Identification of Persistent Cycles in Non‐Gaussian Long‐Memory Time Series

Abstract. Asymptotic distribution is derived for the least squares estimates (LSE) in the unstable AR(p) process driven by a non‐Gaussian long‐memory disturbance. The characteristic polynomial of the autoregressive process is assumed to have pairs of complex roots on the unit circle. In order to describe the limiting distribution of the LSE, two limit theorems involving long‐memory processes are established in this article. The first theorem gives the limiting distribution of the weighted sum, is a non‐Gaussian long‐memory moving‐average process and (c_n,k,1 ≤ k ≤ n) is a given sequence of weights; the second theorem is a functional central limit theorem for the sine and cosine Fourier transforms      

Likelihood inference for discriminating between long‐memory and change‐point models

We develop a likelihood ratio (LR) test procedure for discriminating between a short‐memory time series with a change‐point (CP) and a long‐memory (LM) time series. Under the null hypothesis, the time series consists of two segments of short‐memory time series with different means and possibly different covariance functions. The location of the shift in the mean is unknown. Under the alternative, the time series has no shift in mean but rather is LM. The LR statistic is defined as the normalized log‐ratio of the Whittle likelihood between the CP model and the LM model, which is asymptotically normally distributed under the null. The LR test provides a parametric alternative to the CUSUM test proposed by Berkes et al. (2006) . Moreover, the LR test is more general than the CUSUM test in the sense that it is applicable to changes in other marginal or dependence features other than a change‐in‐mean. We show its good performance in simulations and apply it to two data examples.     

Bayesian Outlier Detection in Non‐Gaussian Autoregressive Time Series

This work investigates outlier detection and modelling in non‐Gaussian autoregressive time series models with margins in the class of a convolution closed parametric family. This framework allows for a wide variety of models for count and positive data types. The article investigates additive outliers which do not enter the dynamics of the process but whose presence may adversely influence statistical inference based on the data. The Bayesian approach proposed here allows one to estimate, at each time point, the probability of an outlier occurrence and its corresponding size thus identifying the observations that require further investigation. The methodology is illustrated using simulated and observed data sets.     

Quasi‐Maximum Likelihood Estimation for a Class of Continuous‐time Long‐memory Processes

Tsai and Chan (2003) has recently introduced the Continuous‐time Auto‐Regressive Fractionally Integrated Moving‐Average (CARFIMA) models useful for studying long‐memory data. We consider the estimation of the CARFIMA models with discrete‐time data by maximizing the Whittle likelihood. We show that the quasi‐maximum likelihood estimator is asymptotically normal and efficient. Finite‐sample properties of the quasi‐maximum likelihood estimator and those of the exact maximum likelihood estimator are compared by simulations. Simulations suggest that for finite samples, the quasi‐maximum likelihood estimator of the Hurst parameter is less biased but more variable than the exact maximum likelihood estimator. We illustrate the method with a real application.     

Trimming and Tapering Semi‐Parametric Estimates in Asymmetric Long Memory Time Series

Abstract. This paper considers semi‐parametric frequency domain inference for seasonal or cyclical time series with asymmetric long memory properties. It is shown that tapering the data reduces the bias caused by the asymmetry of the spectral density at the cyclical frequency. We provide a joint treatment of different tapering schemes and of the log‐periodogram regression and Gaussian semi‐parametric estimates of the memory parameters. Tapering allows for a less restrictive trimming of frequencies for the analysis of the asymptotic properties of both estimates when allowing for asymmetries. Simple rules for inference are feasible thanks to tapering and their validity in finite samples is investigated in a simulation exercise and for an empirical example.     

Testing Non‐Correlation and Non‐Causality between Multivariate ARMA Time Series

Abstract. Haugh [Journal of the American Statistical Association (1976) Vol. 71, pp. 378–85] developed an approach to the problem of testing non‐correlation (at all leads and lags) between two univariate time series. Haugh's tests however have low power against two series which are related over a long distributed lag when individual lag coefficients are relatively small. As a remedy, Koch and Yang [Journal of the American Statistical Association (1986) Vol. 8, pp. 533–44] proposed an alternative method that performs better than Haugh's under such dependencies. A multivariate extension of Haugh's procedure was proposed by El Himdi and Roy [The Canadian Journal of Statistics (1997) Vol. 25, pp. 233–56], but suffers the same weaknesses as the original univariate method. We develop here an asymptotic test generalizing Koch and Yang's method to the multivariate case. Our method includes El Himdi and Roy's as a special case. Based on the same idea, we also suggest a generalization of the El Himdi and Roy procedure for testing causality in the sense of Granger [Econometrica (1969) Vol. 37, pp. 424–38] between two multivariate series. A Monte Carlo study is conducted, which indicates that our approach performs better than El Himdi and Roy's for a wide range of models. Both procedures are applied to the problem of testing the absence of correlation between Canadian and US economic indicators, and to a brief study of causality between money and income in Canada.     

Negative Binomial Quasi‐Likelihood Inference for General Integer‐Valued Time Series Models

Two negative binomial quasi‐maximum likelihood estimates (NB‐QMLEs) for a general class of count time series models are proposed. The first one is the profile NB‐QMLE calculated while arbitrarily fixing the dispersion parameter of the negative binomial likelihood. The second one, termed two‐stage NB‐QMLE, consists of four stages estimating both conditional mean and dispersion parameters. It is shown that the two estimates are consistent and asymptotically Gaussian under mild conditions. Moreover, the two‐stage NB‐QMLE enjoys a certain asymptotic efficiency property provided that a negative binomial link function relating the conditional mean and conditional variance is specified. The proposed NB‐QMLEs are compared with the Poisson QMLE asymptotically and in finite samples for various well‐known particular classes of count time series models such as the Poisson and negative binomial integer‐valued GARCH model and the INAR(1) model. Application to a real dataset is given.     

Likelihood‐based Analysis of a Class of Generalized Long‐Memory Time Series Models

Abstract. This article introduces a family of ‘generalized long‐memory time series models’, in which observations have a specified conditional distribution, given a latent Gaussian fractionally integrated autoregressive moving‐average (ARFIMA) process. The observations may have discrete or continuous distributions (or a mixture of both). The family includes existing models such as ARFIMA models themselves, long‐memory stochastic volatility models, long‐memory censored Gaussian models and others. Although the family of models is flexible, the latent long‐memory process poses problems for analysis. Therefore, we introduce a Markov chain Monte Carlo sampling algorithm and develop a set of recursions which makes it feasible. This makes it possible, among other things, to carry out exact likelihood‐based analysis of a wide range of non‐Gaussian long‐memory models without resorting to the use of likelihood approximations. The procedure also yields predictive distributions that take into account model parameter uncertainty. The approach is demonstrated in two case studies.     

Testing for Change in Long‐Memory Stochastic Volatility Time Series

In this article, change‐point problems for long‐memory stochastic volatility (LMSV) models are considered. A general testing problem which includes various alternative hypotheses is discussed. Under the hypothesis of stationarity the limiting behavior of CUSUM‐ and Wilcoxon‐type test statistics is derived. In this context, a limit theorem for the two‐parameter empirical process of LMSV time series is proved. In particular, it is shown that the asymptotic distribution of CUSUM test statistics may not be affected by long memory, unlike Wilcoxon test statistics which are typically influenced by long‐range dependence. To avoid the estimation of nuisance parameters in applications, the usage of self‐normalized test statistics is proposed. The theoretical results are accompanied by an analysis of Standard & Poor's 500 daily closing indices with respect to structural changes and by simulation studies which characterize the finite sample behavior of the considered testing procedures when testing for changes in mean and in variance.     

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.     

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.     

Signal Extraction for Non‐Stationary Multivariate Time Series with Illustrations for Trend Inflation

This article advances the theory and methodology of signal extraction by developing the optimal treatment of difference stationary multivariate time‐series models. Using a flexible time‐series structure that includes co‐integrated processes, we derive and prove formulas for minimum mean square error estimation of signal vectors in multiple series, from both a finite sample and a bi‐infinite sample. As an illustration, we present econometric measures of the trend in total inflation that make optimal use of the signal content in core inflation.     

A Bayesian Non‐Parametric Dynamic AR Model for Multiple Time Series Analysis

In this article, we propose a Bayesian non‐parametric model for the analysis of multiple time series. We consider an autoregressive structure of order p for each of the series and borrow strength across the series by considering a common error population that is also evolving in time. The error populations (distributions) are assumed non‐parametric whose law is based on a series of dependent Polya trees with zero median. This dependence is of order q and is achieved via a dependent beta process that links the branching probabilities of the trees. We study the prior properties and show how to obtain posterior inference. The model is tested under a simulation study and is illustrated with the analysis of the economic activity index of the 32 states of Mexico.     

Broadband semi‐parametric estimation of long‐memory time series by fractional exponential models

This article proposes broadband semi‐parametric estimation of a long‐memory parameter by fractional exponential (FEXP) models. We construct the truncated Whittle likelihood based on FEXP models in a semi‐parametric setting to estimate the parameter and show that the proposed estimator is more efficient than the FEXP estimator by Moulines and Soulier (1999) in linear processes. A Monte Carlo simulation suggests that the proposed estimation is more preferable than the existing broadband semi‐parametric estimation.     

Time‐Dependent Dual‐Frequency Coherence in Multivariate Non‐Stationary Time Series

Coherence is one common metric for cross‐dependence in multichannel signals. However, standard coherence does not sufficiently model many biological signals with complex dependence structures such as cross‐oscillatory interactions between a low‐frequency component in one signal and a high‐frequency component in another. The notion of cross‐dependence between low‐ and high‐frequency components, as defined in classical harmonizable processes, is still inadequate because it assumes time invariance and thus cannot capture cross‐frequency interactions that evolve over time. We construct a novel framework for modeling and estimating these dependencies under the replicated time series setting. Under this framework, we establish the novel concept of evolutionary dual‐frequency coherence and develop time‐localized estimators based on dual‐frequency local periodograms. The proposed nonparametric estimation procedure does not suffer from model misspecification. It uses the localized fast Fourier transform and hence is able to handle massive data. When applied to electroencephalogram data recorded in a motor intention experiment, the proposed method uncovers new and interesting cross‐oscillatory interactions that have been overlooked by the standard approaches.     

Testing Parameter Change in General Integer‐Valued Time Series

We consider the structural change in a class of discrete valued time series, which the conditional distribution belongs to the one‐parameter exponential family. We propose a change point test based on the maximum likelihood estimator of the model's parameter. Under the null hypothesis (of no change), the test statistic converges to a well‐known distribution, allowing the calculation of the critical value of the test. The test statistic diverges to infinity under the alternative, meaning that the test has asymptotic power one. Some simulation results and real data applications are reported to show the effectiveness of the proposed procedure.     

Correction of second‐order‐diffraction errors in spectrophotometry

Conventional diffraction gratings are designed so that most of the light striking the grating is diffracted toward the “first‐order beam” direction, but in practice, some of the light may be diffracted towards the “second‐order beam” direction, so that the light striking a particular sensor pixel may be a combination of both first‐ and second‐order components. This causes errors in the output of diffraction‐grating based UV–visible spectrometers. A new method has been developed to correct such errors by calibrating the optical system with a UV‐excluded standard (such as an orange tile) and a UV‐included standard (such as a white tile). These two calibrations can predict the relationship between first‐order‐only signals and first‐and‐second‐order combined signals at sensor pixels (nominal wavelengths) where there is second‐order light, and thus help correct the second‐order diffraction error. This compensation method applied on the reflectance measurement of a set of color ceramic tiles showed significant improvement on the accuracy of the result. © 2016 Wiley Periodicals, Inc. Col Res Appl, 42, 189–192, 2017     

The Effect of the Estimation on Goodness‐of‐Fit Tests in Time Series Models

Abstract. We analyze, by simulation, the finite‐sample properties of goodness‐of‐fit tests based on residual autocorrelation coefficients (simple and partial) obtained using different estimators frequently used in the analysis of autoregressive moving‐average time‐series models. The estimators considered are unconditional least squares, maximum likelihood and conditional least squares. The results suggest that although the tests based on these estimators are asymptotically equivalent for particular models and parameter values, their sampling properties for samples of the size commonly found in economic applications can differ substantially, because of differences in both finite‐sample estimation efficiencies and residual regeneration methods.     

A Time‐Symmetric Self‐Normalization Approach for Inference of Time Series

Self‐normalization has been celebrated as an alternative approach for inference of time series because of its ability to avoid direct estimation of the nuisance asymptotic variance. However, when being applied to quantities other than the mean, the conventional self‐normalizer typically exhibits certain degrees of asymmetry, an undesirable feature especially for time‐reversible processes. This paper considers a new self‐normalizer for time series, which (i) provides a time‐symmetric generalization to the conventional self‐normalizer, (ii) is able to automatically reduce to the conventional self‐normalizer in the mean case where the latter is already time‐symmetric to yield a unified inference procedure, and (iii) possibly leads to narrower confidence intervals when compared with the conventional self‐normalizer. For the proposed time‐symmetric self‐normalizer, we establish the asymptotic theory for its induced inference procedure and examine its finite sample performance through numerical experiments.     

Asymptotic Behavior of Optimal Weighting in Generalized Self‐Normalization for Time Series

Self‐normalization has been celebrated for its ability to avoid direct estimation of the nuisance long‐run variance and its versatility in handling the mean and other quantities. The self‐normalizer in its original form uses only recursive estimators of one direction, and generalizations involving both forward and backward estimators were recently given. Unlike existing results that weigh the forward and backward estimators in a deterministic manner, the current article focuses on a data‐driven weight that corresponds to confidence intervals with minimal lengths. We study the asymptotic behavior of such a data‐driven weight choice, and find an interesting dichotomy between linear and nonlinear quantities. Another interesting phenomenon is that, for nonlinear quantities, the data‐driven weight typically distributes over an uncountable set in finite‐sample problems but in the limit it converges to a discrete distribution with a finite support.