Frequency domain bootstrap for ratio statistics under long-range dependence

A frequency domain bootstrap (FDB) is a common technique to apply Efron’s independent and identically distributed resampling technique (Efron, 1979) to periodogram ordinates – especially normalized periodogram ordinates – by using spectral density estimates. The FDB method is applicable to several classes of statistics, such as estimators of the normalized spectral mean, the autocorrelation (but not autocovariance), the normalized spectral density function, and Whittle parameters. While this FDB method has been extensively studied with respect to short-range dependent time processes, there is a dearth of research on its use with long-range dependent time processes. Therefore, we propose an FDB methodology for ratio statistics under long-range dependence, using semi- and nonparametric spectral density estimates as a normalizing factor. It is shown that the FDB approximation allows for valid distribution estimation for a broad class of stationary, long-range (or short-range) dependent linear processes, without any stringent assumptions on the distribution of the underlying process. The results of a large simulation study show that the FDB approximation using a semi- or nonparametric spectral density estimator is often robust for various values of a long-memory parameter reflecting magnitude of dependence. We apply the proposed procedure to two data examples.     

Root-n-consistent Estimation in Partial Linear Models with Long-memory Errors

We consider estimation of β in the semiparametric regression model y ( i ) - x ^T( i )β + f ( i / n ) + ε( i ) where x ( i ) = g ( i )/ n ) + e ( i , f and g are unknown smooth functions and the processes ε( i ) and e ( i ) are stationary with short- or long-range dependence. For the case of i.i.d. errors, Speckman (1988) proposed a √ n –consistent estimator of β. In this paper it is shown that, under suitable regularity conditions, this estimator is asymptotically unbiased and √ n –consistent even if the errors exhibit long-range dependence. The orders of the finite sample bias and of the required bandwidth depend on the long-memory parameters. Simulations and a data example illustrate the method     

Stochastic volatility modelling in continuous time with general marginal distributions: Inference,prediction and model selection

We compare results for stochastic volatility models where the underlying volatility process having generalized inverse Gaussian (GIG) and tempered stable marginal laws. We use a continuous time stochastic volatility model where the volatility follows an Ornstein–Uhlenbeck stochastic differential equation driven by a Lévy process. A model for long-range dependence is also considered, its merit and practical relevance discussed. We find that the full GIG and a special case, the inverse gamma, marginal distributions accurately fit real data. Inference is carried out in a Bayesian framework, with computation using Markov chain Monte Carlo (MCMC). We develop an MCMC algorithm that can be used for a general marginal model.     

Modeling statistical dependence of Markov chains via copula models

Conditional probability distributions have been commonly used in modeling Markov chains. In this paper we consider an alternative approach based on copulas to investigate Markov-type dependence structures. Based on the realization of a single Markov chain, we estimate the parameters using one- and two-stage estimation procedures. We derive asymptotic properties of the marginal and copula parameter estimators and compare performance of the estimation procedures based on Monte Carlo simulations. At low and moderate dependence structures the two-stage estimation has comparable performance as the maximum likelihood estimation. In addition we propose a parametric pseudo-likelihood ratio test for copula model selection under the two-stage procedure. We apply the proposed methods to an environmental data set.     

Characteristic function estimation of non-Gaussian Ornstein–Uhlenbeck processes

Continuous non-Gaussian stationary processes of the OU-type are becoming increasingly popular given their flexibility in modelling stylized features of financial series such as asymmetry, heavy tails and jumps. The use of non-Gaussian marginal distributions makes likelihood analysis of these processes unfeasible for virtually all cases of interest. This paper exploits the self-decomposability of the marginal laws of OU processes to provide explicit expressions of the characteristic function which can be applied to several models as well as to develop efficient estimation techniques based on the empirical characteristic function. Extensions to OU-based stochastic volatility models are provided.     

Properties of generalized Levinson–Durbin–Whittle sequences

For a discrete time, second-order stationary process the Levinson–Durbin recursion is used to determine best fitting one-step-ahead linear autoregressive predictors of successively increasing order, best in the sense of minimizing the mean square error. Whittle [1963. On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix. Biometrika 50, 129–134] generalized the recursion to the case of vector autoregressive processes. The recursion defines what is termed a Levinson–Durbin–Whittle sequence, and a generalized Levinson–Durbin–Whittle sequence is also defined. Generalized Levinson–Durbin–Whittle sequences are shown to satisfy summation formulas which generalize summation formulas satisfied by binomial coefficients. The formulas can be expressed in terms of the partial correlation sequence, and they assume simple forms for time-reversible processes. The results extend comparable formulas obtained in Shaman [2007. Generalized Levinson–Durbin sequences, binomial coefficients and autoregressive estimation. Working paper] for univariate processes.     

On the effect of long-range dependence on extreme value copula estimation with fixed marginals

ABSTRACT

We establish the existence of multivariate stationary processes with arbitrary marginal copula distributions and long-range dependence. The effect of long-range dependence on extreme value copula estimation is illustrated in the case of known marginals, by deriving functional limit theorems for a standard non parametric estimator of the Pickands dependence function and related parametric projection estimators. The asymptotic properties turn out to be very different from the case of iid or short-range dependent observations. Simulated and real data examples illustrate the results.     

Testing for changes in the covariance structure of linear processes

We consider several procedures to detect changes in the mean or the covariance structure of a linear process. The tests are based on the weighted CUSUM process. The limit distributions of the test statistics are derived under the no change null hypothesis. We develop new strong and weak approximations for the sample mean as well as the sample correlations of linear processes. A small Monte Carlo simulation illustrates the applicability of our results.     

Periodograms asymptotic distributions in periodically correlated processes and multivariate stationary processes: An alternative approach

Discrete time periodically correlated (PC) processes are viewed as the processes with time-dependent spectra. This, together with an auxiliary operator which is defined here is employed to apply classical results on the asymptotic distribution of the periodogram of the univariate white noise (innovations) to derive the asymptotic distributions of the periodograms for the PC processes and also for the multivariate stationary processes. We assume only the continuity and positive definiteness of the spectral densities together with the independence of the innovations.     

Asymptotic distributions of statistics and parameter estimates for mixed Poisson processes

Mixed Poisson processes have been used as natural models for events occurring in continuous or discrete time. Our main result is the derivation of the joint asymptotic distributions of statistics, including parameter estimators, computed in different time intervals from data generated by mixed Poisson processes. These distributions can be used, for example, to test the hypothesis about the adequacy of the mixed Poisson process against data. We provide some simulation results and test the model on actual market research data.     

Mutual information in the frequency domain

Coefficients of mutual information (MI) can provide powerful extensions of classical coefficients of correlation. In particular, they have the property of vanishing if and only if the components involved are statistically independent of each other. This characteristic can prove useful in preparatory work to model building. In this article a frequency domain variant of MI is developed and studied for bivariate stationary time series. As a scientific example an ambient seismic noise data set is studied and a lack of independence of the components inferred. The character of the dependence of the MI on frequency may be used to suggest the nature of the statistical dependence.     

Nonparametric functionals of spectral distributions and their applications to time series analysis

There is a close analogy between empirical distributions of i.i.d. random variables and normalized spectral distributions of wide-sense stationary processes. Herein we make use of this analogy to develop nonparametric comparisons of two spectral distributions and nonparametric tests of stationarity versus change-point alternatives via spectral analysis of a time series.     

Stationary mixture transition distribution (MTD) models via predictive distributions

This paper combines two ideas to construct autoregressive processes of arbitrary order. The first idea is the construction of first order stationary processes described in Pitt et al. [(2002). Constructing first order autoregressive models via latent processes. Scand. J. Statist.29, 657–663] and the second idea is the construction of higher order processes described in Raftery [(1985). A model for high order Markov chains. J. Roy. Statist. Soc. B.47, 528–539]. The resulting models provide appealing alternatives to model non-linear and non-Gaussian time series.     

A note on asymptotic parametric prediction

This paper is devoted to asymptotic behaviour of plug-in statistical predictors obtained by replacing the unknown parameter in a conditional expectation by a suitable estimator. We derive the L²$L^2$-convergence rate and limit in distribution for the predictors. Applications to ARMA processes and diffusion processes are considered.     

Long run variance estimation and robust regression testing using sharp origin kernels with no truncation

A new family of kernels is suggested for use in long run variance (LRV) estimation and robust regression testing. The kernels are constructed by taking powers of the Bartlett kernel and are intended to be used with no truncation (or bandwidth) parameter. As the power parameter (ρ)$(\rho )$ increases, the kernels become very sharp at the origin and increasingly downweight values away from the origin, thereby achieving effects similar to a bandwidth parameter. Sharp origin kernels can be used in regression testing in much the same way as conventional kernels with no truncation, as suggested in the work of Kiefer and Vogelsang [2002a, Heteroskedasticity-autocorrelation robust testing using bandwidth equal to sample size. Econometric Theory 18, 1350–1366, 2002b, Heteroskedasticity-autocorrelation robust standard errors using the Bartlett kernel without truncation, Econometrica 70, 2093–2095] Analysis and simulations indicate that sharp origin kernels lead to tests with improved size properties relative to conventional tests and better power properties than other tests using Bartlett and other conventional kernels without truncation.     

SEMIFAR forecasts, with applications to foreign exchange rates

SEMIFAR models introduced in Beran (1997. Estimating trends, long-range dependence and nonstationarity, preprint) provide a semiparametric modelling framework that enables the data analyst to separate deterministic and stochastic trends as well as short- and long-memory components in an observed time series. A correct distinction between these components, and in particular, the decision which of the components may be present in the data have an important impact on forecasts. In this paper, forecasts and forecast intervals for SEMIFAR models are obtained. The forecasts are based on an extrapolation of the nonparametric trend function and optimal forecasts of the stochastic component. In the data analytical part of the paper, the proposed method is applied to foreign exchange rates from Europe and Asia.     

General linear processes in Hilbert spaces and prediction

We discuss a general definition of linear processes in Hilbert spaces that takes into account the outstanding role played by this model in prediction theory.     

Nonparametric spectrum estimation for spatial data

Smoothed nonparametric kernel spectral density estimates are considered for stationary data observed on a d-dimensional lattice. The implications for edge effect bias of the choice of kernel and bandwidth are considered. Under some circumstances the bias can be dominated by the edge effect. We show that this problem can be mitigated by tapering. Some extensions and related issues are discussed.     

Nonparametric regression for dependent data in the errors-in-variables problem

We consider the nonparametric estimation of the regression functions for dependent data. Suppose that the covariates are observed with additive errors in the data and we employ nonparametric deconvolution kernel techniques to estimate the regression functions in this paper. We investigate how the strength of time dependence affects the asymptotic properties of the local constant and linear estimators. We treat both short-range dependent and long-range dependent linear processes in a unified way and demonstrate that the long-range dependence (LRD) of the covariates affects the asymptotic properties of the nonparametric estimators as well as the LRD of regression errors does.