On an independent and identically distributed mixture bilinear time‐series model

A class of nonlinear time‐series models in which the underlying process follows a finite mixture of bilinear representations is proposed. The mixture feature appears in the conditional distribution of the process which is given as a finite mixture of distributions evaluated at the normed innovations of diagonal bilinear specifications. This class is aimed at capturing special characteristics exhibited by many observed time series such as tail heaviness, multimodality, asymmetry and change in regime. Some probabilistic properties of the proposed model, namely strict and second‐order stationarity, geometric ergodicity, covariance structure, existence of higher order moments, tail behaviour and invertibility, are first studied. Parameter estimation is then performed through the EM algorithm, performance of which is shown via simulation experiments. Applications to some real‐time‐series data are proposed and through which it is shown how neglecting the mixture framework in a bilinear representation results in a loss in adequacy.     

Time‐series clustering via quasi U‐statistics

The problem of time‐series discrimination and classification is discussed. We propose a novel clustering algorithm based on a class of quasi U‐statistics and subgroup decomposition tests. The decomposition may be applied to any concave time‐series distance. The resulting test statistics are proven to be asymptotically normal for either i.i.d. or non‐identically distributed groups of time‐series under mild conditions. We illustrate its empirical performance on a simulation study and a real data analysis. The simulation setup includes stationary vs. stationary and stationary vs. non‐stationary cases. The performance of the proposed method is favourably compared with some of the most common clustering measures available.     

A DISTANCE MEASURE FOR CLASSIFYING ARIMA MODELS

Abstract. In a number of practical problems where clustering or choosing from a set of dynamic structures is needed, the introduction of a distance between the data is an early step in the application of multivariate statistical methods. In this paper a parametric approach is proposed in order to introduce a well-defined metric on the class of autoregressive integrated moving-average (ARIMA) invertible models as the Euclidean distance between their autoregressive expansions. Two case studies for clustering economic time series and for assessing the consistency of seasonal adjustment procedures are discussed. Finally, some related proposals are surveyed and some suggestions for further research are made.     

Bayesian Models for Non-linear Autoregressions

We discuss classes of Bayesian mixture models for nonlinear autoregressive times series, based on developments in semiparametric Bayesian density estimation in recent years. The development involves formal classes of multivariate discrete mixture distributions, providing flexibility in modeling arbitrary nonlinearities in time series structure and a formal inferential framework within which to address the problems of inference and prediction. The models relate naturally to existing kernel and related methods, threshold models and others, although they offer major advances in terms of parameter estimation and predictive calculations. Theoretic al and computational aspects are developed here, the latter involving efficient simulation of posterior and predictive distributions. Various examples illustrate our perspectives on identification and inference using this mixture approach     

Distributions for residual autocovariances in parsimonious periodic vector autoregressive models with applications

The asymptotic distribution of the residual autocovariance matrices in the class of periodic vector autoregressive time series models with structured parameterization is derived. Diagnostic checking with portmanteau test statistics represents a useful application of the result. Under the assumption that the periodic white noise process of the periodic vector autoregressive time series model is composed of independent random variables, we demonstrate that the finite sample distributions of the Hosking‐Li‐McLeod portmanteau test statistics can be approximated by those of weighted sums of independent chi‐square random variables. The quantiles of the asymptotic distribution can be computed using the Imhof algorithm or other exact methods. Thus, using the (single) chi‐square distribution for these test statistics appears inadequate in general, although it is often recommended in practice for diagnostic methods of that kind. A simulation study provides empirical evidence.     

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.     

Marginal Estimation of Parameter Driven Binomial Time Series Models

This article develops asymptotic theory for estimation of parameters in regression models for binomial response time series where serial dependence is present through a latent process. Use of generalized linear model estimating equations leads to asymptotically biased estimates of regression coefficients for binomial responses. An alternative is to use marginal likelihood, in which the variance of the latent process but not the serial dependence is accounted for. In practice, this is equivalent to using generalized linear mixed model estimation procedures treating the observations as independent with a random effect on the intercept term in the regression model. We prove that this method leads to consistent and asymptotically normal estimates even if there is an autocorrelated latent process. Simulations suggest that the use of marginal likelihood can lead to generalized linear model estimates result. This problem reduces rapidly with increasing number of binomial trials at each time point, but for binary data, the chance of it can remain over 45% even in very long time series. We provide a combination of theoretical and heuristic explanations for this phenomenon in terms of the properties of the regression component of the model, and these can be used to guide application of the method in practice.     

DETERMINING THE NUMBER OF REGIMES IN MARKOV SWITCHING VAR AND VMA MODELS

We give stable finite‐order vector autoregressive moving average (p ^* ,q ^* ) representations for M‐state Markov switching second‐order stationary time series whose autocovariances satisfy a certain matrix relation. The upper bounds for p ^* and q ^* are elementary functions of the dimension K of the process, the number M of regimes, the autoregressive and moving‐average orders of the initial model. If there is no cancellation, the bounds become equalities, and this solves the identification problem. Our classes of time series include every M‐state Markov switching multi‐variate moving‐average models and autoregressive models in which the regime variable is uncorrelated with the observable. Our results include, as particular cases, those obtained by Krolzig (1997) and improve the bounds given by Zhang and Stine (2001) and Francq and Zakoïan (2001) for our classes of dynamic models. A Monte Carlo experiment and an application on foreign exchange rates complete the article. Copyright © 2013 John Wiley & Sons, Ltd.     

On Parameter Estimation for Exponential Dispersion Arma Models

Abstract. A class of autoregressive moving‐average (ARMA) models proposed by Jørgensen and Song [Journal of Applied Probability (1998), vol. 35, pp. 78–92] with exponential dispersion model margins are useful to deal with non‐normal stationary time series with high‐order autocorrelation. One property associated with the class of models is that the projection process takes the exact form of the classical Box and Jenkins ARMA representation, leading to considerable ease to establish theories. This paper focuses on the issue of parameter estimation for such models, which has not been thoroughly investigated in Jørgensen and Song's paper. The key of the proposed approach is to treat the residual process associated with the projection essentially as a measurement error, which enables us to formulate directly an ARMA representation for the observed time series. The parameter estimation therefore becomes straightforward using the existing methods for the Box and Jenkins ARMA models such as the quasi‐likelihood method. The approach is illustrated by simulation studies and by an analysis of myoclonic seizure counts.     

Estimation in Functional Lagged Regression

The paper introduces a functional time series (lagged) regression model. The impulse‐response coefficients in such a model are operators acting on a separable Hilbert space, which is the function space L² in applications. A spectral approach to the estimation of these coefficients is proposed and asymptotically justified under a general nonparametric condition on the temporal dependence of the input series. Since the data are infinite‐dimensional, the estimation involves a spectral‐domain dimension‐reduction technique. Consistency of the estimators is established under general data‐dependent assumptions on the rate of the dimension‐reduction parameter. Their finite‐sample performance is evaluated by a simulation study that compares two ad hoc approaches to dimension reduction with an alternative, asymptotically justified method.     

Aggregation and Disaggregation of Structural Time Series Models

The aggregation/disaggregation problem has been widely studied in the time series literature. Some main issues related to this problem are modelling, prediction and robustness to outliers. In this paper we look at the modelling problem with particular interest in the local level and local trend structural time series models together with their corresponding ARIMA(0, 1, 1) and ARIMA(0, 2, 2) representations. Given an observed time series that can be expressed by a structural or autoregressive integrated moving-average (ARIMA) model, we derive the necessary and sufficient conditions under which the aggregate and/or disaggregate series can be expressed by the same class of model. Harvey's cycle and seasonal components models (Harvey, Forecasting, Structural Time Series Models and the Kalman Filter , Cambridge: Cambridge University Press, 1989) are also briefly discussed. Systematic sampling of structural and ARIMA models is also discussed.     

Integer‐Valued Autoregressive Models With Survival Probability Driven By A Stochastic Recurrence Equation

This paper proposes a new class of integer‐valued autoregressive models with a dynamic survival probability. The peculiarity of this class of models lies in the specification of the survival probability through a stochastic recurrence equation. The proposed models can effectively capture changing dependence over time and enhance both the in‐sample and out‐of‐sample performance of integer‐valued autoregressive models. This point is illustrated through an empirical application to a real‐time series of crime reports. Additionally, this paper discusses the reliability of likelihood‐based inference for the class of models. In particular, this study proves the consistency of the maximum likelihood estimator and a plug‐in estimator for the conditional probability mass function in a misspecified model setting.     

Partial autocorrelation parameterization for subset autoregression

Abstract. A new version of the partial autocorrelation plot and a new family of subset autoregressive models are introduced. A comprehensive approach to model identification, estimation and diagnostic checking is developed for these models. These models are better suited to efficient model building of high‐order autoregressions with long time series. Several illustrative examples are given.     

Assessing Persistence In Discrete Nonstationary Time‐Series Models

Abstract. The aim of this paper is to examine the application of measures of persistence in a range of time‐series models nested in the framework of Cramer (1961) . This framework is a generalization of the Wold (1938) decomposition for stationary time‐series which, in addition to accommodating the standard I(0) and I(1) models, caters for a broad range of alternative processes. Two measures of persistence are considered in some detail, namely the long‐run impulse‐response and variance‐ratio functions. Particular emphasis is given to the behaviour of these measures in a range of non‐stationary models specified in discrete time. We document the conflict that arises between different measures, applied to the same model, as well as conflict arising from the use of a given measure in different models. Precisely which persistence measures are time dependent and which are not, is highlighted. The nature of the general representation used also helps to clarify which shock the impulse‐response function refers to in the case of models where more than one random disturbance impinges on the time series.     

A note on the mixture transition distribution and hidden Markov models

We discuss an interpretation of the mixture transition distribution (MTD) for discrete‐valued time series which is based on a sequence of independent latent variables which are occasion‐specific. We show that, by assuming that this latent process follows a first order Markov Chain, MTD can be generalized in a sensible way. A class of models results which also includes the hidden Markov model (HMM). For these models we outline an EM algorithm for the maximum likelihood estimation which exploits recursions developed within the HMM literature. As an illustration, we provide an example based on the analysis of stock market data referred to different American countries.     

On Bayesian analysis of nonlinear continuous‐time autoregression models

Abstract. This article introduces a method for performing fully Bayesian inference for nonlinear conditional autoregressive continuous‐time models, based on a finite skeleton of observations. Our approach uses Markov chain Monte Carlo and involves imputing data from times at which observations are not made. It uses a reparameterization technique for the missing data, and because of the non‐Markovian nature of the models, it is necessary to adopt an overlapping blocks scheme for sequentially updating segments of missing data. We illustrate the methodology using both simulated data and a data set from the S & P 500 index.     

Dynamic spatial Bayesian models for radioactivity deposition

Dynamic spatial Bayesian (DSB) models are proposed for the analytical modelling of radioactivity deposition after a nuclear accident. The proposed models are extensions of the multi‐variate time‐series dynamic linear models of West and Harrison (1997) to Markov random field processes. They combine the outputs from a long‐range atmospheric dispersal model with measured data (and prior information) to provide improved deposition prediction in space and time. Two versions of a Gaussian DSB model were applied to the radioactivity deposition in Bavaria over a 15 days period during the Chernobyl nuclear accident. One version had fixed functional forms for its spatial variances and covariances while the other allowed those to adapt and ‘learn’ from data in the conjugate Bayesian paradigm. There were two main sources of information for radioactivity deposition in our application: radioactivity measurements at a sparse set of 13 monitoring stations, and the numerical deposition evaluation of the atmospheric dispersal K‐model for the points of a 64 × 64 regular grid. We have analysed the temporal predictions (one‐step‐ahead forecasting) of those DSB models to show that the dispersal K‐model tended in general to underestimate the deposition levels at all times while the DSB models corrected for that although with different degrees of adjustment.     

Empirical determination of the frequencies of an almost periodic time series

This article deals with the problem of the determination of the finite or countable set of frequencies belonging to any arbitrary almost periodic (in the sense of Bohr) time series. For this purpose, we present a simple computation procedure based on the local maxima of the modulus of a weighted Fourier transform from finite observation of the time series, computed at frequencies in a finite uniform grid of [0, 2π). We study the convergence of this algorithm as the length of the observation goes to infinity. First non‐random signals are considered. Then we tackle the case of a signal disturbed by an additive noise. Finally we show how the algorithm can be applied to almost periodically correlated random time series.     

Modelling Count Data Time Series with Markov Processes Based on Binomial Thinning

Abstract. We obtain new models and results for count data time series based on binomial thinning. Count data time series may have non‐stationarity from trends or covariates, so we propose an extension of stationary time series based on binomial thinning such that the univariate marginal distributions are always in the same parametric family, such as negative binomial. We propose a recursive algorithm to calculate the probability mass functions for the innovation random variable associated with binomial thinning. This simplifies numerical calculations and estimation for the classes of time series models that we consider. An application with real data is used to illustrate the models.     

Efficient Bayesian PARCOR approaches for dynamic modeling of multivariate time series

A Bayesian lattice filtering and smoothing approach is proposed for fast and accurate modeling and inference in multivariate non-stationary time series. This approach offers computational feasibility and interpretable time-frequency analysis in the multivariate context. The proposed framework allows us to obtain posterior estimates of the time-varying spectral densities of individual time series components, as well as posterior measurements of the time-frequency relationships across multiple components, such as time-varying coherence and partial coherence. The proposed formulation considers multivariate dynamic linear models (MDLMs) on the forward and backward time-varying partial autocorrelation coefficients (TV-VPARCOR). Computationally expensive schemes for posterior inference on the multivariate dynamic PARCOR model are avoided using approximations in the MDLM context. Approximate inference on the corresponding time-varying vector autoregressive (TV-VAR) coefficients is obtained via Whittle's algorithm. A key aspect of the proposed TV-VPARCOR representations is that they are of lower dimension, and therefore more efficient, than TV-VAR representations. The performance of the TV-VPARCOR models is illustrated in simulation studies and in the analysis of multivariate non-stationary temporal data arising in neuroscience and environmental applications. Model performance is evaluated using goodness-of-fit measurements in the time-frequency domain and also by assessing the quality of short-term forecasting.