Carsten Jentsch

The multiple hybrid bootstrap - Resampling multivariate linear processes

The paper reconsiders the autoregressive aided periodogram bootstrap (AAPB) which has been suggested in Kreiss and Paparoditis (2003) [18]. Their idea was to combine a time domain parametric and a frequency domain nonparametric bootstrap to mimic not only a part but as much as possible the complete covariance structure of the underlying time series. We extend the AAPB in two directions. Our procedure explicitly leads to bootstrap observations in the time domain and it is applicable to multivariate linear processes, but agrees exactly with the AAPB in the univariate case, when applied to functionals of the periodogram. The asymptotic theory developed shows validity of the multiple hybrid bootstrap procedure for the sample mean, kernel spectral density estimates and, with less generality, for autocovariances.

Covariance matrix estimation and linear process bootstrap for multivariate time series of possibly increasing dimension

Multivariate time series present many challenges, especially when they are high dimensional. The paper’s focus is twofold. First, we address the subject of consistently estimating the autocovariance sequence; this is a sequence of matrices that we conveniently stack into one huge matrix. We are then able to show consistency of an estimator based on the so-called flat-top tapers; most importantly, the consistency holds true even when the time series dimension is allowed to increase with the sample size. Secondly, we revisit the linear process bootstrap (LPB) procedure proposed by McMurry and Politis (Journal of Time Series Analysis, 2010) for univariate time series. Based on the aforementioned stacked autocovariance matrix estimator, we are able to define a version of the LPB valid for multivariate time series. Under rather general assumptions, we show that our multivariate linear process bootstrap (MLPB) has asymptotic validity for the sample mean in two important cases: (a) when the time series dimension is fixed, and (b) when it is allowed to increase with sample size. As an aside, in case (a) we show that the MLPB works also for spectral density estimators which is a novel result even in the univariate case. We conclude with a simulation study that demonstrates the superiority of the MLPB in some important cases.

Testing equality of spectral densities using randomization techniques

In this paper, we investigate the testing problem that the spectral density matrices of several, not necessarily independent, stationary processes are equal. Based on an

A New Frequency Domain Approach of Testing for Covariance Stationarity and for Periodic Stationarity in Multivariate Linear Processes

L_2

Baxter`s inequality and sieve bootstrap for random fields

-type test statistic, we propose a new nonparametric approach, where the critical values of the tests are calculated with the help of randomization methods. We analyze asymptotic exactness and consistency of these randomization tests and show in simulation studies that the new procedures posses very good size and power characteristics.

Block Bootstrap Theory for Multivariate Integrated and Cointegrated 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 L‐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.

Valid Resampling of Higher-Order Statistics Using the Linear Process Bootstrap and Autoregressive Sieve Bootstrap

The concept of the autoregressive (AR) sieve bootstrap is investigated for the case of spatial processes in Z2. This procedure fits AR models of increasing order to the given data and, via resampling of the residuals, generates bootstrap replicates of the sample. The paper explores the range of validity of this resampling procedure and provides a general check criterion which allows to decide whether the AR sieve bootstrap asymptotically works for a specific statistic of interest or not. The criterion may be applied to a large class of stationary spatial processes. As another major contribution of this paper, a weighted Baxter-inequality for spatial processes is provided. This result yields a rate of convergence for the finite predictor coefficients, i.e. the coefficients of finite-order AR model fits, towards the autoregressive coefficients which are inherent to the underlying process under mild conditions. The developed check criterion is applied to some particularly interesting statistics like sample autocorrelations and standardized sample variograms. A simulation study shows that the procedure performs very well compared to normal approximations as well as block bootstrap methods in finite samples.

A Spectral Domain Test for Stationarity of Spatio-Temporal Data

We develop some asymptotic theory for applications of block bootstrap resampling schemes to multivariate integrated and cointegrated time series. It is proved that a multivariate, continuous-path block bootstrap scheme applied to a full rank integrated process, succeeds in estimating consistently the distribution of the least squares estimators in both, the regression and the spurious regression case. Furthermore, it is shown that the same block resampling scheme does not succeed in estimating the distribution of the parameter estimators in the case of cointegrated time series. For this situation, a modified block resampling scheme, the so-called residual based block bootstrap, is investigated and its validity for approximating the distribution of the regression parameters is established. The performance of the proposed block bootstrap procedures is illustrated in a short simulation study.

How Much Information Does Dependence Between Wavelet Coefficients Contain

We show that the linear process bootstrap (LPB) and the autoregressive sieve bootstrap (AR sieve) are, in general, not valid for statistics whose large-sample distribution depends on moments of order higher than two, irrespective of whether the data come from a linear time series or not. Inspired by the block-of-blocks bootstrap, we circumvent this non-validity by applying the LPB and AR sieve to suitably blocked data and not to the original data itself. In a simulation study, we compare the LPB, AR sieve, and moving block bootstrap applied directly and to blocked data.

A connectedness analysis of German financial institutions during the financial crisis in 2008

Many random phenomena in the environmental and geophysical sciences are functions of both space and time; these are usually called spatio-temporal processes. Typically, the spatio-temporal process is observed over discrete equidistant time and at irregularly spaced locations in space. One important aim is to develop statistical models based on what is observed. While doing so a commonly used assumption is that the underlying spatio-temporal process is stationary. If this assumption does not hold, then either the mean or the covariance function is misspecified. This can, for example, lead to inaccurate predictions. In this article we propose a test for spatio-temporal stationarity. The test is based on the dichotomy that Fourier transforms of stochastic processes are near uncorrelated if the process is second-order stationary but correlated if the process is second-order nonstationary. Using this as motivation, a discrete Fourier transform for spatio-temporal data over discrete equidistant times but on irregularly spaced spatial locations is defined. Two statistics which measure the degree of correlation in the discrete Fourier transforms are proposed. These statistics are used to test for spatio-temporal stationarity. It is shown that the same statistics can also be adapted to test for the one-way stationarity (either spatial or temporal stationarity). The proposed methodology is illustrated with a small simulation study.