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Backward CUSUM for Testing and Monitoring Structural Change with an Application to COVID-19 Pandemic Data
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{\it Keywords:} Sequential tests; Recursive residuals; Open-end monitoring; Local delay; Breakpoint estimation.
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Cumulative sums have become a standard statistical tool for testing and monitoring structural changes in time series models. The CUSUM test was introduced by brown1975 as a test for structural breaks in the coefficients of a linear regression model $y_t = x_t^{\prime} \beta_t + u_t$ with time index $t$, where $\beta_t$ denotes the coefficient vector, $x_t$ is the vector of regressor variables and $u_t$ is a zero mean error term. Under the null hypothesis, there is no structural change in $\beta_t$, while, under the alternative hypothesis, the coefficient vector changes at unknown time $T^* \leq T$.
Sequential tests, such as the CUSUM test, consist of a detector statistic and a critical boundary function. The CUSUM detector sequentially cumulates standardized one-step ahead forecast errors, which are also referred to as recursive residuals. The detector is evaluated for each time point within the testing period, and, if its path crosses the boundary function at least once, the null hypothesis is rejected. If the endpoint of the sample is fixed and the test is applied once to the full sample by comparing the path of the detector with the boundary function, the test is called a retrospective test (henceforth: {\it$R$-test}). A variety of {\it$R$-tests} have been proposed in the literature (for recent reviews, see robbins2011, aue2013, and casini2019c).
Since the seminal work of chu1996, increasing interest has been focused on monitoring structural stability in real time. Sequential monitoring procedures (henceforth: {\it$M$-tests}) consist of a detector statistic and a boundary function that are evaluated for periods beyond some historical time span. The monitoring time span with $t>T$ can either have a fixed endpoint $M < \infty$ or an infinite horizon. In the fixed endpoint setting, the monitoring period starts at $T+1$ and ends at $M$, while the boundary function depends on the ratio $m=M/T$. In case of an infinite horizon, the monitoring time span does not need to be specified before the monitoring procedure starts. These two monitoring schemes are also referred to as closed-end and open-end procedures (see kirch2015). The null hypothesis of no structural change is rejected whenever the path of the detector crosses some critical boundary function for the first time. Monitoring procedures for a fixed endpoint were proposed in leisch2000, zeileis2005, wied2013, and dette2020, whereas chu1996, horvath2004, aue2006, fremdt2015, and gosmann2021 considered an infinite monitoring horizon. In recent years, {\it$M$-tests} have become popular as tests for speculative bubbles in financial markets (e.g. phillips2011, homm2012, astill2018).
A well-known drawback of the conventional CUSUM {\it$R$-test} is its low power, whereas the CUSUM {\it$M$-test} may exhibit large detection delays. This is due to the fact that the pre-break recursive residuals are uninformative, as their expectation is equal to zero up to the break date, while the recursive residuals have a non-zero expectation after the break. Hence, the cumulative sums of the recursive residuals contain a large number of uninformative residuals that only add noise to the statistic. In contrast, if one cumulates the recursive residuals backwardly from the end of the sample to the beginning, the cumulative sum collects the informative residuals first, and the likelihood of exceeding the critical boundary will typically be larger than when cumulating residuals from the beginning onwards. In this paper, we show that backward CUSUM test procedures may indeed have a much higher power and lower detection delays than the conventional CUSUM {\it$R$-} and {\it$M$-tests}. phillips2018 proposed a reverse sample scheme for the PSY procedure that is used for detecting crises (bubble collapses). The main difference with our approach is that their regression is performed in reverse order, while our approach estimates the model in the original time but reverses the order of the (recursive) residuals.
Another way of motivating the backward CUSUM testing approach is to consider the simplest possible situation, where, under the null hypothesis, it is assumed that the process is generated as $y_t= \mu + u_t$, with $\mu$ and $\sigma^2=Var(u_t)$ assumed to be known. To test the hypothesis that the mean changes at $T^*$, we introduce the dummy variable $D_t^*$, which is unity for $t \geq T^*$ and zero elsewhere. The uniformly most powerful test statistic is the $t$-statistic for the hypothesis $\delta=0$ in the regression $y_t-\mu = \delta D_t^*+ u_t$, which is given by $\sigma^{-1} (T-T^*+1)^{-1/2} \sum_{t=T^*}^T (y_t - \mu)$. If $\mu$ is unknown, we may replace it by the full sample mean $\overline y$, resulting in the backward cumulative sum of the OLS residuals from period $T$ through $T^*$. If $T^*$ is unknown, the test statistic is computed for all possible values of $T^*$, whereas the starting point $T$ of the backward cumulative sum remains constant. Since the sum of the OLS residuals is zero, it follows that the test is equivalent to a test based on the forward cumulative sum of the OLS residuals. In contrast, if we replace $\mu$ with the recursive mean $\overline \mu_{t-1}=(t-1)^{-1}\sum_{i=1}^{t-1} y_t$, we obtain a test statistic based on the backward cumulative sum of the recursive residuals (henceforth: backward CUSUM). In this case, however, the test is different from a test based on the forward cumulative sum of the recursive residuals (henceforth: forward CUSUM). This is due to the fact that the sum of the recursive residuals is an unrestricted random variable. Accordingly, the two versions of the test may have quite different properties. In particular, it turns out that the backward CUSUM is much more powerful than the standard forward CUSUM at the end of the sample. Accordingly, this version of the CUSUM test procedure is better suited for the purpose of real-time monitoring, where it is crucial to be powerful at the end of the sample.
An additional problem of the conventional CUSUM test is that it has no power against alternatives that do not affect the unconditional mean of $y_t$ (see kramer1988). For both retrospective testing and monitoring, we propose a multivariate sequential statistic in the fashion of the score-based cumulative sum statistic of hansen1992 and the tests by jiang2019. The maximum vector entry of the multivariate statistic yields a detector and a sequential test that has power against a much larger class of structural breaks than when using conventional CUSUM detectors.
We also suggest a new estimator for the break date based on backwardly cumulated recursive residuals. This estimator outperforms the conventional estimator constructed by the sum of squared residuals whenever the break occurs close to the end of the sample, which is the relevant scenario for on-line monitoring.
This paper is organized as follows. In Section (ref), the limiting distribution of the multivariate CUSUM process is derived under both the null hypothesis and local alternatives. Section (ref) introduces the backward CUSUM and the stacked backward CUSUM tests for both retrospective testing and monitoring. While the backward CUSUM is only defined for $t \leq T$ and can thus be implemented only for retrospective testing, the stacked backward CUSUM cumulates recursive residuals backwardly in a triangular scheme and is therefore suitable for real-time monitoring. The local powers of the tests are compared in Section (ref). In the retrospective setting, the powers of the backward CUSUM and the stacked backward CUSUM tests are substantially higher than that of the conventional forward CUSUM test if a single break occurs after one third of the sample size. In the case of monitoring, the detection delay of the stacked backward CUSUM under local alternatives is shown to be much lower than that of the monitoring CUSUM detector by chu1996. In Section (ref) we present a strong invariance principle for the multivariate CUSUM process and propose an infinite horizon monitoring procedure. Section (ref) considers the estimation of the break date based on backwardly cumulated recursive residuals. We present an estimator, which is more accurate than the conventional maximum likelihood estimator if the break is located at the end of the sample. Section (ref) presents Monte Carlo simulation results, in Section (ref) we provide a real-data example on monitoring SARS-CoV-2 infections during the COVID-19 pandemic, and Section (ref) concludes.
Throughout the paper, we use the following notation: $\| a \| = \max_{i=1,\ldots, k} |a_i|$ denotes the maximum norm and $\| A \|_M = \max_{i=1, \ldots, k} \sum_{j=1}^l |A_{i,j}|$ denotes the maximum absolute row sum norm, where $a \in \mathbb R^k$, and $A \in \mathbb R^{k \times l}$. We use $\overset{p}{\longrightarrow}$ to denote convergence in probability as $T \to \infty$, $\overset{d}{\longrightarrow}$ for convergence in distribution, and $\overset{d}{=}$ to indicate that two random variables have the same distribution. The space of right continuous functions with left limits (c\`{a}dl\`{a}g) on $[0,m]$, where $0 < m < \infty$, is denoted as $D([0,m])$, and its $k$-fold product space is $D([0,m])^k = D([0,m]) \times \ldots \times D([0,m])$. The space is equipped with the Skorokhod metric (see billingsley1999), and the symbol “$\Rightarrow$” denotes weak convergence with respect to this metric.
We consider the multiple linear regression model
where $y_t$ is the dependent variable, and $x_t = (1, x_{t2}, \dots, x_{tk})^{\prime}$ is the vector of regressor variables including a constant. The $k \times 1$ vector of regression coefficients $\beta_t$ depends on the time index $t$, and $u_t$ is an error term. The time point $T$ divides the time horizon into the retrospective time period $t \leq T$ and the monitoring period $t > T$. We impose the following assumptions on the regressors and the error term.
Model (ref) allows for conditionally heteroskedastic errors and local non-stationary regressors, provided that a global long-run covariance matrix $\Omega$ exists. The regressors can contain lagged dependent variables such as in autoregressive distributed lag models. We focus on models with a correctly specified dynamic structure and uncorrelated errors.
Following brown1975 this assumption rules out autocorrelated error processes. In practice this may require a dynamic specification with a suitable lag distribution of the variables. In Remark (ref) we show that autocorrelated errors can be accommodated by replacing the ordinary covariance matrix by a (consistent estimate of the) long-run covariance matrix. Since the estimation of long-run covariances can lead to finite sample size distortions (see e.g.\ casini2021b), Assumption (ref) is a common and convenient assumption in practice. The expression of the global covariance matrix simplifies to $\Omega = \sigma^2 C$ under Assumption (ref).
Recursive residuals for linear regression models were introduced by brown1975 as standardized one-step ahead forecast errors, and are defined as
and $w_t = 0$ for $t = 1, \ldots, k$, where $\widehat{ \beta}_{t-1} = (\sum_{i=1}^{t-1} x_i x_i^{\prime} )^{-1} \sum_{i=1}^{t-1} x_i y_i$. Using recursive residuals instead of ordinary OLS residuals as in ploberger1992 has a number of advantages. First, the recursive residuals behave exactly as under the null hypothesis until the parameters change, whereas a structural break affects all OLS residuals in a different manner. Second, under Assumptions (ref) and (ref), the recursive residuals form a martingale difference sequence regardless of the estimation error in the recursive residuals. By contrast, the OLS residuals are (slightly) autocorrelated, which only disappears if the sample size gets large.
The conventional CUSUM detector is given by $S_{t,T} = \widehat \sigma_T^{-1} T^{-1/2} \sum_{i=1}^t w_i$, where $\widehat \sigma_T^2$ denotes the sample variance of $\{ w_{k+1}, \ldots, w_T\}$. Under the null hypothesis $H_0: \beta_t = \beta_0$ for all $t$, the univariate CUSUM process obeys the functional central limit theorem $S_{\lfloor r T \rfloor,T} \Rightarrow W(r)$, where $W(r)$ is a standard Brownian motion (see sen1982). The univariate CUSUM {\it$R$-test} of brown1975 rejects the null hypothesis if the path of $|S_{t,T}|$ exceeds the linear critical boundary function $b_t = \lambda_\alpha d_{\text{lin}}(t/T)$ for at least one time index $t=1, \ldots, T$, where
The critical value $\lambda_\alpha$ is the $(1-\alpha)$ quantile of $\sup_{0\leq r \leq 1} |W(r)|/d_{\text{lin}}(r)$ and determines the significance level $\alpha$, which accounts for the multiplicity issue of the sequential test procedure. In the monitoring context, chu1996 considered the radical type boundary function $b_{\text{rad}}(r) = r^{1/2} (\log(r) - \log(\alpha^2))^{1/2}$, which is derived from the boundary crossing probability for a Brownian motion (see robbins1970). The conventional univariate CUSUM {\it$M$-test} rejects the null hypothesis if the detector statistic $|S_{t,T} - S_{T,T}|$ exceeds $b_t = b_{\text{rad}}(t/T)$ for some $t > T$.
A weakness of univariate CUSUM tests is that they focus on breaks in the intercept. ploberger1990 studied local alternatives of the form $\beta_t = \beta_0 + T^{-1/2} g(t/T)$, where $g: \mathbb R \to \mathbb R^k$ is piecewise constant and bounded. The authors showed that $S_{\lfloor r T \rfloor ,T} \Rightarrow W(r) + \pi^{\prime} h(r)$, where $\pi = e_1' C$, $e_1 = (1,0, \ldots, 0)'$, and
Consequently, univariate CUSUM tests have no power if $g(r)$ is orthogonal to $\pi$. To sidestep this difficulty, we follow jiang2019 and consider the multivariate statistic
Under Assumption 1, the multivariate series $x_t u_t$ obeys a multivariate functional central limit theorem (see phillips1986), which also applies to the multivariate CUSUM process of recursive residuals.
This is an extension of the results in jiang2019, who considered slightly stronger assumptions and no local alternatives. Note that the function $g(r)$ is constant if and only if $\beta_t$ is constant. If $\beta_t = \beta_0$ for all $t$, we have $h(r) = 0$. By contrast, under a local alternative with a non-constant break function $g(r)$, it follows that $h(r)$ is non-zero, and, consequently, $C^{1/2} h(r)$ is non-zero, since $C^{1/2}$ is positive definite. Hence, sequential tests that are based on $Q_T(r)$ have power against a larger class of alternatives than the tests of brown1975 and chu1996.
Therefore, we consider {\it$R$-} and {\it$M$-tests} that are based on the multivariate detector $Q_{t,T} = Q_T(t/T)$. Note that $Q_{t,T} = S_{t,T}$ if there is only an intercept in the model. The multivariate forward CUSUM {\it$R$-test} is defined by the following rule: the null hypothesis is rejected if the path of $\| Q_{t,T} \|$ exceeds the boundary function $b_t = \lambda_\alpha d(t/T)$ for at least one index $t=1, \ldots, T$. Equivalently, we can express this sequential test as a one-shot test, where $H_0$ is rejected if the maximum statistic $\mathcal Q_T = \max_{t=1, \ldots, T} \|Q_{t,T} \|/d(t/T)$ exceeds the critical value $\lambda_\alpha$, which is the $(1-\alpha)$ quantile of its limiting null distribution.
By Theorem (ref) and the continuous mapping theorem it follows that
under the null hypothesis. The multivariate forward CUSUM {\it $M$-test} with fixed endpoint $M=\lfloor m T \rfloor$ rejects $H_0$ if the path of $\| Q_{t,T} - Q_{T,T} \|$ exceeds the boundary function $b_t= \lambda_\alpha d((t-T)/T)$ for at least one index $t=T+1, \ldots, \lfloor m T \rfloor$, where $1 < m < \infty$. The corresponding maximum statistic is $\mathcal Q_{T,m} = \max_{t=T+1, \ldots, \lfloor m T \rfloor} \|Q_{t,T} - Q_{T,T} \|/d((t-T)/T)$, where, under $H_0$,
An alternative approach is to cumulate the recursive residuals in reversed order. Suppose there is a single break in $\beta_t$ at time $t=T^*$. Then, $\{ w_t, \ t < T^* \}$ are the residuals from the pre-break period, and $\{ w_t, \ t \geq T^* \}$ are those from the post-break period. As the pre-break recursive residuals are not affected by a violation of the null hypothesis, they do not provide useful information about a subsequent break. Accordingly, the partial sum process $T^{-1/2} \sum_{j=1}^t w_j$ behaves like a pure random walk for $t < T^*$ and cumulating those residuals brings nothing but noise to the detector statistic. In contrast, the post-break residuals have nonzero mean and reveal relevant information about a possible break. In order to focus on the post-break residuals, we therefore consider backwardly cumulated partial sums of the form $T^{-1/2} \sum_{j=0}^{t-1} w_{T-j}$. We define the retrospective backward CUSUM detector as
The null hypothesis is rejected if $\|BQ_{t,T}\|$ exceeds the boundary $b_t = \lambda_\alpha d((T-t-1)/T)$ for at least one time index $t$. The maximum statistic is given by
and, under the local alternatives defined in Theorem (ref), the continuous mapping theorem implies
Hence, the limiting distribution of $\mathcal{BQ}_T$ under $H_0$ coincides with that of $\mathcal{Q}_T$. Simulated asymptotic critical values under the linear boundary (ref) are presented in Table (ref). Under local alternatives, the limiting distributions of $\mathcal{BQ}_T$ and $\mathcal{Q}_T$ differ. A simple illustrative example of the detector paths together with the linear boundary (ref) of brown1975 are depicted in Figure (ref), in which two processes with $k=1$ and a single break in the mean at $3/4$ and $1/4$ of the sample are simulated.
Unlike the forward CUSUM detector, the backward CUSUM detector is not measurable with respect to the filtration of available information at time $t$ and is therefore not suitable for a monitoring procedure. The path of $\|BQ_{t,T}\|$ cannot be monitored in real-time, as it is only defined for $t \leq T$ with fixed endpoint $T$. To obtain a feasible {\it $M$-test} in practice, we resort to a triangular backward inspection scheme of recursive residuals, which is discussed below.
Let $\mathcal{BQ}_T(t) = \max_{s=1, \ldots, t} \|Q_T(t/T) - Q_T((s-1)/T)\|/d((t-s+1)/T)$ be the backward CUSUM maximum statistic with endpoint $t$. The idea of the stacked backward CUSUM scheme is to compute this statistic sequentially for each $t$, yielding $\mathcal{BQ}_T(1), \ldots, \mathcal{BQ}_T(T)$. The corresponding maximum statistic $\mathcal{SBQ}_T$ is the maximum among this sequence of backward CUSUM statistics. An important feature is that this sequence is measurable with respect to the filtration of information at time $t$, so that $\mathcal{BQ}_T(t)$ is itself a sequential statistic. Stacking all backward CUSUM statistics on one another leads to a triangular array structure given by
which is denoted as the stacked backward CUSUM detector. We reject $H_0$ if $\|SBQ_{s,t,T}\|$ exceeds the triangular boundary $b_{s,t} = b(\frac{t}{T}, \frac{s-1}{T})$ for some $t=1, \ldots, T$ and $s = 1, \ldots, t$.
The stacked backward CUSUM {\it $R$-test} can be equivalently expressed in terms of a maximum statistic. $H_0$ is rejected if the double maximum statistic
exceeds $\lambda_\alpha$. Under the local alternatives defined in Theorem (ref), it follows that
Since the triangular detector (ref) is measurable with respect to the information set at time $t$, it can be monitored on-line across all time points $t > T$. The null hypothesis is rejected if $\|SBQ_{s,t,T}\|$ exceeds $b_{s,t} = b(\frac{t}{T}, \frac{s-1}{T})$ at least once for some $t \geq T+1$ and $s = T+1, \ldots, t$. The {\it $M$-test} maximum statistic with a fixed horizon $m < \infty$ is given by
where, analogously to (ref),
Simulated critical values for the stacked backward CUSUM $R$- and $M$-tests under the linear triangular boundary
are presented in Table (ref).
In order to illustrate the advantages of the backward CUSUM tests, we consider the simple local break model $\beta_t = \beta_0 + T^{-1/2} g(t/T)$ with $g(r) = c 1_{\{ r \geq \tau^* \}}$, where $c \in \mathbb{R}^k$, and $\tau^*$ denotes the break location. From (ref) it follows that
and, under the linear boundaries (ref) and (ref), the $R$-tests satisfy
Asymptotic local power curves from the limiting distributions in (ref) for the case $k=1$ are presented in Figure (ref). The $(2,3)$-element of the panel of figures shows that for a fixed break size the backward CUSUM and the stacked backward CUSUM outperform the forward CUSUM if a single break $\tau^*$ is located after $15\%$ of the sample size. If the break date $\tau^*$ tends to the end of the sample, the power gain of $\mathcal{BQ}_T$ and $\mathcal{SBQ}_T$ increases substantially.
For the $M$-test statistics with fixed endpoint $m=2$, the limiting distributions of $\mathcal Q_{T,m}$ and $\mathcal{SBQ}_{T,m}$ for a break at $\tau^* \in (1,2)$ coincide with those for the $R$-tests presented in (ref) for a break at $\tau^* \in (0,1)$. Hence, the power of $\mathcal{SBQ}_{T,m}$ is higher than that of $\mathcal Q_{T,m}$ if breaks are located after 15% of the pre-monitoring sample.
Another important performance measure for {\it $M$-tests} is the delay between the actual break and the detection time point. aue2004 and aue2009 derived the asymptotic distribution of the detection stopping time of CUSUM $M$-tests that are based on OLS residuals. For $\mathcal Q_{T,m}$ and $\mathcal{SBQ}_{T,m}$, the detection stopping times are given by
where $\lambda_{\alpha,m,Q}$ and $\lambda_{\alpha,m,SBQ}$ are the corresponding critical values. Under the same setting as in (ref), the relative detection stopping times satisfy
as $T \to \infty$, where the limiting relative stopping times are denoted as $\tau_{D,Q}$ and $\tau_{D,SBQ}$, respectively. The asymptotic mean local delays $E ( \tau_{D} \mid \tau^* \leq \tau_{D} \leq m ) - \tau^*$, where $\tau_{D} \in \{\tau_{D,Q}, \tau_{D,SBQ} \}$, are presented in the bottom panels of Figure (ref) for $m=4$ and different break locations. The asymptotic mean local delay of $\mathcal{SBQ}_{T,m}$ is much lower than that of $\mathcal{Q}_{T,m}$. Moreover, the asymptotic mean local delay of $\mathcal{SBQ}_{T,m}$ slowly decreases in $\tau^*$ and is much lower than that of $\mathcal{Q}_{T,m}$, except for early breaks.
The functional central limit theorem given by Theorem (ref) is not suitable for analyzing the asymptotic behavior of an infinite horizon monitoring statistic, since the variance of $Q_T(r)$ is unbounded as $r \to \infty$, and $\sup_{r \geq 1} \| Q_T(r) - W^{(k)}(r) \|$ might not converge in general. Instead, we need an almost sure invariance principle, which is specified by the following high level condition:
Almost sure invariance principles were first studied by strassen1967, who verified Assumption (ref) under the additional assumption that $x_t u_t$ is a stationary and ergodic martingale difference sequence. Optimal rates were first derived by komlos1975. aue2004 and aue2009 present examples where Assumption (ref) is satisfied, which include martingale difference sequences and linear processes with GARCH-type innovations under mild regularity conditions. For more general dependent processes under suitable regularity conditions, Assumption (ref) was shown by wu2007 and berkes2014 with respect to a physical dependence measure (see also berkes2011).
This result is the key tool to establish the limiting distributions of infinite horizon monitoring statistics under $H_0$ and indicates the need of further restrictions on the boundary function. In the Appendix, we also show that this result remains valid if we replace $Q_T(r)$ by the autocorrelation robust statistic $\widetilde Q_T(r)$ under additional mixing assumptions. The infinite horizon forward CUSUM and stacked backward CUSUM maximum statistics are defined as
Analogously to the fixed horizon case, $H_0$ is rejected if the statistic exceeds the $(1-\alpha)$ quantile of its limiting null distribution. Since a maximum over a non-compact set can be unbounded, we need further restrictions on the boundary functions for infinite horizon monitoring.
Under this assumption we show the following theorem:
The linear boundary $d_\text{lin}(r)$ (see equation (ref)) satisfies Assumption (ref), whereas for the linear triangular boundary $d_{\text{sbq}}(r,s)$ (see equation (ref)) it is not satisfied. Instead, the boundary must be at least of order $\sqrt r$ uniformly among all $s$, which motivates the alternative boundary
Simulated critical values for the {\it $M$-tests} under the boundaries $d_\text{lin}(r)$ and $d_\text{inf}(r,s)$ are presented in Table (ref).
As soon as the testing procedure has indicated a structural instability in the coefficient vector, the next step is to locate the break point. In the single break model with $\beta_t = \beta_0 + \delta 1_{\{t \geq T^*\}}$, where $\delta \neq 0$, horvath1995 suggested to estimate the relative break date $\tau^* = T^* / T$ by the relative time index for which the likelihood ratio statistic is maximized. As an asymptotically equivalent estimator, bai1997 proposed the maximum likelihood estimator
where $R_1(t)$ is the OLS residual sum of squares when using observations until time point $t$ and $R_2(t)$ is the OLS residual sum of squares when using observations from time $t+1$ onwards. In case of monitoring, chu1996 considered
to estimate $\tau^*_{\text{mon}} = T^* / T_d$, where $T_d$ denotes the detection time point, which is the stopping time at which the detector statistic exceeds the boundary function for the first time. The maximum likelihood estimator is very accurate if the breakpoint is located in the middle of the sample. However, by construction, the true breakpoint $T^*$ tends to be close to the stopping time $T_d$, and $R_2(T^*)$ is computed from very few observations, which may lead to a large finite sample estimation error for the maximum likelihood estimator. A theoretical explanation for this effect is given in casini2018a,casini2021a, where the finite-sample distribution of the least squares estimator is investigated using a continuous record asymptotic framework.
To bypass this problem, we use backwardly cumulated recursive residuals to estimate the relative break location. In the single break model, $\|BQ_{\lfloor r T \rfloor,T}\|$ is asymptotically proportional to $\| h(1) - h(r) \|$, which is constant in the pre-break period and decreases to zero in the post-break period. When scaled by its asymptotic standard deviation, the detector is asymptotically proportional to $\|h(1) - h(r)\|/\sqrt{1-r}$, which in turn (see equation (ref)) is proportional to
where the maximum is attained at $r = \tau^*$. Accordingly, we consider
This result implies that the breakpoint estimators (ref) are consistent, as $T \to \infty$.
We illustrate the finite sample performance of the $R$-tests and $M$-tests for the models
where $\gamma_t = 0.8 \cdot 1_{\{t/T \geq \tau^*\}}$, $u_t$ and $e_t$ are independent and $NID(0,1)$, and $z_t = (1-0.5L) e_t$, where $L$ is the lag operator. For (ref) and (ref) we consider the full structural break tests, and for (ref) partial break tests with $H=(1,0)$ are considered (see Remark (ref)).
In Table (ref) the empirical sizes and powers of the retrospective tests are compared with that of the sup-Wald test of andrews1993. First, we observe that $\mathcal{BQ}_T$ and $\mathcal{SBQ}_T$ outperform $\mathcal{Q}_T$, except for the case $\tau^* = 0.1$. Second, while $\mathcal{Q}_T$ has much lower power than the sup-Wald test, the reversed order cumulation structure in $\mathcal{BQ}_T$ and $\mathcal{SBQ}_T$ seems to compensate for the weakness of $\mathcal{Q}_T$. andrews1993 showed that the sup-Wald test is weakly optimal in the sense that, in the case of a single structural break, its asymptotic local power curve approaches the power curve from the infeasible point optimal maximum likelihood test, as the significance level tends to zero. Within the framework of the considered models, $\mathcal{BQ}_T$ performs similarly well as the sup-Wald test and thus has comparably good power properties as the weakly optimal test. In contrast to $\mathcal{SBQ}_T$, the the sup-Wald test is not suitable for monitoring since its statistic is not measurable with respect to the filtration of information at time $t$.
Fixed endpoint {\it $M$-tests} are particularly useful when the monitoring period is short. {\it $M$-tests} with infinite horizon can be used for long monitoring periods of arbitrary length. To evaluate the performance of the $M$-tests for finite samples, we simulate the same models as for the {\it $R$-tests} for time points $t=1, \ldots, mT$, where we specify $m = 2$ for the fixed endpoint tests and $m=20$ for the infinite horizon tests. The results in Table (ref) show that the mean delay for $\mathcal{SBQ}_{T,m}$ is much lower than that of $\mathcal{Q}_{T,m}$ and is almost constant across the breakpoint locations.
For infinite horizon monitoring, $\mathcal{SBQ}_{T,\infty}$ performs similarly well compared to conventional tests (see Table (ref)). The detection delay of $\mathcal{Q}_{T,\infty}$ is much higher than that of $\mathcal{SBQ}_{T,\infty}$, and the gap increases further with increasing $\tau^*$. Compared to the tests of chu1996 and fremdt2015, we find a similar picture. Note that the two alternative tests have no power in (ref) and are therefore omitted for this case.
To compare the breakpoint estimator in equation (ref) with its maximum likelihood benchmark in (ref) and (ref), we present Monte Carlo simulation results in Table (ref). If the break $\tau^*$ is located after $85 \%$ of the sample, the estimator based on backwardly cumulated recursive residuals has a much lower bias and root mean squared error than the maximum likelihood estimator, which is due to the fact that the post-break entails only few observations.
We consider the time series $y_t$ of daily new cases of SARS-CoV-2 infections in the US during the first months of the COVID-19 pandemic. The starts of the first three waves of infections are observed in March, June, and September 2020. To monitor for a second and third wave of infections, we consider pre-monitoring training samples of six weeks ($T=46$) starting at the times after the first and second peak. The training sample periods are given by April 10 until May 21, and July 20 until August 30. To account for a seasonal unit root in $y_t$ we consider the seasonally differenced series $\widetilde y_t = y_t - y_{t-7}$. Weekly differences are used to account for the seasonalities resulting from the weekly reporting pattern of COVID-19 data with lower numbers on weekends. We estimate the dynamic model
where $x_t=(1,\widetilde y_{t-1}, \widetilde y_{t-7})^{\prime}$, and $\beta=(\phi_0,\phi_1,\phi_2)^{\prime}$. The parameters for lags 2 and 7 are the only significant autoregressive parameters.
Both the Ljung-Box and the robust Q-statistic of dalla2020 do not indicate any significant autocorrelation in the residuals for the pre-break training periods (see Figure (ref)). We consider the infinite-horizon stacked backward CUSUM statistic for a break in the intercept and the infinite horizon forward CUSUM of chu1996. We are interested in detecting positive changes in the intercept $\phi_0$ and apply one-sided infinite horizon monitoring statistics with a significance level of $5 \%$. The critical values for the partial right-sided tests are given by those of the full test with $\alpha = 0.1$ and $k=1$ (see Remark (ref)). We consider the infinite-horizon stacked backward CUSUM statistic for a break in the intercept and the infinite horizon forward CUSUM of chu1996. To compare the detector statistics, we scale them by their boundaries and critical values, so that $H_0$ is rejected in favor of a positive change in $\phi_0$ if the detector exceeds unity, respectively. An alternative to the dynamic modeling in (ref) is to apply autocorrelation robust {\it $M$-tests} to the model $\widetilde y_t = \phi_0 + u_t$, where we replace $\widehat{\sigma}_T$ with the long-run variance estimator by newey1987 (see Remark (ref)).
The results are presented in Figure (ref). Both monitoring procedures find an indication for a rise in SARS-CoV-2 infections in the US at the end of June and the end of September. The stacked backward CUSUM detects the breaks much earlier and becomes significant between 2 and 16 days before the forward CUSUM becomes significant. This confirms our theoretical analysis and shows that precious time can be saved by applying the backward monitoring scheme.
In this paper we propose two alternatives to the conventional CUSUM detectors by brown1975 and chu1996. It has been demonstrated that cumulating the recursive residuals backwardly results in much higher power than using forwardly cumulated recursive residuals, in particular if the break is located at the end of the sample. Accordingly, the backward scheme is especially attractive for on-line monitoring. To this end, the stacked triangular array of backwardly cumulated recursive residuals is employed and we find that this approach yields a much lower detection delay than the monitoring procedure by chu1996. Due to the multivariate nature of our tests, they also have power against structural breaks that do not affect the unconditional mean of the dependent variable. We also propose a new break date estimator which outperforms conventional estimators if the break is located at the end of the sample.
We are thankful to Holger Dette, Josua Gösmann, Alexander Mayer, Dominik Wied and three referees for their very helpful comments and suggestions which helped to improve the paper a lot. Furthermore, the usage of the CHEOPS HPC cluster for parallel computing is gratefully acknowledged.
An accompanying R-package for all methods presented in this article is available online at https://github.com/ottosven/backCUSUM.