Liudas Giraitis

Rescaled variance and related tests for long memory in volatility and levels

This paper studies properties of tests for long memory for general fourth order stationary sequences. We propose a rescaled variance test based on V/S statistic which is shown to have a simpler asymptotic distribution and to achieve a somewhat better balance of size and power than Los (Econometrica 59 (1991) 1279) modified R/S test and the KPSS test of Kwiatkowski et al. (J. Econometrics 54 (1992) 159). We investigate theoretical performance of R/S, KPSS and V/S tests under short memory hypotheses and long memory alternatives, providing a Monte Carlo study and a brief empirical example. Assumptions of the same type are used in both short and long memory cases, covering all persistent dependence scenarios. We show that the results naturally apply and the assumptions are well adjusted to linear sequences (levels) and to squares of linear ARCH sequences (volatility).

STATIONARY ARCH MODELS: DEPENDENCE STRUCTURE AND CENTRAL LIMIT THEOREM

This paper studies a broad class of nonnegative ARCH(∞) models. Sufficient conditions for the existence of a stationary solution are established and an explicit representation of the solution as a Volterra type series is found. Under our assumptions, the covariance function can decay slowly like a power function, falling just short of the long memory structure. A moving average representation in martingale differences is established, and the central limit theorem is proved.

Large sample inference for long memory processes

Introduction Estimation Some Inference Problems Residual Empirical Processes Regression Models Nonparametric Regression with Heteroscedastic Errors Model Checking under Long Memory Long Memory under Infinite Variance.

Asymptotic normality of regression estimators with long memory errors

This paper discusses asymptotic normality of certain classes of M- and R-estimators of the slope parameter vector in linear regression models with long memory moving average errors, extending recent results of Koul (1992) and Koul and Mukherjee (1993). Like in the case of the long memory Gaussian errors, it is observed that all these estimators are asymptotically equivalent to the least squares estimator, a fact that is in sharp contrast with the i.i.d. errors case.

Recent Advances in ARCH Modelling

The purpose of this selective review is to present recent theoretical findings on the modelling of ARCH type non-linear times series. We provide an overview of recent theoretical results on the existence and the structure of stationary solutions to ARCH(∞), LARCH, bilinear ARCH, EGARCH, IARCH and random coefficient ARCH models, and investigate their second order dependence (memory) structure. The topics discussed in the review are: existence of a stationary solution, the presence of the short memory and long memory in ARCH type models, leverage effect, asymptotic behavior of the sums (sample mean), aggregation, parameter estimation and testing for the change-points.

ARCH-type bilinear models with double long memory

Abstract We discuss the covariance structure and long-memory properties of stationary solutions of the bilinear equation X t = ζ t A t + B t ,(★), where ζ t , t∈ Z are standard i.i.d. r.v.s, and A t , B t are moving averages in X s , s . Stationary solution of (★) is obtained as an orthogonal Volterra expansion. In the case A t ≡1, X t is the classical AR(∞) process, while B t ≡0 gives the LARCH model studied by Giraitis et al. (Ann. Appl. Probab. 10 (2000) 1002). In the general case, X t may exhibit long memory both in conditional mean and in conditional variance, with arbitrary fractional parameters 0 1 1 2 and 0 2 1 2 , respectively. We also discuss the hyperbolic decay of auto- and/or cross-covariances of X t and X t 2 and the asymptotic distribution of the corresponding partial sums’ processes.

Central limit theorem for the empirical process of a linear sequence with long memory

Abstract We discuss the functional central limit theorem (FCLT) for the empirical process of a moving-average stationary sequence with long memory. The cases of one-sided and double-sided moving averages are discussed. In the case of one-sided (causal) moving average, the FCLT is obtained under weak conditions of smoothness of the distribution and the existence of (2+ δ )-moment of i.i.d. innovations, by using the martingale difference decomposition due to Ho and Hsing (1996, Ann. Statist. 24, 992–1014). In the case of double-sided moving average, the proof of the FCLT is based on an asymptotic expansion of the bivariate probability density.

Rate optimal semiparametric estimation of the memory parameter of the Gaussian time series with long-range dependence

There exist several estimators of the memory parameter in long- memory time series models with the spectrum specified only locally near zero frequency. In this paper we give an asymptotic lower bound for the minimax risk of any estimator of the memory parameter as a function of the degree of local smoothness of the spectral density at zero. The lower bound allows one to evaluate and compare different estimators by their asymptotic behaviour, and to claim the rate optimality for any estimator attaining the bound. A log-periodogram regression estimator, analysed by Robinson (Log-periodogram regression of time series with long range dependence. Ann. Stat. 23 (1995), 1048--72), is then shown to attain the lower bound, and is thus rate optimal.

On the Optimal Segment Length for Parameter Estimates for Locally Stationary Time Series

We discuss the behaviour of parameter estimates when stationary time series models are fitted locally to non‐stationary processes which have an evolutionary spectral representation. A particular example is the estimation for an autoregressive process with time‐varying coefficients by local Yule–Walker estimates. The bias and the mean squared error for the parameter estimates are calculated and the optimal length of the data segment is determined.

A TEST FOR STATIONARITY VERSUS TRENDS AND UNIT ROOTS FOR A WIDE CLASS OF DEPENDENT ERRORS

We suggest a rescaled variance type of test for the null hypothesis of stationarity against deterministic and stochastic trends (unit roots). The deterministic trend can be represented as a general function in time (e.g., nonparametric, linear, or polynomial regression, abrupt changes in the mean). Under the null, the asymptotic distribution of the test is derived, and critical values are tabulated for a wide class of stationary processes with short, long, or negative dependence structure. A simulation study examines the performance of the test in terms of size and power. The empirical performance of the test is illustrated using the S&P 500 data.The authors thank the editor, the referees, and Karim Abadir for helpful comments and Alfredas RaA kauskas for drawing our attention to the criterion of Cremers and Kadelka (1986). The first authors work was supported by the ESRC grants R000238212 and R000239538. The last two authors were supported by a cooperation agreement CNRS/LITHUANIA (4714) and by a bilateral Lithuania-France research project Gilibert.