Donatas Surgailis

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.

The change-point problem for dependent observations

Abstract We consider the change-point problem for the marginal distribution function of a strictly stationary time series. Asymptotic behavior of Kolmogorov-Smirnov type tests and estimators of the change point is studied under the null hypothesis and converging alternatives. The discussion is based on a general empirical process approach which enables a unified treatment of both short-memory (weakly dependent) and long-memory time series. In particular, the case of long-memory moving-average process Xj = Σs ⩽ j bj−sξs is studied, using the recent results of Giraitis and Surgailis (1994).

Asymptotics of M-estimators in two-phase linear regression models

This paper discusses the consistency and limiting distributions of a class of M-estimators in two-phase random design linear regression models where the regression function is discontinuous at the change-point with a fixed jump size. The consistency rate of an M-estimator for the change-point parameter r is shown to be n while it is n1/2 for the coefficient parameter estimators, where n denotes the sample size. The normalized M-process is shown to be uniformly locally asymptotically equivalent to the sum of a quadratic form in the coefficient parameter vector and a jump point process in the change-point parameter, in probability. These results are then used to obtain the joint weak convergence of the M-estimators. In particular, is shown to converge weakly to a random variable which minimizes a compound Poisson process, a suitably standardized coefficient parameter M-estimator vector is shown to be asymptotically normal, and independent of .

Asymptotics of empirical processes of long memory moving averages with infinite variance

This paper obtains a uniform reduction principle for the empirical process of a stationary moving average time series {Xt} with long memory and independent and identically distributed innovations belonging to the domain of attraction of symmetric [alpha]-stable laws, 1

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.

Asymptotic Expansion of the Empirical Process of Long Memory Moving Averages

Moving averages in i.i.d. variables form one of the most important classes of long memory time series. The paper reviews various results on the asymptotic distribution of empirical processes of long memory moving averages with finite and infinite variance. It also discusses some interesting applications to goodness-of-fit testing for the marginal stationary error distribution in linear regression models and M-estimation in the one sample location model.

Variance-type estimation of long memory

The aggregation procedure when a sample of length N is divided into blocks of length m=o(N), m-->[infinity] and observations in each block are replaced by their sample mean, is widely used in statistical inference. Taqqu et al. (1995, Fractals, 3, 785-798), and Teverovsky and Taqqu (1997, J. Time Ser. Anal., 18, 279-304) introduced an aggregated variance estimator of the long-memory parameter of a stationary sequence with long range dependence and studied its empirical performance. With respect to autocovariance structure and marginal distribution, the aggregated series is closer to Gaussian fractional noise than the initial series. However, the variance type estimator based on aggregated data is seriously biased. A refined estimator, which employs least-squares regression across varying levels of aggregation, has much smaller bias, permitting deriviation of limiting distributional properties of suitably centered estimates, as well as of a minimum-mean squared error choice of bandwidth m. The results vary considerably with the actual value of the memory parameter.