CUSUM of Squares‐Based Tests for a Change in Persistence

Abstract. Using standardized cumulative sums of squared sub‐sample residuals, we propose a new ratio‐based test of the null hypothesis that a time series exhibits no change in its persistence structure [specifically that it displays constant I(1) behaviour] against the alternative of a change in persistence from trend stationarity to difference stationarity, or vice versa. Neither the direction nor location of any possible change under the alternative hypothesis need be assumed known. A key feature of our proposed test which distinguishes it from extant tests for persistence change [certain of which test the null hypothesis of constant I(0) behaviour while others, like our proposed test, test the null hypothesis of constant I(1) behaviour] is that it displays no tendency to spuriously over‐reject when applied to series which, although not constant I(1) series, do not display a change in persistence [specifically are constant I(0) processes]. Moreover, where our ratio test correctly rejects the null of no persistence change, the tail in which the rejection occurs can also be used to identify the direction of change since, even in relatively small samples, the test almost never rejects in the right [left] tail when there is a change from I(0) to I(1) [I(1) to I(0)]. Again this useful property is not shared by existing tests. As a by‐product of our analysis, we also propose breakpoint estimators which are consistent where the timing of the change in persistence is unknown.     

DETERMINING THE NUMBER OF REGIMES IN MARKOV SWITCHING VAR AND VMA MODELS

We give stable finite‐order vector autoregressive moving average (p ^* ,q ^* ) representations for M‐state Markov switching second‐order stationary time series whose autocovariances satisfy a certain matrix relation. The upper bounds for p ^* and q ^* are elementary functions of the dimension K of the process, the number M of regimes, the autoregressive and moving‐average orders of the initial model. If there is no cancellation, the bounds become equalities, and this solves the identification problem. Our classes of time series include every M‐state Markov switching multi‐variate moving‐average models and autoregressive models in which the regime variable is uncorrelated with the observable. Our results include, as particular cases, those obtained by Krolzig (1997) and improve the bounds given by Zhang and Stine (2001) and Francq and Zakoïan (2001) for our classes of dynamic models. A Monte Carlo experiment and an application on foreign exchange rates complete the article. Copyright © 2013 John Wiley & Sons, Ltd.     

Signal Extraction for Non‐Stationary Multivariate Time Series with Illustrations for Trend Inflation

This article advances the theory and methodology of signal extraction by developing the optimal treatment of difference stationary multivariate time‐series models. Using a flexible time‐series structure that includes co‐integrated processes, we derive and prove formulas for minimum mean square error estimation of signal vectors in multiple series, from both a finite sample and a bi‐infinite sample. As an illustration, we present econometric measures of the trend in total inflation that make optimal use of the signal content in core inflation.     

Adaptive wavelet decompositions of stationary time series‡

A general and flexible framework for the wavelet‐based decompositions of stationary time series in discrete time, called adaptive wavelet decompositions (AWDs), is introduced. It is shown that several particular AWDs can be constructed with the aim of providing decomposition (approximation and detail) coefficients that exhibit certain nice statistical properties, where the latter can be chosen based on a range of theoretical or applied considerations. AWDs make use of a Fast Wavelet Transform‐like algorithm whose filters ‐ in contrast with their counterparts in Orthogonal Wavelet Decompositions (OWDs) – may depend on the scale. As with OWDs, this algorithm has good properties such as computational efficiency and invariance to polynomial trends. A property whose pursuit plays a central role in this work is the decorrelation of the detail coefficients. For many time series models (e.g., FARIMA(0,δ,0)), the AWD filters can be defined so that the resulting AWD detail coefficients are all (exactly) decorrelated. The corresponding AWDs, called Exact AWDs (EAWDs), are particularly useful in simulation of Gaussian stationary time series, if the associated filters have a fast decay. The proposed simulation methods generalize and improve upon existing wavelet‐based ones. AWDs for which the detail coefficients are not exactly decorrelated, but still more decorrelated than those of OWDs, are referred to as approximate AWDs (AAWDs). They can be obtained by truncating EAWD filters, or by adopting some of the existing approaches to modeling the dependence structure of the OWD detail coefficients (e.g., Craigmile et al., 2005 ). AAWDs naturally lead to new wavelet‐based Maximum Likelihood estimators. The performance of these estimators is investigated through simulations and from some theoretical standpoints. The focus in estimation is also on Gaussian stationary series, though the method is expected to work for non‐Gaussian stationary series as well.     

Bootstrap procedures for detecting multiple persistence shifts in heteroskedastic time series

This article proposes new bootstrap procedures for detecting multiple persistence shifts in a time series driven by non-stationary volatility. The assumed volatility process can accommodate discrete breaks, smooth transition variation as well as trending volatility. We develop wild bootstrap sup-Wald tests of the null hypothesis that the process is either stationary [I(0)] or has a unit root [I(1)] throughout the sample. We also propose a sequential procedure to estimate the number of persistence breaks based on ordering the regime-specific bootstrap p-values. The asymptotic validity of the advocated procedures is established both under the null of stability and a variety of persistence change alternatives. A comparison with existing tests that assume homoskedasticity illustrates the finite sample improvements offered by our methods. An application to OECD inflation rates highlights the empirical relevance of the proposed approach and weakens the case for persistence change relative to existing procedures.     

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.     

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.     

Boundary Limit Theory for Functional Local to Unity Regression

This article studies functional local unit root models (FLURs) in which the autoregressive coefficient may vary with time in the vicinity of unity. We extend conventional local to unity (LUR) models by allowing the localizing coefficient to be a function which characterizes departures from unity that may occur within the sample in both stationary and explosive directions. Such models enhance the flexibility of the LUR framework by including break point, trending, and multidirectional departures from unit autoregressive coefficients. We study the behavior of this model as the localizing function diverges, thereby determining the impact on the time series and on inference from the time series as the limits of the domain of definition of the autoregressive coefficient are approached. This boundary limit theory enables us to characterize the asymptotic form of power functions for associated unit root tests against functional alternatives. Both sequential and simultaneous limits (as the sample size and localizing coefficient diverge) are developed. We find that asymptotics for the process, the autoregressive estimate, and its t‐statistic have boundary limit behavior that differs from standard limit theory in both explosive and stationary cases. Some novel features of the boundary limit theory are the presence of a segmented limit process for the time series in the stationary direction and a degenerate process in the explosive direction. These features have material implications for autoregressive estimation and inference which are examined in the article.     

Modelling long‐run trends and cycles in financial time series data

This article proposes a general time series framework to capture the long‐run behaviour of financial series. The suggested approach includes linear and segmented time trends, and stationary and non‐stationary processes based on integer and/or fractional degrees of differentiation. Moreover, the spectrum is allowed to contain more than a single pole or singularity, occurring at both zero but non‐zero (cyclical) frequencies. This framework is used to analyse five annual time series with a long span, namely dividends, earnings, interest rates, stock prices and long‐term government bond yields. The results based on several likelihood criteria indicate that the five series exhibit fractional integration with one or two poles in the spectrum, and are quite stable over the sample period examined.     

A Gaussian Mixture Autoregressive Model for Univariate Time Series

The Gaussian mixture autoregressive model studied in this article belongs to the family of mixture autoregressive models, but it differs from its previous alternatives in several advantageous ways. A major theoretical advantage is that, by the definition of the model, conditions for stationarity and ergodicity are always met and these properties are much more straightforward to establish than is common in nonlinear autoregressive models. Another major advantage is that, for a pth‐order model, explicit expressions of the stationary distributions of dimension p + 1 or smaller are known and given by mixtures of Gaussian distributions with constant mixing weights. In contrast, the conditional distribution given the past observations is a Gaussian mixture with time‐varying mixing weights that depend on p lagged values of the series in a natural and parsimonious way. Because of the known stationary distribution, exact maximum likelihood estimation is feasible and one can assess the applicability of the model in advance by using a non‐parametric estimate of the stationary density. An empirical example with interest rate series illustrates the practical usefulness and flexibility of the model, particularly in allowing for level shifts and temporary changes in variance. Copyright © 2014 Wiley Publishing Ltd     

Testing for nonlinear deterministic components when the order of integration is unknown

We consider testing for the presence of nonlinearities in the deterministic component of a time series, approximating the potential nonlinear behaviour using a Fourier function expansion. In contrast to procedures that are currently available, we develop tests that are robust to the order of integration, in the sense that they are asymptotically correctly sized regardless of whether the stochastic component of the series is stationary or contains a unit root. The tests we propose take the form of Wald statistics based on cumulated series, together with a correction factor to line up the asymptotic critical values across the I(0) and I(1) environments. The local asymptotic power and finite sample properties of the tests are evaluated using various different correction factors. We envisage that the testing procedure we recommend should be very useful to applied researchers wishing to draw robust inference regarding the presence of nonlinear deterministic components in a series.     

Time‐series clustering via quasi U‐statistics

The problem of time‐series discrimination and classification is discussed. We propose a novel clustering algorithm based on a class of quasi U‐statistics and subgroup decomposition tests. The decomposition may be applied to any concave time‐series distance. The resulting test statistics are proven to be asymptotically normal for either i.i.d. or non‐identically distributed groups of time‐series under mild conditions. We illustrate its empirical performance on a simulation study and a real data analysis. The simulation setup includes stationary vs. stationary and stationary vs. non‐stationary cases. The performance of the proposed method is favourably compared with some of the most common clustering measures available.     

Local Whittle estimation of multi‐variate fractionally integrated processes

This article derives a semi‐parametric estimator of multi‐variate fractionally integrated processes covering both stationary and non‐stationary values of d. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the multi‐variate local Whittle estimator of Shimotsu (2007) to cover non‐stationary values of d. Consistency and asymptotic normality is shown for d ∈ (?1/2,∞). A simulation study illustrates the performance of the proposed estimator for relevant sample sizes. Empirical justification of the proposed estimator is shown through an empirical analysis of log spot exchange rates. We find that the log spot exchange rates of Germany, United Kingdom, Japan, Canada, France, Italy and Switzerland against the US Dollar for the period January 1974 until December 2001 are well decribed as I(1) processes.     

A Bayesian regime‐switching time‐series model

This article provides a new Bayesian approach for AR(2) time‐series models with multiple regime‐switching points. Our formulation of the regime‐switching model involves a binary discrete variable that indicates the regime change. This variable is specified to be detected by data in each regime. The model is estimated using Stochastic approximation Monte Carlo method proposed by Liang et al. [JASA (2007)]. This methodology is quite useful since it allows for fitting of more complex regime‐switching models without transition constraint. The proposed model is illustrated using simulated and real data such as GNP and US interest rate data.     

The sampling properties of conditional independence graphs for I(1) structural VAR models

Abstract. Structural vector autoregressions allow dependence among contemporaneous variables. If such models have a recursive structure, the relationships among the variables can be represented by directed acyclic graphs. The identification of these relationships for stationary series may be enabled by the examination of the conditional independence graph constructed from sample partial autocorrelations of the observed series. In this article, we extend this approach to the case when the series follows an I(1) vector autoregression. For such a model, estimated regression coefficients may have non‐standard asymptotic distributions and in small samples this affects the distribution of sample partial autocorrelations. We show that, nevertheless, in large samples, exactly the same inference procedures may be applied as in the stationary case.     

A Note on Non‐Negative Arma Processes

Abstract. Recently, there has been much research on developing models suitable for analysing the volatility of a discrete‐time process. Since the volatility process, like many others, is necessarily non‐negative, there is a need to construct models for stationary processes which are non‐negative with probability one. Such models can be obtained by driving autoregressive moving average (ARMA) processes with non‐negative kernel by non‐negative white noise. This raises the problem of finding simple conditions under which an ARMA process with given coefficients has a non‐negative kernel. In this article, we derive a necessary and sufficient condition. This condition is in terms of the generating function of the ARMA kernel which has a simple form. Moreover, we derive some readily verifiable necessary and sufficient conditions for some ARMA processes to be non‐negative almost surely.     

The Periodogram of fractional processes1

Abstract. We analyse asymptotic properties of the discrete Fourier transform and the periodogram of time series obtained through (truncated) linear filtering of stationary processes. The class of filters contains the fractional differencing operator and its coefficients decay at an algebraic rate, implying long‐range‐dependent properties for the filtered processes when the degree of integration α is positive. These include fractional time series which are nonstationary for any value of the memory parameter (α ≠ 0) and possibly nonstationary trending (α ≥ 0.5). We consider both fractional differencing or integration of weakly dependent and long‐memory stationary time series. The results obtained for the moments of the Fourier transform and the periodogram at Fourier frequencies in a degenerating band around the origin are weaker compared with the stationary nontruncated case for α > 0, but sufficient for the analysis of parametric and semiparametric memory estimates. They are applied to the study of the properties of the log‐periodogram regression estimate of the memory parameter α for Gaussian processes, for which asymptotic normality could not be showed using previous results. However, only consistency can be showed for the trending cases, 0.5 ≤ α < 1. Several detrending and initialization mechanisms are studied and only local conditions on spectral densities of stationary input series and transfer functions of filters are assumed.     

A new frequency domain approach of testing for covariance stationarity and for periodic stationarity in multivariate linear processes

In modelling seasonal time series data, periodically (non‐)stationary processes have become quite popular over the last years and it is well known that these models may be represented as higher‐dimensional stationary models. In this article, it is shown that the spectral density matrix of this higher‐dimensional process exhibits a certain structure if and only if the observed process is covariance stationary. By exploiting this relationship, a new L₂‐type test statistic is proposed for testing whether a multivariate periodically stationary linear process is even covariance stationary. Moreover, it is shown that this test may also be used to test for periodic stationarity. The asymptotic normal distribution of the test statistic under the null is derived and the test is shown to have an omnibus property. The article concludes with a simulation study, where the small sample performance of the test procedure is improved by using a suitable bootstrap scheme.     

Conditional and Marginal Mutual Information in Gaussian and Hyperbolic Decay Time Series

We consider the amount of available information about an arbitrary future state of a Gaussian stochastic process. We derive an infinite series for the marginal mutual information in terms of the autocorrelation function. We derive an infinite series for the newly available information for prediction, the conditional mutual information, in terms of the moving average parameters, and directly characterize predictability in terms of sensitivity to random shocks. We apply our results to long memory, or more generally, hyperbolic decay models, and give information‐theoretic characterizations of the transition from persistence to anti‐persistence, stationary long memory to nonstationarity, and a stationary regime where the mutual information is not summable.     

Transformation to approximate independence for locally stationary Gaussian processes

We provide new approximations for the likelihood of a time series under the locally stationary Gaussian process model. The likelihood approximations are valid even in cases when the evolutionary spectrum is not smooth in the rescaled time domain. We describe a broad class of models for the evolutionary spectrum for which the approximations can be computed particularly efficiently. In developing the approximations, we extend to the locally stationary case the idea that the discrete Fourier transform is a decorrelating transformation for stationary time series. The approximations are applied to fit non‐stationary time‐series models to high‐frequency temperature data. For these data, we fit evolutionary spectra that are piecewise constant in time and use a genetic algorithm to search for the best partition of the time interval.