Real-time monitoring with RCA models

Real-time monitoring with RCA models

\email{[email removed]} \email{[email removed]} \address{$^1$Department of Mathematics, University of Utah, Salt Lake City, UT 84112--0090 USA } \address{$^2$School of Business and Economics, University of Leicester, Leicester, U.K.; Department of Economics and Management, University of Pavia, Pavia, Italy} \subjclass[2020]{60F17}

\doublespacing

Introduction

Economic history is full of events where stationary observations suddenly become explosively increasing, or - vice versa - where explosive trends crash. A classical example are financial bubbles, where periods of \textquotedblleft tame\textquotedblright\ fluctuations are followed by exuberant growth, in turn then followed by\ a collapse which is typically modelled as a stationary regime (see e.g. harvey2017improving; and phillips2018financial). Historical examples include the \textquotedblleft Tulipmania\textquotedblright\ in the Netherlands in the 1630's and the \textquotedblleft South Sea\textquotedblright\ bubble of 1720. More recent instances include the Japan's real estate and stock market bubble of the 1980's, which after the collapse turned into a negative period, the so called \textquotedblleft Lost Decade\textquotedblright ; and, in the US, the \textquotedblleft Dotcom\textquotedblright\ bubble of the 1990's and the housing bubble between 1996 and 2006. Similar phenomena, with observations undergoing a change in persistence or even more radical changes in their nature (from stationary to explosive and vice versa), are also encountered in applied sciences. For example, in epidemiology, the onset of a pandemic is characterised by a sudden, explosive growth in the number of cases, followed by a return to a stationary regime when the pandemic subsides. Hence the importance of developing tools allowing to test for changes in persistence, or a switch between regimes.

In particular, a research area where this issue has been paid particular attention is financial econometrics, where several methodologies have been developed to tests (ex-ante or ex-post) for the onset or collapse of a financial bubble. In the interest of a concise discussion, we refer the reader to the seminal articles on ex-post detection by phillips2011, and phillips2015testing2, and also to skrobotov2023testing for a review. In addition, several contributions deal with the real-time detection of bubble episodes - see, inter alia, homm2012testing, phillips2015testing, and a recent contribution by whitehouse2023real. A common trait to this literature is that it relies on an AutoRegressive (AR) framework as the workhorse model. Using an AR specification has several advantages: it is a parsimonious and well-studied set-up, and it naturally lends itself to modelling both stationary and nonstationary regimes. phillips2011dating show that an AR model with explosiveness is an adequate representation for bubble behaviour under very general assumptions. Moreover, an AR specification lends itself to constructing tests for bubble behaviour using the Dickey-Fuller test and its variants- e.g., the Augmented DF test (ADF; see diba1988theory), the sup-ADF test ( phillips2011; phillips2011dating), and the generalized sup-ADF test (phillips2015testing2; phillips2015testing ). From a technical viewpoint, however, using an AR model is fraught with difficulties when monitoring for changes from an explosive to a stationary regime, although several promising solutions have been proposed such as the reverse regression approach by phillips2018financial. In addition to the ability of detecting changes, ensuring a timely detection is important. To the best of our knowledge, no optimality results exist; indeed, typical procedures are based on unweighted CUSUM (or related) statistics, which are not designed to ensure optimal detection timings (see e.g. aue2004delay).

Main contributions of this paper

We address the general issue of online, real-time detection of changes in persistence and also between regimes (i.e., from stationarity to nonstationarity and vice versa) by using a Random Coefficient Autoregressive (RCA) model, viz.

where $y_{0}$ denotes an initial value. 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). The RCA\ model is particularly well-suited to modelling time series with potentially explosive behaviour; for example, in the seminal paper by diba1988theory, a rational bubble is modelled as having an RCA dynamics (see their equation (17)). Model ((ref)) also nests, as a special case corresponding to having ${\Greekmath 010C} _{i}=1$, the so-called Stochastic Unit Root model (STUR; see granger1997introduction), and it 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 the RCA setup, which therefore can be viewed as a second-order equivalent. Finally, a major advantage of ((ref)) compared to standard AR models is that it is possible to construct estimators of ${\Greekmath 010C} _{0}$ that are always asymptotically normal, irrespective of whether $y_{i}$ is stationary or nonstationary (aue2011). Although the literature has developed several contributions on ex-post changepoint detection using an RCA model (e.g. horvath2022changepoint), to the best of our knowledge there are still significant limitations in the context of online detection - e.g., na2010, li2015 and li2015b all consider sequential changepoint detection using unweighted CUSUM-based statistics, but only under the maintained assumption of stationarity, which precludes the ability to detect changes from stationary to exponential behaviour, and vice versa.

In this paper, we fill the gaps discussed above, by proposing a family of statistics based on the Weighted Least Squares (WLS) estimator, and in particular on the weighted CUSUM\ process of the WLS residuals, designed to ensure a faster detection than unweighted statistics. We study the standard\ CUSUM process, and the so-called \textquotedblleft Page-CUSUM\textquotedblright\ (fremdt2015page); and both \textquotedblleft open-ended\textquotedblright\ and \textquotedblleft closed-ended\textquotedblright\ procedures, where sequential monitoring goes on indefinitely, or stops at a pre-specified point in time, respectively. We make at least five contributions to the current literature on online changepoint detection. First, we derive the limiting distribution of our statistics under the closed-ended case; typically, in the literature, the critical values obtained under the open-ended case are employed even for the closed-ended case, as a (conservative) choice, ultimately leading to loss of power. Second, we study the case of a closed-ended procedure with a \textquotedblleft very short\textquotedblright\ monitoring horizon, adapting the boundary function for this case, and deriving the corresponding limiting distribution; this case has not been considered in the literature before, but it is of pratical relevance because the researcher may prefer to carry out monitoring for a short time, and then - in the absence of changepoints - restart the procedure afresh. Third, in the case of the standardised CUSUM,\footnote{ We define \textquotedblleft standardised CUSUM\textquotedblright\ the CUSUM\ process weighted using weights proportional to its standard deviation, as opposed to the \textquotedblleft weighted CUSUM\textquotedblright\ which uses lighter weights - see also below for a formal definition.} we propose an approximation to compute critical values which offers a superior alternative to asymptotic critical values, whose accuracy is marred by the notoriously weak convergence to the Extreme Value Distribution; simulations show that our approximation works extremely well even in small samples, and for all cases considered, offering size control and short detection delays. Fourth, we derive the limiting distribution of the weighted Page-CUSUM statistics, showing also that such a limit is a well-defined random variable. Fifth, as well as studying the basic RCA\ model, we also develop the full-blown theory (irrespective of whether $y_{i}$ is stationary or not) for the case where there are covariates in ((ref)); to the best of our knowledge, this is the first contribution to deal with this case. As a final remark, we would like to add that a major advantage of the RCA\ set-up is that our statistics can be employed under both stationarity and explosiveness with no modifications required; indeed, no prior knowledge on the stationarity or not of the observations is required. Hence, our methodology can be applied to detect changes in the persistence of a stationary series, or changes from stationarity to a non-stationary regime, or vice versa from a non-stationary regime to a stationary one. This has important practical consequences, allowing for e.g. faster responses from public health authorities in the presence of a pandemic, or from policy makers in the presence of inflationary shocks, or a more accurate date stamping of bubble onsets and collapses.

The remainder of the paper is organised as follows. We introduce our main model, and the test statistics, in Section (ref). The asymptotics under the null and the alternative is reported in Section (ref). We consider the extension of the RCA model to the case of exogenous regressors in Section (ref). Monte Carlo evidence is in Section (ref); Section (ref) contains two empirical applications, to Covid-19 hospitalisation data and house prices. Section (ref) concludes.

NOTATION. Henceforth, we use: \textquotedblleft $\rightarrow $ \textquotedblright\ for the ordinary limit; \textquotedblleft $\overset{ \mathcal{P}}{\rightarrow }$\textquotedblright\ for convergence in probability; \textquotedblleft a.s.\textquotedblright\ for \textquotedblleft almost surely\textquotedblright ; \textquotedblleft $\overset{a.s.}{ \rightarrow }$\textquotedblright\ for almost sure convergence; and \textquotedblleft $\left\Vert \cdot \right\Vert $\textquotedblright\ for the Euclidean norm of vectors and matrices. Finally, $\left\{ W\left( t\right) ,0\leq t\leq 1\right\} $ denotes a standard Wiener process. Other notation is introduced later on in the paper.

Sequential monitoring of RCA\ models

Recall the RCA model

It is well-known (aue2006) that, in ((ref)), the stationarity or lack thereof of $y_{i}$ is determined by the value of $E\log \left\vert {\Greekmath 010C} _{0}+{\Greekmath 010F} _{0,1}\right\vert $:

The following assumption must hold under all three cases above.

We further assume that the autoregressive parameter ${\Greekmath 010C} _{i}$ is constant over the training segment $\left\{ y_{i},1\leq i\leq m\right\} $

The requirement in Assumption (ref) is known as the non-contamination assumption (see CSW96). We subsequently test for the null hypothesis that, as new data come in after $m$, ${\Greekmath 010C} _{i} $ remains constant, viz.

The WLS\ estimator using the observations on the training segment $\left\{ y_{i},1\leq i\leq m\right\} $ is

Standard weighted CUSUM-based detectors

We define the CUSUM\ process of the WLS\ residuals as

Heuristically, under the null of no change, the residuals have zero mean; hence, the partial sum process $Z_{m}\left( k\right) $ - customarily known as the detector - should also fluctuate around zero with increasing variance. Conversely, in the presence of a break (at, say, $k^{\ast }$), $ \widehat{{\Greekmath 010C} }_{m}$ is a biased estimator for the \textquotedblleft new\textquotedblright\ autoregressive parameter ${\Greekmath 010C} _{m+k^{\ast }+1}$; thus, $Z_{m}\left( k\right) $ should have a drift term. Hence, a break is flagged if $Z_{m}\left( k\right) $ exceeds a threshold. We call such a threshold the boundary function, and propose the following family

In ((ref)), $0\leq {\Greekmath 0120} \leq 1/2$, and $\mathscr{s}$ is defined as

with ${\Greekmath 011B} _{1}^{2}$ and ${\Greekmath 011B} _{2}^{2}$ defined in Assumption (ref) , and

where recall that $\left\{ \overline{y}_{i},-\infty <i<\infty \right\} $ is the stationary solution of ((ref)).

On account of ((ref)) and ((ref)), a changepoint is found at a stopping time ${\Greekmath 011C} _{m,{\Greekmath 0120} }$ defined as

The constant $c_{{\Greekmath 010B} ,{\Greekmath 0120} }$ in ((ref)) is chosen so as to ensure that: (a) under the null, the procedure-wise probability of Type I\ Error does not exceed a user-chosen value ${\Greekmath 010B} $, viz. $ \lim_{m\rightarrow \infty }P\left\{ {\Greekmath 011C} _{m,{\Greekmath 0120} }=\right. $ $\left. \infty |H_{0}\right\} ={\Greekmath 010B} $; and (b) under the alternative, $\lim_{m\rightarrow \infty }P\left\{ {\Greekmath 011C} _{m,{\Greekmath 0120} }<\infty |H_{A}\right\} =1$.

In ((ref)), the monitoring goes on indefinitely (\textquotedblleft open-ended\textquotedblright ). On the other hand, in some applications it may be desirable to stop the sequential monitoring procedure after $m^{\ast } $ period (\textquotedblleft closed-ended\textquotedblright ), e.g. in order to extend or update the training period. In this case, we use the same detector $Z_{m}\left( k\right) $ defined in ((ref)), but we modify ((ref)) as

with boundary function

The boundary function $g_{m,{\Greekmath 0120} }^{\ast }\left( k\right) $ is suitable for the case where the monitoring horizon $m^{\ast }$ is \textquotedblleft long\textquotedblright , i.e. when it goes on for a period which is at least proportional (or more than proportional) to the size of the training sample $ m$. If a shorter monitoring horizon is considered, where $m^{\ast }=o\left( m\right) $, then we use the same detector as above, $Z_{m}\left( k\right) $, but the boundary function needs to be modified as

In this case, the stopping rule is defined as

Page-CUSUM detectors

In a series of recent contributions, fremdt2015page, kirch2022asymptotic and kirch2022sequential study a different family of detectors, known as the \textquotedblleft Page-CUSUM\textquotedblright\ processes, designed to offer a shorter detection delay:

Intuitively, this family of detectors searches for the \textquotedblleft worst-case scenario\textquotedblright\ at each point in time $k$, and therefore should guarantee faster detection in the presence of a changepoint. Consistently with the approach studied in this paper, we consider weighted versions of $Z_{m}^{\dagger }\left( k\right) $; the corresponding stopping rules are defined as:

with $g_{m,{\Greekmath 0120} }\left( k\right) $ defined in ((ref)), for an open-ended procedure (replacing the critical value $c_{{\Greekmath 010B} ,{\Greekmath 0120} }$ with $ c_{{\Greekmath 010B} ,{\Greekmath 0120} }^{\dagger }$);

with $g_{m,{\Greekmath 0120} }^{\ast }\left( k\right) $ defined in ((ref)), for the case of a closed-ended procedure with a long monitoring horizon (replacing the critical value $c_{{\Greekmath 010B} ,{\Greekmath 0120} }^{\ast }$ with $c_{{\Greekmath 010B} ,{\Greekmath 0120} }^{\ast \dagger }$); and

with $\overline{g}_{m,{\Greekmath 0120} }\left( k\right) $ defined in ((ref)), for the case of a closed-ended procedure with a short monitoring horizon (replacing the critical value $\overline{c}_{{\Greekmath 010B} ,{\Greekmath 0120} }$ with $\overline{ c}_{{\Greekmath 010B} ,{\Greekmath 0120} }^{\dagger }$).

Asymptotics

We begin by listing a set of technical assumptions, which complement Assumption (ref) and are required in the case of nonstationarity, $ E\log \left\vert {\Greekmath 010C} _{0}+{\Greekmath 010F} _{0,1}\right\vert \geq 0$.

Assumption (ref) is required specifically in order to derive the anti-concentration bound in ((ref)). Assumption (ref) ensures that $\left\vert y_{i}\right\vert \overset{a.s.}{\rightarrow }\infty $, ruling out that it could be identically equal to zero (see berkes2009).

Asymptotics under the null

We derive the weak limits of our test statistics. Results differ depending on the choice of the weight ${\Greekmath 0120} $, on whether an open-ended or a closed-ended monitoring scheme is used (and, in the latter case, on the length of the monitoring horizon), and on whether the detector is constructed using the standard CUSUM or the Page-CUSUM. In all cases, we derive nuisance free limiting distributions, from which critical values can be obtained by simulation for a given desired nominal significance level, $ {\Greekmath 010B} $.

Weighted standard CUSUM-based detectors

We begin by studying the standard CUSUM detectors defined in Section (ref), starting with the open-ended case, also studied in lajos04 in the context of a linear regression.

We now turn to the closed-ended case. We begin by considering the case of ( (ref)), where the monitoring goes on for a sufficient amount of time:

In this case, we use the boundary function $g_{m,{\Greekmath 0120} }^{\ast }\left( k\right) $\ defined in ((ref)), and the corresponding stopping rule ((ref)).\ Define $m_{\ast }=m_{0}/\left( 1+m_{0}\right) $ if $ m_{0}<\infty $, and $m_{\ast }=1$ if $m_{0}=\infty $.

We now study the case of ((ref)), in a closed-ended set-up where monitoring stops after very few steps, viz.

In this case, we use the boundary function $\overline{g}_{m,{\Greekmath 0120} }\left( k\right) $ defined in ((ref)), and the corresponding stopping rule defined in ((ref)).

We now turn to the case ${\Greekmath 0120} =1/2$, studying the closed-ended monitoring procedure under both ((ref)) and ((ref)). Define

and

Upon inspecting the proofs of Theorems (ref)-(ref) and (ref), the limiting distributions of the weighted CUSUM with ${\Greekmath 0120} <1/2$, and of the standardised CUSUM with ${\Greekmath 0120} =1/2$ are asymptotically independent,\footnote{ Intuitively, this is because $W\left( \cdot \right) $ has independent increments, and the limiting distribution, in the case of $0\leq {\Greekmath 0120} <1/2$, is determined by the \textquotedblleft central\textquotedblright\ values of $ W\left( \cdot \right) $; conversely, when ${\Greekmath 0120} =1/2$, the limiting law is determined by values at the very beginning of $W\left( \cdot \right) $.} and therefore the two procedures, in principle, can be combined.

Theorem (ref) offers an explicit formula to compute asymptotic critical values; however, these are bound to be inaccurate due to the slow convergence to the Extreme Value distribution. In particular, simulations show that, in finite samples, asymptotic critical values overstate the true values thus leading to low power. A possible correction can be proposed, similarly to gombay, as follows. Define $ h_{m^{\ast }}=h_{m^{\ast }}\left( m^{\ast }\right) $ such that, as $m^{\ast }\rightarrow \infty $

and let ${\Greekmath 011E} _{m}=\left( m^{\ast }+h_{m^{\ast }}\right) /\left( 2h_{m^{\ast }}\right) $. Let also $\widehat{c}_{{\Greekmath 010B} ,0.5}\rightarrow \infty $ denote the solution of

The choice of $h_{m^{\ast }}$\ is a matter of tuning; qualitatively, as $ h_{m^{\ast }}$ increases, critical values become smaller (thus making the procedure more conservative) and vice versa. Our simulations indicate that $ h_{m^{\ast }}=\left( \log m^{\ast }\right) ^{1/2}$ yields the best results in terms of size and power.

Weighted Page-CUSUM detectors

We now consider the use of the Page-CUSUM detector $Z_{m}^{\dagger }\left( k\right) $ defined in ((ref)), using the stopping rules defined in ((ref))-((ref)). Let $\left\{ W_{1}\left( x\right) ,x\geq 0\right\} $\ and $\left\{ W_{2}\left( x\right) ,x\geq 0\right\} $\ denote two independent standard Wiener processes.

LRV\ estimation

In order to apply all the results above, we require an estimate of $ \mathscr{s}^{2}$ defined in ((ref)). This can be constructed using the data in the training sample as

We note that $\widehat{\mathscr{s}}_{m}^{2}$ does not depend on ${\Greekmath 0120} $, so the result in Corollary (ref) can be applied for all $0\leq {\Greekmath 0120} \leq 1/2$, and for both open-ended and closed-ended procedures.

Asymptotics under the alternative

We consider the following alternative, where the deterministic part of the autoregressive coefficient of ((ref)) undergoes a change

where ${\Greekmath 010C} _{0}\neq {\Greekmath 010C} _{A}$ and $k^{\ast }$ is the time of change. In ( (ref)), we do not put any constraints on the values of ${\Greekmath 010C} _{0}$ and ${\Greekmath 010C} _{A}$. Hence, under ((ref)), the observations could transition from a stationary regime to another stationary regime (if both $E\log \left\vert {\Greekmath 010C} _{0}+{\Greekmath 010F} _{0,1}\right\vert <0$ and $E\log \left\vert {\Greekmath 010C} _{A}+{\Greekmath 010F} _{0,1}\right\vert <0$); from a nonstationary regime to another nonstationary regime (if both $E\log \left\vert {\Greekmath 010C} _{0}+{\Greekmath 010F} _{0,1}\right\vert \geq 0$ and $E\log \left\vert {\Greekmath 010C} _{A}+{\Greekmath 010F} _{0,1}\right\vert \geq 0$); or either regime can be stationary and the other one nonstationary (which arises if $E\log \left\vert {\Greekmath 010C} _{0}+{\Greekmath 010F} _{0,1}\right\vert <0$ and $E\log \left\vert {\Greekmath 010C} _{A}+{\Greekmath 010F} _{0,1}\right\vert \geq 0$, implying a switch from stationarity to nonstationarity, or if $E\log \left\vert {\Greekmath 010C} _{0}+{\Greekmath 010F} _{0,1}\right\vert \geq 0$ and $E\log \left\vert {\Greekmath 010C} _{A}+{\Greekmath 010F} _{0,1}\right\vert <0$, implying a switch from nonstationarity to stationarity). We entertain the possibility that the amplitude of change may depend on the (training) sample size $m$, thus defining

Condition ((ref)) entails that $\left\vert \Delta _{m}\right\vert $ can drift to zero, but not too fast (i.e., power is ensured as long as breaks are \textquotedblleft not too small\textquotedblright ), for all ${\Greekmath 0120} <1/2$. When using ${\Greekmath 0120} =1/2$, condition ((ref)) suggests that there is a (minor) loss of power. A straightforward extension of the proofs in aue2004delay and aue2008monitoring would yield that the detection delay is proportional to $m^{\left( 1-2{\Greekmath 0120} \right) /\left( 2\left( 1-{\Greekmath 0120} \right) \right) }$ when ${\Greekmath 0120} <1/2$, and to $\log \log m$ when ${\Greekmath 0120} =1/2$, thus indicating that heavier weights ensure faster detection.

Monitoring the RCA\ model with covariates

In a recent contribution, astill2023using argue in favour of adding covariates to the basic AR specification, showing theoretically and empirically that this results in better (and quicker) detection of bubble episodes. Hence, we modify ((ref)) as

where $y_{0}$ is an initial value and $\mathbf{x}_{i}\in \mathbb{R}^{p}$. Equation ((ref)) is, essentially, a dynamic model with exogenous covariates, with $\mathbf{{\Greekmath 0115} }_{0}$ constant over time.

In order to monitor for the stability of the autoregressive coefficient, we propose again a detector based on the WLS loss function

The estimators of ${\Greekmath 010C} _{0}$ and $\mathbf{{\Greekmath 0115} }_{0}$ are defined as $ \left( \widehat{{\Greekmath 010C} }_{m},\widehat{\mathbf{{\Greekmath 0115} }}_{m}\right) =\arg \min_{{\Greekmath 010C} ,\mathbf{{\Greekmath 0115} }}\mathcal{G}_{m}\left( {\Greekmath 010C} ,\mathbf{{\Greekmath 0115} } \right) $, and satisfy

which suggests the following detector

The following assumptions complement Assumptions (ref)-(ref).

Assumption (ref) states that the regressors $\mathbf{x}_{i}$ form a decomposable Bernoulli shift - i.e., a weakly dependent, stationary process which can be well-approximated by an $m$-dependent sequence. The concepts of Bernoulli shift and decomposability appeared first in ibragimov1962some; see also wu2005 and berkeshormann . Bernoulli shifts have proven a convenient way to model dependent time series, mainly due to their generality and to the fact that they are much easier to verify than e.g. mixing conditions: aue09 and linliu, inter alia, provide numerous examples of such DGPs, which include ARMA models, ARCH/GARCH sequences, and other nonlinear time series models (such as e.g. random coefficient autoregressive models and threshold models). Indeed, under stationarity, $y_{i}$ itself can be approximated by a decomposable Bernoulli shift ( horvath2022changepoint). Assumption (ref) states that the exogenous variables $\mathbf{x}_{i}$ are independent of the innovations $ {\Greekmath 010F} _{i,1}$ and ${\Greekmath 010F} _{i,2}$, and Assumption (ref) is similar to Assumption (ref), ensuring $\left\vert y_{i}\right\vert \overset{a.s.}{\rightarrow }\infty $.

Finally, we note that, in the presence of covariates, we need to exclude the boundary case $E\log \left\vert {\Greekmath 010C} _{0}+{\Greekmath 010F} _{0,1}\right\vert =0$; this is because we would need an exact (and large enough) rate of divergence for $\left\vert y_{i}\right\vert $ as $i\rightarrow \infty $, but this result is not available in the case $E\log |{\Greekmath 010C} _{0}+{\Greekmath 010F} _{0,1}|=0$ (see also Theorem 4 in HT2019, and the discussion thereafter). \newline The boundary function is defined as

where $c_{{\Greekmath 010B} ,{\Greekmath 0120} }^{\left( x\right) }$ is a critical value, and

with

where $\mathbf{a}$, $\mathbf{Q}$ and $\mathbf{C}$ are defined in ((ref))-( (ref)) in the Supplement. The stopping rule is

and

for an open-ended and a closed-ended monitoring procedure respectively.

Along the same lines as in Section (ref), it is possible to construct weighted monitoring schemes based on the detector

In all the results above, the limiting behaviour of the stopping time is the same as in the absence of covariates; however, this does not mean that these do not play a role, since they build into the recursion that defines $y_{i}$ . As far as Theorem (ref) is concerned, the same approximation for critical values as in ((ref)) can be used.

Under the alternative that the deterministic part of the autoregressive root changes,

the same results as in Section (ref) hold, as summarised in the following theorem.

Simulations

We provide some Monte Carlo evidence and guidelines on implementation; further details and results are reported in Section (ref) in the Supplement. We consider the following Data Generating Process, based on ((ref))

for $1\leq i\leq m+1,000$, where we simulate ${\Greekmath 010F} _{i,1}$ and ${\Greekmath 010F} _{i,2}$ as independent of one another and i.i.d. with distributions $N\left( 0,{\Greekmath 011B} _{1}^{2}\right) $ and $N\left( 0,{\Greekmath 011B} _{2}^{2}\right) $ respectively, discarding the first $1,000$ observations to avoid dependence on initial conditions. We used ${\Greekmath 011B} _{1}^{2}=0.01$ in all experiments, and we have considered three cases: in Case I, we set ${\Greekmath 010C} _{0}=0.5$, with $E\log \left\vert {\Greekmath 010C} _{0}+{\Greekmath 010F} _{i,1}\right\vert =-0.717$, corresponding to a stationary regime; in Case II, we set $ {\Greekmath 010C} _{0}=1.05$, with $E\log \left\vert {\Greekmath 010C} _{0}+{\Greekmath 010F} _{i,1}\right\vert =0.044$, corresponding to a mildly explosive regime; and in Case III, we set ${\Greekmath 010C} _{0}=1$, with $E\log \left\vert {\Greekmath 010C} _{0}+{\Greekmath 010F} _{i,1}\right\vert =-0.007$, indicating a stationary, but \textquotedblleft nearly non-stationary\textquotedblright\ process (this case corresponds to the STUR model). The variance of the idiosyncratic shock is ${\Greekmath 011B} _{2}^{2}=0.5$ in Case I, and ${\Greekmath 011B} _{2}^{2}=0.1$ in Cases II and III; in unreported simulations, using different values does not result in any significant changes, save for the (expected) fact that tests have better properties (in terms of size and power) for smaller values of ${\Greekmath 011B} _{2}^{2}$. When covariates are used, we set $ {\Greekmath 0115} _{0}=1$ and generate $x_{i}$ as i.i.d.$N\left( 0,1\right) $ . Critical values (for a nominal level equal to $5\%$) are computed using the results in Theorem (ref) when using the weighted CUSUM with ${\Greekmath 0120} <1/2$, and Theorem (ref)(ii) when using Page-CUSUM statistics. When using the weighted CUSUM with ${\Greekmath 0120} =1/2$, we use both the asymptotic critical values $c_{{\Greekmath 010B} ,0.5}$\ defined in ((ref)), and the approximation $\widehat{c}_{{\Greekmath 010B} ,0.5}$\ defined as the solution of ((ref)), with $h_{m^{\ast }}=\left( \log m^{\ast }\right) ^{1/2}$ . Results are based on $1,000$ replications.

Empirical rejection frequencies are in Tables (ref)-(ref), where we consider the case where, in ((ref)), ${\Greekmath 0115} _{0}=0$ - i.e., a \textquotedblleft pure\textquotedblright\ RCA without covariates. In the case of stationarity (Table (ref)), the procedure-wise probability of Type I Errors is always controlled when using ${\Greekmath 0120} =1/2$; the asymptotic critical values $c_{{\Greekmath 010B} ,0.5}$\ defined in ((ref)) lead to under-rejection, as can be expected in light of the slow convergence to the asymptotic distribution; conversely, their approximation using $\widehat{c} _{{\Greekmath 010B} ,0.5}$\ defined as the solution of ((ref)) achieves size control in all cases considered. When using ${\Greekmath 0120} <1/2$, both in the case of the CUSUM and the Page-CUSUM, the procedure-wise probability of Type I Errors is also controlled, but larger sample sizes $m$ are required. In the explosive case (Table (ref)), our procedures tend to under-reject whenever using ${\Greekmath 0120} =1/2$ (both with $c_{{\Greekmath 010B} ,0.5}$\ and $\widehat{c} _{{\Greekmath 010B} ,0.5}$), although this seems to slowly improve as $m$ increases, for all lengths of the monitoring horizon $m^{\ast }$; results are less conservative when using ${\Greekmath 0120} <1/2$, again both in the case of the CUSUM and the Page-CUSUM. On the other hand, in the STUR case (Table (ref)), empirical rejection frequencies tend to be higher than in the other cases. The procedure-wise probability of Type I Errors is controlled in all cases when using ${\Greekmath 0120} =1/2$ and $c_{{\Greekmath 010B} ,0.5}$, whereas using $\widehat{c} _{{\Greekmath 010B} ,0.5}$\ requires either large $m$ ($\geq 200$), or not overly long monitoring horizons when $m<200$ (in those cases, using $m^{\ast }\leq m$ always results in size control). When ${\Greekmath 0120} <1/2$, size control is more problematic and the procedure tends to be oversized, unless $m^{\ast }$ is \textquotedblleft small\textquotedblright\ compared to $m$. Whilst results are broadly similar for the CUSUM and the Page-CUSUM, we note that the latter yields marginally improved size control over the former.

The main message of Tables (ref)-(ref) is that, broadly speaking, using the CUSUM with ${\Greekmath 0120} =1/2$ seems the preferred solution across all cases considered. The choice between $c_{{\Greekmath 010B} ,0.5}$\ and $ \widehat{c}_{{\Greekmath 010B} ,0.5}$ essentially depends on $m$ (and $m^{\ast }$), but in the vast majority of the cases considered, using $\widehat{c}_{{\Greekmath 010B} ,0.5}$ offers excellent size control without having an overly conservative procedure. Our results suggest avoiding overly long monitoring horizons when $m\leq 100$. In these cases, monitoring can be carried out firstly over a short horizon and, if no changepoints are detected, the procedure can be started afresh including the previous monitoring horizon within the training sample. In Tables (ref)-(ref) in the Supplement, we report empirical rejection frequencies in the presence of covariates in ((ref)). Results are broadly similar, but empirical rejection frequencies increase in all cases considered. In the stationary case (Table (ref)), results are essentially the same, and, in the explosive case (Table (ref)), the increase in empirical rejection frequencies leads to improvements, making the procedure less conservative. The latter result is interesting, and it suggests that sequential monitoring during a bubble phase may benefit from including control variables in the price dynamics. On the other hand, results worsen in the STUR case (Table (ref)), and the procedures tend to (sometimes massively) over-reject. The best results are obtained with ${\Greekmath 0120} =1/2$ and $c_{{\Greekmath 010B} ,0.5}$, which achieves size control at least when $m>50$ (or when $m\leq 50$, but the monitoring horizon is not too long, i.e. $m^{\ast }\leq m$). This suggests that, when covariates are included in the basic RCA specification, ${\Greekmath 0120} =1/2 $ and $c_{{\Greekmath 010B} ,0.5}$ should be employed - especially when trying to detect the inception of a bubble from a near stationary regime.

We would like to emphasize that all the cases considered above are fully under the researcher's control: whether to include covariates or not, and the length of the monitoring horizon $m^{\ast }$, can be decided a priori; as mentioned above, preliminary analysis on $y_{i}$ during the training period offers further help.

Under the alternative, monitoring is carried out for $m+1\leq i\leq m+m^{\ast }$, with the same specifications as in Cases I-III and

We set: $\Delta =0.5$ under Case I, with ${\Greekmath 010C} _{A}=0.75$ and $E\log \left\vert {\Greekmath 010C} _{A}+{\Greekmath 010F} _{i,1}\right\vert =-0.298$, thus having a change in persistence with $y_{i}$ stationary under the null and the alternative; $\Delta =-0.1$ under Case II, with ${\Greekmath 010C} _{A}=0.969$ and $ E\log \left\vert {\Greekmath 010C} _{A}+{\Greekmath 010F} _{i,1}\right\vert =-0.063$, thus having a change from nonstationarity to stationarity; and $\Delta =0.1$ under Case III, with ${\Greekmath 010C} _{A}=1.1$ and $ E\log \left\vert {\Greekmath 010C} _{A}+{\Greekmath 010F} _{i,1}\right\vert =0.089$, thus having a change from stationarity to nonstationarity. Median delays and empirical rejection frequencies for $m=200$, under the case of no covariates (${\Greekmath 0115} _{0}=0$), are in Table (ref).

Under stationarity, using ${\Greekmath 0120} =0.45$ yields the best results in terms of delay and power, with little difference between the CUSUM and the Page-CUSUM (with the latter occasionally delivering higher power, if not shorter detection delays); results are anyway satisfactory also when using ${\Greekmath 0120} =1/2 $ with $\widehat{c}_{{\Greekmath 010B} ,0.5}$, which is always either the best or the second best in terms of detection delay. The same results are observed in the explosive and in the STUR cases. The latter is particularly remarkable: using ${\Greekmath 0120} =1/2$ with $\widehat{c}_{{\Greekmath 010B} ,0.5}$ delivers essentialy the same performance as using ${\Greekmath 0120} =0.45$; however, according to Table (ref), the latter choice yields an oversized procedure, thus making its performance under the alternative less reliable. Hence, Table (ref) essentially indicates that using ${\Greekmath 0120} =1/2$ with $\widehat{c} _{{\Greekmath 010B} ,0.5}$ yields the best results in terms of timeliness of detection and power. As a final remark, detection delays worsen as $m^{\ast }$ increases; this is a natural phenomenon, since critical values (see e.g. Corollary (ref)) increase with $m^{\ast }$, thus making detection, ceteris paribus, more infrequent. In Table (ref) in the Supplement, we report median delays and empirical rejection frequencies in the presence of covariates in the basic RCA specification. Our guidelines based on size control suggest using ${\Greekmath 0120} =1/2$ with $c_{{\Greekmath 010B} ,0.5}$ in this case; median delays appear very satisfactory, although the cases ${\Greekmath 0120} =1/2$ with $\widehat{c}_{{\Greekmath 010B} ,0.5}$ and especially ${\Greekmath 0120} =0.45$ perform better in several cases.

In conclusion, the results in this section show that online changepoint detection in an RCA\ model based on weighted CUSUM statistics seems to work very well both under stationarity and nonstationarity. Broadly speaking, in the latter case using the standardised CUSUM (i.e., ${\Greekmath 0120} =1/2$) is the preferred choice; in the former case, choosing ${\Greekmath 0120} $ close to (but smaller than) $1/2$, appears to yield the best compromise between size control and timely detection.

Empirical applications: finding bubbles in RCA\ models

We validate our approach through two empirical applications, both involving (potentially) nonstationary data. In Section (ref), we consider a \textquotedblleft pure\textquotedblright\ RCA model with no covariates, and use it to detect daily Covid-19 hospitalisations; in Section (ref), we consider and RCA model with covariates, and use it to detect changes in the dynamics of house prices.

Sequential monitoring of Covid-19 hospitalisations

We consider daily data on Covid-19 hospitalisations. There is a huge literature on the application of time series methods to the reduced form of epidemiological models; 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; and jeremy apply the RCA model to a similar dataset, detecting several changepoints. In this context, it is 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 they change from a stationary to an explosive regime (corresponding to a surge in the epidemic). Our dataset consists of daily data for England recorded between March 19th, 2020, and January 30th, 2021, i.e. spanning the period from approximately the first lockdown (which was announced on March 23rd, 2020), until after the third and last lockdown of January 6th, 2021. We transform the series into logs (plus one since on some days the number of hospitalisation is zero); no further pre-processing is applied.\footnote{ The data are available from \url{https://ourworldindata.org/grapher/uk-daily-covid-admissions?tab=chart &stackMode=absolute&time=2020-03-29..latest&region=World} See also \url{https://www.instituteforgovernment.org.uk/sites/default/files/2022-12/timeline-coronavirus-lockdown-december-2021.pdf} , for a timeline of the UK Government decisions on lockdown and closures.}

We use a \textquotedblleft pure\textquotedblright\ RCA\ specification with no covariates. The purpose of our exercise is to ascertain whether public health authorities, back in 2020, could have benefited from the use of a sequential monitoring procedure to flag changes in the dynamics of daily hospitalisations, thus informing their decisions. We have used ${\Greekmath 0120} =1/2$ in $g_{m,{\Greekmath 0120} }\left( k\right) $ in ((ref)); based on the empirical rejection frequencies reported in Tables (ref)-(ref), we use the critical values $\widehat{c}_{{\Greekmath 010B} ,0.5}$ defined in ((ref));\footnote{ Results with different values of ${\Greekmath 0120} $ - and with ${\Greekmath 0120} =1/2$ and asymptotic critical values - are available upon request.} in all scenarios we set the length of the monitoring horizon always equal to the training sample size, i.e. $m=m^{\ast }$, which, according to the results in Tables (ref)-(ref), always ensures size control.

By way of preliminary analysis (and also to assess how our online detection methodology fares), in Section (ref) in the Supplement we apply the test by horvath2022changepoint to the whole sample (Table (ref)). In particular, after the closure of the education and hospitality sectors in all UK nations announced on March 23rd, 2020 (and implemented 3 days later), a changepoint is found at April 10th, 2020, indicating that the lockdown had started to \textquotedblleft bite\textquotedblright\ after that date; subsequently, another changepoint is found towards the end of August, which can be naturally interpreted as the beginning of the second wave in the UK, after an increase in travelling during the holiday season. Hence, we use, as training sample, the period between April 11th, 2020 and August 15th, 2020, with $m=127$. We find evidence of a changepoint on August 28th, 2020. As can be seen in Table (ref) in the Supplement, the ex-post test by horvath2022changepoint finds a break at August 26th. Clearly, a direct comparison between the two tests is not meaningful, since ex-post detection uses the information contained in the full sample, whereas online detection cannot make use of it - thus putting the latter at a disadvantage compared to the former. Still, there is (only) a two days delay between the two procedures. Moreover, the August changepoint was officially acknowledged by the PM on September 18th, 2020; hence, our sequential monitoring procedure could have brought forward public health decisions. The value of $ {\Greekmath 010C} $ after the break is above $1$, indicating an explosive dynamics in Covid-19 hospitalisations. Table (ref) in the Supplement indicates that there was a further break on October 29th, 2020, corresponding to the lockdown at the end of that month. Hence, we restart our monitoring procedure, and we use the training sample August 29th, 2020 till October 29th, 2020, with $m=61$; under an explosive regime, this should suffice to ensure size control and short detection delay, according to the results in Tables (ref) and (ref). A changepoint is flagged on November 4th, 2020, with a decline in ${\Greekmath 010C} $ indicating the effect of the closures and also of the growing concerns about a second wave. The break is detected very close to (and indeed before) the lockdown, suggesting that lockdowns tended to occur when a turning point in hospitalisations had already occurred, or was \textquotedblleft in the making\textquotedblright . Such evidence can be read in conjunction with the results in Table (ref) in the Supplement, and also with wood2021, who, albeit with a different methodology and focus, finds similar results for hospital deaths. Finally, we carry out monitoring using a training sample between November 5th, 2020, till December 31st, 2020, thus having $m=58$. We find a changepoint on January 27th, 2021, i.e. two weeks later the break found with the ex-post test by horvath2022changepoint , and three weeks later after the national lockdown announced on January 6th, 2021.

In conclusion, the empirical evidence presented above shows that our RCA-based approach to online changepoint detection is suitable to detect the onset (and the receding) of a pandemic with short delays, thus being a recommended item in the toolbox of public health decision-makers.

Sequential monitoring of housing prices

We apply our sequential detection procedures to the online detection of a change in housing prices in Los Angeles at the end of the first decade of the century. horvath2022changepoint apply an RCA-based, ex-post changepoint test, and find evidence of such a bubble starting around February 4th, 2009, when, after a period of \textquotedblleft hard landing\textquotedblright , prices stabilised;\footnote{ See Section (ref) for a plot of the data.} we also refer to a related paper by zenhya, who use monthly data. We use Los Angeles as a case study to check how timely online detection is, and also to assess the robustness of our results to the choice of the training and monitoring sample sizes, and the benefit of adding covariates. In our application, following horvath2022changepoint, we use (logs of) daily housing prices.\footnote{ We use the daily data constructed by bollerslev, and we refer to that paper for a description of the datasets.}

All our monitoring exercises start at January 15th, 2009; horvath2022changepoint find no changepoints between May 4th, 2006, and February 3rd, 2009, which entails that the non-contamination assumption during the training period is satisfied in all cases considered. We monitor for changepoints using several alternative models. In addition to the basic RCA specification, with no covariates, we also use different combinations of regressors, including: two variables that are closely related to the risk-free interest rate, taking into account the opportunity cost of capital (the Moody's Seasoned Aaa Corporate Bond Yield, and the 10 Year US Treasury Constant Maturity Rate); a measure of volatility (namely, the VXO volatility index); and a measure of real economic activity (we use the Lewis-Mertens-Stock Weekly Economic Indicator, WEI\footnote{ See lewis2022measuring for a thorough description.}). All regressors are taken from the FRED St Louis website. The risk-free interest rate proxies and the volatility measure are transformed into logs. Applying standard unit root tests to these explanatory variables, we find overwhelming evidence of a unit root in all of them during the period March 28th, 2008, corresponding to the earliest starting point for the training sample, and October 30th, 2009, corresponding to the latest ending point for the monitoring horizon (see Section (ref) in the Supplement for details); hence, we employ their first differences. As far as the WEI is concerned, it comes at a weekly frequency, and we use its weekly value for each day by way of disaggregation; other disaggregation methods could be employed following the Mixed-Data Sampling literature (MIDAS; see e.g. ghysels2018mixed), e.g. using a weighted average interpolation, but we found that these did not make virtually any difference in our results (in fact, marginally worsening the detection delay). No further pre-processing is carried out. As far as implementation is concerned, we have used ${\Greekmath 0120} =1/2$ in $g_{m,{\Greekmath 0120} }^{\left( x\right) }\left( k\right) $ in ((ref)); in the case considered, the data are \textquotedblleft nearly\textquotedblright\ non-stationary during the training period, and therefore, based on Table (ref), we use the asymptotic critical values $c_{{\Greekmath 010B} ,0.5}$.

Results in Table (ref) show that adding covariates can potentially lead to meaningful improvements in terms of detection delay: a measure of economic activity such as the WEI index, in particular, seems to contain relevant information to model the dynamics of house prices. We also note that detection delays seem to be around $4$ months, which may be a consequence of the relatively small change in ${\Greekmath 010C} $ before and after the change. Interestingly, results are unaffected by the size of the training sample $m$, but they do differ if $m^{\ast }$ changes. This is a purely mechanical effect, due to the increase in the asymptotic critical values as $ m^{\ast }$ increases, as also noted in Section (ref).

Discussion and conclusions

We propose a family of weighted CUSUM statistics for the online detection of changepoints in a Random Coefficient Autoregressive model. Our statistics are, in particular, based on the CUSUM\ process of the WLS\ residuals, and we study both the standard CUSUM, and the so-called Page-CUSUM monitoring schemes, which is designed to offer higher power/shorter detection delay. In the case of the standard CUSUM, we also study the standardised version using ${\Greekmath 0120} =1/2$, obtaining, under the null, a Darling-Erd\H{o}s limit theorem. In this case, seeing as the asymptotic critical values are a poor approximation leading to an overly conservative procedure, we also propose an approximation which works well in finite samples, avoiding overrejections. As our simulations show, the use of weighted statistics is particularly beneficial under the alternative, with detection delays decreasing as ${\Greekmath 0120} $ increases. Whilst, for the ease of exposition, we focus on the RCA\ case with no covariates, we also extend our theory to include exogenous regressors, which are allowed to be weakly dependent according to a very general definition of dependence. Simulations show that our procedures broadly guarantee size control; indeed, our experiments indicate that, for any given training sample size $m$ and monitoring horizon $m^{\ast }$, it is always possible to choose the appropriate weighing scheme to ensure the best balance between size control and timely detection. This is reinforced by our empirical illustrations, showing that when our methodology is applied to the online detection of epidemiological and housing data, it manages to find breaks very quickly, even when compared against ex-post detection methodologies.

Importantly, building on the well-known fact that, in an RCA\ model, WLS\ inference on the deterministic part of the autoregressive parameter is always Gaussian, irrespective of whether the observations form a stationary or nonstationary sequence, our monitoring procedures can be applied to virtually any type of data: stationary, or with an explosive dynamics, or on the boundary between the two regimes, with no modifications required. Hence, our methodology is particularly suited, as a leading example, to the detection of both the onset, or the collapse, of a bubble (when using financial data), or of a pandemic (when using epidemiological data). Other cases also can be considered, e.g. monitoring for changes in the persistence of a stationary series such as inflation, and the extension of the basic RCA specification to include covariates should enhance the applicability of our methodology.

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Further Monte Carlo evidence and guidelines

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Further empirical evidence

Further empirical evidence on UK Covid-19 hospitalisation data

We report a graph of the logs of (one plus) the daily hospitalisation data with the identified changepoints in Figure (ref).

We have conducted, by way of comparison, an ex-post changepoint detection exercise, applying the techniques developed in horvath2022changepoint to the whole sample. We only report results obtained using R\'{e}nyi statistics (corresponding to using a weighted version of the CUSUM process with weights ${\Greekmath 0114} =0.51$, $0.55$, $0.65$, $0.75$, $0.85$ and $1$ - see horvath2022changepoint for details).\footnote{ Using other weighing schemes 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 ${\Greekmath 0114} $, although discrepancies are, when present, in the region of few days ($2-5$ at most). We use binary segmentation, as also discussed in horvath2022changepoint, to detect multiple breaks.

Further empirical evidence on housing data

We report some preliminary information on our data. In Table (ref), we report the outcome of a standard unit root test on the three covariates used in our exercise.

In Figure (ref), we plot housing prices in Los Angeles between March 28th, 2008, and October 30th, 2009.

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Preliminary lemmas

We will use the following facts and notation: \textquotedblleft $\overset{ \mathcal{D}}{\rightarrow }$\textquotedblright\ denotes convergence in distribution; \textquotedblleft $\overset{\mathcal{D}}{=}$ \textquotedblright\ denotes equality in distribution; $c_{1}$, $c_{2}$, ... denote positive, finite constants which do not depend on sample sizes, and whose values may change from line to line.

Recall ((ref)) under $H_{0}$

If $E\log \left\vert {\Greekmath 010C} _{0}+{\Greekmath 010F} _{0,1}\right\vert <0$, define the stationary solution

with the convention that $\prod_{\oslash }=0$. Finally, let

Consider now the decomposition

where $\widehat{{\Greekmath 010C} }_{m}$ is the WLS\ estimator computed using $\left\{ y_{1},...,y_{m}\right\} $.

In the next lemma, we obtain asymptotic representations for the terms in ( (ref)).

Let

We now report some preliminary results to prove the main results in Section (ref). We begin by studying the WLS\ estimator $\widehat{{\Greekmath 010C} } _{m}$. Define

and the diagonal matrix

Then it holds that

where $\mathbf{Y}_{m}=\left( y_{2},y_{3},...,y_{m}\right) ^{\intercal }$. Using the recursion defined in ((ref)), we obtain

having defined $\mathbf{b}_{0}=\left( {\Greekmath 010C} _{0},\mathbf{{\Greekmath 0115} } _{0}^{\intercal }\right) ^{\intercal }$ and

Based on the above, under the condition for stationarity $E\log \left\vert {\Greekmath 010C} _{0}+{\Greekmath 010F} _{0,1}\right\vert <0$, it can be verified that

has a unique stationary, causal solution. Consider the variables

and define

and

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Proofs

In order to prove the next set of results, we will make use of the following decomposition, valid under ((ref))

with the convention that $\sum_{a}^{b}=0$ whenever $b<a$.