Error Correction Models for Fractionally Cointegrated Time Series

This paper provides a proof of Granger's (1986 ) error correction model for fractionally cointegrated variables and points out a necessary assumption that has not been noted before. Moreover, a simpler, alternative error correction model is proposed which can be employed to estimate fractionally cointegrated systems in three steps.     

Higher-Order Accurate Spectral Density Estimation of Functional Time Series

Under the frequency domain framework for weakly dependent functional time series, a key element is the spectral density kernel which encapsulates the second-order dynamics of the process. We propose a class of spectral density kernel estimators based on the notion of a flat-top kernel. The new class of estimators employs the inverse Fourier transform of a flat-top function as the weight function employed to smooth the periodogram. It is shown that using a flat-top kernel yields a bias reduction and results in a higher-order accuracy in terms of optimizing the integrated mean square error (IMSE). Notably, the higher-order accuracy of flat-top estimation comes at the sacrifice of the positive semi-definite property. Nevertheless, we show how a flat-top estimator can be modified to become positive semi-definite (even strictly positive definite) in finite samples while retaining its favorable asymptotic properties. In addition, we introduce a data-driven bandwidth selection procedure realized by an automatic inspection of the estimated correlation structure. Our asymptotic results are complemented by a finite-sample simulation where the higher-order accuracy of flat-top estimators is manifested in practice.     

Regression Models with Time Series Errors

In models of the form Y_t = r ( X_t ) + Z_t , where r is an unknown function and { X_t } is a covariate process independent of the stationary error { Z_t }, we give conditions under which estimators based on residuals Z ₁, ..., Z _n obtained from linear smoothers are asymptotically equivalent to those based on the actual errors Z ₁, ..., Z_n .     

Partial Likelihood Inference For Time Series Following Generalized Linear Models

Abstract.  The present article offers a certain unifying approach to time series regression modelling by combining partial likelihood (PL) inference and generalized linear models. An advantage gained by resorting to PL is that the joint distribution of the response and the covariates is left unspecified, and furthermore, PL allows for temporal or sequential conditional inference with respect to a filtration generated by all that is known to the observer at the time of observation. Two real data examples illustrate the methodology.     

Generation Of Time Series Models With Given Spectral Properties

Abstract.  We give a method for generation of periodically correlated and multivariate ARIMA models whose dynamic characteristics are partially or fully specified in terms of spectral poles and zeroes or their equivalents in the form of eigenvalues/eigenvectors of associated model matrices. Our method is based on the spectral decomposition of multi-companion matrices and their factorization into products of companion matrices. Generated models are needed in simulation but may also be used in estimation, e.g. to set sensible initial values of parameters for nonlinear optimization.
We are not aware of any other general method for multivariate linear systems of comparable generality and control over the spectral properties of the generated model.     

Robust Hypothesis Testing and Robust Time Series Interpolation And Regression

Abstract. Recent results on minimax robust time series interpolation and regression coefficient estimation are generalized and extended through a relationship with robust hypothesis testing. The spectral uncertainty classes in the time series problems are assumed to be convex and to satisfy an integral constraint such as on the variance of the process. It is shown that robust solutions in such cases can always be obtained from the least-favourable probability density functions for corresponding hypothesis testing problems. A specific class, the bounded spectral densities from the band model, is considered to illustrate the results.     

Generalized Resampling Scheme With Application to Spectral Density Matrix in Almost Periodically Correlated Class of Time Series

The aim of this article is to introduce new resampling scheme for nonstationary time series, called generalized resampling scheme (GRS). The proposed procedure is a generalization of well known in the literature subsampling procedure and is simply related to existing block bootstrap techniques. To document the usefulness of GRS, we consider the example of model with almost periodic phenomena in mean and variance function, where the consistency of the proposed procedure was examined. Finally, we prove the consistency of GRS for the spectral density matrix for nonstationary, multivariate almost periodically correlated time series. We consider both zero mean and non‐zero mean case. The consistency holds under general assumptions concerning moment and α‐mixing conditions for multivariate almost periodically correlated time series. Proving the consistency in this case poses a difficulty since the estimator of the spectral density matrix can be interpreted as a sum of random matrixes whose dependence grow with the sample size.     

Data Driven Order Selection for Projection Estimator of the Spectral Density of Time Series with Long Range Dependence

Fractional exponential (FEXP) models have been introduced by Robinson (1991) and Beran (1993) to model the spectral density of a covariance stationary long-range dependent process. In this class of models, the spectral density f ( x ) of the process is decomposed as f ( x ) = |1 − exp( ix )|^{−2 d} f _*( x ), where f _*( x ) accounts for the short-memory component. In this contribution, FEXP models are used to construct semi-parametric estimates of the fractional differencing coefficient and of the spectral density, by considering an infinite Fourier series expansion of log f _*( x ). A data-driven order selection procedure, adapted from the Mallows' C _p procedure, is proposed to determine the order of truncation. The optimality of the data-driven procedure is established, under mild assumptions on the short-memory component f _*( x ). A limited Monte-Carlo experiment is presented to support our claims.     

ESTIMATION OF THE LONG-MEMORY PARAMETER, BASED ON A MULTIVARIATE CENTRAL LIMIT THEOREM

Abstract. Long memory is known to occur in many fields of statistical application. Stationary processes whose correlations decay asymptotically like ‖ k ‖^{2 H -2}, where k is the lag and H ε (0.5, 1), provide useful parsimonious models with long memory. The parameter H characterizes the long-memory features of the data. For long time series, maximum likelihood estimation of H can be costly in terms of CPU time. In this paper, we show that, for disjoint stretches of the data, estimates of H and other parameters that characterize the dependence structure are asymptotically independent. Averaging these estimates leads to a fast and efficient approximate maximum likelihood method.     

A Frequency‐Domain Test to Check Equality in Spectral Densities of Multiple Time Series With Unequal Lengths

In this paper, a new frequency‐domain test is proposed to check the equality of spectral densities of two or more stationary time series. The proposed test is able to deal with multiple independent time series of different lengths naturally, based on some regression models of log periodograms. The asymptotic null distribution of the proposed test is obtained. The consistency is shown under any fixed alternative and a sequence of local alternatives. A simulation study is conducted to examine the finite sample performance of the test. By jointly modeling all log periodograms, the test is empirically robust when multiple time series are mutually dependent to some extent. It also works well for non‐Gaussian time series. The proposed test is applied to compare several vibrational signals for damage detection of a mechanical system.     

Bayesian Inference on Periodicities and Component Spectral Structure in Time Series

We detail and illustrate time series analysis and spectral inference in autoregressive models with a focus on the underlying latent structure and time series decompositions. A novel class of priors on parameters of latent components leads to a new class of smoothness priors on autoregressive coefficients, provides for formal inference on model order, including very high order models, and leads to the incorporation of uncertainty about model order into summary inferences. The class of prior models also allows for subsets of unit roots, and hence leads to inference on sustained though stochastically time-varying periodicities in time series. Applications to analysis of the frequency composition of time series, in both time and spectral domains, is illustrated in a study of a time series from astronomy. This analysis demonstrates the impact and utility of the new class of priors in addressing model order uncertainty and in allowing for unit root structure. Time-domain decomposition of a time series into estimated latent components provides an important alternative view of the component spectral characteristics of a series. In addition, our data analysis illustrates the utility of the smoothness prior and allowance for unit root structure in inference about spectral densities. In particular, the framework overcomes supposed problems in spectral estimation with autoregressive models using more traditional model-fitting methods.     

Discriminant Analysis of Time Series in the Presence of Within‐Group Spectral Variability

Many studies record replicated time series epochs from different groups with the goal of using frequency domain properties to discriminate between the groups. In many applications, there exists variation in cyclical patterns from time series in the same group. Although a number of frequency domain methods for the discriminant analysis of time series have been explored, there is a dearth of models and methods that account for within‐group spectral variability. This article proposes a model for groups of time series in which transfer functions are modelled as stochastic variables that can account for both between‐group and within‐group differences in spectra that are identified from individual replicates. An ensuing discriminant analysis of stochastic cepstra under this model is developed to obtain parsimonious measures of relative power that optimally separate groups in the presence of within‐group spectral variability. The approach possesses favourable properties in classifying new observations and can be consistently estimated through a simple discriminant analysis of a finite number of estimated cepstral coefficients. Benefits in accounting for within‐group spectral variability are empirically illustrated in a simulation study and through an analysis of gait variability.     

On the Spectral Density of the Wavelet Coefficients of Long‐Memory Time Series with Application to the Log‐Regression Estimation of the Memory Parameter

Abstract. In recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semi‐parametric asymptotic theory, comparable with the one developed for Fourier methods, is still lacking. In this article, we adapt to the wavelet setting, the classical semi‐parametric framework introduced by Robinson and his co‐authors for estimating the memory parameter of a (possibly) non‐stationary process. Our results apply to a class of wavelets with bounded supports, which include but are not limited to Daubechies wavelets. We derive an explicit expression of the spectral density of the wavelet coefficients and show that it can be approximated, at large scales, by the spectral density of the continuous‐time wavelet coefficients of fractional Brownian motion. We derive an explicit bound for the difference between the spectral densities. As an application, we obtain minimax upper bounds for the log‐scale regression estimator of the memory parameter for a Gaussian process and we derive an explicit expression of its asymptotic variance.     

Polynomial Cointegration Between Stationary Processes With Long Memory

Abstract. In this article we consider polynomial cointegrating relationships between stationary processes with long range dependence. We express the regression functions in terms of Hermite polynomials and consider a form of spectral regression around frequency zero. For these estimates, we establish consistency by means of a more general result on continuously averaged estimates of the spectral density matrix at frequency zero.     

Semiparametric Estimation in Time‐Series Regression with Long‐Range Dependence

Abstract. We consider semiparametric estimation in time‐series regression in the presence of long‐range dependence in both the errors and the stochastic regressors. A central limit theorem is established for a class of semiparametric frequency domain‐weighted least squares estimates, which includes both narrow‐band ordinary least squares and narrow‐band generalized least squares as special cases. The estimates are semiparametric in the sense that focus is on the neighbourhood of the origin, and only periodogram ordinates in a degenerating band around the origin are used. This setting differs from earlier studies on time‐series regression with long‐range dependence, where a fully parametric approach has been employed. The generalized least squares estimate is infeasible when the degree of long‐range dependence is unknown and must be estimated in an initial step. In that case, we show that a feasible estimate which has the same asymptotic properties as the infeasible estimate, exists. By Monte Carlo simulation, we evaluate the finite‐sample performance of the generalized least squares estimate and the feasible estimate.