On the detection of changes in autoregressive time series I. Asymptotics (original) (raw)
Application of Block Sieve Bootstrap to Change-Point detection in time series
2010
Since the introduction of CUSUM statistic by E.S. Page (1951), detection of change or a structural break in time series has gained significant interest as its applications span across various disciplines including economics, industrial applications, and environmental data sets. However, many of the early suggested statistics, such as CUSUM or MOSUM, lose their effectiveness when applied to time series data. Either the size or power of the test statistic gets distorted, especially for higher order autoregressive moving average processes. We use the test statistic from Gombay and Serban (2009) for detecting change in the mean of an autoregressive process and show how the application of Sieve Bootstrap to the time series data can improve the performance of our test to detect change. The effectiveness of the proposed method is illustrated by applying it to econometric data sets.
Monitoring changes in the error distribution of autoregressive models based on Fourier methods
TEST, 2012
We develop a procedure for monitoring changes in the error distribution of autoregressive time series while controlling the overall size of the sequential test. The proposed procedure, unlike standard procedures which are also referred to, utilizes the empirical characteristic function of properly estimated residuals. The limit behavior of the test statistic is investigated under the null hypothesis as well as under alternatives. Since the asymptotic null distribution contains unknown parameters, a bootstrap procedure is proposed in order to actually perform the test, and corresponding results on the finite-sample performance of the new method are presented. As it turns out, the procedure is not only able to detect distributional changes but also changes in the regression coefficient.
Monitoring procedure for parameter change in causal time series
Journal of Multivariate Analysis, 2014
We propose a new sequential procedure to detect change in the parameters of a process X = (X t ) t∈Z belonging to a large class of causal models (such as AR(∞), ARCH(∞), TARCH(∞), ARMA-GARCH processes). The procedure is based on a difference between the historical parameter estimator and the updated parameter estimator, where both these estimators are based on a quasi-likelihood of the model. Unlike classical recursive fluctuation test, the updated estimator is computed without the historical observations. The asymptotic behavior of the test is studied and the consistency in power as well as an upper bound of the detection delay are obtained. Some simulation results are reported with comparisons to some other existing procedures exhibiting the accuracy of our new procedure. The procedure is also applied to the daily closing values of the Nikkei 225, S&P 500 and FTSE 100 stock index. We show in this real-data applications how the procedure can be used to solve off-line multiple breaks detection.
Ratio tests for change point detection
Collections, 2008
We propose new tests to detect a change in the mean of a time series. Like many existing tests, the new ones are based on the CUSUM process. Existing CUSUM tests require an estimator of a scale parameter to make them asymptotically distribution free under the no change null hypothesis. Even if the observations are independent, the estimation of the scale parameter is not simple since the estimator for the scale parameter should be at least consistent under the null as well as under the alternative. The situation is much more complicated in case of dependent data, where the empirical spectral density at 0 is used to scale the CUSUM process. To circumvent these difficulties, new tests are proposed which are ratios of CUSUM functionals. We demonstrate the applicability of our method to detect a change in the mean when the errors are AR(1) and GARCH(1,1) sequences.
SSRN Electronic Journal, 2019
In this paper we discuss the general application of the bootstrap as a tool for statistical inference in econometric time series models. We do this by considering the implementation of bootstrap inference in the class of double-autoregressive [DAR] models discussed in Ling (2004). DAR models are particularly interesting to illustrate implementation of the bootstrap to time series: …rst, standard asymptotic inference is usually di¢ cult to implement due to the presence of nuisance parameters under the null hypothesis; second, inference involves testing whether one or more parameters are on the boundary of the parameter space; third, under the alternative hypothesis, fourth or even second order moments may not exist. In most of these cases, the bootstrap is not considered an appropriate tool for inference. Conversely, and taking testing (non-) stationarity to illustrate, we show that although a standard bootstrap based on unrestricted parameter estimation is invalid, a correct implementation of a bootstrap based on restricted parameter estimation (restricted bootstrap) is …rst-order valid; that is, it is able to replicate, under the null hypothesis, the correct limiting null distribution. Importantly, we also show that the behaviour of this bootstrap under the alternative hypothesis may be di¤erent because of possible lack of …nite second-order moments of the bootstrap innovations. This features makes-for some parameter con…gurationsthe restricted bootstrap unable to replicate the null asymptotic distribution when the null is false. We show that this drawback can be …xed by using a new 'hybrid' bootstrap, where the parameter estimates used to construct the bootstrap data are obtained with the null imposed, while the bootstrap innovations are sampled with replacement from the unrestricted residuals. We show that this bootstrap, novel in this framework, mimics the correct asymptotic null distribution, irrespetively of the null to be true or false. Throughout the paper, we use a number of examples from the bootstrap time series literature to illustrate the importance of properly de…ning and analyzing the bootstrap generating process and associated bootstrap statistics.
Journal of Time Series Analysis, 2018
We derive tests of stationarity for continuous univariate time series by combining changepoint tests sensitive to changes in the contemporary distribution with tests sensitive to changes in the serial dependence. Rank-based cumulative sum tests based on the empirical distribution function and on the empirical autocopula at a given lag are considered first. The combination of their dependent p-values relies on a joint dependent multiplier bootstrap of the two underlying statistics. Conditions under which the proposed combined testing procedure is asymptotically valid under stationarity are provided. After discussing the choice of the maximum lag to investigate, extensions based on tests solely focusing on secondorder characteristics are proposed. The finite-sample behaviors of all the derived statistical procedures are investigated in large-scale Monte Carlo experiments and illustrations on two real data sets are provided. Extensions to multivariate time series are briefly discussed as well.
Specification testing in nonlinear and nonstationary time series autoregression
The Annals of Statistics, 2009
This paper considers a class of nonparametric autoregressive models with nonstationarity. We propose a nonparametric kernel test for the conditional mean and then establish an asymptotic distribution of the proposed test. Both the setting and the results differ from earlier work on nonparametric autoregression with stationarity. In addition, we develop a new bootstrap simulation scheme for the selection of a suitable bandwidth parameter involved in the kernel test as well as the choice of a simulated critical value. The finitesample performance of the proposed test is assessed using one simulated example and one real data example.
A method for detecting changes in long time series
1995
Modern scientific activities, both physical and computational, can result in time series of many thousands or even millions of data values. Here we describe a statistically motivated algorithm for quick screening of very long time series data for the presence of potentially interesting but arbitrary changes. The basic data model is a stationary Gaussian stochastic process, and the approach to detecting a change is the comparison of two predictions of the series at a time point or contiguous collection of time points. One prediction is a "forecast", Le. based on data from earlier times, while the other a "backcast", Le. based on data from later times. The statistic is the absolute value of the log-likelihood ratio for these two predictions, evaluated at the observed data. A conservative procedure is suggested for specifying critical values for the statistic under the null hypothesis of "no change".
Testing for Shifts in Trend With an Integrated or Stationary Noise Component
Journal of Business & Economic Statistics, 2009
This paper considers the problem of testing for structural changes in the trend function of a univariate time series without any prior knowledge as to whether the noise component is stationary or contains an autoregressive unit root. We propose a new approach that builds on the work of Perron and Yabu (2005), based on a Feasible Quasi Generalized Least Squares procedure that uses a superefficient estimate of the sum of the autoregressive parameters α when α = 1. In the case of a known break date, the resulting Wald test has a chi-square limit distribution in both the I(0) and I(1) cases. When the break date is unknown, the Exp functional of Andrews and Ploberger (1994) yields a test with nearly identical limit distributions in the two cases so that a testing procedure with nearly the same size in the I(0) and I(1) cases can be obtained. To improve the finite sample properties of the tests, we use the bias corrected version of the OLS estimate of α proposed by . We show our procedure to be substantially more powerful than currently available alternatives and also to have a power function that is close to that attainable if we knew the true value of α in many cases. The extension to the case of multiple breaks is also discussed. JEL Classification Number: C22.