Heavy-tailed Time Series Research Articles

Testing nonlinearity of heavy-tailed time series

A test statistic for nonlinearity of a given heavy-tailed time series process is constructed, based on the sub-sample stability of Gini-based sample autocorrelations. The finite-sample performance of the proposed test is evaluated in a Monte Carlo study and compared to a similar test based on the sub-sample stability of a heavy-tailed analogue of the conventional sample autocorrelation function. In terms of size and power properties, the quality of our test outperforms a nonlinearity test for heavy-tailed time series processes proposed by [S.I. Resnick and E. Van den Berg, A test for nonlinearity of time series with infinite variance, Extremes 3 (2000), pp. 145–172.]. A nonlinear Pareto-type autoregressive process and a nonlinear Pareto-type moving average process are used as alternative specifications when comparing the power of the proposed test statistic. The efficacy of the test is illustrated via the analysis of a heavy-tailed actuarial data set and two time series of Ethernet traffic.

Extremal quantile autoregression for heavy-tailed time series

The conventional quantile estimator of the extreme conditional quantiles under quantile autoregression models is relatively unstable due to data sparsity in the tails. Extreme value theory provides a mathematical foundation of extrapolation to estimate extreme quantiles. However, the asymptotic distributions of existing estimators are often complicated making it inconvenient to apply for statistical inference. We develop a new adaptive estimation procedure based on a generalized Hill estimator of the extreme value index to estimate extreme conditional quantiles in autoregression models. We establish the asymptotic normality of the proposed estimators under some regularity conditions by applying the martingale central limit theorem. The asymptotic variances have closed expressions and can be directly estimated. Based on these results, we propose a procedure to construct confidence intervals for the extreme conditional quantiles. We also provide a diagnostic tool to check the heavy-tailedness of the distribution. Simulation studies and a real data analysis are carried out to demonstrate the advantages of the proposed methods over existing approaches.

Tail risk inference via expectiles in heavy-tailed time series

Expectiles define the only law-invariant, coherent and elicitable risk measure apart from the expectation. The popularity of expectile-based risk measures is steadily growing and their properties have been studied for independent data, but further results are needed to establish that extreme expectiles can be applied with the kind of dependent time series models relevant to finance. In this paper we provide a basis for inference on extreme expectiles and expectile-based marginal expected shortfall in a general β-mixing context that encompasses ARMA and GARCH models with heavy-tailed innovations. Our methods allow the estimation of marginal (pertaining to the stationary distribution) and dynamic (conditional on the past) extreme expectile-based risk measures. Simulations and applications to financial returns show that the new estimators and confidence intervals greatly improve on existing ones when the data are dependent.

Distribution-Based Entropy Weighting Clustering of Skewed and Heavy Tailed Time Series

The goal of clustering is to identify common structures in a data set by forming groups of homogeneous objects. The observed characteristics of many economic time series motivated the development of classes of distributions that can accommodate properties, such as heavy tails and skewness. Thanks to its flexibility, the skewed exponential power distribution (also called skewed generalized error distribution) ensures a unified and general framework for clustering possibly skewed and heavy tailed time series. This paper develops a clustering procedure of model-based type, assuming that the time series are generated by the same underlying probability distribution but with different parameters. Moreover, we propose to optimally combine the estimated parameters to form the clusters with an entropy weighing k-means approach. The usefulness of the proposal is shown by means of application to financial time series, demonstrating also how the obtained clusters can be used to form portfolio of stocks.

Robust Test for Detecting Changes in the Autocovariance Function of a Time Series

We propose a new robust test to detect changes in the autocovariance function of a time series. The test is based on empirical autocovariances of a robust transformation of the original time series. Because of the transformation, we do not require any finite moments of the original time series, making the test especially suitable for heavy tailed time series. We furthermore propose a lag weighting scheme, which puts emphasis on changes of the autocovariance at smaller lags. Our approach is compared to existing ones in some simulations.

Lasso guarantees for $\beta$-mixing heavy-tailed time series

Many theoretical results for lasso require the samples to be i.i.d. Recent work has provided guarantees for lasso assuming that the time series is generated by a sparse Vector Autoregressive (VAR) model with Gaussian innovations. Proofs of these results rely critically on the fact that the true data generating mechanism (DGM) is a finite-order Gaussian VAR. This assumption is quite brittle: linear transformations, including selecting a subset of variables, can lead to the violation of this assumption. In order to break free from such assumptions, we derive nonasymptotic inequalities for estimation error and prediction error of lasso estimate of the best linear predictor without assuming any special parametric form of the DGM. Instead, we rely only on (strict) stationarity and geometrically decaying $\beta$-mixing coefficients to establish error bounds for lasso for sub-Weibull random vectors. The class of sub-Weibull random variables that we introduce includes sub-Gaussian and subexponential random variables but also includes random variables with tails heavier than an exponential. We also show that, for Gaussian processes, the $\beta$-mixing condition can be relaxed to summability of the $\alpha$-mixing coefficients. Our work provides an alternative proof of the consistency of lasso for sparse Gaussian VAR models. But the applicability of our results extends to non-Gaussian and nonlinear times series models as the examples we provide demonstrate.

The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails

We consider a $p$-dimensional time series where the dimension $p$ increases with the sample size $n$. The resulting data matrix $X$ follows a stochastic volatility model: each entry consists of a positive random volatility term multiplied by an independent noise term. The volatility multipliers introduce dependence in each row and across the rows. We study the asymptotic behavior of the eigenvalues and eigenvectors of the sample covariance matrix $XX'$ under a regular variation assumption on the noise. In particular, we prove Poisson convergence for the point process of the centered and normalized eigenvalues and derive limit theory for functionals acting on them, such as the trace. We prove related results for stochastic volatility models with additional linear dependence structure and for stochastic volatility models where the time-varying volatility terms are extinguished with high probability when $n$ increases. We provide explicit approximations of the eigenvectors which are of a strikingly simple structure. The main tools for proving these results are large deviation theorems for heavy-tailed time series, advocating a unified approach to the study of the eigenstructure of heavy-tailed random matrices.

Asymptotic Theory for QMLE for Real-Time GARCH(1,1) Model

We investigate the asymptotic properties of the Gaussian Quasi-Maximum-Likelihood estimator (QMLE) for the Real-time GARCH(1,1) model of Smetanina (2017). The developed theory relies on the new dependence measure developed in Wu (2005) and is substantially different to the standard asymptotic theory for GARCH models. We prove consistency and asymptotic normality for the parameter vector at the usual √T rate. Finally, as part of the developed theory we also demonstrate how convergence rates of uniform laws of large numbers can be established via the powerful maximal inequalities for high-dimensional heavy-tailed time series using uniform functional dependence measure.

Inference in heavy-tailed vector error correction models

This paper first studies the full rank least squares estimator (FLSE) of the heavy-tailed vector error correction (VEC) models. It is shown that the rate of convergence of the FLSE related to the long-run parameters is n (sample size) and its limiting distribution is a stochastic integral in terms of two stable random processes when the tail index α∈(0,2). Furthermore, we show that the rate of convergence of the FLSE related to the short-term parameters is n1∕αL̃(n) and its limiting distribution is a functional of two stable processes when α∈(1,2), where L̃(n) is a slowly varying function. However, when α∈(0,1), we show that the rate of convergence of the FLSE related to the short-term parameters is n and its limiting distribution not only depends on the stationary component itself but also depends on the unit root component. Based on the FLSE, we then study the limiting behavior of the reduced rank LSE (RLSE). The results related to the short-term parameters of both FLSE and RLSE are significantly different from those of heavy-tailed time series in the literature, and it may provide new insights in the area for future research. Simulation study is carried out to demonstrate the performance of both estimators. A real example with application to 3-month Treasury Bill rate, 1-year Treasury Bill rate and Federal Fund rate is given.

Testing Serial Correlation and ARCH Effect of High-Dimensional Time-Series Data

This article proposes several tests for detecting serial correlation and ARCH effect in high-dimensional data. The dimension of data may go to infinity when the sample size . It is shown that the sample autocorrelations and the sample rank autocorrelations (Spearman’s rank correlation) of the L1-norm of data are asymptotically normal. Two portmanteau tests based, respectively, on the norm and its rank are shown to be asymptotically χ2-distributed, and the corresponding weighted portmanteau tests are shown to be asymptotically distributed as a linear combination of independent χ2 random variables. These tests are dimension-free, that is, independent of p, and the norm rank-based portmanteau test and its weighted counterpart can be used for heavy-tailed time series. We further discuss two standardized norm-based tests. Simulation results show that the proposed test statistics have satisfactory sizes and are powerful even for the case of small n and large p. We apply the tests to two real datasets. Supplementary materials for this article are available online.

Inference for multiple change points in heavy-tailed time series via rank likelihood ratio scan statistics

This paper proposes a rank likelihood ratio scan method for estimating multiple change points in piecewise heavy-tailed time series. It can effectively improve the estimate accuracy and solve the problem that the likelihood ratio scan method overestimates the change points in such a time series. A simulation and analyses of two sets of real data illustrate the efficiency of the method.

Parameter Estimation of Heavy-Tailed AR Model With Missing Data Via Stochastic EM

The autoregressive (AR) model is a widely used model to understand time series data. Traditionally, the innovation noise of the AR is modeled as Gaussian. However, many time series applications, for example, financial time series data, are non-Gaussian, therefore, the AR model with more general heavy-tailed innovations is preferred. Another issue that frequently occurs in time series is missing values, due to system data record failure or unexpected data loss. Although there are numerous works about Gaussian AR time series with missing values, as far as we know, there does not exist any work addressing the issue of missing data for the heavy-tailed AR model. In this paper, we consider this issue for the first time, and propose an efficient framework for parameter estimation from incomplete heavy-tailed time series based on a stochastic approximation expectation maximization (SAEM) coupled with a Markov Chain Monte Carlo (MCMC) procedure. The proposed algorithm is computationally cheap and easy to implement. The convergence of the proposed algorithm to a stationary point of the observed data likelihood is rigorously proved. Extensive simulations and real datasets analyses demonstrate the efficacy of the proposed framework.

Random Matrix Theory for Heavy-Tailed Time Series

This paper is a review of recent results for large random matrices with heavy-tailed entries. First, we outline the development of and some classical results in random matrix theory. We focus on large sample covariance matrices, their limiting spectral distributions, and the asymptotic behavior of their largest and smallest eigenvalues and their eigenvectors. The limits significantly depend on the finite or infiniteness of the fourth moment of the entries of the random matrix. We compare the results for these two regimes which give rise to completely different asymptotic theories. Finally, the limits of the extreme eigenvalues of sample correlation matrices are examined.

Editorial, September 2018

On behalf of both the editorial board and the readership of the Journal of Time Series Analysis, I would like to take this opportunity to thank Professor Andrew Harvey for his many years of dedicated service to our journal. Andrew stepped down as a co-editor with effect from 31 May 2018. I am, however, delighted to announce that Andrew has agreed to remain on the editorial board as an advisory editor, as from 1 June 2018. Andrew was first appointed to the editorial board of the Journal of Time Series Analysis in 1989. I have greatly benefitted from his support, help and advice since I took over as editor of the journal. I would like to personally thank Andrew for all he has done for me and for our journal. Shiqing Ling is Professor of at the Hong Kong University of Science and Technology. His research involves change-point problems, empirical processes, financial econometrics, large sample theory, long memory time series, non-stationary time series and nonlinear time series. He is a leading expert in the field of time series analysis and has made a number of fundamental contributions to the area, including asymptotic theory of unit-root GARCH process, inference of GARCH-type models and threshold time series model, and proposed the self-weight estimation method for heavy-tailed time series models.

Modelling of low count heavy tailed time series data consisting large number of zeros and ones

In this paper, we construct a new mixture of geometric INAR(1) process for modeling over-dispersed count time series data, in particular data consisting of large number of zeros and ones. For some real data sets, the existing INAR(1) processes do not fit well, e.g., the geometric INAR(1) process overestimates the number of zero observations and underestimates the one observations, whereas Poisson INAR(1) process underestimates the zero observations and overestimates the one observations. Furthermore, for heavy tails, the PINAR(1) process performs poorly in the tail part. The existing zero-inflated Poisson INAR(1) and compound Poisson INAR(1) processes have the same kind of limitations. In order to remove this problem of under-fitting at one point and over-fitting at others points, we add some extra probability at one in the geometric INAR(1) process and build a new mixture of geometric INAR(1) process. Surprisingly, for some real data sets, it removes the problem of under and over-fitting over all the observations up to a significant extent. We then study the stationarity and ergodicity of the proposed process. Different methods of parameter estimation, namely the Yule-Walker and the quasi-maximum likelihood estimation procedures are discussed and illustrated using some simulation experiments. Furthermore, we discuss the future prediction along with some different forecasting accuracy measures. Two real data sets are analyzed to illustrate the effective use of the proposed model.

Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices with general growth rates: The iid case

In this paper we study the joint distributional convergence of the largest eigenvalues of the sample covariance matrix of a p-dimensional time series with iid entries when p converges to infinity together with the sample size n. We consider only heavy-tailed time series in the sense that the entries satisfy some regular variation condition which ensures that their fourth moment is infinite. In this case, Soshnikov (2004, 2006) and Auffinger et al. (2009) proved the weak convergence of the point processes of the normalized eigenvalues of the sample covariance matrix towards an inhomogeneous Poisson process which implies in turn that the largest eigenvalue converges in distribution to a Fréchet distributed random variable. They proved these results under the assumption that p and n are proportional to each other. In this paper we show that the aforementioned results remain valid if p grows at any polynomial rate. The proofs are different from those in Auffinger et al. (2009) and Soshnikov (2004, 2006); we employ large deviation techniques to achieve them. The proofs reveal that only the diagonal of the sample covariance matrix is relevant for the asymptotic behavior of the largest eigenvalues and the corresponding eigenvectors which are close to the canonical basis vectors. We also discuss extensions of the results to sample autocovariance matrices.

A large deviations approach to limit theory for heavy-tailed time series

In this paper we propagate a large deviations approach for proving limit theory for (generally) multivariate time series with heavy tails. We make this notion precise by introducing regularly varying time series. We provide general large deviation results for functionals acting on a sample path and vanishing in some neighborhood of the origin. We study a variety of such functionals, including large deviations of random walks, their suprema, the ruin functional, and further derive weak limit theory for maxima, point processes, cluster functionals and the tail empirical process. One of the main results of this paper concerns bounds for the ruin probability in various heavy-tailed models including GARCH, stochastic volatility models and solutions to stochastic recurrence equations. 1. Preliminaries and basic motivation In the last decades, a lot of efforts has been put into the understanding of limit theory for dependent sequences, including Markov chains (Meyn and Tweedie [42]), weakly dependent sequences (Dedecker et al. [21]), long-range dependent sequences (Doukhan et al. [23], Samorodnitsky [54]), empirical processes (Dehling et al. [22]) and more general structures (Eberlein and Taqqu [25]), to name a few references. A smaller part of the theory was devoted to limit theory under extremal dependence for point processes, maxima, partial sums, tail empirical processes. Resnick [49, 50] started a systematic study of the relations between the convergence of point processes, sums and maxima, see also Resnick [51] for a recent account. He advocated the use of multivariate regular variation as a flexible tool to describe heavy-tail phenomena combined with advanced continuous mapping techniques. For example, maxima and sums are understood as functionals acting on an underlying point process, if the point process converges these functionals converge as well and their limits are described in terms of the points of the limiting point process. Davis and Hsing [13] recognized the power of this approach for limit theory of point processes, maxima, sums, and large deviations for dependent regularly varying processes, i.e., stationary sequences whose finite-dimensional distributions are regularly varying with the same index. Before [13], limit theory for particular regularly varying stationary sequences was studied for the sample mean, maxima, sample autocovariance and autocorrelation functions of linear and bilinear processes with iid regularly varying noise and extreme value theory was considered for regularly varying ARCH processes and solutions to stochastic recurrence equation, see Rootz\'en [53], Davis and 1991 Mathematics Subject Classification. Primary 60F10, 60G70, secondary 60F05. Key words and phrases. Large deviation principle, regularly varying processes, central limit theorem, ruin probabilities, GARCH.

Heavy tailed time series with extremal independence

We consider heavy tailed time series whose finite-dimensional distributions are extremally independent in the sense that extremely large values cannot be observed consecutively. This calls for methods beyond the classical multivariate extreme value theory which is convenient only for extremally dependent multivariate distributions. We use the Conditional Extreme Value approach to study the effect of an extreme value at time zero on the future of the time series. In formal terms, we study the limiting conditional distribution of future observations given an extreme value at time zero. To this purpose, we introduce conditional scaling functions and conditional scaling exponents. We compute these quantities for a variety of models, including Markov chains, exponential autoregressive models, stochastic volatility models with heavy tailed innovations or volatilities.

Simultaneous sparse model selection and coefficient estimation for heavy-tailed autoregressive processes

We propose a sparse coefficient estimation and automated model selection procedure for autoregressive processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator satisfies a strong consistency, OP(N−1/2) consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator and smoothly clipped average deviation, are compared. The proposed method provides a distribution-based penalized inference to AR models, which is especially useful when the other estimation methods fail or under perform for AR processes with heavy-tailed innovations [Feigin, Resnick. Pitfalls of fitting autoregressive models for heavy-tailed time series. Extremes. 1999;1:391–422]. A simulation study confirms our theoretical results. At the end, we apply our method to a historical price data of the US Industrial Production Index for consumer goods, and obtain very promising results.

Measures of serial extremal dependence and their estimation

The goal of this paper is two-fold: (1) We review classical and recent measures of serial extremal dependence in a strictly stationary time series as well as their estimation. (2) We discuss recent concepts of heavy-tailed time series, including regular variation and max-stable processes.Serial extremal dependence is typically characterized by clusters of exceedances of high thresholds in the series. We start by discussing the notion of extremal index of a univariate sequence, i.e. the reciprocal of the expected cluster size, which has attracted major attention in the extremal value literature. Then we continue by introducing the extremogram which is an asymptotic autocorrelation function for sequences of extremal events in a time series. In this context, we discuss regular variation of a time series. This notion has been useful for describing serial extremal dependence and heavy tails in a strictly stationary sequence. We briefly discuss the tail process coined by Basrak and Segers to describe the dependence structure of regularly varying sequences in a probabilistic way. Max-stable processes with Fréchet marginals are an important class of regularly varying sequences. Recently, this class has attracted attention for modeling and statistical purposes. We apply the extremogram to max-stable processes. Finally, we discuss estimation of the extremogram both in the time and frequency domains.