Robust empirical likelihood for time series

This article introduces a robust frequency domain empirical likelihood inference procedure for the parametric component in the spectral densities of stationary processes. We construct the empirical likelihood function by using a new spectral estimating function to achieve robustness against contamination in the spectral density. Simulation studies demonstrate the good performance of the proposed robust frequency domain empirical likelihood method, which produces more accurate confidence regions than the ordinary empirical likelihood counterpart.     

CONSISTENT ESTIMATION OF THE FOURTH-ORDER CUMULANT SPECTRAL DENSITY

Abstract. In this paper we consider the estimation of the fourth-order cumulant spectral density. Indeed this is the first case where the cumulant depends on lower-order product moments for a mean-zero stationary process. The proposed estimator of the fourth-order cumulant spectral density is constructed by replacing product moments with appropriately weighted estimates of product moments according to the definition of the fourth-order cumulant spectral density. Asymptotic unbiasedness and consistency are shown to hold for these estimators under stationarity and absolute summability of cumulants up to various orders with no restrictions on the frequencies. An expression for the asymptotic variance is also obtained.     

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.     

Locally stationary harmonizable complex improper stochastic processes

This article concerns continuous‐time second‐order complex‐valued improper stochastic processes that are harmonizable and locally stationary in Silverman's sense. We study necessary and sufficient conditions for the property of local stationarity in the time and frequency domains. A sufficient condition by Silverman is generalized and extended to the improper case. We obtain a result on the absolute continuity of the complementary spectral measure with respect to the spectral measure, which is related to a spectral characterization of improper wide‐sense stationary processes.     

Empirical Likelihood for a Long Range Dependent Process Subordinated to a Gaussian Process

This article develops empirical likelihood methodology for a class of long range dependent processes driven by a stationary Gaussian process. We consider population parameters that are defined by estimating equations in the time domain. It is shown that the standard block empirical likelihood (BEL) method, with a suitable scaling, has a non‐standard limit distribution based on a multiple Wiener–Itô integral. Unlike the short memory time series case, the scaling constant involves unknown population quantities that may be difficult to estimate. Alternative versions of the empirical likelihood method, involving the expansive BEL (EBEL) methods are considered. It is shown that the EBEL renditions do not require an explicit scaling and, therefore, remove this undesirable feature of the standard BEL. However, the limit law involves the long memory parameter, which may be estimated from the data. Results from a moderately large simulation study on finite sample properties of tests and confidence intervals based on different empirical likelihood methods are also reported.     

A FREQUENCY DOMAIN APPROACH FOR THE ESTIMATION OF PARAMETERS OF SPATIO‐TEMPORAL STATIONARY RANDOM PROCESSES

A frequency domain methodology is proposed for estimating parameters of covariance functions of stationary spatio‐temporal processes. Finite Fourier transforms of the processes are defined at each location. Based on the joint distribution of these complex valued random variables, an approximate likelihood function is constructed. The sampling properties of the estimators are investigated. It is observed that the expectation of these transforms can be considered to be a frequency domain analogue of the classical variogram. We call this measure frequency variogram. The method is applied to simulated data and also to Pacific wind speed data considered earlier by Cressie and Huang (1999). The proposed method does not depend on the distributional assumptions about the process.     

Transformation to approximate independence for locally stationary Gaussian processes

We provide new approximations for the likelihood of a time series under the locally stationary Gaussian process model. The likelihood approximations are valid even in cases when the evolutionary spectrum is not smooth in the rescaled time domain. We describe a broad class of models for the evolutionary spectrum for which the approximations can be computed particularly efficiently. In developing the approximations, we extend to the locally stationary case the idea that the discrete Fourier transform is a decorrelating transformation for stationary time series. The approximations are applied to fit non‐stationary time‐series models to high‐frequency temperature data. For these data, we fit evolutionary spectra that are piecewise constant in time and use a genetic algorithm to search for the best partition of the time interval.     

Inference for the Fourth‐Order Innovation Cumulant in Linear Time Series

The rescaled fourth‐order cumulant of the unobserved innovations of linear time series is an important parameter in statistical inference. This article deals with the problem of estimating this parameter. An existing nonparametric estimator is first discussed, and its asymptotic properties are derived. It is shown how the autocorrelation structure of the underlying process affects the behaviour of the estimator. Based on our findings and on an important invariance property of the parameter of interest with respect to linear filtering, a pre‐whitening‐based nonparametric estimator of the same parameter is proposed. The estimator is obtained using the filtered time series only; that is, an inversion of the pre‐whitening procedure is not required. The asymptotic properties of the new estimator are investigated, and its superiority is established for large classes of stochastic processes. It is shown that for the particular estimation problem considered, pre‐whitening can reduce the variance and the bias of the estimator. The finite sample performance of both estimators is investigated by means of simulations. The new estimator allows for a simple modification of the multiplicative frequency domain bootstrap, which extends its considerable range of validity. Furthermore, the problem of testing hypotheses about the rescaled fourth‐order cumulant of the unobserved innovations is also considered. In this context, a simple test for Gaussianity is proposed. Some real‐life data applications are presented.     

Moving Fourier Analysis for Locally Stationary Processes with the Bootstrap in View

We introduce a moving Fourier transformation for locally stationary time series, which captures the time‐varying spectral density in a similar manner as the classical Fourier transform does for stationary time series. In particular, the resulting Fourier coefficients as well as moving local periodograms are shown to be (almost all) asymptotically uncorrelated. The moving local periodogram is obtained by thinning the local periodogram to avoid multiple information present at different but close points in time. We obtain consistent estimators for the local spectral density at each point in time by smoothing the moving local periodogram. Furthermore, the moving Fourier coefficients, respectively periodograms, are well suited to adapt stationary frequency domain bootstrap methods to the locally stationary case. For the wild time frequency toggle bootstrap, it is shown that the corresponding bootstrap covariance of a global locally stationary bootstrap samples captures the time‐varying covariance structure of the underlying locally stationary time series correctly. Furthermore, this bootstrap in addition to adaptations of other frequency domain bootstrap methods is used in a simulation study to obtain uniform confidence bands for the time‐varying autocorrelation at lag 1. Finally, this methodology is applied to a wind data set.     

Stability conditions for heteroscedastic factor models with conditionally autoregressive betas

We characterize the stability properties of a heteroscedastic multi‐factor model of financial asset returns, with conditionally known factors and beta coefficients driven by general conditionally autoregressive processes. These processes generalize existing structures and address a number of empirical issues of current concern. Our analysis derives closed‐form sufficient conditions for the existence of strict stationary solutions for the composite asset conditional variances and covariances, not known previously in the literature. It is shown that stability is guaranteed when individual‐process and cross‐process restrictions hold simultaneously. Our results are also applicable to the study of the co‐movement between volatility and beta coefficients as well as between beta coefficients themselves.     

ON THE COVARIANCE OF THE PERIODOGRAM

Abstract. The paper discusses the covariance of the periodogram from a zero mean fourth order stationary stochastic process. The fourth order cumulant term appearing in the covariance is shown to be a convolution between the fourth order cumulant spectrum and a bounded approximate identity, and this gives precise results about its asymptotic behaviour. The covariance is also studied both pointwise and as a measure of two variables. This leads to necessary and sufficient conditions for mean square consistency of estimates of the spectral moments and related parameters.     

Bartlett Correction of Empirical Likelihood for Non‐Gaussian Short‐Memory Time Series

Bartlett correction, which improves the coverage accuracies of confidence regions, is one of the desirable features of empirical likelihood. For empirical likelihood with dependent data, previous studies on the Bartlett correction are mainly concerned with Gaussian processes. By establishing the validity of Edgeworth expansion for the signed root empirical log‐likelihood ratio statistics, we show that the Bartlett correction is applicable to empirical likelihood for short‐memory time series with possibly non‐Gaussian innovations. The Bartlett correction is established under the assumptions that the variance of the innovation is known and the mean of the underlying process is zero for a single parameter model. In particular, the order of the coverage errors of Bartlett‐corrected confidence regions can be reduced from O(n^?1) to O(n^?2).     

Estimation of stationary autoregressive models with the Bayesian LASSO

This article explores the problem of estimating stationary autoregressive models from observed data using the Bayesian least absolute shrinkage and selection operator (LASSO). By characterizing the model in terms of partial autocorrelations, rather than coefficients, it becomes straightforward to guarantee that the estimated models are stationary. The form of the negative log‐likelihood is exploited to derive simple expressions for the conditional likelihood functions, leading to efficient algorithms for computing the posterior mode by coordinate‐wise descent and exploring the posterior distribution by Gibbs sampling. Both empirical Bayes and Bayesian methods are proposed for the estimation of the LASSO hyper‐parameter from the data. Simulations demonstrate that the Bayesian LASSO performs well in terms of prediction when compared with a standard autoregressive order selection method.     

SPECTRAL RATIO DISCRIMINANTS AND INFORMATION THEORY

Abstract. We examine the problem of discriminating in the frequency domain between zero mean stationary Gaussian processes. Use of the Kullback-Liebler information measure enables the selection of the important frequency bands, and a special case relevant to seismic records leads to a justification for the use of spectral ratios in discrimination. Methods are given for combining the ratios when there are more than two relevant bands, and for estimating the error rates when using spectral ratios. The results are extended to a special class of non-stationary models.     

Empirical Likelihood for Outlier Detection and Estimation in Autoregressive Time Series

Identification and estimation of outliers in time series is proposed by using empirical likelihood methods. Theory and applications are developed for stationary autoregressive models with outliers distinguished in the usual additive and innovation types. Some other useful outlier types are considered as well. A simulation experiment is used for studying the behaviour of the empirical likelihood‐based method in finite samples and indicates that the proposed methods are preferable when dealing with the non‐Gaussian data. Our simulations suggest that the usual sequential procedure for multiple outlier detection is suitable also for the methods based on empirical likelihood.     

UNIFIED INTERVAL ESTIMATION FOR RANDOM COEFFICIENT AUTOREGRESSIVE MODELS

The consistency of the quasi‐maximum likelihood estimator for random coefficient autoregressive models requires that the coefficient be a non‐degenerate random variable. In this article, we propose empirical likelihood methods based on weighted‐score equations to construct a confidence interval for the coefficient. We do not need to distinguish whether the coefficient is random or deterministic and whether the process is stationary or non‐stationary, and we present two classes of equations depending on whether a constant trend is included in the model. A simulation study confirms the good finite‐sample behaviour of our resulting empirical likelihood‐based confidence intervals. We also apply our methods to study US macroeconomic data.     

Difference Equations for the Higher-Order Moments and Cumulants of the INAR(1) Model

Recently, as a result of the growing interest in modelling stationary processes with discrete marginal distributions, several models for integer value time series have been proposed in the literature. One of these models is the INteger-AutoRegressive (INAR) model. Here we consider the higher-order moments and cumulants of the INAR(1) process and show that they satisfy a set of Yule–Walker type difference equations. We also obtain the spectral and bispectral density functions, thus characterizing the INAR(1) process in the frequency domain. We use a frequency domain approach, namely the Whittle criterion, to estimate the parameters of the model. The estimation theory and associated asymptotic theory of this estimation method are illustrated numerically.     

TESTS FOR COMPARING TWO ESTIMATED SPECTRAL DENSITIES

Abstract. This paper was motivated by a problem in the gas industry and describes a number of periodogram-based tests of the hypothesis that two independent time-series are realizations of the same stationary process. Non-parametric tests analogous to the maximum periodogram ordinate and cumulative periodogram tests for white noise are compared with a likelihood ratio test based on a postulated quadratic model for the log spectral ratio. The latter is found to be generally more powerful against alternatives in which the two series are realizations of different low order AR processes. The operation of the likelihood ratio test is illustrated by two sets of data, the classic Beveridge wheat price series and a set of data supplied by British Gas.     

Effect of temporal aggregation on multiple time series in the frequency domain

The effect of temporal aggregation on bivariate spectral measures is investigated. First, the low‐frequency regression coefficient turns out to be invariant under aggregation irrespective of differencing, with the exception of when the aggregation of flow and stock variables is combined. Second, the long‐run squared coherency is invariant with respect to aggregation irrespective of differencing. Third, for frequencies different from zero, limiting results for a growing aggregation level m are obtained equal to those at frequency 0 of the underlying basic series. Hence, all frequency domain information is distorted by aggregation apart from the long‐run one. This also holds true for the phase angle that always approaches zero with growing aggregation level m. The sole exception to these findings is the case of the skip sampling stationary series. Moreover, for finite aggregation level, one may exactly quantify the aggregational effect on each cycle of interest. Numerical examples illustrate our results.     

A Gaussian Mixture Autoregressive Model for Univariate Time Series

The Gaussian mixture autoregressive model studied in this article belongs to the family of mixture autoregressive models, but it differs from its previous alternatives in several advantageous ways. A major theoretical advantage is that, by the definition of the model, conditions for stationarity and ergodicity are always met and these properties are much more straightforward to establish than is common in nonlinear autoregressive models. Another major advantage is that, for a pth‐order model, explicit expressions of the stationary distributions of dimension p + 1 or smaller are known and given by mixtures of Gaussian distributions with constant mixing weights. In contrast, the conditional distribution given the past observations is a Gaussian mixture with time‐varying mixing weights that depend on p lagged values of the series in a natural and parsimonious way. Because of the known stationary distribution, exact maximum likelihood estimation is feasible and one can assess the applicability of the model in advance by using a non‐parametric estimate of the stationary density. An empirical example with interest rate series illustrates the practical usefulness and flexibility of the model, particularly in allowing for level shifts and temporary changes in variance. Copyright © 2014 Wiley Publishing Ltd