Oracle M‐Estimation for Time Series Models

We propose a thresholding M‐estimator for multivariate time series. Our proposed estimator has the oracle property that its large‐sample properties are the same as of the classical M‐estimator obtained under the a priori information that the zero parameters were known. We study the consistency of the standard block bootstrap, the centred block bootstrap and the empirical likelihood block bootstrap distributions of the proposed M‐estimator. We develop automatic selection procedures for the thresholding parameter and for the block length of the bootstrap methods. We present the results of a simulation study of the proposed methods for a sparse vector autoregressive VAR(2) time series model. The analysis of two real‐world data sets illustrate applications of the methods in practice.     

On the Consistency of Bootstrap Testing for a Parameter on the Boundary of the Parameter Space

It is well known that with a parameter on the boundary of the parameter space, such as in the classic cases of testing for a zero location parameter or no autoregressive conditional heteroskedasticity (ARCH) effects, the classic nonparametric bootstrap – based on unrestricted parameter estimates – leads to inconsistent testing. In contrast, we show here that for the two aforementioned cases, a nonparametric bootstrap test based on parameter estimates obtained under the null – referred to as ‘restricted bootstrap’ – is indeed consistent. While the restricted bootstrap is simple to implement in practice, novel theoretical arguments are required in order to establish consistency. In particular, since the bootstrap is analysed both under the null hypothesis and under the alternative, non‐standard asymptotic expansions are required to deal with parameters on the boundary. Detailed proofs of the asymptotic validity of the restricted bootstrap are given and, for the leading case of testing for no ARCH, a Monte Carlo study demonstrates that the bootstrap quasi‐likelihood ratio statistic performs extremely well in terms of empirical size and power for even remarkably small samples, outperforming the standard and bootstrap Lagrange multiplier tests as well as the asymptotic quasi‐likelihood ratio test.     

Generalized Resampling Scheme With Application to Spectral Density Matrix in Almost Periodically Correlated Class of Time Series

The aim of this article is to introduce new resampling scheme for nonstationary time series, called generalized resampling scheme (GRS). The proposed procedure is a generalization of well known in the literature subsampling procedure and is simply related to existing block bootstrap techniques. To document the usefulness of GRS, we consider the example of model with almost periodic phenomena in mean and variance function, where the consistency of the proposed procedure was examined. Finally, we prove the consistency of GRS for the spectral density matrix for nonstationary, multivariate almost periodically correlated time series. We consider both zero mean and non‐zero mean case. The consistency holds under general assumptions concerning moment and α‐mixing conditions for multivariate almost periodically correlated time series. Proving the consistency in this case poses a difficulty since the estimator of the spectral density matrix can be interpreted as a sum of random matrixes whose dependence grow with the sample size.     

Bootstrap Unit‐Root Tests: Comparison and Extensions

Abstract. In this article, we study and compare the properties of several bootstrap unit‐root tests recently proposed in the literature. The tests are Dickey–Fuller (DF) or Augmented DF, based either on residuals from an autoregression and the use of the block bootstrap or on first‐differenced data and the use of the stationary bootstrap or sieve bootstrap. We extend the analysis by interchanging the data transformations (differences vs. residuals), the types of bootstrap and the presence or absence of a correction for autocorrelation in the tests. We show that two sieve bootstrap tests based on residuals remain asymptotically valid. In contrast to the literature which focuses on a comparison of the bootstrap tests with an asymptotic test, we compare the bootstrap tests among themselves using response surfaces for their size and power in a simulation study. This study leads to the following conclusions: (i) augmented DF tests are always preferred to standard DF tests; (ii) the sieve bootstrap performs better than the block bootstrap; (iii) difference‐based tests appear to have slightly better size properties, but residual‐based tests appear more powerful.     

Moving Fourier Analysis for Locally Stationary Processes with the Bootstrap in View

We introduce a moving Fourier transformation for locally stationary time series, which captures the time‐varying spectral density in a similar manner as the classical Fourier transform does for stationary time series. In particular, the resulting Fourier coefficients as well as moving local periodograms are shown to be (almost all) asymptotically uncorrelated. The moving local periodogram is obtained by thinning the local periodogram to avoid multiple information present at different but close points in time. We obtain consistent estimators for the local spectral density at each point in time by smoothing the moving local periodogram. Furthermore, the moving Fourier coefficients, respectively periodograms, are well suited to adapt stationary frequency domain bootstrap methods to the locally stationary case. For the wild time frequency toggle bootstrap, it is shown that the corresponding bootstrap covariance of a global locally stationary bootstrap samples captures the time‐varying covariance structure of the underlying locally stationary time series correctly. Furthermore, this bootstrap in addition to adaptations of other frequency domain bootstrap methods is used in a simulation study to obtain uniform confidence bands for the time‐varying autocorrelation at lag 1. Finally, this methodology is applied to a wind data set.     

Bootstrapping confidence intervals for the change‐point of time series

Abstract. We study an at‐most‐one‐change time‐series model with an abrupt change in the mean and dependent errors that fulfil certain mixing conditions. We obtain confidence intervals for the unknown change‐point via bootstrapping methods. Precisely, we use a block bootstrap of the estimated centred error sequence. Then, we reconstruct a sequence with a change in the mean using the same estimators as before. The difference between the change‐point estimator of the resampled sequence and the one of the original sequence can be used as an approximation of the difference between the real change‐point and its estimator. This enables us to construct confidence intervals using the empirical distribution of the resampled time series. A simulation study shows that the resampled confidence intervals are usually closer to their target levels and at the same time smaller than the asymptotic intervals.     

A bootstrap test for additive outliers in non‐stationary time series

In this paper we propose a new procedure for detecting additive outliers in a univariate time series based on a bootstrap implementation of the test of Perron and Rodríguez (2003, Journal of Time Series Analysis 24, 193‐220). This procedure is used to test the null hypothesis that a time series is uncontaminated by additive outliers against the alternative that one or more additive outliers are present. We demonstrate that the existing tests of, inter alia, Vogelsang (1999, Journal of Time Series Analysis 20, 237–52) Perron and Rodríguez (2003) and Burridge and Taylor (2006, Journal of Time Series Analysis 27, 685–701) are unable to strike a balance between size and power when the order of integration of a time series is unknown and the time series is driven by innovations drawn from an unknown distribution. We show that the proposed bootstrap testing procedure is able to control size to such an extent that its size properties are comparable with the robust test of Burridge and Taylor (2006) when the distribution of the innovations is not assumed known, whilst maintaining power in the Gaussian environment close to that of the test of Perron and Rodríguez (2003).     

A Structural‐Factor Approach to Modeling High‐Dimensional Time Series and Space‐Time Data

This article considers a structural‐factor approach to modeling high‐dimensional time series and space‐time data by decomposing individual series into trend, seasonal, and irregular components. For ease in analyzing many time series, we employ a time polynomial for the trend, a linear combination of trigonometric series for the seasonal component, and a new factor model for the irregular components. The new factor model simplifies the modeling process and achieves parsimony in parameterization. We propose a Bayesian information criterion to consistently select the order of the polynomial trend and the number of trigonometric functions, and use a test statistic to determine the number of common factors. The convergence rates for the estimators of the trend and seasonal components and the limiting distribution of the test statistic are established under the setting that the number of time series tends to infinity with the sample size, but at a slower rate. We study the finite‐sample performance of the proposed analysis via simulation, and analyze two real examples. The first example considers modeling weekly PM_2.5 data of 15 monitoring stations in the southern region of Taiwan and the second example consists of monthly value‐weighted returns of 12 industrial portfolios.     

Testing equality of autocovariance operators for functional time series

We consider strictly stationary stochastic processes of Hilbert space-valued random variables and focus on fully functional tests for the equality of the lag-zero autocovariance operators of several independent functional time series. A moving block bootstrap (MBB)-based testing procedure is proposed which generates pseudo random elements that satisfy the null hypothesis of interest. It is based on directly bootstrapping the time series of tensor products which overcomessome common difficulties associated with applications of the bootstrap to related testing problems. The suggested methodology can be potentially applied to a broad range of test statistics of the hypotheses of interest. As an example, we establish validity for approximating the distribution under the null of a test statistic based on the Hilbert–Schmidt distance of the corresponding sample lag-zero autocovariance operators, and show consistency under the alternative. As a prerequisite, we prove a central limit theorem for the MBB procedure applied to the sample autocovariance operator which is of interest on its own. The finite sample size and power performance of the suggested MBB-based testing procedure is illustrated through simulations and an application to a real-life dataset is discussed.     

A HYBRID BOOTSTRAP APPROACH TO UNIT ROOT TESTS

This article proposes a hybrid bootstrap approach to approximate the augmented Dickey–Fuller test by perturbing both the residual sequence and the minimand of the objective function. Since innovations can be dependent, this allows the inclusion of conditional heteroscedasticity models. The new bootstrap method is also applied to least absolute deviation‐based unit root test statistics, which are efficient in handling heavy‐tailed time‐series data. The asymptotic distributions of resulting bootstrap tests are presented, and Monte Carlo studies demonstrate the usefulness of the proposed tests.     

Subsampling inference for the mean of heavy‐tailed long‐memory time series

In this article, we revisit a time series model introduced by MCElroy and Politis (2007a) and generalize it in several ways to encompass a wider class of stationary, nonlinear, heavy‐tailed time series with long memory. The joint asymptotic distribution for the sample mean and sample variance under the extended model is derived; the associated convergence rates are found to depend crucially on the tail thickness and long memory parameter. A self‐normalized sample mean that concurrently captures the tail and memory behaviour, is defined. Its asymptotic distribution is approximated by subsampling without the knowledge of tail or/and memory parameters; a result of independent interest regarding subsampling consistency for certain long‐range dependent processes is provided. The subsampling‐based confidence intervals for the process mean are shown to have good empirical coverage rates in a simulation study. The influence of block size on the coverage and the performance of a data‐driven rule for block size selection are assessed. The methodology is further applied to the series of packet‐counts from ethernet traffic traces.     

A Generalised Fractional Differencing Bootstrap for Long Memory Processes

A bootstrap methodology suitable for use with stationary and non‐stationary fractionally integrated time series is further developed in this article. The resampling algorithm involves estimating the degree of fractional integration, applying the fractional differencing operator, resampling the resulting approximation to the underlying short memory series and, finally, cumulating to obtain a resample of the original fractionally integrated process. This approach extends existing methods in the literature by allowing for general bootstrap schemes including blockwise bootstraps. Furthermore, we show that it can also be validly used for non‐stationary fractionally integrated processes. We establish asymptotic validity results for the general method and provide simulation evidence which highlights a number of favourable aspects of its finite sample performance, relative to other commonly used bootstrap methods.     

Time series analysis based on running Mann‐Whitney Z Statistics

A time series analysis method based on the calculation of Mann–Whitney U statistics is described. This method samples data rankings over running time windows, converts those samples to Mann–Whitney U statistics, and then normalizes the U statistics to Z statistics using Monte‐Carlo generated null parameters. Based on the Z statistics’ magnitudes this algorithm can identify time windows containing significant incidences of low or high data rankings, where the window length is determined by the sample size. By repeating this process with sampling windows of varying duration ranking regimes of arbitrary onset and duration can be objectively identified in a time series. The simplicity of the procedure's output – a time series’ most significant non‐overlapping ranking sequences – makes it possible to graphically identify common temporal breakpoints and patterns of variability in the analyses of multiple time series. This approach is demonstrated using United States annual temperature data during 1896–2008.     

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.     

Bootstrapping Time Series Regressions with Integrated Processes

This paper studies the bootstrap procedures for time series regressions with integrated processes. Both estimation and hypothesis testing are studied. It is shown that the suggested bootstrap approximations to the distribution of the least squares estimator and the regression test statistic are asymptotically valid. A Monte Carlo experiment is conducted to evaluate the finite sample performance of these bootstrap procedures. The simulation results indicate that the bootstrap method provides reasonably good approximation to the distribution of the least squares estimator, and gives proper size and satisfactory power.     

Testing the Null of Co‐integration in the Presence of Variance Breaks

Abstract. We show that changes in the innovation covariance matrix of a vector of series can generate spurious rejections of the null hypothesis of co‐integration when applying standard residual‐based co‐integration tests. A bootstrap solution to the inference problem is suggested which is shown to perform well in practice, redressing the size problems associated with the standard test but not losing power relative to the standard test under the alternative.     

Bootstrap Approximation to Prediction MSE for State–Space Models with Estimated Parameters

Abstract. We propose simple parametric and nonparametric bootstrap methods for estimating the prediction mean square error (PMSE) of state vector predictors that use estimated model parameters. As is well known, substituting the model parameters by their estimates in the theoretical PMSE expression that assumes known parameter values results in underestimation of the true PMSE. The parametric method consists of generating parametrically a large number of bootstrap series from the model fitted to the original series, re‐estimating the model parameters for each series using the same method as used for the original series and then estimating the separate components of the PMSE. The nonparametric method generates the series by bootstrapping the standardized innovations estimated for the original series. The bootstrap methods are compared with other methods considered in the literature in a simulation study that also examines the robustness of the various methods to non‐normality of the model error terms. Application of the bootstrap method to a model fitted to employment ratios in the USA that contains 18 unknown parameters, estimated by a three‐step procedure yields unbiased PMSE estimators.     

Semiparametric Modeling of Seasonal Time Series

Often, in practice, one may regard an observed time series as being composed of a function that is smooth over years, with additive seasonal effects. As a modification, we formulate a particular multiplicative model that expresses the observed data as a yearly trend function with additive amplitude-modulated seasonal factors. Without smoothness restrictions on the yearly trend and modulation components, the least squares solutions for the seasonal components are shown to be proportional to the eigenvector corresponding to the maximum eigen value of the within-season covariance matrix. If the trend and seasonal modulations are modeled as smooth splines, we give the comparable estimators for the smooth functions and the seasonal factors. We show consistency for the trend, modulation and seasonal factors as well as asymptotic normality for the seasonal estimates. Model selection, fitting and forecasting are considered for a quarterly earnings series that exhibits extreme nonlinear and nonstationary behavior. We compare the results with those obtained using a competing nonstationary multiplicative ARIMA model     

A Goodness‐of‐Fit Test for Integer‐Valued Autoregressive Processes

For autoregressive count data time series, a goodness‐of‐fit test based on the empirical joint probability generating function is considered. The underlying process is contained in a general class of Markovian models satisfying a drift condition. Asymptotic theory for the test statistic is provided, including a functional central limit theorem for the non‐parametric estimation of the stationary distribution and a parametric bootstrap method. Connections between the new approach and existing tests for count data time series based on moment estimators appear in limiting scenarios. Finally, the test is applied to a real data set.     

Implementing Residual‐Based KPSS Tests for Cointegration with Data Subject to Temporal Aggregation and Mixed Sampling Frequencies

We show how different data types (stocks and flows) and temporal aggregation affect the size and power of the dynamic ordinary least squares residual‐based Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test of the null of cointegration. Size may be more effectively controlled by setting the minimum number of leads equal to one – as opposed to zero – when selecting the lag/lead order of the dynamic ordinary least squares regression using aggregated data, but at a cost to power. If high‐frequency data for one or more series are available – that is, the model has mixed sampling frequencies – we show how to effectively utilize the high‐frequency data to increase power while controlling size.