Robust discrimination between long‐range dependence and a change in mean

In this article we introduce a robust to outliers Wilcoxon change‐point testing procedure, for distinguishing between short‐range dependent time series with a change in mean at unknown time and stationary long‐range dependent time series. We establish the asymptotic distribution of the test statistic under the null hypothesis for L₁ near epoch dependent processes and show its consistency under the alternative. The Wilcoxon‐type testing procedure similarly as the CUSUM‐type testing procedure (of Berkes I., Horváth L., Kokoszka P. and Shao Q. 2006. Ann.Statist. 34:1140–1165), requires estimation of the location of a possible change‐point, and then using pre‐ and post‐break subsamples to discriminate between short and long‐range dependence. A simulation study examines the empirical size and power of the Wilcoxon‐type testing procedure in standard cases and with disturbances by outliers. It shows that in standard cases the Wilcoxon‐type testing procedure behaves equally well as the CUSUM‐type testing procedure but outperforms it in presence of outliers. We also apply both testing procedure to hydrologic data.     

Likelihood inference for discriminating between long‐memory and change‐point models

We develop a likelihood ratio (LR) test procedure for discriminating between a short‐memory time series with a change‐point (CP) and a long‐memory (LM) time series. Under the null hypothesis, the time series consists of two segments of short‐memory time series with different means and possibly different covariance functions. The location of the shift in the mean is unknown. Under the alternative, the time series has no shift in mean but rather is LM. The LR statistic is defined as the normalized log‐ratio of the Whittle likelihood between the CP model and the LM model, which is asymptotically normally distributed under the null. The LR test provides a parametric alternative to the CUSUM test proposed by Berkes et al. (2006) . Moreover, the LR test is more general than the CUSUM test in the sense that it is applicable to changes in other marginal or dependence features other than a change‐in‐mean. We show its good performance in simulations and apply it to two data examples.     

A Self‐Normalized Semi‐Parametric Test to Detect Changes in the Long Memory Parameter

We consider the problem of testing for change points in the long memory parameter. The test relies on semi‐parametric estimation of the long memory parameter, which does not require the complete parametric specification of the whole spectrum. A self‐normalizer utilizing a sequence of recursive semi‐parametric estimators is used to make the asymptotic distribution of the test statistic free of the nuisance scale parameter. We study the asymptotic behavior of the proposed test for situations when there is at most one change point and also when there are an unknown number of change points. Monte Carlo simulations are carried out to examine the finite‐sample performance of the proposed test.     

Testing for parameter stability in nonlinear autoregressive models

In this article we develop testing procedures for the detection of structural changes in nonlinear autoregressive processes. For the detection procedure, we model the regression function by a single layer feedforward neural network. We show that CUSUM‐type tests based on cumulative sums of estimated residuals, that have been intensively studied for linear regression, can be extended to this case. The limit distribution under the null hypothesis is obtained, which is needed to construct asymptotic tests. For a large class of alternatives, it is shown that the tests have asymptotic power one. In this case, we obtain a consistent change‐point estimator which is related to the test statistics. Power and size are further investigated in a small simulation study with a particular emphasis on situations where the model is misspecified, i.e. the data is not generated by a neural network but some other regression function. As illustration, an application on the Nile data set as well as S&P log‐returns is given.     

Robust Wilcoxon‐Type Estimation of Change‐Point Location Under Short‐Range Dependence

We introduce a robust estimator of the location parameter for the change‐point in the mean based on Wilcoxon statistic and establish its consistency for L₁ near‐epoch dependent processes. It is shown that the consistency rate depends on the magnitude of the change. A simulation study is performed to evaluate the finite sample properties of the Wilcoxon‐type estimator under Gaussianity as well as under heavy‐tailed distributions and disturbances by outliers, and to compare it with a CUSUM‐type estimator. It shows that the Wilcoxon‐type estimator is equivalent to the CUSUM‐type estimator under Gaussianity but outperforms it in the presence of heavy tails or outliers in the data.     

Testing for Change‐Points in Long‐Range Dependent Time Series by Means of a Self‐Normalized Wilcoxon Test

We propose a testing procedure based on the Wilcoxon two‐sample test statistic in order to test for change‐points in the mean of long‐range dependent data. We show that the corresponding self‐normalized test statistic converges in distribution to a non‐degenerate limit under the hypothesis that no change occurred and that it diverges to infinity under the alternative of a change‐point with constant height. Furthermore, we derive the asymptotic distribution of the self‐normalized Wilcoxon test statistic under local alternatives, that is, under the assumption that the height of the level shift decreases as the sample size increases. Regarding the finite sample performance, simulation results confirm that the self‐normalized Wilcoxon test yields a consistent discrimination between hypothesis and alternative and that its empirical size is already close to the significance level for moderate sample sizes.     

Structural breaks in time series

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.     

Mean shift testing in correlated data

Several tests for detecting mean shifts at an unknown time in stationary time series have been proposed, including cumulative sum (CUSUM), Gaussian likelihood ratio (LR), maximum of F(F_max) and extreme value statistics. This article reviews these tests, connects them with theoretical results, and compares their finite sample performance via simulation. We propose an adjusted CUSUM statistic which is closely related to the LR test and which links all tests. We find that tests based on CUSUMing estimated one‐step‐ahead prediction residuals from a fitted autoregressive moving average perform well in general and that the LR and F_max tests (which induce substantial computational complexities) offer only a slight increase in power over the adjusted CUSUM test. We also conclude that CUSUM procedures work slightly better when the changepoint time is located near the centre of the data, but the adjusted CUSUM methods are preferable when the changepoint lies closer to the beginning or end of the data record. Finally, an application is presented to demonstrate the importance of the choice of method.     

Tests for Volatility Shifts in Garch Against Long‐Range Dependence

Many empirical findings show that volatility in financial time series exhibits high persistence. Some researchers argue that such persistency is due to volatility shifts in the market, while others believe that this is a natural fluctuation explained by stationary long‐range dependence models. These two approaches confuse many practitioners, and forecasts for future volatility are dramatically different depending on which models to use. In this article, therefore, we consider a statistical testing procedure to distinguish volatility shifts in generalized AR conditional heteroscedasticity (GARCH) model against long‐range dependence. Our testing procedure is based on the residual‐based cumulative sum test, which is designed to correct the size distortion observed for GARCH models. We examine the validity of our method by providing asymptotic distributions of test statistic. Also, Monte Carlo simulations study shows that our proposed method achieves a good size while providing a reasonable power against long‐range dependence. It is also observed that our test is robust to the misspecified GARCH models.     

Testing for a break in persistence under long-range dependencies

Abstract.  We show that tests for a break in the persistence of a time series in the classical I (0)/ I (1) framework have serious size distortions when the actual data-generating process (DGP) exhibits long-range dependencies. We prove that the limiting distribution of a CUSUM of squares-based test depends on the true memory parameter if the DGP exhibits long memory. We propose adjusted critical values for the test and give finite sample response curves that allow easy implementation of the test by the practitioner and also ease in computing the relevant critical values. We furthermore prove the consistency of the test for a simple breakpoint estimator also under long memory. We show that the test has satisfying power properties when the correct critical values are used.     

A FIXED‐ b TEST FOR A BREAK IN LEVEL AT AN UNKNOWN TIME UNDER FRACTIONAL INTEGRATION

In this paper, we propose a test for a break in the level of a fractionally integrated process when the timing of the putative break is not known. This testing problem has received considerable attention in the literature in the case where the time series is weakly autocorrelated. Less attention has been given to the case where the underlying time series is allowed to be fractionally integrated. Here, valid testing can only be performed if the limiting null distribution of the level break test statistic is well defined for all values of the fractional integration exponent considered. However, conventional sup‐Wald type tests diverge when the data are strongly autocorrelated. We show that a sup‐Wald statistic, which is standardized using a non‐parametric kernel‐based long‐run variance estimator, does possess a well‐defined limit distribution, depending only on the fractional integration parameter, provided the recently developed fixed‐b asymptotic framework is applied. We give the appropriate asymptotic critical values for this sup‐Wald statistic and show that it has good finite sample size and power properties.     

Statistical tests for a single change in mean against long‐range dependence

Statistical tests are introduced for distinguishing between short‐range dependent time series with a single change in mean, and long‐range dependent time series, with the former making the null hypothesis. The tests are based on estimation of the self‐similarity parameter after removing the change in mean from the series. The focus is on the GPH (Geweke and Porter‐Hudak, 1983) and local Whittle estimation methods in the spectral domain. Theoretical properties of the resulting estimators are established when testing for a single change in mean, and small sample properties of the tests are examined in simulations. The introduced tests improve on the BHKS ( Berkes et al., 2006 ) test which is the only other available test for the considered problem. It is argued that the BHKS test has a low power against long‐range dependence alternatives and that this happens because the BHKS test statistic involves estimation of the long‐run variance. The BHKS test could be improved readily by considering its R/S‐like regression version which estimates the self‐similarity parameter and which does not involve the long‐run variance. Yet better alternatives are to use more powerful estimation methods (such as GPH or local Whittle) and lead to the tests introduced here.     

Parameter change test for random coefficient integer-valued autoregressive processes with application to polio data analysis

Abstract.  In this paper, we consider the problem of testing for a parameter change in a first-order random coefficient integer-valued autoregressive [RCINAR(1)] model. We employ the cumulative sum (CUSUM) test based on the conditional least-squares and modified quasi-likelihood estimators. It is shown that under regularity conditions, the CUSUM test has the same limiting distribution as the supremum of the squares of independent Brownian bridges. The CUSUM test is then applied to the analysis of the monthly polio counts data set.     

Tests for Comparing Time‐Invariant and Time‐Varying Spectra Based on the Pearson Statistic

Two tests are proposed in this paper for comparing spectra of two univariate time series. One is a Pearson‐like statistic based only on periodograms of the compared time series and applicable for testing the equality of two time‐invariant spectra of two independent or dependent time series, with an asymptotic chi‐squared distribution under the null hypothesis. The other is based on the maximum of the Pearson‐like statistics. Not only does this test, again, depend only on periodograms but also approximately equals the maximum of a chi‐squared distribution of the same degrees of freedom under the null. It can be used to test the equality of spectra of two locally stationary time series regardless of whether they are dependent or independent. Multiple simulation examples show that both statistics achieve good performance. The proposed approach is illustrated by an application to longitudinal vibration data from a container ship.     

A simulation method to analyze chemo‐mechanical behavior of swelling‐induced shape‐memory polymer in response to solvent

A network of thermally responsive shape‐memory polymers (SMPs) could imbibe a quantity of solvent molecules to swell, and subsequently induces a chemical potential change in polymer. When an equilibrium is reached between the mechanical load and the chemical potential of polymer network and solvent, the SMP polymer usually swells with a field of inhomogeneous and anisotropic deformation, which is considered to be equivalent to a hyperelastic field. We implement this theory in the free‐energy function equation, and analyze examples of swelling‐induced deformation and shape recovery behavior. This work may provide a powerful tool to study complex swelling‐induced shape‐memory behavior of SMPs in response to the immersing solvents. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2011     

Detecting Tail Risk Differences in Multivariate Time Series

We derive functional central limit theory for tail index estimates in multivariate time series under mild conditions on the extremal dependence between the components. We use this result to also derive convergence results for extreme value‐at‐risk and extreme expected shortfall estimates. This allows us to construct tests for equality of ‘tail risk’ in multivariate data, which can be useful in a number of empirical contexts. In constructing test statistics, we avoid estimating long‐run variances by using self‐normalization. Size and power of the tests for equal ‘tail risk’ are assessed in simulations. An empirical application to exchange returns illustrates the practical usefulness of the tests.     

High Moment Partial Sum Processes of Residuals in ARMA Models and their Applications

Abstract. In this article, we study high moment partial sum processes based on residuals of a stationary autoregressive moving average (ARMA) model with known or unknown mean parameter. We show that they can be approximated in probability by the analogous processes which are obtained from the i.i.d. errors of the ARMA model. However, if a unknown mean parameter is used, there will be an additional term that depends on model parameters and a mean estimator. When properly normalized, this additional term will vanish. Thus the processes converge weakly to the same Gaussian processes as if the residuals were i.i.d. Applications to change‐point problems and goodness‐of‐fit are considered, in particular, cumulative sum statistics for testing ARMA model structure changes and the Jarque–Bera omnibus statistic for testing normality of the unobservable error distribution of an ARMA model.     

Range Unit‐Root (RUR) Tests: Robust against Nonlinearities,Error Distributions,Structural Breaks and Outliers

Abstract. Since the seminal paper by Dickey and Fuller in 1979, unit‐root tests have conditioned the standard approaches to analysing time series with strong serial dependence in mean behaviour, the focus being placed on the detection of eventual unit roots in an autoregressive model fitted to the series. In this paper, we propose a completely different method to test for the type of long‐wave patterns observed not only in unit‐root time series but also in series following more complex data‐generating mechanisms. To this end, our testing device analyses the unit‐root persistence exhibited by the data while imposing very few constraints on the generating mechanism. We call our device the range unit‐root (RUR) test since it is constructed from the running ranges of the series from which we derive its limit distribution. These nonparametric statistics endow the test with a number of desirable properties, the invariance to monotonic transformations of the series and the robustness to the presence of important parameter shifts. Moreover, the RUR test outperforms the power of standard unit‐root tests on near‐unit‐root stationary time series; it is invariant with respect to the innovations distribution and asymptotically immune to noise. An extension of the RUR test, called the forward–backward range unit‐root (FB‐RUR) improves the check in the presence of additive outliers. Finally, we illustrate the performances of both range tests and their discrepancies with the Dickey–Fuller unit‐root test on exchange rate series.     

Moment tests for window length selection in singular spectrum analysis of short– and long–memory processes

In this paper, we propose a new methodology for selecting the window length in Singular Spectral Analysis in which the window length is determined from the data prior to the commencement of modelling. The selection procedure is based on statistical tests designed to test the convergence of the autocovariance function. A classical time series portmanteau type statistic and two test statistics derived using a conditional moment principle are considered. The first two are applicable to short–memory processes, and the third is applicable to both short– and long–memory processes. We derive the asymptotic null and alternative distributions of the statistics under fairly general regularity conditions. Consistency of the tests implies that the selection criteria will identify true convergence with a finite window length with probability arbitrarily close to one as the sample size increases. Results obtained using Monte Carlo simulation point to the relevance of the asymptotic theory and show that the conditional moment tests will choose a window length consistent with the Whitney embedding theorem. Application to observations on the Southern Oscillation Index shows how observed experimental behaviour can be reflected in features seen with real world data sets.     

Inference in Autoregression under Heteroskedasticity

Abstract. A scalar pth‐order autoregression (AR(p)) is considered with heteroskedasticity of the unknown form delivered by a transition function of time. A limit theory is developed and three heteroskedasticity‐robust test statistics are proposed for inference, one of which is based on the nonparametric estimation of the variance function. The performance of the resulting testing procedures in finite samples is compared in simulations and some suggestions for practical application are given.