E-mail: aaue@ucdavis.edu

Review Article

# Structural breaks in time series

Article first published online: 14 SEP 2012

DOI: 10.1111/j.1467-9892.2012.00819.x

© 2012 Wiley Publishing Ltd

Additional Information

#### How to Cite

Aue, A. and Horváth, L. (2013), Structural breaks in time series. Journal of Time Series Analysis, 34: 1–16. doi: 10.1111/j.1467-9892.2012.00819.x

#### Publication History

- Issue published online: 21 DEC 2012
- Article first published online: 14 SEP 2012
- First version received December 2011

- Abstract
- Article
- References
- Cited By

### Keywords:

- Change-points;
- CUSUM;
- long memory;
- mean change;
- unit-root;
- variance change

### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. CUSUM procedures under dependence
- 3. Likelihood ratio statistics
- 4. Estimating the number of breaks
- 5. Discriminating break points, long memory and unit roots
- 6. Miscellanea
- Acknowledgements
- References

This paper gives an account of some of the recent work on structural breaks in time series models. In particular, we show how procedures based on the popular cumulative sum, CUSUM, statistics can be modified to work also for data exhibiting serial dependence. Both structural breaks in the unconditional and conditional mean as well as in the variance and covariance/correlation structure are covered. CUSUM procedures are nonparametric by design. If the data allows for parametric modeling, we demonstrate how likelihood approaches may be utilized to recover structural breaks. The estimation of multiple structural breaks is discussed. Furthermore, we cover how one can disentangle structural breaks (in the mean and/or the variance) on one hand and long memory or unit roots on the other. Several new lines of research are briefly mentioned.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. CUSUM procedures under dependence
- 3. Likelihood ratio statistics
- 4. Estimating the number of breaks
- 5. Discriminating break points, long memory and unit roots
- 6. Miscellanea
- Acknowledgements
- References

The analysis of structural breaks, or change-points, has its origins in quality control (Page, 1954, 1955) but has since become an integral part of a wide variety of fields with a significant statistical component, among them economics (Perron, 2006), finance (Andreou and Ghysels, 2009), climatology (Reeves *et al.*, 2007) and engineering (Stoumbos *et al.* 2000). Much of the methodology was first developed for independent observations, so naturally procedures to detect instabilities in mean and variance have played a dominant role. The relevant lines of research containing results on simple location shift and scale shift models, and on more complex regression models for the independent case may be found in Brodsky and Darkhovsky (1993), Carlstein *et al.* (1994) and Csőrgő and Horvath (1997).

In many applications, however, it appears to be of interest to incorporate serial dependence of the observations into the statistical analysis. It is this case that the paper is concerned with. Two approaches to deal with the time series effect have emerged. One aims at quantifying the effect of dependence on the test statistics developed for the independent setting and then to extend their reach to include also the second-order properties as given for example in the autocorrelation function. In this case, the fitting of a particular parametric time series model may be avoided. This appears to be advantageous whenever ambiguity arises at the model fitting stage and model misspecification becomes an issue. The first approach then leads to establishing functional central limit theorems for the dependent case and, most crucially, to deriving appropriate estimators for the long-run variances. The second approach utilizes particular time series models and seeks to explicitly describe the dependence structure concurrently with potential structural breaks in the observations. Most popular are the classes of linear ARMA and nonlinear GARCH-type models. Since parametric assumptions are being made, likelihood methods are available and can be used to design relevant test statistics.

Traditionally, structural break problems have been phrased as hypothesis tests. The null is set up to describe structural stability, the alternative contains one or multiple structural break(s). The test statistics may be viewed as two-sample tests adjusted for the unknown break location, thus leading to max-type procedures. Often asymptotic relationships are derived to obtain critical values for the tests. After the null hypothesis is rejected, the location(s) of the break(s) need(s) to be estimated. This is the setting covered, for example, in Bai and Perron (1998), and Csőrgő and Horváth 1997. Lately, there have been attempts to view structural break estimation as a model selection problem, see Davis *et al.* (2006), Lu *et al.* (2010), Robbins *et al.* (2011). Besides these contributions to historical methods, sequential detection procedures have been developed. This is in line with the original interest in quality control (Page, 1954), where one monitors the output of a production line and wishes to signal deviations from the null hypothesis (the in-control scenario) quickly. These methods have been extended to other areas of applications, notably economics and finance in a number of recent publications, see Aue *et al.* (2009e), Chu *et al.* (1996).

During the past decade or so, structural breaks research has focused more on how to disentangle structural breaks from other forms of departures from the null model such as long memory and unit roots as they have a similar effect on the second-order properties of a time series. For example, all induce a slow decay in the autocorrelation function. Empirical research findings by Bhattacharya *et al.* (1983) and others were subsequently made rigorous by a number of authors, among them Giraitis *et al.* (2003), Berkes *et al.* (2006), Aue *et al.* (2009d), Harvey *et al.* (2010).

The contents of this paper are selective. We did not have the space to cover, for example, frequency domain and wavelet-based methods. The interested reader may refer Picard (1985), Adak (1998), Lavielle and Ludena (2000) and Ombao *et al.* (2001) for more information. Bayesian change-point methods can be found in Barry and Hartigan (1993) and Chib (1998). Additional references for contributions in econometrics and finance may be found in Andreou and Ghysels (2009), Perron (2006).

The paper is organized as follows. In Section 2, we discuss how the popular CUSUM procedures may be adjusted to serial dependence in the observations. Section 3 covers likelihood procedures for parametric time series models such as the popular ARMA processes. Section 4 deals with the estimation of multiple break points. Section 5 contains information on how to distinguish structural breaks from other departures of the null model, namely long memory and unit roots. Section 6 summarizes miscellanea, such as sequential procedures for time series and on the detection of breaks in functional data. We tried to keep the presentation of results illustrative. The interested reader may find a more in-depth analysis of results in the original research papers, many of which are given in the extensive yet still selective list of references.

### 2. CUSUM procedures under dependence

- Top of page
- Abstract
- 1. Introduction
- 2. CUSUM procedures under dependence
- 3. Likelihood ratio statistics
- 4. Estimating the number of breaks
- 5. Discriminating break points, long memory and unit roots
- 6. Miscellanea
- Acknowledgements
- References

Let denote the set of integers. One often used model is the signal-plus-noise model

- (1)

where is the signal and the noise component, which has *E*[*ɛ*_{t}] = 0 and . In many instances it is of interest to test for the structural stability of the signal. Structural stability may mean one of the following:

- •The signal is constant, that is, all
*μ*_{t}≡*μ*are identical. This is equivalent to saying that the*unconditional*mean of the*Y*_{t}’s does not change over time. - •If one has additional information on the form of the signal, expressed through an
*r*-dimensional covariate , one can use , with^{′}signifying transposition, and work in a linear regression framework. Structural stability then refers to the constancy of the regression coefficients, that is, all*β*_{t}≡are identical, which is equivalent to saying that the form of the*β**conditional*mean does not change over time. - •In some cases, one is also interested in testing for the structural stability of the (conditional) variance. Here may potentially be a function of time. More generally, it is of interest in the time series context to look into the stability of the second-order structure as expressed through the autocovariance function , , assuming zero-mean stationarity of .

It is the first two bullet points that this section is mainly concerned with, but some remarks will be added in the end on the stability of variances and covariances/correlations. Testing for the constancy of the unconditional mean is one of the most often studied problems in change-point analysis. Csőrgő and Horváth (1997) provide a survey of various methods for this case if the data can be assumed independent. Most prominent among these methods are versions of what is known as the cumulative sum (CUSUM) procedure. In this section, we show how CUSUM procedures may be adjusted to time series data.

#### 2.1. Structural breaks in the unconditional mean

To be specific, assume that *n* observations have been taken from the real-valued stochastic process in (1), and that we are interested in testing the null hypothesis of constant (unconditional) means

against the alternative that *H*_{0} is not true and that the mean has changed at least once during the observation period. The (rescaled) CUSUM process of the observations is given by

- (2)

where ⌊·⌋ denotes integer part. Note that, under *H*_{0}, , so that the value of the CUSUM is then independent of the unknown (but common) mean *μ*. It is now important to quantify the large-sample behavior of the partial sums of . Since the observations are possibly correlated, the classical Functional Central Limit Theorem (FCLT) as described, for example, in Billingsley's (1968) monograph cannot be applied directly. Define the standardized partial sum process by

Much recent research effort has then been devoted to providing conditions under which the weak convergence

- (3)

in the Skorohod space *D*[0, 1] holds. Here *W* = (*W*(*t*):*t* ∈ [0,1]) denotes a standard Brownian motion and *ω* > 0 is a scaling parameter to be discussed further below. Recent contributions discussing the weak convergence of dependent random variables are Bradley (2009b), Dedecker *et al.* (2007), Wu (2007) and Aue *et al.* (2007). If (3) holds, one immediately gets also that

- (4)

where *B* = (*B*(*t*) : *t* ∈ [0,1]), with *B*(*t*) = *W*(*t*) − *tW*(1), is a standard Brownian bridge. It can be seen from (4) that the CUSUM process limits for independent observations and time series data have the same form but differ in the scaling parameter *ω*. Since we have the relation , *ω*^{2} is typically referred to as the long-run variance. In applications, *ω*^{2} is unknown and has to be estimated from the data with an estimator . If this estimator is weakly consistent, (4) implies that

- (5)

Finding accurate estimates of *ω*^{2} under dependence can be a difficult task that is commonly approached through the use of kernel estimators of the Bartlett or flat-top type. For the most up-to-date, and essentially optimal, long-run variance estimation results we refer to the recent paper Liu and Wu (2010).

With the preceding, we can now construct a test procedure for *H*_{0}. Evaluated in the argument *x* = *k*/*n*, the CUSUM process *Z*_{n}(*x*) basically compares the sample mean of the observations up to lag *k* with the global sample mean of all observations. Since the timing of the break is unknown, one checks all possible choices *k* ∈ {1,…, *n*}. This leads to the max-type test statistics

whose values should be ‘small’ if *H*_{0} holds and ‘large’ if *H*_{0} is violated. To quantify this statement, one utilizes (5) and the Continuous Mapping Theorem to obtain

where signifies convergence in distribution. The distribution of *M* has been tabulated, for example in Shorack and Wellner (1986), and can therefore be used to construct tests that hold a pre-specified asymptotic level *α*. In the Example 1, we discuss the application of *M*_{n} in the case of autoregressive processes of order one (AR(1) processes).

**Example 1.** **(CUSUM for the Mean of AR(1) Processes)** Here, we highlight the CUSUM procedure if the innovations form an AR(1) process, so that

where is a sequence of independent standard normal random variables. The left panel of Figure 1 shows how the CUSUM process *Z*_{n} behaves under the null hypothesis of no mean break. It can be clearly seen how variability increases with increasing autoregressive parameter *φ*. The long-run variance in this example is computed as and needs to be estimated to appropriately rescale *Z*_{n}. If one is reasonably confident in the AR(1) assumption, one can do this parametrically. Otherwise, the lag-window estimator

can be used, where are the sample covariances, *w*_{h} = 1 − *h*/*n* are the Bartlett weights and is a bandwidth selector. Andrews (1991) recommended the choice with . Using the Bartlett estimator with this bandwidth, the scaled CUSUM processes are plotted in the right panel of Figure 1. For the simulated data, a visual inspection seems to be confirm that the approach works well. There are, however, other results in the literature that point to certain disadvantages using the Bartlett estimator for strong positive and negative correlations, among them Robbins *et al.* (2011) who recommended innovations-based CUSUM procedures (see also below). Note that the 10% critical value of the CUSUM statistic is 1.225 and that unadjusted correlation will clearly inflate the empirical levels.

The Examples 2 and 3 deal with non-obvious applications of the CUSUM method to checking the stability of the correlation in AR(1) processes and the stability of the expected volatility in GARCH(1,1) processes. To keep the exposition readable, we restrict the discussion to simple models. Similar results, however, are also expected to hold for higher-order counterparts of the AR(1) and GARCH(1,1) setting. It should be stressed that CUSUM-based procedures work reasonably well if the parameter of interest is a moment, for example, covariances as in Aue *et al.* (2009 b).

**Example 2.** **(CUSUM for the Correlation of AR(1) Processes)** Let us now assume that the data are generated by the AR(1) process , where are i.i.d. with *E*[*ɛ*_{1}] = 0, and . The interest here is in testing

against the alternative

for an unknown 1 ≤ *k*^{*} < *n*. Assume that under *H*_{0}, |*φ*| < 1, so that constitutes a stationary sequence. Also under *H*_{0}, we have that and a change in *φ* implies therefore a change also in the second moment. Thus, functionals of the CUSUM process *Z*_{n} in (2) with may be used. Since the fourth moments are finite, under *H*_{0}, satisfies (3) and (5).

Assume that, under *H*_{A}, the parameter *φ* changes to *φ*_{*} at *k*^{*} = ⌊*κ**n*⌋ for some *κ* ∈ (0, 1), where |*φ*|, |*φ*_{*}| < 1. Then, it can be shown that

- (6)

where signifies convergence in probability. Since the estimator for the long-run variance *ω*^{2} associated with the squared process remains bounded also under *H*_{0}, the asymptotic consistency of *M*_{n} is established, that is, *M*_{n} ∞ in probability as *n* ∞.

Figure 2 shows in the left panel the time series plot of *n* = 500 observations of which the first *k*^{*} = 250 were generated according to an AR(1) process with parameter *φ* = 0.6 and the remaining *n* − *k*^{*} = 250 observations from an AR(1) process with parameter *φ*_{*} = −0.2. Independent standard normal variates were used as innovations. The first half of the simulated data exhibits a smooth sample path, indicating positive correlation, while the second half is rougher, indicating negative correlation. The right panel shows the scaled CUSUM process of (6). Indicated as a horizontal line is also the limit in (6), which can be computed as 0.130 using the simulation parameters. The data gives that . This maximum is reached for the 210th observation. The second largest peak in the sample path of is reached at lag 254 and returns a value 0.131. Since the transition between pre-break and post-break sample is smooth, greater difficulties in locating the break time have to be expected.

**Example 3.** **(CUSUM for the Volatility of GARCH(1,1) Processes)** Let be given by

with *ν*_{t} > 0, and i.i.d. with *E*[*ɛ*_{t}] = 0 and . Assume that have been observed. This is the famous GARCH(1,1) process that parametrizes the conditional variance in terms of an ARMA(1,1) relation, see Engle (1982) and Bollerslev (1986). The relevant parameter vectors are and the null hypothesis of interest is

implying that the conditional variances are a stationary process. Note that, under *H*_{0}, is strictly stationary if . For additional information on variants of GARCH processes and their applications we refer to Aue *et al.* (2006a), and Francq and Zakoïan (2010).

Assume that, instead of testing *H*_{0} directly, we are interested in testing the stability of the expected volatilities . Notice that are unobservable quantities but that, by construction, , so that testing procedures may be set up using the CUSUM process *Z*_{n} in (2) with . According to Berkes *et al.* (2008), the partial sums of obey a FCLT, so (3) and (5) are satisfied.

The testing procedures based on *M*_{n} work best if the structural breaks occur close to the sample center, so that pre-break and post-break sample are of similar size. Power naturally decays if the breaks appear early or late. To improve detection ability for this latter case, weighted versions of the CUSUM procedure have been introduced. This leads to test statistics based on functionals of the type , where the supremum is taken over an interval and *q* is a weight function satisfying certain conditions (see Csőrgő and Horváth (1997) for details). Most prominent is the weight function as it is the standard deviation of the Brownian bridge *B*(*x*). These functionals are related to the maximally selected likelihood ratio test assuming normality, see Section 3 below. Notice that, if , then the Continuous Mapping Theorem implies that

Critical values for an application of the truncated CUSUM test can now be obtained via simulation of the limit process, whose distribution depends on the amount of truncation *ε*. If, however, *ε*_{n} = 0 or *ε*_{n} 0 as *n* ∞, then even under *H*_{0} and the test statistic has to be renormalized in order to obtain a non-degenerate limit distribution, namely the Gumbel or double exponential distribution. The resulting non-standard limit theorems are called Darling–Erdős-type results after Darling and Erdős (1956), see also Example 4. For these, standard FCLTs are not applicable and stronger assumptions on the underlying processes are needed such as strong approximations (see Einmahl (1989) for the independent case) and rates of convergence for .

Similar to the discussion in Example 2, one can show that the CUSUM-based test statistic *M*_{n} has asymptotic power one under the one-break alternative of exactly one mean change. The power may, however, not be monotone in the sense that a larger break size leads to a more powerful test. This is mainly due to the behaviour of the long-run variance estimator under the alternative. More specifically, the issue occurs because the bandwidth, chosen to work well under the null hypothesis, can be severely flawed under the alternative. Remedies have been offered by Altissimo and Corradi (2003), Juhl and Xiao (2009), and Shao and Zhang (2010).

Once a structural break has been detected, the time of its occurrence has to be estimated. The CUSUM can also be used to locate the break points. We continue to assume for now that, under *H*_{A}, there is exactly one break at time *k*^{*} = ⌊*n**κ*⌋ with some *κ* ∈ (0, 1). Notice that *k*^{*} itself cannot be estimated consistently, but that the break point fraction *κ* can be consistently estimated by

If the number of breaks in the underlying observations is equal to *m* and the observations can thus be divided into *m* + 1 homogeneous subsequences, then still converges in probability to one of the *m* distinct break locations. Therefore CUSUM-based testing procedures are consistent also against the *m*-break alternative. Results on the estimation of multiple break points can be found, for example, in Bai (1999), Banerjee and Urga (2005), Bernard *et al.* (2007) and Döring (2011). We return to the estimation of multiple breaks in Section 4 below.

The weak convergence in (5) has a multivariate extension that will be discussed in the remainder of this section. Assume that we change the signal-plus-noise model (1) to its *d*-dimensional counterpart

where is a process taking values in and are stationary time series innovations with *E*[*ɛ*_{t}] = 0 and . This setting is similar to the one discussed in Horváth *et al.* (1999). Based on observations , we wish to test the null hypothesis

against the at-least-one-change alternative. Following the steps leading to (2) above, one defines analogously the *d*-dimensional CUSUM process by

- (7)

Then, as in the univariate case, we have that under *H*_{0} the value of **Z**_{n}(*t*) is independent of the unknown but common mean parameter ** μ**. Let (

**S**

_{n}(

*x*) :

*x*∈ [0, 1]) be the

*d*-dimensional partial sum process

Aue *et al.* (2009b) and Bradley (2007) provide conditions under which the weak convergence

- (8)

in the *d*-dimensional Skorohod space *D*^{d}[0, 1] holds. The limit in the previous display is a *d*-dimensional Brownian motion with covariance matrix Ω, that is, **W**_{Ω} is Gaussian with *E*[**W**_{Ω}(*x*)] = 0 and . Notice that Ω is termed the long-run covariance matrix (or, a multiple of the spectral density matrix at the origin). It is assumed that one can construct a weakly consistent estimator for Ω, that is,

- (9)

Sufficient conditions for (9) to hold may be found, for example, in Liu and Wu (2010). If (8) and (9) are satisfied and if Ω is non-singular, one obtains that

- (10)

where denote independent standard Brownian bridges. Functionals of the quadratic form process may now be used to construct test statistics. Using (10) the Continuous Mapping Theorem yields that

The limit distribution M has been tabulated (see Shorack and Wellner (1986)) and thus asymptotic test procedures can now be constructed. Weighted versions of the multivariate CUSUM process, such as

with lower bound *l*_{n} = 1/(*n* + 1) and upper bound *u*_{n} = *n*/(*n* + 1), which are more sensitive to find structural breaks in the beginning and the end of the sample, have also been considered in the literature. More information on this may be found in Csőrgő and Horváth (1997).

#### 2.2. Structural breaks in the conditional mean

The CUSUM methodology introduced in the Section 2.1 can be modified to cover linear regression models. To this end, we return to the model (1) and assume that the signal component has the form with some *r*-dimensional covariate process . This gives the model

Assume that observations are available for . The null hypothesis then becomes

which has to be tested against the alternative that *H*_{0} is not true. To construct the test, one computes first an estimator for ** β**, typically using least squares techniques, and defines then the estimated residuals . If

*H*

_{0}is true, the estimated residuals will be ‘close’ to the innovations. If

*H*

_{0}is violated, the estimated residuals should systematically deviate from the innovations. This can be exploited if the CUSUM process

*Z*

_{n}in (2) is built with in place of the observations . This scenario has been considered (including extensions) by Bai and Perron (1998). These authors also provide a set of non-restrictive conditions on the covariates and the innovations that allow for an asymptotic theory akin to the one outlined in Section 2.1. Both are notably allowed to exhibit time series character. Extensions to the multivariate regression setting may be found in Qu and Perron (2007). The use of weighted residuals is discussed in Hušková

*et al.*(2007) for the case that is an AR process.

Other choices for regressors that may be used include polynomials as in Aue *et al.* (2008, 2009), Kulperger (1985), Tang and MacNeill (1993), as well as more general smooth functions satisfying certain regularity conditions, see Aue *et al.* (2012). Importantly, lagged response variables have been used as regressors and therefore provided a structural breaks framework for autoregressive time series that is based on innovations rather than the observations themselves. This approach has been advocated by Bai (1993) and Yu (2007) in a more general ARMA setting and more recently by Robbins *et al.* (2011), who show that empirical levels of innovations-based test statistics tend to be less inflated than those of observation-based test statistics, and closer to the nominal levels.

#### 2.3. Structural breaks in the variance and second-order characteristics

Testing for structural breaks in the variance has been of interest as well. Important contributions for the independent setting are due to Inclán and Tiao (1994), who developed an iterative cumulative sums of squares algorithm and applied it to IBM stock data. Related work was done by Gombay *et al.* (1996), who studied the detection of possible breaks in the variance with and without concurrent breaks in the mean in a sequence of independent observations.

In the time series case, it often makes sense to assume a linear process structure. Owing to the Wold decomposition, any purely non-deterministic zero-mean stationary process can be represented in the form

- (11)

where are coefficients determining the dependence structure. To enable more parsimonious modeling, these coefficients are typically approximated by ARMA process fitting. Lee and Park (2001) showed how Inclán and Tiao's (1994) test can be applied to linear processes by computing the appropriate scaling. This is similar to the adjustment of CUSUM procedures for the mean.

More generally, the linear process in (11) can be utilized to detect structural breaks in covariance and correlation structure. This was done in Berkes *et al.* (2009b), who gave results related to the stability of the covariances based on weighted approximations. In a nonlinear time series setting parametric procedures were utilized by Kokoszka and Leipus (2000) to detect breaks in the parameters of ARCH processes, and by Berkes *et al.* (2009b) to sequentially monitor for breaks in the parameters of GARCH processes.

The multivariate variance-covariance structure of a large class of time series was investigated by Aue *et al.* (2009b). Wied *et al.* (2012) use a similar approach to test for the constancy of cross-correlations in a bivariate setting. Andreou and Ghysels (2002) were concerned with the dynamic evolution of financial market volatilities.

### 3. Likelihood ratio statistics

- Top of page
- Abstract
- 1. Introduction
- 2. CUSUM procedures under dependence
- 3. Likelihood ratio statistics
- 4. Estimating the number of breaks
- 5. Discriminating break points, long memory and unit roots
- 6. Miscellanea
- Acknowledgements
- References

The CUSUM procedures discussed in Section 2 are non-parametric and do not make use of a particular time series model fit. The dependence of the observations enters only in form of the long-run variance estimators and . In many instances, parametric time series assumptions are being made that describe explicitly the dependence structure found in the data. As a consequence, forecasting procedures may be easily implemented and can be based on existing algorithms as long as the observations satisfy the underlying no structural break null hypothesis. Here, discuss structural break procedures based on likelihood methods.

Let us assume that we are interested in the *d*-dimensional process and that the distribution of **Y**_{t} depends on an unknown parameter vector *θ*_{t}. Assume further that have been observed. In this case the null hypothesis of structural stability becomes

which is to be tested against the alternative that there is exactly one unknown break point *k*^{*}, that is,

To do so, suppose for the moment that the break was located at lag *k*. Then, one can split the data into the two subsamples and and costruct the likelihood function taking into account that these subsamples have different generating parameters *θ*_{0} and . The likelihood can now be compared to the likelihood *L*_{n}(** θ**) coming from the null model, assuming that the parameter

*θ*_{0}generated the observations. The comparison is done via the likelihood ratio

where, is the maximum likelihood estimator for *θ*_{0} for the first subsample and is the maximum likelihood estimator for for the second subsample. Since the location of the break is unknown, one rejects the null hypothesis *H*_{0} when the maximally selected log-likelihood ratio statistic

is large. The distribution of *Z*_{n} and other functionals of −2 log Λ_{k}, *k* = 1,…, *n* − 1, were derived in a series of papers by Gombay and Horváth (1990, 1994, 1996) and Horváth (1993) in the case of independent observations.

In time series models, it can be difficult to compute the joint distributions of and . In this case, the quasi-likelihood method can be used. Usually the normality of the observations is assumed to get an explicit expression for Λ_{k}, but the properties of Λ_{k} and *Z*_{n} are studied without using normality. The following example shows how the likelihood-based methodology can be applied to autoregressive time series.

**Example 4.** **(AR Processes)** We focus on the case of univariate observations and follow Davis *et al.* (1995), who considered the segmented autoregressive model

with and i.i.d. innovations satisfying *E*[*ɛ*_{1}] = 0 and . Under the null hypothesis *H*_{0} of structural stability, we have thus *k*^{*} ≥ *n* and all observations are generated from the parameter vector . Under the one structural break alternative *H*_{A}, we have that 1 ≤ *k*^{*} < *n* and the observations are generated from the parameter vector prior to *k*^{*} and by thereafter.

Assuming that the errors follow a normal distribution, the likelihood ratio Λ_{k} can be constructed and the maximally selected log-likelihood ratio statistic *Z*_{n} computed. If *H*_{0} should hold, then one obtains an extreme value limit distribution, in accordance with the discussion in Section 2. More specifically, it is shown in Davis (1995) that, under additional regularity assumptions not stated here,

where

and Γ(·) denotes the Gamma function. It should be noted again that this limit result does not require the normality of the innovations. For the application of the test procedure *σ*^{2} has to be replaced with a consistent estimator converging at a fast enough rate.

The extreme value asymptotics in Example 4 appears because *Z*_{n} is, by definition, built from a maximum over all possible break locations. Since this includes also the very early and late lags, standard limit results based on FCLTs do not apply anymore and additional normalization in form of the centering sequence *b*_{n} and scaling sequence *a*_{n} are required to obtain a non-degenerate limit distribution. Many authors (among them Andrews (1993), Bai (1999), Bai and Perron (1998), Ghysels *et al.* (1997) and Hansen (2000)) have, instead of applying extreme value theory, resorted to a truncation of *Z*_{n} and thus enabled standard limit theory.

It is known (see Hall (1979)), that the convergence to the extreme value limit can be slow and asymptotic tests often tend to be too conservative in finite samples. To circumvent this issue, one can use resampling and bootstrap methods that lead to a better approximation of the test levels. Related literature, including both retrospective and sequential methods, is Aue *et al.* (2012), Hušková (2004), Hušková and Picek (2005), Kirch (2008), and Hušková and Kirch (2012).

### 4. Estimating the number of breaks

- Top of page
- Abstract
- 1. Introduction
- 2. CUSUM procedures under dependence
- 3. Likelihood ratio statistics
- 4. Estimating the number of breaks
- 5. Discriminating break points, long memory and unit roots
- 6. Miscellanea
- Acknowledgements
- References

Here, we briefly discuss several approaches to estimating and locating multiple break points in the observations. This problem has since its origins, for example in Yao (1988), been treated as a model selection problem. In this framework, one can view the segmented, piecewise stationary time series model as the one that best matches the observations, even without explicitly assuming that the underlying model be true for the data.

For most of this section, we focus on the form

- (12)

where , *E*[*ɛ*_{t}] = 0 and . Equation (12) describes a model which has piecewise constant means and variances. All mean and standard deviation parameters, *μ*_{ℓ} and *σ*_{ℓ}, are unknown, as are the number of breaks *m*^{0} and the break locations . Assuming that the observations are normal and changes occur only in the mean, Yao (1988) suggested to use Schwarz’ (1978) criterion to estimate *m*^{0}. Yao's result was modified by Serbinowska (1996) to cover the case of binomial observations. Her method was applied to determine the number of authors of the Lindisfare Gospels. Kühn (2001) extended the Schwarz’ criterion to the time series case using strong invariance principles.

To estimate the number of breaks in observations given by (12), one computes, for *m* ≤ *M* and a given candidate segmentation , the residual sum of squares

where denotes the sample mean of the ℓth segment. Define

with the minimum being taken over the candidate segmentations . Yao (1988) suggested to estimate *m*^{0} with

Notice that the first term in SC is related to the log-likelihood and is thus a measure for the goodness of the fit, while the second term is a penalty term that is applied to avoid overfitting. Roughly speaking, *d*_{n} must be larger than the rate of convergence of the partial sums on the segments (without breaks) to standard Brownian motions. Kühn (2001) has proved that for time series satisfying strong invariance principles one obtains weak consistency,

Vostrikova's binary segmentation procedure works well for estimating the number of breaks only if *m*^{0} is rather small. This procedure can be based, for example, on the maximum-type CUSUM procedures discussed in Section 2, since they reach their largest value in the neighbourhood of a (true) break point. Using this largest value to split the sample into two, and repeating the same steps on the subsamples, one obtains a multiple testing procedure. While the binary segmentation can be used under fairly general assumptions on the underlying process, it is consistent only of the significance levels chosen in each step converge to zero with increasing sample size. The interested reader may confer Bai (1997, 1999), Bai and Perron (1998), and Qu and Perron (2005) for more details.

In a piecewise stationary autoregressive time series setting, Davis *et al.* (2006) proposed an estimator for *m*^{0} based on the minimum description length principle (which amounts to selecting a different penalty term *d*_{n}). The consistency of the procedure is assessed empirically and a consistency result for the relative break locations *κ*_{ℓ}, where , is proved in the case that *m*^{0} is known. Aue and Lee (2011) used a similar approach for image segmentation. Kurozumi and Tuvaandorj (2011) returned to modifications of the classical model selection criteria to estimate *m*^{0}. They use variants of Akaike's information criterion, the Bayes information criterion and Mallow's *C*_{p} criterion and discuss conditions for consistency of these procedures.

### 5. Discriminating break points, long memory and unit roots

- Top of page
- Abstract
- 1. Introduction
- 2. CUSUM procedures under dependence
- 3. Likelihood ratio statistics
- 4. Estimating the number of breaks
- 5. Discriminating break points, long memory and unit roots
- 6. Miscellanea
- Acknowledgements
- References

Tests for structural stability are not robust against other deviations from the null such as long memory and unit roots. All three of these phenomena would, for example, inherit an autocorrelation function with many significant lags that decay only slowly. Differences can be hard to detect based on finite samples. In this section, we summarize the research in this still active area, starting with the breaks vs. long memory case in Section 5.1 and continuing with the breaks vs. unit roots case in Section 5.2.

#### 5.1. Structural breaks and long memory

A stationary process is said to exhibit long memory if its autocovariance function *γ* is not absolutely summable, that is, . An example of long memory processes is the class of fractionally integrated ARMA time series introduced in Granger and Joyeux (1980) and Hosking (1981), in which the difference operator (1−*B*)^{d} is applied for fractional values of *d*. Stationarity is given if . It can be seen that the spectral density of a long memory process is unbounded at the origin. Popular applications of long memory processes may be found in hydrology, environmental sciences and notably macroeconomics.

Research in the past two decades has, however, revealed that the features described above, typically regarded as indicative of long memory, can also occur in short memory processes, for which , affected by structural breaks in the mean or trend. For example, Bhattacharya *et al.* (1983) showed that the so-called Hurst effect (to achieve convergence, partial sums have to be scaled with *n*^{H}, being the Hurst parameter, and *d* > 0) can also be explained in a short memory scenario if structural breaks in the mean are allowed. Giraitis *et al.* (2001) followed up on these findings by showing that a number of popular test statistics used to detect long memory diverge to infinity also in the presence of structural breaks in a short memory sequence. Attention was consequently given to the development of statistical procedures that could discriminate between these situations.

Berkes *et al.* (2006) phrased this problem in the form of a hypothesis test with the null hypothesis corresponding to structural breaks in the mean of a short memory time series. More precisely, the models under consideration are the following.

- •Under the null hypothesis
*H*_{0}, the observations come from the structural break model of Section 2, namelywhere*μ*≠*μ*^{*}. The innovations are assumed to satisfy the FCLT (3) and be fourth-order stationary. - •Under the alternative
*H*_{A}, the fourth-order stationary observations satisfy , where the normalized partial sum process given by

satisfies the weak convergence

- (13)

for some and *c*_{H} > 0. Here denotes a fractional Brownian motion with Hurst parameter *H*, that is, a Gaussian process with zero mean and covariances .

The test statistic for these hypotheses is constructed as follows. Let be an estimator for the time of change as discussed in Sections 2 and 4. Use this estimate to split the observations into the two subsamples and . Recall the definition of the CUSUM process *Z*_{n} in (2) and define the max-type test statistics

where denotes the CUSUM process for the second subsample, assuming further that and are consistent estimators of the long-run variance, based on the two subsamples, of under *H*_{0}. The test statistic is then defined as

- (14)

Let *B*^{(1)} and *B*^{(2)} denote two independent standard Brownian bridges. Under additional regularity assumptions on the timing and the magnitude of the break, one can show with the help of the weak convergence in (2.5) that, under *H*_{0},

while, under *H*_{A}, , thereby showing consistency.

**Example 5.** The sample ACF plotted in the left panel of Figure 3 comes from *n* = 500 observations of the process , where *μ*_{t} = 0 if *t*≤*k*^{*} = 250 and *μ*_{t} = 1 if *t* > *k*^{*} = 250, and are from an AR(1) time series with parameter *φ* = 0.8. This is an example of a time series observed under *H*_{0}. In the right panel of Figure 3, we see the sample ACF of *n* = 500 observations of the long-memory process

where and (*ξ*_{t}) are independent standard normals. For the simulations, has been truncated at an upper limit *N*=10,000. Both sample ACF plots display the same features and it seems possible to mistake long-memory for short-memory subject to mean breaks, and vice versa. An application of separates the two phenomena. First, consider the short-memory series. The break point is estimated at . Then, and are computed. The corresponding plot in the left panel of Figure 4 shows that now the null of short memory can no longer be rejected and long memory is ruled out. Second, consider the long-memory series. The break point is estimated at . Then, and are computed and shown in the right panel of Figure 4. Short memory with a mean break is rejected and the series is identified to have strong dependence. Note that the 10% asymptotic critical value is 1.36.

Other contributions in the literature reversed the roles of *H*_{0} and *H*_{A}. Among them are Ohanissian *et al.* (2008) who used an aggregation approach, Qu (2011) who utilized a frequency domain test statistic, and Shao (2011) who worked with a CUSUM-type statistics. Baek and Pipiras (2012) designed tests based on an estimation of the self-similarity parameter. Additionally, Hidalgo and Robinson (1996), Lazarová (2005) and Yoon (2005) provided tests for structural breaks in the mean if the errors exhibit long memory. The effects of persistence and breaks in volatility were examined in Giraitis *et al.* (2003). Finally, Yamaguchi (2011) estimated the time of change when the long-memory parameter is subject to change.

#### 5.2. Structural breaks and unit roots

Fractionally integrated processes as discussed in the previous section were introduced to bridge the gap between stationary time series and non-stationary time series with unit roots such as the random walk. The latter are important in econometrics. In this section, we discuss how to separate breaks in the mean from random walks and also from innovations that switch from a stationary to a random walk behaviour. In particular, we are interested in the following.

- •Under the null hypothesis
*H*_{0}, the observations come from the structural break model of Section 2, namelywhere*μ*≠*μ*^{*}. The innovations are assumed to satisfy the FCLT (3). - •Under the first alternative , the observation have a constant mean
*μ*and the process satisfies the weak convergencewith some scaling parameter- (15)

*ω*and a Brownian motion*W*= (*W*(*t*):*t*∈ [0,1]). - •The second alternative has constant means
*μ*, but at lag*k*^{*}the stationary observations start to display unit-root behaviour, that is, are such that the innovations satisfy (3) and satisfy (15).

Several test statistics will be considered. The first procedure is the CUSUM procedure, adjusted for a mean break, in (14) and therefore the prototype of statistics used to detect breaks in level. The second procedure is based on the modified adjusted range statistic

as proposed by Lo (1991). Using an estimator for the time of change, one can proceed as in the previous section and modify *R*_{n} for a potential mean break and obtain

where denotes the sample mean of . This gives the test statistic .

A modification (by a mean correction) of the KPSS test (Kwiatkowski *et al.*, 1992) is

This rescaled variance statistic was considered by (Giraitis *et al.*, 2001, 2003). Mimicking the above steps, one constructs again two statistics on the subsamples that are split at the estimated break point to obtain

from which the test statistics is computed. If *B*^{(1)} and *B*^{(2)} denote again two independent standard Brownian bridges, and if other regularity conditions are met, one gets the following limit distributions under the null hypothesis. It holds, under *H*_{0} and if *n*∞,

The limit distributions can be easily obtained as they are maxima of two independent random variables with known distribution functions, see Shorack and Wellner (1986). It can be shown that all tests are weakly consistent against both and . The results presented here follow Aue *et al.* (2009d). All omitted details can be found there.

**Example 6.** Suppose that we generate *n* = 500 observations of the random walk (*Y*_{t}), where and (*ξ*_{t}) a sequence of independent standard normals. The left panel of Figure 5 shows a typical ACF plot, which is similar as those in Figure 3. An application of first estimates the break point location at . Then, and are computed. The corresponding plot in the right panel of Figure 5 shows that now the null of short memory is rejected at the 10% significance level for which the critical value equals 1.36. One should note the greater smoothness in this plot (compared to the plots in Figure 4). Similar behaviours can be shown for and , but are omitted here.

Related contributions are Kim *et el.* (2002). Cavaliere and Taylor (2008). Harvey *et al.* (2009, 2010). These authors study the possibility of breaks in the mean and/or the variance of the innovations and the unit-root problem. Their methodology proceeds by first attending to the mean/variance breaks and then to test whether the errors are stationary or exhibit unit roots. An interesting application can be found in King and Ramlogan-Dobson (2011).

### 6. Miscellanea

- Top of page
- Abstract
- 1. Introduction
- 2. CUSUM procedures under dependence
- 3. Likelihood ratio statistics
- 4. Estimating the number of breaks
- 5. Discriminating break points, long memory and unit roots
- 6. Miscellanea
- Acknowledgements
- References

We briefly discuss several other lines of research in the structural break field that have some impact. First, we mention some of the contributions to sequential methodologies as they apply to time series in Section 6.1. Second we discuss in Section 6.2 some recent publications in the field of functional data analysis, where contributions discussing the temporal dependence are still sparse.

#### 6.1. Sequential procedures

While the body of literature concerned with retrospective break point tests and estimation procedures is rich, this has as of late not been true to the same degree for the corresponding sequential procedures. Starting with the seminal paper Chu *et al.* (1996) this has slowly changed. These authors have developed fluctuation tests that are based on the general paradigm that an initial time period of length *n* is used to estimate a model with the goal to monitor for parameter changes on-line.The asymptotic analysis is carried out for *n*∞. To test the null hypothesis of structural stability sequentially, one defines a stopping time *τ*_{n} that rejects the null as soon as a suitably constructed detector function Γ_{n} crosses an appropriate threshold *g*_{n} (measuring the growth of the detector under the null), that is,

Sequential tests based on CUSUM-based detector functions were considered in Aue *et al.* (2006, 2009) and Horváth *et al.* (2004) for linear and time series regressions, in Gombay and Serban (2009), Hušková*et al.* (2007) for AR processes, and in Berkes *et al.* (2004) for GARCH processes. Hušková and Kirch (2012) and Kirch (2008) developed bootstrap techniques for sequential CUSUM procedures. The detection of breaks in counting processes is covered in Gut and Steinebach (2002, 2009). Aue *et al.* (2012) designed sequential monitoring procedures to test for the stability of the betas in a functional version of the Capital Asset Pricing Model. Other contributions in the literature utilize moving sum, MOSUM, procedures. These have the advantage of faster detection times when compared to CUSUM procedures. A unifying view based on generalized fluctuation tests, incorporating CUSUM and MOSUM procedures as special cases, was offered in Chu *et al.* (1999a, b), Leisch *et al.* (2000), Kuan and Hornik (2005), and Zeileis *et al.* (2005). Starting with Aue and Horváth (2004), there have been a number of contributions deriving the limit distribution of the stopping time *τ*_{n} under the alternative of a structural break, see for example Aue *et al.* (2008, 2009). Steland (2007) reversed the roles of null and alternative and monitors under the unit-root null hypothesis. Pawlak *et al.* (2010) designed nonparametric methods based on the vertical box control chart.

#### 6.2. Structural breaks for functional data

A field that has seen increased research output is functional data analysis. Research in this area assumes that data can be described by smooth curves rather than as discrete observations. This approach has become popular both in situations of dense data (roughly, many observations per curve) and sparse data (roughly, few observations per curve). The relevant lines of research are presented in Ramsay and Silverman (2005), Horváth and Kokoszka (2012). Early contributions on structural breaks in functional data are Berkes *et al.* (2009), who developed a functional CUSUM-type test exploiting functional principal components analysis and applied it to temperature records viewing annual profiles as one functional observation. Aue *et al.* (2009e) analyzed the limit distribution of a break point estimator in the same setting. The time series character of functional data was for the first time systematically treated in Hörmann and Kokoszka (2010), who also highlighted how structural break procedures are affected by serial functional correlation. Horváth *et al.* (2010) investigated how one might test for structural stability of the autoregressive operator in a Hilbert-space valued autoregressive process. Many open problems remain.

### Acknowledgements

- Top of page
- Abstract
- 1. Introduction
- 2. CUSUM procedures under dependence
- 3. Likelihood ratio statistics
- 4. Estimating the number of breaks
- 5. Discriminating break points, long memory and unit roots
- 6. Miscellanea
- Acknowledgements
- References

This research was partially supported by NSF grant DMS 0905400

### References

- Top of page
- Abstract
- 1. Introduction
- 2. CUSUM procedures under dependence
- 3. Likelihood ratio statistics
- 4. Estimating the number of breaks
- 5. Discriminating break points, long memory and unit roots
- 6. Miscellanea
- Acknowledgements
- References

- 1998) Time dependent spectral analysis of non-stationary time series. Journal of the American Statistical Association 93, 1488–1501. (
- 2003) Strong rules for detecting the number of breaks in a time series. Journal of Econometrics 117, 207–44. and (
- 2002) Detecting multiple breaks in financial market volatility dynamics. Journal of Applied Econometrics 17, 579–600. and (
- 2009) Structural breaks in financial time series. In: Handbook of Financial Time Series, (eds T. G. Andersen
*et al.*). Berlin: Springer-Verlag, pp. 839–70. and ( - 1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–58. (
- 1993) Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–56. (
- 2006a) Strong approximation for the sums of squares of augmented GARCH sequences. Bernoulli 12, 583–608. , and (
- 2009a) Estimation of a change-point in the mean function of functional data. Journal of Multivariate Analysis 100, 2254–69. , , and (
- Sequential testing for the stability of high frequency portfolio betas. Econometric Theory 28, 804–37. , , , and
- 2009b) Break detection in the covariance structure of multivariate nonlinear time series models. The Annals of Statistics 37, 4046–87. , , and (
- 2004) Delay time in sequential detection of change. Statistics & Probability Letters 67, 221–31. and (
- 2009c) Extreme value theory for stochastic integrals of Legendre polynomials. Journal of Multivariate Analysis 100, 1029–43. , , and (
- 2012) Segmenting mean-nonstationary time series via trending regressions. Journal of Econometrics 168, 367–81. , and (
- 2006b) Change-point monitoring in linear models. Econometrics Journal 9, 373–403. , , and (
- 2008a) Testing for changes in polynomial regression. Bernoulli 14, 637–60. , , and (
- 2009d) On distinguishing between random walk and change in the mean alternatives. Econometric Theory 25, 411–41. , , and (
- 2008b) Monitoring shifts in mean: asymptotic normality of stopping times. Test 17, 515–30. , , and (
- 2009e) Delay times of sequential procedures for multiple time series regression models. Journal of Econometrics 149, 174–90. , and (
- 2011) On image segmentation using information theoretic criteria. The Annals of Statistics 39, 2912–35. , and (
- 2012) Statistical tests for a single change in mean against long-range dependence. Journal of Time Series Analysis 33, 131–51. and (
- 1993) On the partial sums of residuals in autoregressive and moving average models. Journal of Time Series Analysis 14, 247–60. Direct Link: (
- 1997) Estimating multiple breaks one at a time. Econometric Theory 13, 315–52. (
- 1999) Likelihood ratio tests for multiple structural changes. Journal of Econometrics 91, 299–323. (
- 1998) Estimating and testing linear models with multiple structural changes. Econometrica 66, 47–78. , and (
- 2005) Modelling structural breaks, long memory and stock market volatility: an overview. Journal of Econometrics 129, 1–34. , and (
- 1993) A Bayesian analysis for change point problems. Journal of the American Statistical Association 88, 309–19. and (
- 2009a) Detecting changes in the mean of functional observations. Journal of the Royal Statistical Society, Series B 71, 927–46. , , and (
- 2009b) Testing for changes in the covariance structure of linear processes. Journal of Statistical Planning and Inference 139, 2044–63. , and (
- 2004) Sequential change-point detection in GARCH(p,q) models. Econometric Theory 20, 1140–67. , , and (
- 2008) The functional central limit theorem for a family of GARCH observations with applications. Statistics & Probability Letters 78, 2725–30. , and (
- 2006) On discriminating between long-range dependence and changes in the mean. The Annals of Statistics 34, 1140–65. , , and (
- 2007) Finite sample multivariate structural change tests with application to energy demand models. Journal of Econometrics 141, 1229–44. , , and (
- 1983) The Hurst effect under trends. Journal of Applied Probability 20, 649–67. , and (
- 1968) Convergence of Probability Measures. New York: Wiley. (
- 1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–27. (
- 2007) Introduction to Strong Mixing Conditions Vol.1–3.. Heber City, UT: Kendrick Press. (
- 1993) Nonparametric Methods in Change-Point Problems. Dordrecht: Kluwer. and (
- 1994) Change-Point Problems. IMS Lecture Notes–Monograph Series, Vol. 23. Hayward, CA. , and (
- 2008) Testing for a change in persistence in the presence of non-stationary volatility. Journal of Econometrics 147, 84–98. and (
- 1998) Estimation and comparison of multiple change-point models. Journal of Econometrics 86, 221–41. (
- 1995) MOSUM tests for parameter constancy. Biometrika 82, 603–17. , and (
- 1995) The moving-estimates test for parameter stability. Econometric Theory 11, 699–720. , and (
- 1996) Monitoring structural change. Econometrica 64, 1045–65. , and (
- 1997) Limit Theorems in Change-Point Analysis. Chichester: Wiley. and (
- 1956) A limit theorem for the maximum of normalized sums of independent random variables. Duke Mathematics Journal 23, 143–55. and (
- 1995) Testing for a change in the parameter values and order of an autoregressive model. The Annals of Statistics 23, 282–304. , and (
- 2006) Structural break estimation for nonstationary time series models. Journal of the American Statistical Association 101, 223–39. , and (
- 2007) Weak Dependence with Applications and Examples. Lecture Notes in Statistics 190. New York: Springer. , , , , and . (
- 2011) Convergence in distribution of multiple change point estimators. Journal of Statistical Planning and Inference 141, 2238–48. (
- 1989) The Darling-Erdős theorem for sums of i.i.d. random variables. Probability Theory and Related Fields 82, 241–57. (
- 1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007. (
- 2010) GARCH Models. Chichester: Wiley. and (
- 1997) Predictive tests for structural change with unknown breakpoint. Journal of Econometrics 82, 209–33. , , and (
- 2001) Testing for long memory in the presence of a general trend. Journal of Applied Probability 38, 1033–54. , and (
- 2003) Rescaled variance and related tests for long memory in volatility and levels. Journal of Econometrics 112, 265–94. , , and (
- 1990) Asymptotic distributions of maximum likelihood tests for change in the mean. Biometrika 77, 411–14. and (
- 1994) An application of the maximum likelihood test to the change-point problem. Stochastic Processes and Their Applications 50, 161–71. and (
- 1996) On the rate of approximations for maximum likelihood tests in change-point models. Journal of Multivariate Analysis 56, 120–52. and (
- 2009) Monitoring parameter change in AR(p) time series models. Journal of Multivariate Analysis 100, 715–25. and (
- 1996) Estimators and tests for change in variances. Statistics & Decisions 14, 145–59. , and (
- 1980) An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis 1, 15–29. Direct Link: and (
- 2002) Truncated sequential change-point detection based on renewal counting processes. Scandinavian Journal of Statistics 29, 693–719. and (
- 2009) Truncated sequential change-point detection based on renewal counting processes II. Journal of Statistical Planning and Inference 139, 1921–36. and (
- 1979) On the rate of convergence of normal extremes. Journal of Applied Probability 16, 433–39. (
- 2000) Testing for structural change in conditional models. Journal of Econometrics 97, 93–115. (
- 2009) Simple, robust, and powerful tests of the breaking trend hypothesis. Econometric Theory 25, 995–1029. , and (
- 2010) Robust methods for detecting multiple level breaks in autocorrelated time series. Journal of Econometrics 157, 342–58. , and (
- 1996) Testing for structural change in a long-memory environment. Journal of Econometrics 70, 159–74. and (
- 2010) Weakly dependent functional data. The Annals of Statistics 38, 1845–84. and (
- 1993) The maximum likelihood method for testing changes in the parameters of normal observations. The Annals of Statistics 2, 671–80. (
- 2010) Testing the stability of the functional autoregressive process. Journal of Multivariate Analysis 101, 352–67. , and (
- 2004) Monitoring changes in linear models. Journal of Statistical Planning and Inference 126, 225–51. , , and (
- 2012) Inference for Functional Data with Applications. New York: Springer. and (
- 1999) Testing for changes in multivariate dependent observations with an application to temperature changes. Journal of Multivariate Analysis 68, 96–119. , and (
- 1981) Fractional differencing. Biometrika 68, 165–76. (
- 2004) Permutation principle and bootstrap in change point analysis. In Asymptotic Methods in Stochastics. (eds L. Horváth, and B. Szyszkowicz,). Providence, RI: Fields Institute Communications 44, American Mathematical Society, pp. 273–91. (
- 2012) Bootstrapping sequential cange-point tests for linear regression. Metrika 75, 673–708. , and (
- 2005) Bootstrap in detection of changes in linear regression. Sankhy: The Indian Journal of Statistics 67, 1–27. and (
- 2007) On the detection of changes in autoregressive time series I. Asymptotics. Journal of Statistical Planning and Inference 137, 1243–59. , and (
- 1994) Use of cumulative sums of squares for retrospective detection of change of variance. Journal of the American Statistical Association 89, 913–23. and (
- 2009). Testing for changing mean monotonic power. Journal of Econometrics 48, 14–24. and (
- 2002) Unit root tests with a break in innovation variance. Journal of Econometrics 109, 365–87. , and (
- 2011) Nonlinear time-series convergence: The role of structural breaks. Economics Letters 110, 238–40. and (
- 2008) Bootstrapping sequential change-point tests. Sequential Analysis 27, 330–49. (
- 2000) Change-point estimation in ARCH models. Bernoulli 6, 513–39. and (
- 2005) The generalized fluctuation test: A unifying view. Econometric Reviews 82, 603–17. and (
- 2001) An estimator of the number of change-points based on a weak invariance principle. Statistics & Probability Letters 51, 189–96. (
- 1985) On the residuals of autoregressive processes and polynomial regression. Stochastic Processes and their Applications 21, 107–18. (
- 2011) Model selection criteria in multivariate models with multiple structural changes. Journal of Econometrics 164, 218–38. and (
- 1992) Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root?. Journal of Econometrics 54, 159–78. , , and (
- 2000) The multiple change-points problem for the spectral distribution. Bernoulli 6, 845–69. and (
- 2005) Testing for structural change in regression with long memory errors. Journal of Econometrics 129, 329–72. (
- 2001) The cusum of squares test for scale changes in infinite order moving average processes. Scandinavian Journal of Statistics 28, 625–44. and (
- 2000) Monitoring structural changes with the generalized fluctuation test. Econometric Theory 16, 835–54. , and (
- 2010) Asymptotics of spectral density estimates. Econometric Theory 26, 1218–45. and (
- 1991) Long-term memory in stock market prices. Econometrica 59, 1279–1313. (
- 2010) An MDL approach to the climate segmentation problem. The Annals of Applied Statistics 4, 299–319. , and (
- 2008) True or spurious long memory? A new test. Journal of Business & Economic Statistics 26, 161–75. , and (
- 2001) Automatic statistical analysis of bivariate nonstationary time series. Journal of the American Statistical Association 96, 543–60. , , and (
- 1954) Continuous inspection schemes. Biometrika 41, 100–5. (
- 1955) A test for a change in a parameter occurring at an unknown point. Biometrika 42, 523–27. (
- 2010). Nonparametric sequential change-point detection by a vertically trimmed box method. IEEE Transactions on Information Theory 56, 3621–3634 . , and (
- 2006) Dealing with structural breaks. In Palgrave Handbook of Econometrics, Vol. 1, (eds K. Patterson, and T. C. Mills, ). xxxx: Palgrave Macmillan, pp. 278–352. (
- 1985) Testing and estimating change-points in time series. Advances in Applied Probability 17, 841–67. (
- 2011) A test against spurious long memory. Journal of Business & Economic Statistics 29, 423–38. (
- 2007) Estimating and testing structural changes in multivariate regressions. Econometrica 75, 459–502. and (
- 2005) Functional Data Analysis, 2nd ed. New York: Springer. , and (
- 2007) A review and comparison of changepoint detection techniques for climate data. Journal of Applied Meteorology and Climatology 46, 900–15. , , , and (
- 2011) Mean shift testing in correlated data. Journal of Time Series Analysis 32, 498–511. , , and (
- 2011) Changepoints in the North Atlantic tropical cyclone record. Journal of the American Statistical Association 106, 89–99. , , and (
- 1978) Estimating the dimension of a model. The Annals of Statistics 6, 461–64. (
- 1996) Consistency of an estimator of the number of changes in binomial observations. Statistics & Probability Letters 29, 337–44. (
- 2011) A simple test of changes in the mean in the possible presence of long-range dependence. Journal of Time Series Analysis 32, 598–606. (
- 2010) Testing for change points in time series. Journal of the American Statistical Association 105, 1228–40. and (
- 1986) Empirical Processes with Applications to Statistics. New York: Wiley. and (
- 2007) Monitoring procedures to detect unit roots and stationarity. Econometric Theory 23, 1108–35. (
- 2000) The state of statistical process control as we proceed into the 21st century. Journal of the American Statistical Association 95, 992–8. , , , (
- 1993) The effect of serial correlation on tests for parameter change at unknown time. The Annals of Statistics 21, 552–75. and (
- 1981) Detection of ‘‘disorder’’ in a Wiener process. Theory of Probability and its Applications 26, 356–62. (
- 2012)Testing for a change in correlation at an unknown point in time using an extended functional delta method. Econometric Theory 28, 570–89. , and (
- 2007) Strong invariance principles for dependent random variables. The Annals of Probability 35, 2294–320. (
- 2011) Estimating a change point in the long memory parameter. Journal of Time Series Analysis 32, 304–14. (
- 1988) Estimating the number of change-points via Schwarz’ criterion. Statistics & Probability Letters 6, 181–9. (
- 2005) Long-memory property of nonlinear transformations of break processes. Economics Letters 87, 373–7. (
- 2007) High moment partial sum processes of residuals in ARMA models and their applications. Journal of Time Series Analysis 28, 72–91. (
- 2005) Monitoring structural change in dynamic econometric models. Journal of Applied Econometrics 20, 99–121. , , and (