A Wavelet Characterization of Continuous‐Time Periodically Correlated Processes with Application to Simulation

We introduce a wavelet characterization of continuous‐time periodically correlated processes based on a linear combination of infinite‐dimensional stationary processes. The finite version of this linear combination converges to the main process. The first‐order and second‐order estimators based on the wavelets are presented. Under a simple and easy algorithm, the periodically correlated process is simulated for a given autocovariance function. The proposed algorithm has two main advantages: first, it is fast, and second, it is distribution free. We indicate through four examples that the simulated data are periodically correlated with the desired period.     

Maximum entropy for periodically correlated processes from nonconsecutive autocovariance coefficients

Abstract.  We consider the maximum entropy extension of a partially specified autocovariance sequence of a periodically correlated process. The sequence may be specified on a non-contiguous set. We give a method which solves the problem completely – it gives the positive definite solution when it exists and reports that it does not exist otherwise. The method is numerically reliable even when the solution is 'almost' semidefinite. It also works when only positive semidefinite extension(s) exist.     

An Extension Problem For Discrete-Time Periodically Correlated Stochastic Processes

In the context of wide-sense stationary processes, the so-called Caratheodory–Fejer problem of extending a finite non-negative sequence of matrices has been much studied. We here investigate a similar extension problem in the setting of wide-sense periodically correlated processes: given the first N coefficients of T scalar-valued sequences, we study under which condition(s) it is possible to find T extensions which are the cyclocorrelaion sequences of a periodically correlated process with period T . Using a result of Gladysev, the problem is shifted to a Caratheodory–Fejer problem with symmetry constraints. The existence of extensions is proved. In nondegenerate cases, the set of all solutions is given in terms of a homographic transformation of some Schur function G . The choice G =0 leads to the maximum entropy solution. The associated Gaussian processes are then proved to have a periodic autoregressive structure.     

Evolutive instantaneous spectrum associated with partial autocorrelation function

Several approaches have been developed for the spectral analysis of nonstationary processes in the literature. Otherwise, it has been shown recently that, as in the stationary case, the partial autocorrelation function characterizes, like the autocovariance function, the second-order properties of the process. Our main result is the introduction of a new time-dependent power spectrum clearly related to this function. At each time, this spectrum describes a stationary situation in which the present is correlated with the past in the same way as our nonstationary process at this time. The properties of this spectrum are analysed. In particular, it is defined for all nonstationary processes and is in a one-to-one correspondence with the autocovariance function. Unfortunately, no spectral representation of the process is actually associated with it. This spectrum is also compared with two similar other spectra. Some examples of theoretical spectra and an estimated spectrum are considered for illustration.     

ASYMPTOTIC RESULTS FOR PERIODIC AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. This paper is concerned with the derivation of asymptotic distributions for the sample autocovariance and sample autocorrelation functions of periodic autoregressive moving-average processes, which are useful in modelling periodically stationary time series. In an effort to obtain a parsimonious model representing a periodically stationary time series, the asymptotic properties of the discrete Fourier transform of the estimated periodic autocovariance and autocorrelation functions are presented. Application of the asymptotic results to some specific models indicates their usefulness for model identification analysis.     

ON VECTOR AUTOCORRELATIONS AND GENERALIZED SECOND-ORDER FUNCTIONS FOR TIME SERIES

Abstract. Some structural properties of certain vector generalizations of second-order functions of a stationary stochastic process based on determinantial functions of autocovariances are discussed. In particular, a generalized autocovariance function which retains all properties of the ordinary autocovariance function is considered and the linear dependence structure of certain scalar stochastic processes associated with this function is investigated. Properties of the normalized function are discussed and a duality property is found, according to which this function also generalizes in a natural way the ordinary partial autocorrelation function of stochastic processes.     

Banded and tapered estimates for autocovariance matrices and the linear process bootstrap

We address the problem of estimating the autocovariance matrix of a stationary process. Under short range dependence assumptions, convergence rates are established for a gradually tapered version of the sample autocovariance matrix and for its inverse. The proposed estimator is formed by leaving the main diagonals of the sample autocovariance matrix intact while gradually down‐weighting off‐diagonal entries towards zero. In addition, we show the same convergence rates hold for a positive definite version of the estimator, and we introduce a new approach for selecting the banding parameter. The new matrix estimator is shown to perform well theoretically and in simulation studies. As an application, we introduce a new resampling scheme for stationary processes termed the linear process bootstrap (LPB). The LPB is shown to be asymptotically valid for the sample mean and related statistics. The effectiveness of the proposed methods are demonstrated in a simulation study.     

On processes with hyperbolically decaying autocorrelations

We discuss some relations between autocorrelations (ACFs) and partial autocorrelations (PACFs) of weakly stationary processes. First, we construct an extension of a process ARIMA(0,d,0) for d ∈ (?∞, 0), which enjoys non‐summable partial autocorrelations and autocorrelations decaying as rapidly as ρ_n ? n^?1+2d. Such a situation is impossible if the absolute sum of autocorrelations is sufficiently small. We show that then the PACF is less than the ACF up to a multiplicative constant. Our second result complements a similar result of Baxter (1962).     

Integration of CARMA processes and spot volatility modelling

Continuous‐time autoregressive moving average (CARMA) processes with a non‐negative kernel and driven by a non‐decreasing Lévy process constitute a useful and very general class of stationary, non‐negative continuous‐time processes which have been used, in particular for the modelling of stochastic volatility. In the celebrated stochastic volatility model of Barndorff‐Nielsen and Shephard (2001) , the spot (or instantaneous) volatility at time t, V(t), is represented by a stationary Lévy‐driven Ornstein‐Uhlenbeck process. This has the shortcoming that its autocorrelation function is necessarily a decreasing exponential function, limiting its ability to generate integrated volatility sequences, , with autocorrelation functions resembling those of observed realized volatility sequences. (A realized volatility sequence is a sequence of estimated integrals of spot volatility over successive intervals of fixed length, typically 1 day.) If instead of the stationary Ornstein–Uhlenbeck process, we use a CARMA process to represent spot volatility, we can overcome the restriction to exponentially decaying autocorrelation function and obtain a more realistic model for the dependence observed in realized volatility. In this article, we show how to use realized volatility data to estimate parameters of a CARMA model for spot volatility and apply the analysis to a daily realized volatility sequence for the Deutsche Mark/ US dollar exchange rate.     

Embedding a Gaussian discrete‐time autoregressive moving average process in a Gaussian continuous‐time autoregressive moving average process

Abstract. Embedding a discrete‐time autoregressive moving average (DARMA) process in a continuous‐time ARMA (CARMA) process has been discussed by many authors. These authors have considered the relationship between the autocovariance structures of continuous‐time and related discrete‐time processes. In this article, we treat the problem from a slightly different point of view. We define embedding in a more rigid way by taking account of the probability structure. We consider Gaussian processes. First we summarize the necessary and sufficient condition for a DARMA process to be able to be embedded in a CARMA process. Secondly, we show a concrete condition such that a DARMA process can be embeddable in a CARMA process. This condition is new and general. Thirdly, we show some special cases including new examples. We show how we can examine embeddability for these special cases.     

Some Notes on Mutual Information Between Past and Future

Abstract. We present some new results on the mutual information between past and future for Gaussian stationary sequences. We provide several formulae to calculate this quantity. As a by‐product, we establish the so‐called reflectrum identity that links partial autocorrelation coefficients and cepstrum coefficients. So as to obtain these results, we provide an account of several regularity conditions for Gaussian stationary processes in terms of properties of the associated Toeplitz and Hankel operators. We discuss conditions under which the mutual information is finite. These results lead us to an interesting perspective towards the definition of long‐memory processes. Our result implies that zeros on the unit circle can cause mutual information to be infinite. Examples include fractional autoregressive integrated moving average (ARIMA) models. In addition, we consider a finite sample from a Gaussian stationary sequence. In the expansion of the determinant of its covariance matrix, the Toeplitz matrix, the first and second term are, entropy and mutual information respectively. A form of approximation to the likelihood using entropy and mutual information is presented.     

Bootstrapping the Local Periodogram of Locally Stationary Processes

Abstract. Locally stationary processes are non‐stationary stochastic processes the second‐order structure of which varies smoothly over time. In this paper, we develop a method to bootstrap the local periodogram of a locally stationary process. Our method generates pseudo local periodogram ordinates by combining a parametric time and non‐parametric frequency domain bootstrap approach. We first fit locally a time varying autoregressive model so as to capture the essential characteristics of the underlying process. A locally calculated non‐parametric correction in the frequency domain is then used so as to improve upon the locally parametric autoregressive fit. As an application, we investigate theoretically the asymptotic properties of the bootstrap method proposed applied to the class of local spectral means, local ratio statistics and local spectral density estimators. Some simulations demonstrate the ability of our method to give accurate estimates of the quantities of interest in finite sample situations and an application to a real‐life data‐set is presented.     

Subsampling inference for the autocovariances and autocorrelations of long‐memory heavy‐ tailed linear time series

We provide a self‐normalization for the sample autocovariances and autocorrelations of a linear, long‐memory time series with innovations that have either finite fourth moment or are heavy‐tailed with tail index 2 < α < 4. In the asymptotic distribution of the sample autocovariance there are three rates of convergence that depend on the interplay between the memory parameter d and α, and which consequently lead to three different limit distributions; for the sample autocorrelation the limit distribution only depends on d. We introduce a self‐normalized sample autocovariance statistic, which is computable without knowledge of α or d (or their relationship), and which converges to a non‐degenerate distribution. We also treat self‐normalization of the autocorrelations. The sampling distributions can then be approximated non‐parametrically by subsampling, as the corresponding asymptotic distribution is still parameter‐dependent. The subsampling‐based confidence intervals for the process autocovariances and autocorrelations are shown to have satisfactory empirical coverage rates in a simulation study. The impact of subsampling block size on the coverage is assessed. The methodology is further applied to the log‐squared returns of Merck stock.     

A Note on Non‐Negative Arma Processes

Abstract. Recently, there has been much research on developing models suitable for analysing the volatility of a discrete‐time process. Since the volatility process, like many others, is necessarily non‐negative, there is a need to construct models for stationary processes which are non‐negative with probability one. Such models can be obtained by driving autoregressive moving average (ARMA) processes with non‐negative kernel by non‐negative white noise. This raises the problem of finding simple conditions under which an ARMA process with given coefficients has a non‐negative kernel. In this article, we derive a necessary and sufficient condition. This condition is in terms of the generating function of the ARMA kernel which has a simple form. Moreover, we derive some readily verifiable necessary and sufficient conditions for some ARMA processes to be non‐negative almost surely.     

On Bartlett's Formula for Non-linear Processes

Bartlett's formula is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocovariances. However, it is derived under precise assumptions (namely linearity of the underlying process and vanishing of its fourth-order cumulants) and effectiv e computations show that the value given by this formula can deviate markedly from the true asymptotic covariance when the requirements on the underlying process are not satisfied. This is the case for a large class of models, for instance bilinear and autoregressive conditionally heteroscedastic processes. For these reasons we investigate the behaviour of smoothed empirical estimates of the covariance between two sample autocovariance s. We prove L ² and strong consistency for strongly mixing stationary processes and define for the covariance matrix of a vector of sample autocovariances a consistent estimate which is a non-negative definite matrix. The choice of the parameters is discussed, applications are given and comparisons are made through a simulation study     

A new frequency domain approach of testing for covariance stationarity and for periodic stationarity in multivariate linear processes

In modelling seasonal time series data, periodically (non‐)stationary processes have become quite popular over the last years and it is well known that these models may be represented as higher‐dimensional stationary models. In this article, it is shown that the spectral density matrix of this higher‐dimensional process exhibits a certain structure if and only if the observed process is covariance stationary. By exploiting this relationship, a new L₂‐type test statistic is proposed for testing whether a multivariate periodically stationary linear process is even covariance stationary. Moreover, it is shown that this test may also be used to test for periodic stationarity. The asymptotic normal distribution of the test statistic under the null is derived and the test is shown to have an omnibus property. The article concludes with a simulation study, where the small sample performance of the test procedure is improved by using a suitable bootstrap scheme.     

HIGHER ORDER MOMENTS OF SAMPLE AUTOCOVARIANCES AND SAMPLE AUTOCORRELATIONS FROM AN INDEPENDENT TIME SERIES

Abstract. Given length- n sampled time series, generated by an independent distributed process, in this paper we treat the problem of determining the greatest order, in n , that moments of the sample autocovariances and sample autocorrelations can attain. For the sample autocovariance moments, we achieve quite general results; but, for the autocorrelation moments, we restrict study to Gaussian white noise (normal, independent and identically distributed). Our main theorem relates to the cross-moments of the non-centred sample autocovariances, but we also establish a relation between these and the corresponding moments for the centred sample autocovariances.     

Simulation of Real‐Valued Discrete‐Time Periodically Correlated Gaussian Processes with Prescribed Spectral Density Matrices

Abstract. In this article, we provide a spectral characterization for a real‐valued discrete‐time periodically correlated process, and then proceed on to establish a simulation procedure to simulate such a Gaussian process for a given spectral density. We also prove that the simulated process, at each time index, converges to the actual process in the mean square.     

A LIMITING PROPERTY OF SAMPLE AUTOCOVARIANCES OF PERIODICALLY CORRELATED PROCESSES WITH APPLICATION TO PERIOD DETERMINATION

Abstract. It is shown that the sample autocovariance of a periodically correlated process converges to a limit which reveals the same periodicity as the process. A theorem is proved relating to the rate of almost sure convergence, which is uniform in the lag up to some orders of observation length. Based on the limiting property, a strongly consistent estimate of hidden period is proposed.