Testing for Stationarity in Multivariate Locally Stationary Processes

In this article, we propose a nonparametric procedure for validating the assumption of stationarity in multivariate locally stationary time series models. We develop a bootstrap‐assisted test based on a Kolmogorov–Smirnov‐type statistic, which tracks the deviation of the time‐varying spectral density from its best stationary approximation. In contrast to all other nonparametric approaches, which have been proposed in the literature so far, the test statistic does not depend on any regularization parameters like smoothing bandwidths or a window length, which is usually required in a segmentation of the data. We additionally show how our new procedure can be used to identify the components where non‐stationarities occur and indicate possible extensions of this innovative approach. We conclude with an extensive simulation study, which shows finite‐sample properties of the new method and contains a comparison with existing approaches.     

Stationary bootstrapping for non‐parametric estimator of nonlinear autoregressive model

We consider stationary bootstrap approximation of the non‐parametric kernel estimator in a general kth‐order nonlinear autoregressive model under the conditions ensuring that the nonlinear autoregressive process is a geometrically Harris ergodic stationary Markov process. We show that the stationary bootstrap procedure properly estimates the distribution of the non‐parametric kernel estimator. A simulation study is provided to illustrate the theory and to construct confidence intervals, which compares the proposed method favorably with some other bootstrap methods.     

Maximum likelihood estimation for nearly non‐stationary stable autoregressive processes

The maximum likelihood estimate (MLE) of the autoregressive coefficient of a near‐unit root autoregressive process Y_t = ρ_nY_t?1 + ?_t with α‐stable noise {?_t} is studied in this paper. Herein ρ_n = 1 ? γ/n, γ ≥ 0 is a constant, Y₀ is a fixed random variable and ε_t is an α‐stable random variable with characteristic function φ(t,θ) for some parameter θ. It is shown that when 0 < α < 1 or α > 1 and E?₁ = 0, the limit distribution of the MLE of ρ_n and θ are mixtures of a stable process and Gaussian processes. On the other hand, when α > 1 and E?₁ ≠ 0, the limit distribution of the MLE of ρ_n and θ are normal. A Monte Carlo simulation reveals that the MLE performs better than the usual least squares procedures, particularly for the case when the tail index α is less than 1.     

On residual empirical processes of GARCH‐SM models: application to conditional symmetry tests

Abstract. Considering the generalized autoregressive conditionally heteroskedastic with stochastic mean (GARCH‐SM) model, we establish in this article the consistency and the weak representation of a functional of its residual empirical process. Based on this result, a symmetry test for GARCH‐SM model is developed. Simulations are given to show the asymptotic behaviour and normality of the test statistic.     

Wavelet‐Based Tests for Comparing Two Time Series with Unequal Lengths

Test procedures for assessing whether two stationary and independent time series with unequal lengths have the same spectral density (or same auto‐covariance function) are investigated. A new test statistic is proposed based on the wavelet transform. It relies on empirical wavelet coefficients of the logarithm of two spectral densities' ratio. Under the null hypothesis that two spectral densities are the same, the asymptotic normal distribution of the empirical wavelet coeffcients is derived. Furthermore, these empirical wavelet coefficients are asymptotically uncorrelated. A test statistic is proposed based on these results. The performance of the new test statistic is compared to several recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed test is very comparable to the current test statistics in most cases. The main advantage of our proposed test statistic is that it is constructed very simply and is easy to implement.     

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.     

Wavelet change‐point estimation for long memory non‐parametric random design models

For a random design regression model with long memory design and long memory errors, we consider the problem of detecting a change point for sharp cusp or jump discontinuity in the regression function. Using the wavelet methods, we obtain estimators for the change point, the jump size and the regression function. The strong consistencies of these estimators are given in terms of convergence rates.     

Tests for the Equality of Two Processes' Spectral Densities with Unequal Lengths Using Wavelet Methods

Testing procedures for assessing whether two stationary and independent linear processes with unequal lengths have the same spectral densities or same auto‐covariance functions are investigated. New test statistics are proposed based on the difference of the two wavelet‐based estimates of the two spectral densities. The asymptotic normal distributions of the empirical wavelet coefficients are derived based on Bartlett type approximation of a quadratic form with dependent variables by the corresponding quadratic form with independent and identically distributed (i.i.d.) random variables. The limit distributions of the proposed test statistics are derived from those asymptotic results, and they asymptotically follow known chi‐square distributions. The advantage of those new procedures is that those test statistics are constructed very simply and can be used for two time series with arbitrary lengths. The performance of those new tests is compared with some recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed tests are very comparable to the current tests.     

High‐frequency sampling of a continuous‐time ARMA process

Continuous‐time autoregressive moving average (CARMA) processes have recently been used widely in the modelling of non‐uniformly spaced data and as a tool for dealing with high‐frequency data of the form ,n = 0, 1, 2,…, where Δ is small and positive. Such data occur in many fields of application, particularly in finance and in the study of turbulence. This article is concerned with the characteristics of the process , when Δ is small and the underlying continuous‐time process is a specified CARMA process.     

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.     

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.     

On the Spectral Density of the Wavelet Coefficients of Long‐Memory Time Series with Application to the Log‐Regression Estimation of the Memory Parameter

Abstract. In recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semi‐parametric asymptotic theory, comparable with the one developed for Fourier methods, is still lacking. In this article, we adapt to the wavelet setting, the classical semi‐parametric framework introduced by Robinson and his co‐authors for estimating the memory parameter of a (possibly) non‐stationary process. Our results apply to a class of wavelets with bounded supports, which include but are not limited to Daubechies wavelets. We derive an explicit expression of the spectral density of the wavelet coefficients and show that it can be approximated, at large scales, by the spectral density of the continuous‐time wavelet coefficients of fractional Brownian motion. We derive an explicit bound for the difference between the spectral densities. As an application, we obtain minimax upper bounds for the log‐scale regression estimator of the memory parameter for a Gaussian process and we derive an explicit expression of its asymptotic variance.     

Residual Empirical Processes and Weighted Sums for Time‐Varying Processes with Applications to Testing for Homoscedasticity

In the context of heteroscedastic time‐varying autoregressive (AR)‐process we study the estimation of the error/innovation distributions. Our study reveals that the non‐parametric estimation of the AR parameter functions has a negligible asymptotic effect on the estimation of the empirical distribution of the residuals even though the AR parameter functions are estimated non‐parametrically. The derivation of these results involves the study of both function‐indexed sequential residual empirical processes and weighted sum processes. Exponential inequalities and weak convergence results are derived. As an application of our results we discuss testing for the constancy of the variance function, which in special cases corresponds to testing for stationarity.     

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.     

Akaike’s information criterion correction for the least‐squares autoregressive spectral estimator

In this article we propose a new correction for the penalty term of the Akaike’s information criterion (AIC), when it is used in the context of order selection for an autoregressive fit of the spectral density of a stationary time series. The classical AIC penalty term may be viewed as an approximation of an appropriate target quantity. Simulations show that the quality of this approximation strongly depends on the type of autoregressive estimator used, as well as on the discrepancy used. Therefore here we consider the least squares autoregressive estimator and the Whittle discrepancy only. In this context we propose a closed formula correction of the AIC penalty term. We also develop asymptotic theory which justifies this proposal: an asymptotically valid second‐order expansion of a stochastic approximation of the target quantity. This expansion assumes a non‐parametric framework: it does not assume gaussianity of the process and only requires its spectral density to be smooth enough. Simulations show that, as compared to previously introduced corrections, this new correction performs similarly to finite sample information criterion, while they both outperform AIC corrected and AIC.     

A Generalized Portmanteau Test For Independence Of Two Infinite‐Order Vector Autoregressive Series

Abstract. In many situations, we want to verify the existence of a relationship between multivariate time series. Here, we propose a semiparametric approach for testing the independence between two infinite‐order vector autoregressive (VAR(∞)) series, which is an extension of Hong's [Biometrika (1996c) vol. 83, 615–625] univariate results. We first filter each series by a finite‐order autoregression and the test statistic is a standardized version of a weighted sum of quadratic forms in the residual cross‐correlation matrices at all possible lags. The weights depend on a kernel function and on a truncation parameter. Using a result of Lewis and Reinsel [Journal of Multivariate Analysis (1985) Vol. 16, pp. 393–411], the asymptotic distribution of the test statistic is derived under the null hypothesis and its consistency is also established for a fixed alternative of serial cross‐correlation of unknown form. Apart from standardization factors, the multivariate portmanteau statistic proposed by Bouhaddioui and Roy [Statistics and Probability Letters (2006) vol. 76, pp. 58–68] that takes into account a fixed number of lags can be viewed as a special case by using the truncated uniform kernel. However, many kernels lead to a greater power, as shown in an asymptotic power analysis and by a small simulation study in finite samples. A numerical example with real data is also presented.     

Asymptotic self‐similarity and wavelet estimation for long‐range dependent fractional autoregressive integrated moving average time series with stable innovations

Abstract. Methods for parameter estimation in the presence of long‐range dependence and heavy tails are scarce. Fractional autoregressive integrated moving average (FARIMA) time series for positive values of the fractional differencing exponent d can be used to model long‐range dependence in the case of heavy‐tailed distributions. In this paper, we focus on the estimation of the Hurst parameter H = d + 1/α for long‐range dependent FARIMA time series with symmetric α‐stable (1 < α < 2) innovations. We establish the consistency and the asymptotic normality of two types of wavelet estimators of the parameter H. We do so by exploiting the fact that the integrated series is asymptotically self‐similar with parameter H. When the parameter α is known, we also obtain consistent and asymptotically normal estimators for the fractional differencing exponent d = H ? 1/α. Our results hold for a larger class of causal linear processes with stable symmetric innovations. As the wavelet‐based estimation method used here is semi‐parametric, it allows for a more robust treatment of long‐range dependent data than parametric methods.