Flexible and Robust Mixed Poisson INGARCH Models

In this article, we propose a general class of INteger‐valued Generalized AutoRegressive Conditional Heteroskedastic (INGARCH) models based on a flexible family of mixed Poisson (MP) distributions. Our proposed class of count time series models contains the negative binomial (NB) INGARCH process as particular case and open the possibility to introduce new models such as the Poisson‐inverse Gaussian (PIG) and Poisson generalized hyperbolic secant processes. In particular, the PIG INGARCH model is an interesting and robust alternative to the NB model. We explore first‐order and second‐order stationary properties of our MPINGARCH models and provide expressions for the autocorrelation function and mean and variance marginals. Conditions to ensure strict stationarity and ergodicity properties for our class of INGARCH models are established. We propose an Expectation‐Maximization algorithm to estimate the parameters and obtain the associated information matrix. Further, we discuss two additional estimation methods. Monte Carlo simulation studies are considered to evaluate the finite‐sample performance of the proposed estimators. We illustrate the flexibility and robustness of the MPINGARCH models through two real‐data applications about number of cases of Escherichia coli and Campylobacter infections. This article contains a Supporting Information.     

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.     

First‐order integer valued AR processes with zero inflated poisson innovations

The first‐order nonnegative integer valued autoregressive process has been applied to model the counts of events in consecutive points of time. It is known that, if the innovations are assumed to follow a Poisson distribution then the marginal model is also Poisson. This model may however not be suitable for overdispersed count data. One frequent manifestation of overdispersion is that the incidence of zero counts is greater than expected from a Poisson model. In this paper, we introduce a new stationary first‐order integer valued autoregressive process with zero inflated Poisson innovations. We derive some structural properties such as the mean, variance, marginal and joint distribution functions of the process. We consider estimation of the unknown parameters by conditional or approximate full maximum likelihood. We use simulation to study the limiting marginal distribution of the process and the performance of our fitting algorithms. Finally, we demonstrate the usefulness of the proposed model by analyzing some real time series on animal health laboratory submissions.     

Poisson QMLE of Count Time Series Models

Regularity conditions are given for the consistency of the Poisson quasi‐maximum likelihood estimator of the conditional mean parameter of a count time series model. The asymptotic distribution of the estimator is studied when the parameter belongs to the interior of the parameter space and when it lies at the boundary. Tests for the significance of the parameters and for constant conditional mean are deduced. Applications to specific integer‐valued autoregressive (INAR) and integer‐valued generalized autoregressive conditional heteroscedasticity (INGARCH) models are considered. Numerical illustrations, Monte Carlo simulations and real data series are provided.     

MCMC for Integer‐Valued ARMA processes

Abstract. The classical statistical inference for integer‐valued time‐series has primarily been restricted to the integer‐valued autoregressive (INAR) process. Markov chain Monte Carlo (MCMC) methods have been shown to be a useful tool in many branches of statistics and is particularly well suited to integer‐valued time‐series where statistical inference is greatly assisted by data augmentation. Thus in this article, we outline an efficient MCMC algorithm for a wide class of integer‐valued autoregressive moving‐average (INARMA) processes. Furthermore, we consider noise corrupted integer‐valued processes and also models with change points. Finally, in order to assess the MCMC algorithms inferential and predictive capabilities we use a range of simulated and real data sets.     

A PARAMETER‐DRIVEN LOGIT REGRESSION MODEL FOR BINARY TIME SERIES

We consider a parameter‐driven regression model for binary time series, where serial dependence is introduced by an autocorrelated latent process incorporated into the logit link function. Unlike in the case of parameter‐driven Poisson log‐linear or negative binomial logit regression model studied in the literature for time series of counts, generalized linear model (GLM) estimation of the regression coefficient vector, which suppresses the latent process and maximizes the corresponding pseudo‐likelihood, cannot produce a consistent estimator. As a remedial measure, in this article, we propose a modified GLM estimation procedure and show that the resulting estimator is consistent and asymptotically normal. Moreover, we develop two procedures for estimating the asymptotic covariance matrix of the estimator and establish their consistency property. Simulation studies are conducted to evaluate the finite‐sample performance of the proposed procedures. An empirical example is also presented.     

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.     

Local asymptotic normality and efficient estimation for INAR(p) models

Abstract. Integer‐valued autoregressive (INAR) processes have been introduced to model non‐negative integer‐valued phenomena that evolve in time. The distribution of an INAR(p) process is determined by two parameters: a vector of survival probabilities and a probability distribution on the non‐negative integers, called an immigration distribution. This paper provides an efficient estimator of the parameters, and in particular, shows that the INAR(p) model has the Local Asymptotic Normality property.     

A p‐Order signed integer‐valued autoregressive (SINAR(p)) model

In this article, we propose an extension of integer‐valued autoregressive INAR models. Using a signed version of the thinning operator, we define a larger class of ‐valued processes, called SINAR, which can have positive as well as negative correlations. Using a Markov chain method, conditions for stationarity and the existence of moments are investigated. In particular, it is shown that the autocorrelation function of any real‐valued AR process can be recovered with a SINAR process, which improves INAR modeling.     

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.     

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

Abstract. Here we obtain difference equations for the higher order moments and cumulants of a time series {X_t} satisfying an INAR(p) model. These equations are similar to the difference equations for the higher order moments and cumulants of the bilinear time series model. We obtain the spectral and bispectral density functions for the INAR(p) process in state–space form, thus characterizing it in the frequency domain. We consider a frequency domain method – the Whittle criterion – to estimate the parameters of the INAR(p) model and illustrate it with the series of the number of epilepsy seizures of a patient.     

Measuring nonlinear dependence in time‐series,a distance correlation approach

We extend the concept of distance correlation of Szekely et al. (2007) and propose the auto distance correlation function (ADCF) to measure the temporal dependence structure of time‐series. Unlike the classic measures of correlations such as the autocorrelation function, the proposed measure is zero if and only if the measured time‐series components are independent. In this article, we propose and theoretically verify a subsampling methodology for the inference of sample ADCF for dependent data. Our methodology provides a useful tool for exploring nonlinear dependence structures in time‐series.     

Testing for Stationarity in Multivariate Locally Stationary Processes

In this article, we propose a nonparametric procedure for validating the assumption of stationarity in multivariate locally stationary time series models. We develop a bootstrap‐assisted test based on a Kolmogorov–Smirnov‐type statistic, which tracks the deviation of the time‐varying spectral density from its best stationary approximation. In contrast to all other nonparametric approaches, which have been proposed in the literature so far, the test statistic does not depend on any regularization parameters like smoothing bandwidths or a window length, which is usually required in a segmentation of the data. We additionally show how our new procedure can be used to identify the components where non‐stationarities occur and indicate possible extensions of this innovative approach. We conclude with an extensive simulation study, which shows finite‐sample properties of the new method and contains a comparison with existing approaches.     

Infinitely Divisible Distributions in Integer‐Valued Garch Models

We propose an integer‐valued stochastic process with conditional marginal distribution belonging to the class of infinitely divisible discrete probability laws. With this proposal, we introduce a wide class of models for count time series that includes the Poisson integer‐valued generalized autoregressive conditional heteroscedastic (INGARCH) model (Ferland et al., 2006) and the negative binomial and generalized Poisson INGARCH models (Zhu, 2011, 2012a). The main probabilistic analysis of this process is developed stating, in particular, first‐order and second‐order stationarity conditions. The existence of a strictly stationary and ergodic solution is established in a subclass including the Poisson and generalized Poisson INGARCH models.     

FIRST-ORDER INTEGER-VALUED AUTOREGRESSIVE (INAR(1)) PROCESS

Abstract. A simple model for a stationary sequence of integer-valued random variables with lag-one dependence is given and is referred to as the integer-valued autoregressive of order one (INAR(1)) process. The model is suitable for counting processes in which an element of the process at time t can be either the survival of an element of the process at time t - 1 or the outcome of an innovation process. The correlation structure and the distributional properties of the INAR(1) model are similar to those of the continuous-valued AR(1) process. Several methods for estimating the parameters of the model are discussed, and the results of a simulation study for these estimation methods are presented.     

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.     

A Structural‐Factor Approach to Modeling High‐Dimensional Time Series and Space‐Time Data

This article considers a structural‐factor approach to modeling high‐dimensional time series and space‐time data by decomposing individual series into trend, seasonal, and irregular components. For ease in analyzing many time series, we employ a time polynomial for the trend, a linear combination of trigonometric series for the seasonal component, and a new factor model for the irregular components. The new factor model simplifies the modeling process and achieves parsimony in parameterization. We propose a Bayesian information criterion to consistently select the order of the polynomial trend and the number of trigonometric functions, and use a test statistic to determine the number of common factors. The convergence rates for the estimators of the trend and seasonal components and the limiting distribution of the test statistic are established under the setting that the number of time series tends to infinity with the sample size, but at a slower rate. We study the finite‐sample performance of the proposed analysis via simulation, and analyze two real examples. The first example considers modeling weekly PM_2.5 data of 15 monitoring stations in the southern region of Taiwan and the second example consists of monthly value‐weighted returns of 12 industrial portfolios.     

Oracle M‐Estimation for Time Series Models

We propose a thresholding M‐estimator for multivariate time series. Our proposed estimator has the oracle property that its large‐sample properties are the same as of the classical M‐estimator obtained under the a priori information that the zero parameters were known. We study the consistency of the standard block bootstrap, the centred block bootstrap and the empirical likelihood block bootstrap distributions of the proposed M‐estimator. We develop automatic selection procedures for the thresholding parameter and for the block length of the bootstrap methods. We present the results of a simulation study of the proposed methods for a sparse vector autoregressive VAR(2) time series model. The analysis of two real‐world data sets illustrate applications of the methods in practice.     

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 geometric time series model with dependent Bernoulli counting series

A new stationary first‐order integer‐valued autoregressive process with geometric marginal distribution based on the generalized binomial thinning is introduced. The model involves dependent count variables. Some properties of the process are determined. A set of estimators are obtained, and their asymptotic distributions are considered. Some numerical results of the estimates are presented. Possible application of the process is discussed through the real data example.