Remigijus Leipus

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.

Change-point estimation in ARCH models

This paper studies the change-point problem and the cross-covariance function for ARCH models. Bounds for the cross-covariance function are derived and explicit formulae are obtained in special cases. Consistency of a CUSUM type change-point estimator is proved and its rate of convergence is established. A Haijek-R6nyi type inequality is also proved. Results are obtained under weak moment assumptions.

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.

Change-point in the mean of dependent observations

We prove the consistency of a family of CUSUM-type estimators of the point of change in the mean of dependent observations and derive the rates of convergence. The result is valid under weak assumptions on the dependence structure.

Testing for parameter changes in ARCH models

The paper develops the asymptotic theory for CUSUM-type tests for a change point in parameters of an ARCH(∞) model. Special attention is given to asymptotics under local alternatives.

Random coefficient autoregression, regime switching and long memory

We discuss long-memory properties and the partial sums process of the AR(1) process {X t , t ∈ 𝕫} with random coefficient {a t , t ∈ 𝕫} taking independent values A j ∈ [0,1] on consecutive intervals of a stationary renewal process with a power-law interrenewal distribution. In the case when the distribution of generic A j has either an atom at the point a=1 or a beta-type probability density in a neighborhood of a=1, we show that the covariance function of {X t } decays hyperbolically with exponent between 0 and 1, and that a suitably normalized partial sums process of {X t } weakly converges to a stable Lévy process.

ON STATIONARITY IN THE ARCH([infty infinity]) MODEL

We continue investigation of the ARCH(∞) model begun in Giraitis, Kokoszka, and Leipus (2000, Econometric Theory 16, 3–22). Nonrestrictive conditions for the existence of a strictly stationary solution are established. The paper generalizes the results of Nelson (1990, Econometric Theory 6, 318–334) and Bougerol and Picard (1992, Journal of Econometrics 52, 115–127) to the ARCH(∞) model.

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.

Stability of random coefficient ARCH models and aggregation schemes

In this paper, we consider ARCH(∞) models with nonnegative random coefficients. Necessary and sufficient conditions for the existence of first and second moments are established. The ARCH models with deterministic coefficients are shown to be always short memory processes. The effect of small perturbations of the parameters is investigated. The results obtained are applied to some aggregation schemes.