
\mathfrak{g}(t,s)$ is given in (\ref{rcadefg}).
\end{theorem}
Some comments on the practical implementation of the results in Theorem \ref
{thrca5} are in order. Parts \textit{(ii)} and \textit{(iii)} require an
estimate of $\mathfrak{g}(t,t)$; to this end, we use $\widehat{\mathfrak{c}}
_{N,1}(t)$ defined in (\ref{c-hat}) instead of $\mathfrak{c}_{1}(t)$, and we
estimate ${\mathfrak{b}}(t,s)$ as
\begin{equation*}
\widehat{\mathfrak{b}}_{N}(t)=\frac{1}{N}\sum_{i=2}^{\lfloor Nt\rfloor
}\left( \frac{(y_{i}-\widehat{\beta }_{N,1}y_{i-1})y_{i-1}}{1+y_{i-1}^{2}}
\right) ^{2},\quad 0\leq t\leq 1.
\end{equation*}
Then we can define
\begin{equation}
\widehat{{\mathfrak{g}}}(t,s)=\widehat{\mathfrak{c}}_{N,1}^{2}\left(
1\right) \widehat{\mathfrak{b}}_{N}\left( \min \left( t,s\right) \right) -
\widehat{\mathfrak{c}}_{N,1}\left( 1\right) \left( \widehat{\mathfrak{c}}
_{N,1}\left( t\right) \widehat{\mathfrak{b}}_{N}\left( s\right) +\widehat{
\mathfrak{c}}_{N,1}\left( s\right) \widehat{\mathfrak{b}}_{N}\left( t\right)
\right) +\widehat{\mathfrak{c}}_{N,1}\left( t\right) \widehat{\mathfrak{c}}
_{N,1}\left( s\right) \widehat{\mathfrak{b}}_{N}\left( 1\right) .
\label{g-hat}
\end{equation}
The implementation of part \textit{(i)} of Theorem \ref{thrca5} is more
complicated, since the presence of nuisance parameters is not relegated to a
multiplicative function. We reject the null hypothesis in (\ref{rcanu}) if
\begin{equation*}
\sup_{0<t<1}\frac{\left\vert \overline{Q}_{N}(t)\right\vert }{w(t)}\geq
c(\alpha ),
\end{equation*}
with $c(\alpha )$ defined as $P\left\{ \sup_{0<t<1}\frac{\left\vert \Theta
\left( t\right) \right\vert }{w(t)}\geq c(\alpha )\right\} =\alpha $.
Computing the covariance functions, one can verify that $\{\Delta (t),0\leq
t\leq 1\}\;\overset{{\mathcal{D}}}{=}\;\{W(\mathfrak{b}(t)),0\leq t\leq 1\}$
, where $\{W(x),0\leq x<\infty \}$ is a Wiener process. In order to
approximate the critical values, one can simulate independent Wiener
processes $W_{i}(x)$, $1\leq i\leq L$, and compute the empirical
distribution function
\begin{equation}
{\mathcal{F}}_{N,L}(x)=\frac{1}{L}\sum_{i=1}^{L}I\left\{ \sup_{0<t<1}\frac{
\left\vert \widehat{\Theta }_{i}(t)\right\vert }{w(t)}\leq x\right\} ,
\label{cr-val-method}
\end{equation}
where $\widehat{\Theta }_{i}(t)=\widehat{\mathfrak{c}}_{N,2}(t)W_{i}(
\widehat{\mathfrak{b}}_{N}(t))-\widehat{\mathfrak{c}}_{N,1}(t)(W_{i}(
\widehat{\mathfrak{b}}_{N}(1))-W_{i}(\widehat{\mathfrak{b}}_{N}(t)))$.
Let $c_{N,L}(\alpha )$ be defined as $c_{N,L}(\alpha )=\inf \{x:\;{\mathcal{F
}}_{N,L}(x)\geq 1-\alpha \}$.
\begin{corollary}
\label{cnlalpha}We assume that $H_{0}$ of (\ref{rcanu}) holds. Under the
same assumptions as Theorem \ref{thrca5}(i), it holds that
\begin{equation*}
\lim_{\min \left( N,L\right) \rightarrow \infty }P\left\{ \sup_{0<t<1}\frac{
\left\vert \overline{Q}_{N}(t)\right\vert }{w(t)}\geq c_{N,L}(\alpha
)\right\} =\alpha .
\end{equation*}
The results of Theorem \ref{thrca5}(ii)-(iii) remain true if ${\mathfrak{g}}
(t,s)$ is replaced with $\widehat{{\mathfrak{g}}}(t,s)$.
\end{corollary}
\begin{comment}
We point out that we need to estimate the \textquotedblleft long-run
variances\textquotedblright\ $\widehat{{\mathfrak{g}}}(t,s)$; however, this
does not require the specification of tuning parameters such as e.g. a
kernel function or a bandwidth.
\end{comment}
\subsection{Consistency versus alternatives\label{consistency}}
We study the consistency of our tests versus the AMOC alternative\footnote{
In Section \ref{mbreaks} in the Supplement, we also discuss the power of our
tests versus the alternative of multiple breaks.}
\begin{equation*}
y_{i}=\left\{
\begin{array}{ll}
\left( \beta _{0}+\epsilon _{i,1}\right) y_{i-1}+\epsilon _{i,2},\quad
\mbox{if}\;\;\;1\leq i\leq k^{\ast } & \\
\left( \beta _{A}+\epsilon _{i,1}\right) y_{i-1}+\epsilon _{i,2},\quad
\mbox{if}\;\;\;k^{\ast }+1\leq i\leq N &
\end{array}
\right. .
\end{equation*}
Let $\Delta _{N}=\left\vert \beta _{0}-\beta _{A}\right\vert $, and define $
t^{\ast }$ as $\left\lfloor Nt^{\ast }\right\rfloor =k^{\ast }$.
\begin{theorem}
\label{amoc}We assume that $H_{0}$ of (\ref{rcanu}) holds. Under the same
assumptions as Theorem \ref{thrca5}, if, as $N\rightarrow \infty $
\begin{equation}
N^{1/2}\Delta _{N}\frac{\displaystyle \left( \frac{k^{\ast }}{N}\left( \frac{
N-k^{\ast }}{N}\right) \right) }{\displaystyle w\left( t^{\ast }\right) }
\rightarrow \infty , \label{restr-1}
\end{equation}
then it holds that, as $\min \left( N,L\right) \rightarrow \infty $
\begin{equation}
\sup_{0<t<1}\frac{\left\vert \overline{Q}_{N}(t)\right\vert }{w\left(
t\right) }\overset{P}{\rightarrow }\infty . \label{part3}
\end{equation}
Further, if, as $N\rightarrow \infty $
\begin{equation}
\frac{N^{1/2}}{\left( \ln \ln N\right) ^{1/2}}\Delta _{N}\left( \frac{
k^{\ast }}{N}\left( \frac{N-k^{\ast }}{N}\right) \right) ^{1/2}\rightarrow
\infty , \label{restr-2}
\end{equation}
then it holds that
\begin{equation}
a(\ln N)\sup_{0<t<1}\frac{1}{\widehat{\mathfrak{g}}^{1/2}\left( t,t\right) }
\left\vert \overline{Q}_{N}(t)\right\vert -b(\ln N)\overset{P}{\rightarrow }
\infty . \label{part2}
\end{equation}
Finally, if
\begin{equation}
\left( \frac{r_{N}}{N}\right) ^{\kappa -1/2}N^{1/2}\Delta _{N}\left( \frac{
k^{\ast }}{N}\left( \frac{N-k^{\ast }}{N}\right) \right) ^{1-\kappa
}\rightarrow \infty , \label{restr-3}
\end{equation}
then it holds that, for all $\kappa >\frac{1}{2}$
\begin{equation}
\left( \frac{r_{N}}{N}\right) ^{\kappa -1/2}\sup_{t_{1}<t<t_{2}}\frac{
(t(1-t))^{-\kappa +1/2}}{\widehat{\mathfrak{g}}^{1/2}\left( t,t\right) }
\left\vert \overline{Q}_{N}(t)\right\vert \overset{P}{\rightarrow }\infty .
\label{part33}
\end{equation}
\end{theorem}
The theorem ensures that, as long as (\ref{restr-1}), (\ref{restr-2}) and (
\ref{restr-3}) hold, our tests reject the null with probability
(asymptotically) $1$. Conditions (\ref{restr-1}), (\ref{restr-2}) and (\ref
{restr-3}) essentially state that breaks will be detected as long as they
are \textquotedblleft not too small\textquotedblright , and
\textquotedblleft not too close\textquotedblright\ to the endpoints of the
sample.
Consider (\ref{restr-1}). This condition can be understood by considering
two cases. First, when $\frac{k^{\ast }}{N}\rightarrow c\in \left(
0,1\right) $, it is required that $N^{1/2}\Delta _{N}\rightarrow \infty $:
this entails that $\beta _{A}$ may depend on the sample size $N$, so that
even small changes in the regression parameter are allowed. When $\Delta
_{N}>0$, (\ref{restr-1}) holds as long as $k^{\ast }N^{\frac{2\kappa -1}{
2\left( 1-\kappa \right) }}\rightarrow \infty $: tests based on weight
functions $w\left( t\right) =\left( t\left( 1-t\right) \right) ^{\kappa }$
can detect breaks almost as close to the sample endpoints as $O\left( N^{
\frac{1-2\kappa }{2\left( 1-\kappa \right) }}\right) $.
Turning to (\ref{restr-2}), when $\frac{k^{\ast }}{N}\rightarrow c>0$, the
test is powerful as long as $\left( \frac{N}{\ln \ln N}\right) ^{1/2}\Delta
_{N}\rightarrow \infty $: again small changes are allowed for, but these are
now \textquotedblleft less small\textquotedblright\ by a $O\left( \ln \ln
N\right) $ factor. Conversely, when $\Delta _{N}>0$, (\ref{restr-1}) holds
as long as $k^{\ast }\left( \ln \ln N\right) ^{-1/2}\rightarrow \infty $:
breaks that are as close as $O\left( \sqrt{\ln \ln N}\right) $ periods to
the sample endpoints can be detected. This effect is reinforced in the case
of R\'{e}nyi statistics, where, on account of (\ref{restr-3}), the only
requirement is that $k^{\ast }>r_{N}$.
\section{Simulations\label{simulations}}
We provide some Monte Carlo evidence on the performance of the test
statistics proposed in Section \ref{feasible}.\footnote{
In Section \ref{crval} in the Supplement, we also evaluate, as a benchmark,
the performance of our tests under homoskedasticity, based on the theory in
Section \ref{homosk}.}
Data are generated using (\ref{arcnew1}). In all experiments, we use $\beta
_{0}\in \left\{ 0.5,0.75,1,1.05\right\} $ to consider both the cases of
stationary and nonstationary $y_{i}$. We have experimented also with
different values of $\beta _{0}$, but results are essentially the same.
Under the alternative, we consider both a mid-sample and an end-of-sample
break
\begin{align}
y_{i} =&\left( \beta _{0}+\Delta I\left( i\geq 0.5N\right) +\epsilon
_{i,1}\right) y_{i-1}+\epsilon _{i,2}, \label{po1} \\
y_{i} =&\left( \beta _{0}+\Delta I\left( i\geq 0.9N\right) +\epsilon
_{i,1}\right) y_{i-1}+\epsilon _{i,2}. \label{po2}
\end{align}
The shocks $\epsilon _{i,1}$ and $\epsilon _{i,2}$ are simulated as
independent of one another and \textit{i.i.d.} with distributions $N\left(
0,\sigma _{1}^{2}\right) $ and $N\left( 0,\sigma _{2}^{2}\right) $
respectively. We report results for $\sigma _{1}^{2}=0.01$ and $\sigma
_{2}^{2}=0.5$ - the value of $\sigma _{1}^{2}$ is based on \textquotedblleft
typical\textquotedblright\ values as found e.g. in the empirical
applications in \citet{HT16}. We note however that, in unreported
simulations using different values of $\sigma _{1}^{2}$ and $\sigma _{2}^{2}$
, the main results do not change, except for the (expected) fact that tests
have better properties (in terms of size and power) for smaller values of $
\sigma _{2}^{2}$. Similarly, the test performs better (with empirical
rejection frequencies closer to their nominal value) when $\sigma _{1}^{2}$
is larger, and tends to be undersized for smaller values of $\sigma _{1}^{2}$
. Both effects (of $\sigma _{1}^{2}$ and $\sigma _{2}^{2}$) vanish as $N$
increases. When allowing for heteroskedasticity, we generate $\epsilon
_{i,1} $ and $\epsilon _{i,2}$ as \textit{i.i.d.}$N\left( 0,\sigma
_{1}^{2}\right) $ and \textit{i.i.d.}$N\left( 0,\sigma _{2}^{2}\right) $ for
$1\leq i\leq N/2$, and \textit{i.i.d.}$N\left( 0,1.5\sigma _{1}^{2}\right) $
and \textit{i.i.d.}$N\left( 0,1.5\sigma _{2}^{2}\right) $ for $N/2+1\leq
i\leq N$.
Finally, we generate $N+1,000$ values of $y_{i}$ from (\ref{arcnew1}) - with
$y_{0}=0$ - and discard the first $1,000$ values. All our routines are based
on $2,000$ replications, and we use critical values corresponding to a
nominal level equal to $5\%\footnote{
When using (\ref{cr-val-method}), we use $L=200$. Results are however not
particulary sensitive to this specification.}$ - hence, empirical rejection
frequencies under the null have a $95\%$ confidence interval $\left[
0.04,0.06\right] $. \newline
We consider four different cases: \textit{(i)} homoskedasticity in both $
\epsilon _{i,1}$ and $\epsilon _{i,2}$; \textit{(ii)} homoskedasticity in $
\epsilon _{i,1}$ and heteroskedasticity in $\epsilon _{i,2}$; \textit{(iii) }
homoskedasticity in $\epsilon _{i,2}$ and heteroskedasticity in $\epsilon
_{i,1}$; and, finally, \textit{(iv)} heteroskedasticity in both $\epsilon
_{i,1}$ and $\epsilon _{i,2}$.
\begin{table*}[h]
\caption{Empirical rejection frequencies under the null - part 1}
\label{tab:SizeHet1}\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{llllllllllllllllllllll}
\hline\hline
& & & & & & & & & & & & & & & & & & & & & \\
& & $\beta $ & \multicolumn{4}{c}{$0.5$} & \multicolumn{1}{c}{} &
\multicolumn{4}{c}{$0.75$} & \multicolumn{1}{c}{} & \multicolumn{4}{c}{$1$}
& \multicolumn{1}{c}{} & \multicolumn{4}{c}{$1.05$} \\
& & & & & & & & & & & & & & & & & & & & & \\
& $N$ & & $200$ & $400$ & $800$ & $1600$ & & $200$ & $400$ & $800$ & $1600$
& & $200$ & $400$ & $800$ & $1600$ & & $200$ & $400$ & $800$ & $1600$ \\
$\kappa $ & & & & & & & & & & & & & & & & & & & & & \\
& & & & & & & & & & & & & & & & & & & & & \\
$0$ & & & $0.056$ & $0.066$ & $0.048$ & $0.060$ & & $0.058$ & $0.049$ & $0.054$ & $0.058$ & & $0.058$ & $0.059$ & $0.056$ & $0.060$ & & $0.057$ & $0.057$ & $0.063$ & $0.059$ \\
$0.25$ & & & $0.048$ & $0.053$ & $0.046$ & $0.053$ & & $0.046$ & $0.050$
& $0.050$ & $0.058$ & & $0.052$ & $0.060$ & $0.056$ & $0.056$ & & $0.057$
& $0.056$ & $0.062$ & $0.060$ \\
$0.45$ & & & $0.033$ & $0.034$ & $0.036$ & $0.038$ & & $0.035$ & $0.027$
& $0.031$ & $0.040$ & & $0.033$ & $0.042$ & $0.038$ & $0.036$ & & $0.029$
& $0.034$ & $0.038$ & $0.041$ \\
$0.5$ & & & $0.033$ & $0.026$ & $0.033$ & $0.050$ & & $0.035$ & $0.030$ &
$0.034$ & $0.035$ & & $0.037$ & $0.038$ & $0.042$ & $0.041$ & & $0.040$ & $0.039$ & $0.042$ & $0.045$ \\
$0.51$ & & & $0.015$ & $0.020$ & $0.019$ & $0.037$ & & $0.016$ & $0.019$
& $0.028$ & $0.038$ & & $0.016$ & $0.017$ & $0.025$ & $0.038$ & & $0.015$
& $0.025$ & $0.028$ & $0.036$ \\
$0.75$ & & & $0.034$ & $0.047$ & $0.042$ & $0.040$ & & $0.042$ & $0.043$
& $0.047$ & $0.047$ & & $0.039$ & $0.043$ & $0.041$ & $0.051$ & & $0.035$
& $0.042$ & $0.045$ & $0.034$ \\
$0.85$ & & & $0.034$ & $0.050$ & $0.045$ & $0.046$ & & $0.047$ & $0.050$
& $0.050$ & $0.047$ & & $0.043$ & $0.049$ & $0.049$ & $0.053$ & & $0.043$
& $0.060$ & $0.048$ & $0.040$ \\
$1$ & & & $0.038$ & $0.050$ & $0.047$ & $0.045$ & & $0.050$ & $0.048$ & $0.049$ & $0.043$ & & $0.043$ & $0.051$ & $0.046$ & $0.052$ & & $0.043$ & $0.061$ & $0.046$ & $0.042$ \\
& & & & & & & & & & & & & & & & & & & & & \\
\hline\hline
& & & & & & & & & & & & & & & & & & & & & \\
& & $\beta $ & \multicolumn{4}{c}{$0.5$} & \multicolumn{1}{c}{} &
\multicolumn{4}{c}{$0.75$} & \multicolumn{1}{c}{} & \multicolumn{4}{c}{$1$}
& \multicolumn{1}{c}{} & \multicolumn{4}{c}{$1.05$} \\
& & & & & & & & & & & & & & & & & & & & & \\
& $N$ & & $200$ & $400$ & $800$ & $1600$ & & $200$ & $400$ & $800$ & $1600$
& & $200$ & $400$ & $800$ & $1600$ & & $200$ & $400$ & $800$ & $1600$ \\
$\kappa $ & & & & & & & & & & & & & & & & & & & & & \\
& & & & & & & & & & & & & & & & & & & & & \\
$0$ & & & $0.055$ & $0.058$ & $0.051$ & $0.053$ & & $0.054$ & $0.050$ & $0.055$ & $0.057$ & & $0.058$ & $0.052$ & $0.052$ & $0.058$ & & $0.054$ & $0.058$ & $0.058$ & $0.060$ \\
$0.25$ & & & $0.054$ & $0.053$ & $0.053$ & $0.054$ & & $0.057$ & $0.043$
& $0.053$ & $0.055$ & & $0.052$ & $0.056$ & $0.054$ & $0.054$ & & $0.050$
& $0.059$ & $0.058$ & $0.060$ \\
$0.45$ & & & $0.033$ & $0.031$ & $0.033$ & $0.043$ & & $0.034$ & $0.029$
& $0.033$ & $0.037$ & & $0.030$ & $0.038$ & $0.040$ & $0.034$ & & $0.027$
& $0.034$ & $0.034$ & $0.043$ \\
$0.5$ & & & $0.036$ & $0.026$ & $0.043$ & $0.044$ & & $0.038$ & $0.030$ &
$0.036$ & $0.040$ & & $0.039$ & $0.037$ & $0.040$ & $0.040$ & & $0.037$ & $0.040$ & $0.040$ & $0.048$ \\
$0.51$ & & & $0.018$ & $0.019$ & $0.025$ & $0.041$ & & $0.019$ & $0.023$
& $0.032$ & $0.034$ & & $0.020$ & $0.018$ & $0.027$ & $0.040$ & & $0.017$
& $0.029$ & $0.026$ & $0.045$ \\
$0.75$ & & & $0.031$ & $0.045$ & $0.043$ & $0.043$ & & $0.045$ & $0.046$
& $0.052$ & $0.050$ & & $0.046$ & $0.043$ & $0.045$ & $0.054$ & & $0.039$
& $0.050$ & $0.040$ & $0.042$ \\
$0.85$ & & & $0.036$ & $0.050$ & $0.049$ & $0.050$ & & $0.050$ & $0.047$
& $0.053$ & $0.048$ & & $0.046$ & $0.047$ & $0.049$ & $0.051$ & & $0.044$
& $0.055$ & $0.042$ & $0.046$ \\
$1$ & & & $0.037$ & $0.055$ & $0.048$ & $0.051$ & & $0.051$ & $0.050$ & $0.051$ & $0.047$ & & $0.045$ & $0.050$ & $0.046$ & $0.051$ & & $0.044$ & $0.053$ & $0.045$ & $0.047$ \\
& & & & & & & & & & & & & & & & & & & & & \\
\hline\hline
\end{tabular}
}
\par
\begin{tablenotes}
\tiny
\item Empirical rejection frequencies under the null of no change - the first panel refers to homoskedastic innovations; in the second panel we consider heteroskedasticity in $\epsilon_{i,2}$ only.
\end{tablenotes}
\end{table*}
\begin{table*}[h]
\caption{Empirical rejection frequencies under the null - part 2}
\label{tab:SizeHet2}\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{llllllllllllllllllllll}
\hline\hline
& & & & & & & & & & & & & & & & & & & & & \\
& & $\beta $ & \multicolumn{4}{c}{$0.5$} & \multicolumn{1}{c}{} &
\multicolumn{4}{c}{$0.75$} & \multicolumn{1}{c}{} & \multicolumn{4}{c}{$1$}
& \multicolumn{1}{c}{} & \multicolumn{4}{c}{$1.05$} \\
& & & & & & & & & & & & & & & & & & & & & \\
& $N$ & & $200$ & $400$ & $800$ & $1600$ & & $200$ & $400$ & $800$ & $1600$
& & $200$ & $400$ & $800$ & $1600$ & & $200$ & $400$ & $800$ & $1600$ \\
$\kappa $ & & & & & & & & & & & & & & & & & & & & & \\
& & & & & & & & & & & & & & & & & & & & & \\
$0$ & & & $0.046$ & $0.061$ & $0.045$ & $0.057$ & & $0.065$ & $0.045$ & $0.056$ & $0.052$ & & $0.066$ & $0.063$ & $0.056$ & $0.058$ & & $0.058$ & $0.054$ & $0.062$ & $0.059$ \\
$0.25$ & & & $0.044$ & $0.055$ & $0.050$ & $0.056$ & & $0.062$ & $0.047$
& $0.056$ & $0.054$ & & $0.059$ & $0.058$ & $0.054$ & $0.060$ & & $0.051$
& $0.050$ & $0.062$ & $0.059$ \\
$0.45$ & & & $0.023$ & $0.031$ & $0.029$ & $0.036$ & & $0.035$ & $0.030$
& $0.040$ & $0.034$ & & $0.036$ & $0.033$ & $0.041$ & $0.043$ & & $0.027$
& $0.036$ & $0.038$ & $0.045$ \\
$0.5$ & & & $0.027$ & $0.033$ & $0.026$ & $0.040$ & & $0.040$ & $0.026$ &
$0.042$ & $0.040$ & & $0.040$ & $0.029$ & $0.043$ & $0.039$ & & $0.036$ & $0.035$ & $0.039$ & $0.050$ \\
$0.51$ & & & $0.015$ & $0.006$ & $0.017$ & $0.021$ & & $0.008$ & $0.014$
& $0.023$ & $0.020$ & & $0.009$ & $0.013$ & $0.021$ & $0.021$ & & $0.013$
& $0.018$ & $0.020$ & $0.025$ \\
$0.75$ & & & $0.035$ & $0.040$ & $0.041$ & $0.043$ & & $0.031$ & $0.050$
& $0.043$ & $0.045$ & & $0.035$ & $0.046$ & $0.047$ & $0.043$ & & $0.036$
& $0.044$ & $0.040$ & $0.040$ \\
$0.85$ & & & $0.040$ & $0.046$ & $0.043$ & $0.043$ & & $0.035$ & $0.055$
& $0.046$ & $0.048$ & & $0.040$ & $0.058$ & $0.053$ & $0.046$ & & $0.043$
& $0.050$ & $0.042$ & $0.041$ \\
$1$ & & & $0.044$ & $0.045$ & $0.045$ & $0.044$ & & $0.038$ & $0.058$ & $0.045$ & $0.047$ & & $0.042$ & $0.063$ & $0.051$ & $0.047$ & & $0.047$ & $0.054$ & $0.045$ & $0.041$ \\
& & & & & & & & & & & & & & & & & & & & & \\
\hline\hline
& & & & & & & & & & & & & & & & & & & & & \\
& & $\beta $ & \multicolumn{4}{c}{$0.5$} & \multicolumn{1}{c}{} &
\multicolumn{4}{c}{$0.75$} & \multicolumn{1}{c}{} & \multicolumn{4}{c}{$1$}
& \multicolumn{1}{c}{} & \multicolumn{4}{c}{$1.05$} \\
& & & & & & & & & & & & & & & & & & & & & \\
& $N$ & & $200$ & $400$ & $800$ & $1600$ & & $200$ & $400$ & $800$ & $1600$
& & $200$ & $400$ & $800$ & $1600$ & & $200$ & $400$ & $800$ & $1600$ \\
$\kappa $ & & & & & & & & & & & & & & & & & & & & & \\
& & & & & & & & & & & & & & & & & & & & & \\
$0$ & & & $0.053$ & $0.060$ & $0.046$ & $0.059$ & & $0.062$ & $0.043$ & $0.057$ & $0.052$ & & $0.057$ & $0.060$ & $0.060$ & $0.057$ & & $0.058$ & $0.054$ & $0.062$ & $0.059$ \\
$0.25$ & & & $0.047$ & $0.059$ & $0.044$ & $0.055$ & & $0.060$ & $0.044$
& $0.058$ & $0.046$ & & $0.054$ & $0.053$ & $0.060$ & $0.060$ & & $0.051$
& $0.050$ & $0.062$ & $0.059$ \\
$0.45$ & & & $0.026$ & $0.029$ & $0.030$ & $0.035$ & & $0.036$ & $0.029$
& $0.038$ & $0.033$ & & $0.040$ & $0.040$ & $0.047$ & $0.042$ & & $0.027$
& $0.036$ & $0.038$ & $0.045$ \\
$0.5$ & & & $0.030$ & $0.033$ & $0.027$ & $0.041$ & & $0.040$ & $0.029$ &
$0.039$ & $0.031$ & & $0.034$ & $0.027$ & $0.038$ & $0.038$ & & $0.036$ & $0.035$ & $0.039$ & $0.050$ \\
$0.51$ & & & $0.014$ & $0.007$ & $0.020$ & $0.024$ & & $0.007$ & $0.017$
& $0.023$ & $0.022$ & & $0.010$ & $0.013$ & $0.018$ & $0.021$ & & $0.013$
& $0.018$ & $0.020$ & $0.025$ \\
$0.75$ & & & $0.035$ & $0.040$ & $0.040$ & $0.045$ & & $0.029$ & $0.049$
& $0.046$ & $0.043$ & & $0.032$ & $0.041$ & $0.041$ & $0.040$ & & $0.036$
& $0.044$ & $0.040$ & $0.040$ \\
$0.85$ & & & $0.041$ & $0.044$ & $0.045$ & $0.046$ & & $0.035$ & $0.056$
& $0.050$ & $0.044$ & & $0.036$ & $0.047$ & $0.045$ & $0.042$ & & $0.043$
& $0.050$ & $0.042$ & $0.041$ \\
$1$ & & & $0.045$ & $0.050$ & $0.048$ & $0.044$ & & $0.035$ & $0.060$ & $0.052$ & $0.049$ & & $0.039$ & $0.049$ & $0.047$ & $0.042$ & & $0.047$ & $0.054$ & $0.045$ & $0.041$ \\
& & & & & & & & & & & & & & & & & & & & & \\
\hline\hline
\end{tabular}
}
\par
\begin{tablenotes}
\tiny
\item Empirical rejection frequencies under the null of no change - in the first panel we consider heteroskedasticity in $\epsilon_{i,1}$ only; in the second panel we consider heteroskedasticity in both $\epsilon_{i,1}$ and $\epsilon_{i,2}$ only.
\end{tablenotes}
\end{table*}
Empirical rejection frequencies under the null are in Tables \ref
{tab:SizeHet1} and \ref{tab:SizeHet2}. We have used asymptotic critical
values for R\'{e}nyi statistics, and the method described in Section \ref
{feasible} for the cases where $\kappa <0.5$. When using the Darling-Erd\H{o}
s statistic ($\kappa =0.5$), asymptotic critical values yield hugely
undersized tests; we have therefore used the critical values in Table I in
\citet{gombay}. From Tables \ref{tab:SizeHet1} and \ref{tab:SizeHet2}, all
tests work very well in all cases considered, possibly being slightly worse
in the fully homoskedastic case. Tests never over-reject, not even in small
samples - conversely, there are some cases of (severe) under-rejection in
small samples, especially when $\kappa $ is around $0.5$. As $N$ increases,
however, this vanishes and the empirical rejection frequencies all lie
within their $95\%$ confidence interval. The only exception is the R\'{e}nyi
statistic with $\kappa =0.51$, which is severely undersized even in large
samples.
\bigskip
The empirical power of the tests is reported in Figures \ref{fig:FigHo}-\ref
{fig:FigHeEB}, where we only consider a sample size of $N=400$ to save
space. The figures illustrate the robustness of the approach proposed in
Section \ref{feasible}, showing, essentially, the same pattern: tests work
well in all cases considered, with the power increasing monotonically in $
\Delta $. Test statistics with lower $\kappa $ exhibit more power versus
alternatives with \textquotedblleft small\textquotedblright\ values of $
\Delta $: in this case, the power is monotonically decreasing in $\kappa $,
with virtually no exceptions. Figures \ref{fig:FigHo}-\ref{fig:FigHeE},
compared with Figures \ref{fig:FigHeB}-\ref{fig:FigHeEB}, show an
interesting feature: the power of the test is virtually unaffected by the
presence or absence in heteroskedasticity in $\epsilon _{i,2}$; conversely,
heteroskedasticity in $\epsilon _{i,1}$ does have an impact. This is
particularly apparent in the cases where $\beta _{0}\geq 1$, which could be
explained by noting that, in the nonstationary case the asymptotics of the
WLS\ estimator is driven only by $\epsilon _{i,1}$. This can be read in
conjunction with the fact that, as shown in Figures \ref{fig:FigHo}-\ref
{fig:FigHeE}, the value of $\beta _{0}$ has vitually no impact on the power
of our tests when $\epsilon _{i,1}$ is constant.
We also consider the case of end-of-sample breaks (\ref{po2}). Results are
in Figures \ref{fig:FigHo2}-\ref{fig:FigHeEB2}. The results show,
essentially, the same pattern as above: all test statistics have monotonic
power in $\Delta $, and whilst heteroskedasticity in $\epsilon _{i,2}$ does
not affect the whole picture, heteroskedasticity in $\epsilon _{i,1}$ gives
very different results, with its presence increasing power especially for $
\beta _{0}\geq 1$. However, the impact of $\kappa $ here is, as expected,
completely reversed: the power versus breaks that occur at the end of the
sample increases monotonically, \textit{ceteris paribus}, with $\kappa $.
This makes a difference particularly in the case of medium-sized changes -
e.g., when $\Delta =0.35$, increases in power from $\kappa =0$ to $\kappa =1$
are in the region of $10-15\%$.
\bigskip
Finally, in Figures \ref{fig:bubb1}-\ref{fig:bubb2} we report a small scale
exercise where we evaluate the empirical rejection frequencies when $\beta
_{0}$ is close to unity. We only consider heteroskedasticity in $\epsilon
_{i,2}$: results for other cases are available upon request and, in general,
no major differences are noted compared to the other results. These
\textquotedblleft boundary\textquotedblright\ cases should be helpful to
shed more light on the performance of our procedure when detecting changes
from stationarity to nonstationarity (when $\beta _{0}<1$ and changes are
positive), and vice versa (when $\beta _{0}>1$ and changes are negative).
The main message of Figures \ref{fig:bubb1}-\ref{fig:bubb2} is that our
tests work very well in these boundary cases. In particular, the tests are
very effective in detecting changes from stationarity to explosive
behaviour, and vice versa. The power is especially high when $\beta _{0}>1$
- i.e. when the RCA\ process changes from an explosive to a stationary
behaviour. This suggests a possible, effective test to detect e.g. the
collapse of a bubble in financial econometrics applications.
\section{Empirical applications\label{empirics}}
We illustrate our approach through three applications to real data. In
Sections \ref{inflation}-\ref{ibm}, we use economic and financial time
series; in Section \ref{covid}, we use Covid-19 data.
\subsection{Application I: changes in the persistence of US CPI\label
{inflation}}
In his landmark paper, \citet{engle1982} applies an ARCH(1) specification to
monthly inflation data, showing that these exhibit conditional
heteroskedasticity. Inspired by this, and by the fact that the RCA model is
a second-order equivalent to the ARCH\ model, in this section we test for
the presence of changepoints in the dynamics of US CPI over the last
century. There is an increasing literature on testing for changes in the
persistence of inflation: in particular, \citet{benati} carry out a
systematic study, applying tests for structural breaks to an AR$(p)$ model
for inflation for several countries. Their analysis shows that not only the
average level of inflation (as is well-documented), but also its serial
correlation, may be subject to numerous changes.
We use monthly CPI data taken from the FRED dataset over a period spanning
from January 1913 until January 2021, with $N=1297$. We use monthly
inflation rates, calculated as the month-on-month log differences of the
series. Given that the series is quite long, we expect to see more than one
break; hence, we use binary segmentation (as suggested in
\citealp{vostrikova}), reporting the point in time at which the relevant
test statistic is maximised as the breakdate estimate.
\begin{table*}[h]
\caption{Changepoint detection with US CPI data.}
\label{tab:TableCPI}
\resizebox{\textwidth}{!}{
\begin{tabular}{llllllllllllllllllll}
\hline\hline
& & & & & & & & & & & & & & & & & & & \\
\textbf{Breakdate} & & \multicolumn{18}{l}{\textbf{Notes}} \\
& & & & & & & & & & & & & & & & & & & \\
Jan 1918 & & \multicolumn{18}{l}{Date found with $\kappa =0.85$, $1$ (found
at Apr 1918 with other $\kappa >0.51$; not found with $\kappa =0.51$). Found
at $10\%$ level with $\kappa =0.25$ and $0.5$} \\
& & & & & & & & & & & & & & & & & & & \\
Sep 1921 & & \multicolumn{18}{l}{Date found with $\kappa =0.85,1$ (found
Jan 1922 with $\kappa >0.65$; not found with other values of $\kappa $)} \\
& & & & & & & & & & & & & & & & & & & \\
May 1929 & & \multicolumn{18}{l}{Break found at $10\%$ nominal level only,
with $\kappa <0.5$ only} \\
& & & & & & & & & & & & & & & & & & & \\
Jul 1957 & & \multicolumn{18}{l}{Same date found by all R\'{e}nyi-type stats
(also found at Jan 1957 with $\kappa =0.5$). Break not found with $\kappa
<0.5$} \\
& & & & & & & & & & & & & & & & & & & \\
Nov 1966 & & \multicolumn{18}{l}{Same date found by all tests - except $\kappa =0.5$ (found at Feb 1966), and $\kappa =0.25$ (at $10\%$ level)} \\
& & & & & & & & & & & & & & & & & & & \\
Jun 1982 & & \multicolumn{18}{l}{Same date found by all tests (found at Dec
1981 with $\kappa =0.5$) - R\'{e}nyi-type stats at $5\%$, nominal level, all
others at $10\%$} \\
& & & & & & & & & & & & & & & & & & & \\
Nov 1989 & & \multicolumn{18}{l}{Same date found by all R\'{e}nyi-type stats
(found at Mar 1989 with $\kappa =1$). Break not found with $\kappa \leq 0.5$}
\\
& & & & & & & & & & & & & & & & & & & \\ \hline\hline
\end{tabular}
}
\par
\begin{tablenotes}
\tiny
\item We use the monthly log differences of CPI.
\item Detected changepoints, and their estimated date, are presented in \textit{chronological} order; as mentioned in the text, breakdates have been estimated as the earliest detection date across different values of $\kappa$. Whilst details are available upon request, we note that breaks were detected with this order (from the first to be detected to the last one): November 1966; July 1957 and January 1918; June 1982 and September 1921; March 1989 and May 1929.
\end{tablenotes}
\end{table*}
Results in Table XXX differ across tests only marginally, and suggest the
presence of several changepoints in the autoregressive coefficient - see
also Figure \ref{fig:FigCPI} in the Supplement. Some of the estimated
breakdates have a clear economic interpretation. In chronological order, the
first break is found (only by R\'{e}nyi statistics) around January 1918,
which should reflect not only the war effort, but also the increasingly more
comprehensive data collection from the Bureau of Labor Statistics; evidence
in favour of the changepoint increases as $\kappa $ increases, as the theory
would suggest given that this is an early change (occurring circa at $5\%$
of the sample). Indeed, when $\kappa \leq 0.5$, the break is found at $10\%$
, but not at $5\%$ level. Similar considerations, on the detection ability
and timing corresponding to different values of $\kappa $, can be made for
the break found in 1921; in this case, a possible cause is the impact of the
severe recession at the beginning of the decade, as well as the very rapid
deflation which had occurred in 1920. Conversely, there is limited evidence
for a break in the autoregressive parameter of inflation around the Great
Depression - looking at Figure \ref{fig:FigCPI}, this may be explained as a
shift which occurred only in the mean as opposed to the persistence. The
break occurring in 1957 may be viewed as related to inflation reemerging,
albeit modestly, in the spring of 1956 after a long period of price
stability, with the All-Items CPI increasing by 3.6 percent from April 1956
to April 1957 (comparing with the previous period, the All-Items CPI had
risen by $0.2$ percent annualized rate from July 1952 to April 1956). The
changepoint in 1966 (which is the first one to be found, by all tests) can
be explained by noting that food prices had started accelerating early at
the end of 1965; and, by October 1966, the change in the All-Items CPI
reached its highest since 1957. The change in 1981 is documented also in
other studies (\citealp{eo}), and it corresponds to the beginning of an
aggressive FED policy to rein in inflation after the 1970s. Finally, the
evidence for the break in 1989 (which is not picked up by all tests) is less
clear, but it may be the outcome of the cooling off in FED policy towards
the end of the 1980s.
\subsection{Application II: monthly IBM\ returns\label{ibm}}
We use IBM\ monthly returns, over a period spanning January 1962 till March
2021 (corresponding to $T=710$). The data used in this application has also
been employed, in the context of testing for changepoints, by \citet{yauzhao}
, who find evidence of two breaks: one around June 1987 (with confidence
interval between June 1986 and June 1988), and another around October 2002
(with confidence interval between April 2001 and April 2004). As in the
previous section, we have applied the tests using binary segmentation, using
all the test statistics developed above. None of them rejected the null of
no break at $5\%$ level; when applying the tests at $10\%$ level, though,
several breaks were found.
\bigskip
\begin{table}[h!]
\centering
\begin{threeparttable}
{\tiny
\caption{Changepoint detection with IBM data.}
\label{tab:TableIBM}\centering
\par
\begin{tabular}{lllllll}
\hline\hline
& & & & & & \\
& & \multicolumn{1}{c}{\textbf{Changepoint 1}} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{\textbf{Changepoint 2}} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{\textbf{Changepoint 3}} \\
& & & & & & \\
\textit{Date} & & \multicolumn{1}{c}{Sep 1973} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{Nov 1987} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{Oct
1999} \\
& & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} \\
\textit{Notes} & & \multicolumn{1}{c}{R\'{e}nyi stats ($10\%$ nominal level)} &
\multicolumn{1}{c}{} & \multicolumn{1}{c}{R\'{e}nyi stats ($10\%$ nominal level)}
& \multicolumn{1}{c}{} & \multicolumn{1}{c}{Weighted stats ($10\%$ nominal
level)} \\
& & & & & & \\ \hline\hline
\end{tabular}
\begin{tablenotes}
\tiny
\item We use the logs of the original data, with no further transformations.
\item For each estimated changepoint, we indicate which statistic has detected it, and at which (nominal) level. When more than one statistic finds a break, the estimated date is computed as the majority vote across statistics, using the point in time at which the relevant statistic finds a break as estimator. Whilst details are available upon request, we note that breaks were detected with this order (from the first to be detected to the last one): break in 1973; break in 1987; break in 1999.
\end{tablenotes}
}
\end{threeparttable}
\end{table}
We found three changepoints in the whole series (see Table \ref{tab:TableIBM}
). The first one, whose date corresponds to the well-known 1973-74 market
crash (due to the collapse of the Bretton-Woods system, and compounded by
the oil shock), is relatively close to the beginning of the sample, and
indeed it has been identified by the R\'{e}nyi statistics (the other tests
do not identify such break). The second changepoint can also be related to a
specific event, i.e. the Black Monday (the break is found in November 1987,
i.e. one month later the actual event). Finally, R\'{e}nyi statistics do not
find the third changepoint, which occurs mid-sample, confirming the idea
that mid-sample breaks are better detected using milder weight functions
(indeed, not even the Darling-Erd\H{o}s test finds evidence of such a
break); the break is found before the collapse of the dot-com bubble
(traditionally dated around March 2000), reflecting the trouble brewing in
the months leading to the event.
\subsection{Application III: Covid-19 UK hospitalisation data\label{covid}}
In this section, we consider UK data on Covid-19 - in particular, we use
data on hospitalisations rather than cases, as the latter may be less
reliable due to the change in number of tests administered. \citet{shtatland}
\textit{inter alia }advocate using a low-order autoregression as an
approximation of the popular SIR\ model, especially as a methodology for the
early detection of outbreaks. In this context, the autoregressive root is of
crucial importance since, as the authors put it, if \textquotedblleft the
parameter is greater than one, we have an explosive case (an outbreak of
epidemic)\textquotedblright . It is therefore important to check whether the
observations change from an explosive to a stationary regime (meaning that
the epidemic is slowing down), or vice versa whether the change occurs from
a stationary to an explosive regime (i.e., the epidemic undergoes a surge,
or \textquotedblleft wave\textquotedblright ). In this respect, the
empirical exercise in this section should be read in conjunction with
Figures \ref{fig:bubb1}-\ref{fig:bubb2}.
We use (logs of) UK daily data, for the four UK\ nations, and for the
various regions of England\footnote{
The data are available from
\par
https://ourworldindata.org/grapher/uk-daily-covid-admissions?tab=chart
\&stackMode=absolute\&time=2020-03-29..latest\®ion=World}, again using
binary segmentation to detect multiple breaks. We only report results
obtained using R\'{e}nyi statistics (with $\kappa =0.51$, $0.55$, $0.65$, $
0.75$, $0.85$ and $1$); the other tests give very similar results, available
upon request. As far as breakdates are concerned, we pick the ones
corresponding to the \textquotedblleft majority vote\textquotedblright\
across $\kappa $, although discrepancies are, when present, in the region of
few days ($2-5$ at most).
\begin{table*}[h!]
\caption{Changepoint analysis for Covid-19 daily hospitalisation UK data.}
\label{tab:TabCovid}
\par
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{lllllllllllllll}
\hline\hline
& & & & & & & & & & & & & & \\
\textbf{Region} & & \multicolumn{1}{c}{\textit{Start Date}} & \textit{Sample size} & & & \textbf{Changepoint 1} & & \textbf{Changepoint 2} & &
\textbf{Changepoint 3} & & \textbf{Changepoint 4} & & \textbf{Changepoint 5} \\
& & \multicolumn{1}{c}{} & & & & & & & & & & & & \\
East of England & & \multicolumn{1}{c}{19/3/20} & \multicolumn{1}{c}{$317 $} & & & \multicolumn{1}{c}{$\underset{\left[ 1.015;0.990\right] }{12\text{ Apr}}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[
0.993;1.008\right] }{25\text{ Aug}}$} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{} & & \multicolumn{1}{c}{} & & \multicolumn{1}{c}{$\underset{\left[ 1.011;1.000\right] }{08\text{ Jan}}$} \\
London & & \multicolumn{1}{c}{19/3/20} & \multicolumn{1}{c}{$317$} & &
& \multicolumn{1}{c}{$\underset{\left[ 1.014;0.991\right] }{10\text{ Apr}}$}
& \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 0.991;1.006\right] }{08\text{ Aug}}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} &
& \multicolumn{1}{c}{} & & \multicolumn{1}{c}{$\underset{\left[ 1.008;0.999\right] }{12\text{ Jan}}$} \\
Midlands & & \multicolumn{1}{c}{19/3/20} & \multicolumn{1}{c}{$317$} &
& & \multicolumn{1}{c}{$\underset{\left[ 1.013;0.992\right] }{05\text{ Apr}}
$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 0.994;1.005\right] }{12\text{ Aug}}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 1.011;1.000\right] }{14\text{ Nov}}$} & &
\multicolumn{1}{c}{$\underset{\left[ 0.997;1.001\right] }{12\text{ Dec}}$} &
& \multicolumn{1}{c}{$\underset{\left[ 1.03;1.000\right] }{13\text{ Jan}}$}
\\
North East & & \multicolumn{1}{c}{19/3/20} & \multicolumn{1}{c}{$317$} &
& & \multicolumn{1}{c}{$\underset{\left[ 1.018;0.992\right] }{07\text{ Apr}}
$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 0.995;1.005\right] }{19\text{ Aug}}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 1.013;1.000\right] }{13\text{ Nov}}$} & &
\multicolumn{1}{c}{$\underset{\left[ 0.996;1.000\right] }{13\text{ Dec}}$} &
& \multicolumn{1}{c}{$\underset{\left[ 1.002;1.000\right] }{12\text{ Jan}}$}
\\
North West & & \multicolumn{1}{c}{19/3/20} & \multicolumn{1}{c}{$317$} &
& & \multicolumn{1}{c}{$\underset{\left[ 1.023;0.993\right] }{09\text{ Apr}}
$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 1.013;1.000\right] }{18\text{ Aug}}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 1.015,1.000\right] }{29\text{ Oct}}$} & &
\multicolumn{1}{c}{$\underset{\left[ 0.997;1.001\right] }{13\text{ Dec}}$} &
& \multicolumn{1}{c}{$\underset{\left[ 1.003;1.001\right] }{12\text{ Jan}}$}
\\
South East & & \multicolumn{1}{c}{19/3/20} & \multicolumn{1}{c}{$317$} &
& & \multicolumn{1}{c}{$\underset{\left[ 1.016;0.992\right] }{22\text{ Apr}}
$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 0.993;1.007\right] }{21\text{ Aug}}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} &
& \multicolumn{1}{c}{} & & \multicolumn{1}{c}{$\underset{\left[ 1.009;1.000\right] }{07\text{ Jan}}$} \\
South West & & \multicolumn{1}{c}{19/3/20} & \multicolumn{1}{c}{$317$} &
& & \multicolumn{1}{c}{$\underset{\left[ 1.013;0.989\right] }{12\text{ Apr}}
$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 0.992;1.010\right] }{01\text{ Sep}}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 1.027;1.002\right] }{13\text{ Nov}}$} & &
\multicolumn{1}{c}{} & & \multicolumn{1}{c}{$\underset{\left[ 1.002;1.000\right] }{21\text{ Jan}}$} \\
& & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & & & \multicolumn{1}{c}{}
& \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{} & & \multicolumn{1}{c}{} & & \multicolumn{1}{c}{} \\
\hline
& & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & & & & & & &
\multicolumn{1}{c}{} & & \multicolumn{1}{c}{} & & \\
England & & \multicolumn{1}{c}{19/3/20} & \multicolumn{1}{c}{$317$} & &
& \multicolumn{1}{c}{$\underset{\left[ 1.009;0.995\right] }{10\text{ Apr}}$}
& \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 0.996;1.004\right] }{26\text{ Aug}}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 1.010;1.001\right] }{29\text{ Oct}}$} & &
\multicolumn{1}{c}{} & & \multicolumn{1}{c}{$\underset{\left[ 1.002;1.000\right] }{12\text{ Jan}}$} \\
Northern Ireland & & \multicolumn{1}{c}{02/3/20} & \multicolumn{1}{c}{$325$} & & & \multicolumn{1}{c}{$\underset{\left[ 1.071;1.000\right] }{04\text{ Apr}}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[
0.984;1.008\right] }{23\text{ Jul}}$} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{} & & \multicolumn{1}{c}{} & & \multicolumn{1}{c}{$\underset{\left[ 1.009;1.000\right] }{12\text{ Jan}}$} \\
Scotland & & \multicolumn{1}{c}{01/3/20} & \multicolumn{1}{c}{$326$} &
& & \multicolumn{1}{c}{$\underset{\left[ 1.026;0.999\right] }{04\text{ Apr}}
$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 0.984;1.007\right] }{12\text{ Aug}}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} &
& \multicolumn{1}{c}{} & & \multicolumn{1}{c}{$\underset{\left[ 1.009;1.001\right] }{12\text{ Jan}}$} \\
Wales & & \multicolumn{1}{c}{23/3/20} & \multicolumn{1}{c}{$313$} & &
& \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$\underset{\left[ 0.998;1.000\right] }{21\text{ Aug}}$} & \multicolumn{1}{c}{} & & &
& & \multicolumn{1}{c}{} \\
& & & & & & & & & & & & & & \\ \hline\hline
\end{tabular}
}
\par
\begin{tablenotes}
\tiny
\item All series end at 30 January 2021. We use the logs of the original data (plus one, given that, in some days, hospitalisations are equal to zero): no further transformations are used.
\item All changepoints have been detected by all R\'{e}nyi-type tests - no discrepancies were noted. Detected changepoints, and their estimated date, are presented in \textit{chronological} order; breakdates have been estimated as the points in time where the majority of tests identifies a changepoint. Whilst details are available upon request, we note that breaks were detected with this order (from the first to be detected to the last one): breaks in August; breaks in April and January; breaks in October-November; breaks in December.
\item For each changepoint, we report in square brackets, for reference, the left and right WLS estimates of $\beta_0$.
\end{tablenotes}
\end{table*}
The results in Table \ref{tab:TabCovid} suggest that, with the exception of
Wales, there were multiple breaks in all series considered\footnote{
Figure \ref{fig:FigCovid} contains the same information, albeit limited to
the four UK nations only to save space}; we note that Wales is an outlier as
regards hospital admissions, because these are counted in a different way
than the rest of the UK\footnote{
Specifically, Wales reports also \textit{suspected} Covid-19 cases, whereas
all the other nations only report \textit{confirmed} cases; see
https://www.cebm.net/covid-19/the-flaw-in-the-reporting-of-welsh-data-on-covid-hospital-admissions/
}.
\begin{figure}[!b]
\caption{Daily Covid-19 hospitalisations for the four UK nations }
\label{fig:FigCovid}\centering
\hspace{-2.5cm}
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[scale=0.4]{england_fig}
\label{fig:t1}
\end{minipage}
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[scale=0.4]{wales_fig}
\label{fig:t2}
\end{minipage} \\[0.25cm]
\par
\hspace{-2.5cm}
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[scale=0.4]{scotland_fig}
\label{fig:t3}
\end{minipage}
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[scale=0.4]{ni_fig}
\label{fig:t4}
\end{minipage} \\[0.25cm]
\par
\end{figure}
Some breaks occur closely to the sample endpoints, highlighting the
importance of using R\'{e}nyi statistics. Also, all changepoints indicate a
transition of the autoregressive coefficient $\beta _{0}$ around unity.
Differences between pre- and post-break values of $\beta _{0}$ are small,
but sufficient to trigger, or quench, an outbreak - on account of the Monte
Carlo evidence contained in Figures \ref{fig:bubb1}-\ref{fig:bubb2}, we
would not expect spurious detection of breaks when these are absent.
Considering first regions of England, all of these experience a break in
early April as a consequence of the first national lockdown, which started
on March 23rd, 2020, but was preceded by growing concerns, and closures in
the education and hospitality sectors, the week before. Similarly, all
series have a subsequent change (with $\beta _{0}$ exceeding unity after the
breaks) in late August - one exception is London, where the change occurred
in early August. These breaks indicate the beginning of the
\textquotedblleft second wave\textquotedblright\ in the UK, which has been
ascribed (also) to an increase in travelling during the holiday season and
which was officially acknowledge by the PM\ on September 18th, 2020. The
breaks in autumn, where present, can be explained as the effect of the local
and national lockdowns which were implemented at the end of October, and of
the easing of restrictions in early December. Finally, all series have a
change towards stationarity around mid-January, which again can be explained
as the effect of the national lockdown announced on January 4th, 2021, and
of the growing concerns about a third wave voiced before and during the
Christmas holidays.
The same picture applies to England as a whole. Conversely, the other UK\
nations experienced slightly different patterns, likely as a consequence of
different policies implemented by local governments. With the exception of
Wales, which seems to have only one break (but note the caveat about Welsh
data mentioned above), Scotland and Northern Ireland are essentially aligned
with the results for England in terms of the effects of the first lockdown,
the summer holiday, and the third lockdown.
\section{Discussion and conclusions\label{conclusions}}
In this paper, we study changepoint detection in the deterministic part of
the autoregression coefficient of a Random Coefficient AutoRegressive model.
We use the CUSUM\ process based on comparing the left and right WLS
estimators. In order to be able to detect changepoints close to the sample
endpoints, we study \textit{weighted} statistics, where more weight is
placed at the sample endpoints. We consider a very wide class of weighing
functions, studying: \textit{(i)} weighing schemes based on the functions $
w\left( t\right) $, which drift to zero, at sample endpoints, more slowly
than $\left( t\left( 1-t\right) \right) ^{1/2}$; \textit{(ii)} standardised
statistics, with weighting $\left( t\left( 1-t\right) \right) ^{1/2}$; and
\textit{(iii)} R\'{e}nyi statistics, where heavier weights are used. The
last class of statistics is still not fully studied (with the notable
exception of \citealp{horvathmiller}), and looks extremely promising in the
detection of early or late breaks.
From a practical point of view, our tests can be applied in the presence of
heteroskedasticity (requiring no knowledge as to the actual presence, or the
form, thereof), and simulations show that our procedures work very well in
practice - indeed, they work even better than procedures based on asymptotic
critical values in the baseline case of homoskedasticity. Technically, all
our results are based on a (strong) approximation of the weighted maximum of
partial sums. We have developed these both in the stationary and in the
nonstationary case: in the latter case (nonstationary data), our
approximations are entirely novel, and yield the same results as in the
stationary case. This, too, has important practical implications: our tests
can be applied with no prior knowledge as to the stationarity or lack
thereof of the data. This robustness reinforces the case made by
\citet{aue2011} for RCA models, where the authors advocate the use of these
models as an alternative to the AR(1) model, which does not possess the same
property and may require differencing, with the well-known problems attached
to this transformation (\citealp{leybourne1996}). Hence, our procedures lend
themselves to several interesting applications and extensions. As a leading
example, our theory could be used as the building block to develop
procedures for the sequential detection of bubbles starting or collapsing.
This, and other extensions, are under investigation by the authors.
{\footnotesize {\
\bibliographystyle{chicago}
\bibliography{LTbiblio}
} }\newpage