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Change point detection in random coefficient autoregressive models
\address{Lajos Horv\'ath, Department of Mathematics, University of Utah, Salt Lake City, UT 84112--0090 USA} \address{Lorenzo Trapani, School of Economics, University of Nottingham, University Park, Nottingham NG7 2RD U.K.} \subjclass{Primary 62M10; Secondary 62G20}
In this paper we study the stability of the autoregressive parameter of an RCA(1) sequence:
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.
The RCA model was firstly studied by andel and nichollsquinn . It belongs in the wider class of nonlinear models for time series (see 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\ ( akharif2003). Arguably, ((ref)) 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 lieberman2012; leybourne1996); gky argue that a time-varying parameter model like ((ref)) can be viewed as a competitor for a model with an abrupt break in the autoregressive root. Furthermore, equation ((ref) ) also allows for the possibility of (conditional) heteroskedasticity in $ y_{i}$; tsay1987 shows that the widely popular ARCH\ model by engle1982 can be cast into ((ref)), which therefore can be viewed as a second-order equivalent. Finally, a major advantage of ((ref)) 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 leybourne1996).\newline
Given such generality and flexibility, ((ref)) has been used in many applied sciences, including biology (stenseth), medicine ( fryz), and physics (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 regis for a comprehensive review.
The inferential theory for ((ref)) has been studied extensively. schick1996, koul1996 and praskova2004 study Weighted Least Squares (WLS)\ estimation of $\beta _{0}$; berkes2009 and aue2011 study Quasi Maximum Likelihood estimation, and hillpeng2016 develop an Empirical Likelihood estimator. Several tests have also been developed, including tests for stationarity (see e.g. zhao2012; and trapanistrict) and for the randomness of the autoregressive coefficient (akharif2003; nagakura2009 ; and HT16).
In contrast, changepoint detection is still underexplored in the RCA framework. To the best of our knowledge, the only exceptions are lee1998, lee2003cusum and 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 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 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 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, 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.
Contribution of this paper
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 ( 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 harvey2016): with our set-up, it is possible to detect changes from stationary to nonstationary/explosive behaviour (as e.g. in horvath2020sequential; and 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 towards an explosive behaviour have been developed in the literature (see, inter alia, phillips2011 ; phillips2015testing; and the review by homm2012testing ), tests to detect changes 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.
\newline
The remainder of the paper is organised as follows. We present our test statistics in Section (ref), and study their asymptotics in the homoskedastic case, as a benchmark, in Section (ref). The heteroskedastic case is studied in Section (ref). In Section (ref), we report a simulation exercise; applications to real data are in Section (ref). Section (ref) 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 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.
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 (i) avoiding restrictions on the moments of the observations, and (ii) ensuring standard normal asymptotics irrespective of whether $ y_{i}$ is stationary or not. The WLS estimators are
and
Our test statistics will be functionals of the process
A \textquotedblleft natural\textquotedblright\ choice to detect the presence of a possible change is to use the sup-norm of ((ref)), 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:
The functions $w(t)$ satisfying Assumption (ref) 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) ) can be determined based on the finiteness of the integral functional (see csorgo1993)
As we show below, ((ref)) 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.
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 ( darling1956limit); when $\kappa >\frac{1}{2}$, the test statistics defined in ((ref)nyi}) are known as \textquotedblleft R\'{e}nyi statistics\textquotedblright\ (horvathmiller).
We begin by assuming that the errors $\{\epsilon _{i,1},\epsilon _{i,2},-\infty <i<\infty \}$ have constant variance.
In ((ref)), 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 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 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
We require the following notation. In order to study the case $\kappa =\frac{ 1}{2}$, we define
Also, in order to study the case $\kappa >\frac{1}{2}$, let
We start with the stationary case $-\infty \leq E\ln |\beta _{0}+\epsilon _{0,1}|<0$. In this case, the solution of ((ref)) under the null hypothesis is close to $\overline{y}_{i}$, the unique anticipative stationary solution of
We need the following (technical) assumption, to rule out the degenerate case that, under stationarity, the denominator of $\eta ^{2}$ defined in ( (ref)) is zero with probability $1$.
We now turn to the nonstationary case. We need an additional technical condition:
Theorems (ref) and (ref) 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
We use the following estimator for $\eta ^{2}$
Corollary (ref) 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.
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 xu2015, gorecki2017 bardsley2017 and horvath2021 (see also 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.
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.
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
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)-(ref) with
By Assumption (ref), the WLS estimator may have different variances in the various regimes. In order to study the limit theory, consider the following notation:
and
$1\leq \ell \leq M+1$. Also, let
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) and (ref) behave under heteroskedasticity.
Theorem (ref) is only of theoretical interest, but we point out that heteroskedasticity impacts only on part (i). In that case, the limiting distribution of the weighted $Q_{N}(t)$ is given by a Gaussian process with covariance kernel
Parts (ii)-(iii) of the theorem are the same as in the case of homoskedasticity.
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)) 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.
By Theorem (ref), 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
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
Under the null of no change, the same arguments as in the proof of Corollary (ref) guarantee that $\widehat{\mathfrak{c}}_{N,1}(t)$ and $\widehat{ \mathfrak{c}}_{N,2}(t)$ converge to the functions
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) , and $a_{\ell },1\leq \ell \leq M+1$ is defined in ((ref)). In order to present our main results, we define the zero mean Gaussian process
$\{\Delta (t),0\leq t\leq 1\}$ is also a zero mean Gaussian process with $ E\Delta (t)\Delta (s)=\mathfrak{b}(\min (t,s))$, where
Some comments on the practical implementation of the results in Theorem (ref) are in order. Parts (ii) and (iii) require an estimate of $\mathfrak{g}(t,t)$; to this end, we use $\widehat{\mathfrak{c}} _{N,1}(t)$ defined in ((ref)) instead of $\mathfrak{c}_{1}(t)$, and we estimate ${\mathfrak{b}}(t,s)$ as
Then we can define
The implementation of part (i) of Theorem (ref) is more complicated, since the presence of nuisance parameters is not relegated to a multiplicative function. We reject the null hypothesis in ((ref)) if
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
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 $c_{N,L}(\alpha )=\inf \{x:\;{\mathcal{F }}_{N,L}(x)\geq 1-\alpha \}$.
We study the consistency of our tests versus the AMOC alternative\footnote{ In Section (ref) in the Supplement, we also discuss the power of our tests versus the alternative of multiple breaks.}
Let $\Delta _{N}=\left\vert \beta _{0}-\beta _{A}\right\vert $, and define $ t^{\ast }$ as $\left\lfloor Nt^{\ast }\right\rfloor =k^{\ast }$.
The theorem ensures that, as long as ((ref)), ((ref)) and ( (ref)) hold, our tests reject the null with probability (asymptotically) $1$. Conditions ((ref)), ((ref)) and ((ref)) 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)). 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)) 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)), 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)) 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)), the only requirement is that $k^{\ast }>r_{N}$.
We provide some Monte Carlo evidence on the performance of the test statistics proposed in Section (ref).\footnote{ In Section (ref) in the Supplement, we also evaluate, as a benchmark, the performance of our tests under homoskedasticity, based on the theory in Section (ref).}
Data are generated using ((ref)). 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
The shocks $\epsilon _{i,1}$ and $\epsilon _{i,2}$ are simulated as independent of one another and 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 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 i.i.d.$N\left( 0,\sigma _{1}^{2}\right) $ and i.i.d.$N\left( 0,\sigma _{2}^{2}\right) $ for $1\leq i\leq N/2$, and i.i.d.$N\left( 0,1.5\sigma _{1}^{2}\right) $ and 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)) - 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: (i) homoskedasticity in both $ \epsilon _{i,1}$ and $\epsilon _{i,2}$; (ii) homoskedasticity in $ \epsilon _{i,1}$ and heteroskedasticity in $\epsilon _{i,2}$; (iii) homoskedasticity in $\epsilon _{i,2}$ and heteroskedasticity in $\epsilon _{i,1}$; and, finally, (iv) heteroskedasticity in both $\epsilon _{i,1}$ and $\epsilon _{i,2}$.
Empirical rejection frequencies under the null are in Tables (ref) and (ref). We have used asymptotic critical values for R\'{e}nyi statistics, and the method described in Section (ref) 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 gombay. From Tables (ref) and (ref), 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.
The empirical power of the tests is reported in Figures (ref)-(ref), 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), 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)-(ref), compared with Figures (ref)-(ref), 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)-(ref), 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)). Results are in Figures (ref)-(ref). 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, 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 $10-15\%$.
Finally, in Figures (ref)-(ref) 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)-(ref) 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.
We illustrate our approach through three applications to real data. In Sections (ref)-(ref), we use economic and financial time series; in Section (ref), we use Covid-19 data.
In his landmark paper, 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, 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 vostrikova), reporting the point in time at which the relevant test statistic is maximised as the breakdate estimate.
Results in Table XXX differ across tests only marginally, and suggest the presence of several changepoints in the autoregressive coefficient - see also Figure (ref) 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), 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 (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.
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 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.
We found three changepoints in the whole series (see Table (ref) ). 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.
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. shtatland 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)-(ref).
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 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).
The results in Table (ref) suggest that, with the exception of Wales, there were multiple breaks in all series considered\footnote{ Figure (ref) 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 suspected Covid-19 cases, whereas all the other nations only report confirmed cases; see https://www.cebm.net/covid-19/the-flaw-in-the-reporting-of-welsh-data-on-covid-hospital-admissions/ }.
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)-(ref), 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.
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 weighted statistics, where more weight is placed at the sample endpoints. We consider a very wide class of weighing functions, studying: (i) weighing schemes based on the functions $ w\left( t\right) $, which drift to zero, at sample endpoints, more slowly than $\left( t\left( 1-t\right) \right) ^{1/2}$; (ii) standardised statistics, with weighting $\left( t\left( 1-t\right) \right) ^{1/2}$; and (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 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 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 (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.
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