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 ® ¥ and observations in each block are replaced by their sample mean, is widely used in statistical inference. Taqqu, Teverovsky and Willinger (1995), Teverovsky and Taqqu (1997) introduced an aggregate variance estimator of the long memory parameter of a stationary sequence with long range dependence and studied its empirial performance. With respect to autovariance 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 derivation 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

On asymptotic distributions of weighted sums of periodograms

We establish asymptotic normality of weighted sums of periodograms of a stationary linear process where weights depend on the sample size. Such sums appear in numerous statistical applications and can be regarded as a discretized versions of quadratic forms involving integrals of weighted periodograms. Conditions for asymptotic normality of these weighted sums are simple, minimal, and resemble Lindeberg-Feller condition for weighted sums of independent and identically distributed random variables. Our results are applicable to a large class of short, long or negative memory processes. The proof is based on sharp bounds derived for Bartlett type approximation of these sums by the corresponding sums of weighted periodograms of independent and identically distributed random variables.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ456 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Edgeworth expansions for semiparametric Whittle estimation of long memory.

The semiparametric local Whittle or Gaussian estimate of the long memory parameter is known to have especially nice limiting distributional properties, being asymptotically normal with a limiting variance that is completely known. However in moderate samples the normal approximation may not be very good, so we consider a refined, Edgeworth, approximation, for both a tapered estimate, and the original untapered one. For the tapered estimate, our higher-order correction involves two terms, one of order m-1/2 (where m is the bandwidth number in the estimation), the other a bias term, which increases in m; depending on the relative magnitude of the terms, one or the other may dominate, or they may balance. For the untapered estimate we obtain an expansion in which, for m increasing fast enough, the correction consists only of a bias term. We discuss applications of our expansions to improved statistical inference and bandwidth choice. We assume Gaussianity, but in other respects our assumptions seem mild.

On the existence of some ARCH(∞\infty) processes

A new sufficient condition for the existence of a stationary causal solution of an ARCH($\infty$) equation is provided. This condition allows to consider polynomially decaying coefficients, so that it can be applied to the so-called FIGARCH processes, whose existence is thus proved

Edgeworth Expansions for Semiparametric Whittle Estimation of Long Memory

The semiparametric local Whittle or Gaussian estimate of the long memory parameter is known to have especially nice limiting distributional properties, being asymptotically normal with a limiting variance that is completely known. However in moderate samples the normal approximation may not be very good, so we consider a refined, Edgeworth, approximation, for both a tapered estimate, and the original untapered one. For the tapered estimate, our higher-order correction involves two terms, one of order 1/vm (where m is the bandwidth number in the estimation), the other a bias term, which increases in m; depending on the relative magnitude of the terms, one or the other may dominate, or they may balance. For the untapered estimate we obtain an expansion in which, for m increasing fast enough, the correction consists only of a bias term. We discuss applications of our expansions to improved statistical inference and bandwidth choice. We assume Gaussianity, but in other respects our assumptions seem mild.Edgeworth expansion, long memory, semiparametric estimation.

Uniform Limit Theory for Stationary Autoregression

First order autoregression is shown to satisfy a limit theory which is uniform over stationary values of the autoregressive coefficient rho = rho_{n} in [0,1) provided (1 - rho_{n})n approaches infinity. This extends existing Gaussian limit theory by allowing for values of stationary rho that include neighbourhoods of unity provided they are wider than O(n^{1}), even by a slowly varying factor. Rates of convergence depend on rho and are at least squareroot of squareroot of n but less than n. Only second moments are assumed, as in the case of stationary autoregression with fixed rho.Autoregression, Gaussian limit theory, local to unity, uniform limit

Mean and Autocovariance Function Estimation Near the Boundary of Stationarity

We analyze the applicability of standard normal asymptotic theory for linear process models near the boundary of stationarity. The concept of stationarity is refined, allowing for sample size dependence in the array and paying special attention to the rate at which the boundary unit root case is approached using a localizing coefficient around unity. The primary focus of the present paper is on estimation of the the mean, autocovariance and autocorrelation functions within the broad region of stationarity that includes near boundary cases which vary with the sample size. The rate of consistency and the validity of the normal asymptotic approximation for the corresponding estimators is determined both by the sample size n and a parameter measuring the proximity of the model to the unit root boundary. An asymptotic result on the estimation of the localizing coefficient is also presented. To assist in the development of the limit theory in the present case, a suitable asymptotic theory for the behavior of quadratic forms in the vicinity of the boundary of stationarity is provided.Asymptotic normality, Integrated periodogram, Linear process, Local to unity, Localizing coefficient, Moderate deviation, Unit root

Consistent estimation of the memory parameterfor nonlinear time series

For linear processes, semiparametric estimation of the memory parameter, based on the log-periodogramand local Whittle estimators, has been exhaustively examined and their properties are well established.However, except for some specific cases, little is known about the estimation of the memory parameter fornonlinear processes. The purpose of this paper is to provide general conditions under which the localWhittle estimator of the memory parameter of a stationary process is consistent and to examine its rate ofconvergence. We show that these conditions are satisfied for linear processes and a wide class of nonlinearmodels, among others, signal plus noise processes, nonlinear transforms of a Gaussian process ?tandEGARCH models. Special cases where the estimator satisfies the central limit theorem are discussed. Thefinite sample performance of the estimator is investigated in a small Monte-Carlo study.Long memory, semiparametric estimation, local Whittle estimator.

Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle's estimate

We prove uniform convergence results for the integrated periodogram of a weakly dependent time series, namely a law of large numbers and a central limit theorem. These results are applied to Whittle's parametric estimation. Under general weak-dependence assumptions we derive uniform limit theorems and asymptotic normality of Whittle's estimate for a large class of models. For instance the causal $\theta$-weak dependence property allows a new and unified proof of those results for ARCH($\infty$) and bilinear processes. Non causal $\eta$-weak dependence yields the same limit theorems for two-sided linear (with dependent inputs) or Volterra processes