Xiaofeng Shao

Asymptotic spectral theory for nonlinear time series

We consider asymptotic problems in spectral analysis of stationary causal processes. Limiting distributions of periodograms and smoothed periodogram spectral density estimates are obtained and applications to the spectral domain bootstrap are given. Instead of the commonly used strong mixing conditions, in our asymptotic spectral theory we impose conditions only involving (conditional) moments, which are easily verifiable for a variety of nonlinear time series.

The Dependent Wild Bootstrap

We propose a new resampling procedure, the dependent wild bootstrap, for stationary time series. As a natural extension of the traditional wild bootstrap to time series setting, the dependent wild bootstrap offers a viable alternative to the existing block-based bootstrap methods, whose properties have been extensively studied over the last two decades. Unlike all of the block-based bootstrap methods, the dependent wild bootstrap can be easily extended to irregularly spaced time series with no implementational difficulty. Furthermore, it preserves the favorable bias and mean squared error property of the tapered block bootstrap, which is the state-of-the-art block-based method in terms of asymptotic accuracy of variance estimation and distribution approximation. The consistency of the dependent wild bootstrap in distribution approximation is established under the framework of the smooth function model. In addition, we obtain the bias and variance expansions of the dependent wild bootstrap variance estimator for irregularly spaced time series on a lattice. For irregularly spaced nonlattice time series, we prove the consistency of the dependent wild bootstrap for variance estimation and distribution approximation in the mean case. Simulation studies and an empirical data analysis illustrate the finite-sample performance of the dependent wild bootstrap. Some technical details and tables are included in the online supplemental material.

Testing For White Noise Under Unknown Dependence And Its Applications To Diagnostic Checking For Time Series Models

Testing for white noise has been well studied in the literature of econometrics and statistics. For most of the proposed test statistics, such as the well-known Box–Pierce test statistic with fixed lag truncation number, the asymptotic null distributions are obtained under independent and identically distributed assumptions and may not be valid for dependent white noise. Because of recent popularity of conditional heteroskedastic models (e.g., generalized autoregressive conditional heteroskedastic [GARCH] models), which imply nonlinear dependence with zero autocorrelation, there is a need to understand the asymptotic properties of the existing test statistics under unknown dependence. In this paper, we show that the asymptotic null distribution of the Box–Pierce test statistic with general weights still holds under unknown weak dependence as long as the lag truncation number grows at an appropriate rate with increasing sample size. Further applications to diagnostic checking of the autoregressive moving average (ARMA) and fractional autoregressive integrated moving average (FARIMA) models with dependent white noise errors are also addressed. Our results go beyond earlier ones by allowing non-Gaussian and conditional heteroskedastic errors in the ARMA and FARIMA models and provide theoretical support for some empirical findings reported in the literature.

A Simple Test of Changes in Mean in the Possible Presence of Long‐Range Dependence

We propose a simple testing procedure to test for a change point in the mean of a possibly long‐range dependent time series. Under the null hypothesis, the series is stationary with long‐range dependence and our test statistic converges to a non‐degenerate distribution, whereas under the alternative, the series has a change point in the mean and the test statistic diverges to infinity. We demonstrate the good size and power properties of our test via simulations and illustrate its usefulness by analysing two real data sets.

LOCAL WHITTLE ESTIMATION OF FRACTIONAL INTEGRATION FOR NONLINEAR PROCESSES

We study asymptotic properties of the local Whittle estimator of the long memory parameter for a wide class of fractionally integrated nonlinear time series models. In particular, we solve the conjecture posed by Phillips and Shimotsu (2004, Annals of Statistics 32, 656–692) for Type I processes under our framework, which requires a global smoothness condition on the spectral density of the short memory component. The formulation allows the widely used fractional autoregressive integrated moving average (FARIMA) models with generalized autoregressive conditionally heteroskedastic (GARCH) innovations of various forms, and our asymptotic results provide a theoretical justification of the findings in simulations that the local Whittle estimator is robust to conditional heteroskedasticity. Additionally, our conditions are easily verifiable and are satisfied for many nonlinear time series models.We thank Liudas Giraitis for providing the manuscript by Dalla, Giraitis, and Hidalgo (2006). We are grateful to the two referees and the editor for their detailed comments, which led to substantial improvements. We also thank Michael Stein for helpful comments on an earlier version. The work is supported in part by NSF grant DMS-0478704.

Martingale Difference Correlation and Its Use in High-Dimensional Variable Screening

In this article, we propose a new metric, the so-called martingale difference correlation, to measure the departure of conditional mean independence between a scalar response variable V and a vector predictor variable U. Our metric is a natural extension of distance correlation proposed by Székely, Rizzo, and Bahirov, which is used to measure the dependence between V and U. The martingale difference correlation and its empirical counterpart inherit a number of desirable features of distance correlation and sample distance correlation, such as algebraic simplicity and elegant theoretical properties. We further use martingale difference correlation as a marginal utility to do high-dimensional variable screening to screen out variables that do not contribute to conditional mean of the response given the covariates. Further extension to conditional quantile screening is also described in detail and sure screening properties are rigorously justified. Both simulation results and real data illustrations demonstrate the effectiveness of martingale difference correlation-based screening procedures in comparison with the existing counterparts. Supplementary materials for this article are available online.

A LIMIT THEOREM FOR QUADRATIC FORMS AND ITS APPLICATIONS

We consider quadratic forms of martingale differences and establish a central limit theorem under mild and easily verifiable conditions. By approximating Fourier transforms of stationary processes by martingales, our central limit theorem is applied to the smoothed periodogram estimate of spectral density functions. Our results go beyond earlier ones by allowing a variety of nonlinear time series and by avoiding strong mixing and/or summability conditions on joint cumulants.We thank the two reviewers for their detailed comments, which led to substantial improvements. The work is supported in part by NSF grant DMS-0478704.

Testing the structural stability of temporally dependent functional observations and application to climate projections

Abstract: We develop a self-normalization (SN) based test to test the structural stability of temporally dependent functional observations. Testing for a change point in the mean of functional data has been studied in Berkes, Gabrys, Horvath and Kokoszka [4], but their test was developed under the independence assumption. In many applications, functional observations are expected to be dependent, especially when the data is collected over time. Building on the SN-based change point test proposed in Shao and Zhang [23] for a univariate time series, we extend the SNbased test to the functional setup by testing the constant mean of the finite dimensional eigenvectors after performing functional principal component analysis. Asymptotic theories are derived under both the null and local alternatives. Through theory and extensive simulations, our SN-based test statistic proposed in the functional setting is shown to inherit some useful properties in the univariate setup: the test is asymptotically distribution free and its power is monotonic. Furthermore, we extend the SN-based test to identify potential change points in the dependence structure of functional observations. The method is then applied to central England temperature series to detect the warming trend and to gridded temperature fields generated by global climate models to test for changes in spatial bias structure over time.

Nonstationarity-extended whittle estimation

For long memory time series models with uncorrelated but dependent errors, we establish the asymptotic normality of the Whittle estimator under mild conditions. Our framework includes the widely used fractional autoregressive integrated moving average models with generalized autoregressive conditional heteroskedastic-type innovations. To cover nonstationary fractionally integrated processes, we extend the idea of Abadir, Distaso, and Giraitis (2007, Journal of Econometrics 141, 1353–1384) and develop the nonstationarity-extended Whittle estimation. The resulting estimator is shown to be asymptotically normal and is more efficient than the tapered Whittle estimator. Finally, the results from a small simulation study are presented to corroborate our theoretical findings.

Self-Normalization for Time Series: A Review of Recent Developments

This article reviews some recent developments on the inference of time series data using the self-normalized approach. We aim to provide a detailed discussion about the use of self-normalization in different contexts and highlight distinctive feature associated with each problem and connections among these recent developments. The topics covered include: confidence interval construction for a parameter in a weakly dependent stationary time series setting, change point detection in the mean, robust inference in regression models with weakly dependent errors, inference for nonparametric time series regression, inference for long memory time series, locally stationary time series and near-integrated time series, change point detection, and two-sample inference for functional time series, as well as the use of self-normalization for spatial data and spatial-temporal data. Some new variations of the self-normalized approach are also introduced with additional simulation results. We also provide a brief review of related inferential methods, such as blockwise empirical likelihood and subsampling, which were recently developed under the fixed-b asymptotic framework. We conclude the article with a summary of merits and limitations of self-normalization in the time series context and potential topics for future investigation.