Conway–Maxwell–Poisson Autoregressive Moving Average Model for Equidispersed,Underdispersed, and Overdispersed Count Data

In this work, we propose a dynamic regression model based on the ConwayŮMaxwell–Poisson (CMP) distribution with time-varying conditional mean depending on covariates and lagged observations. This new class of ConwayŮMaxwell–Poisson autoregressive moving average (CMP-ARMA) models is suitable for the analysis of time series of counts. The CMP distribution is a two-parameter generalization of the Poisson distribution that allows the modeling of underdispersed, equidispersed, and overdispersed data. Our main contribution is to combine this dispersion flexibility with the inclusion of lagged terms to model the conditional mean response, inducing an autocorrelation structure, usually relevant in time series. We present the conditional maximum likelihood estimation, hypothesis testing inference, diagnostic analysis, and forecasting along with their asymptotic properties. In particular, we provide closed-form expressions for the conditional score vector and conditional Fisher information matrix. We conduct a Monte Carlo experiment to evaluate the performance of the estimators in finite sample sizes. Finally, we illustrate the usefulness of the proposed model by exploring two empirical applications.     

Modeling the Variance of Return Intervals Toward Volatility Prediction

Interval-valued time series has been attracting increasing interest. There have been fruitful results on mean models, but variance models largely remain unexploited. In this article, we propose a conditional heteroskedasticity model for the return interval process, which aims at capturing the underlying variance structure. Under the general framework of random sets, the model properties are investigated. Parameters are estimated by the maximum likelihood method, and the asymptotic properties are established. Empirical application to stocks and financial indices data sets suggests that our model overall outperforms the traditional generalized autoregressive conditional heteroskedasticity for both in-sample estimation and out-of-sample prediction of the volatility.     

Time‐varying multi‐regime models fitting by genetic algorithms

Many time series exhibit both nonlinearity and non‐stationarity. Though both features have been often taken into account separately, few attempts have been proposed for modelling them simultaneously. We consider threshold models, and present a general model allowing for different regimes both in time and in levels, where regime transitions may happen according to self‐exciting, or smoothly varying or piecewise linear threshold modelling. Since fitting such a model involves the choice of a large number of structural parameters, we propose a procedure based on genetic algorithms, evaluating models by means of a generalized identification criterion. The performance of the proposed procedure is illustrated with a simulation study and applications to some real data.     

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.     

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.     

A Space-Time Bilinear Model and its Identification

In this paper we propose a class of space–time bilinear (STBL) models which can be used to model space–time series which exhibit bilinear behavior. The STBL model is shown to be an extension of a space–time autoregressive moving-average model and a special form of the multiple bilinear model. We focus on the identification procedure of the models. Some results about stationarity and the covariance structure of these models are also discussed. An identification procedure based on the squared observations is established for the simplest pure bilinear model and some illustrative examples are provided.     

THRESHOLD TIME SERIES MODELS AS MULTIMODAL DISTRIBUTION JUMP PROCESSES

Abstract. Recent contributions by Tong and others in modelling time series exhibiting threshold points have generally been based on approximating non-linear processes by piecewise linear time series models. In this paper we provide an alternative framework in which to model time series displaying jump behaviour by using a multimodal conditional distribution to capture the jump process. Each subordinate model of the distribution is determined by an autoregressive process, and jump behaviour occurs when the relative heights of the modes of the distribution change whilst the threshold points are identified by the antimodes of the distribution. This class of models is referred to as multipredictor autoregressive time series (MATS).     

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.     

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.     

ON THE SQUARED RESIDUAL AUTOCORRELATIONS IN NON-LINEAR TIME SERIES WITH CONDITIONAL HETEROSKEDASTICITY

Abstract. Time series with a changing conditional variance have been found useful in many applications. Residual autocorrelations from traditional autoregressive moving-average models have been found useful in model diagnostic checking. By analogy, squared residual autocorrelations from fitted conditional heteroskedastic time series models would be useful in checking the adequacy of such models. In this paper, a general class of squared residual autocorrelations is defined and their asymptotic distribution is obtained. The result leads to some useful diagnostic tools for statisticians using conditional heteroskedastic time series models. Some simulation results and an illustrative example are also reported.     

Modeling bivariate long-range dependence with general phase

Bivariate time series models are considered that are suitable for estimation, that have interpretable parameters and that can capture the general semi-parametric formulation of bivariate long-range dependence, including a general phase. The models also allow for short-range dependence and fractional cointegration. A simulation study to test the performance of a conditional maximum likelihood estimation method is carried out, under the proposed models. Finally, an application is presented to the U.S. inflation rates in goods and services where models not allowing for general phase suffer from misspecification.     

Transformations and seasonal adjustment

Abstract.  We address the problem of seasonal adjustment of a nonlinear transformation of the original time series, measured on a ratio scale, which aims at enforcing two essential features: additivity and orthogonality of the components. The posterior mean and variance of the seasonally adjusted series admit an analytic finite representation only for particular values of the transformation parameter, e.g. for a fractional Box–Cox transformation parameter. Even if available, the analytical derivation can be tedious and difficult. As an alternative we propose to compute the two conditional moments of the seasonally adjusted series by means of numerical and Monte Carlo integration. The former is both fast and reliable in univariate applications. The latter uses the algorithm known as the 'simulation smoother' and it is most useful in multivariate applications. We present two case studies dealing with robust seasonal adjustment under the square root and the fourth root transformation. Our overall conclusion is that robust seasonal adjustment under transformations is feasible from the computational standpoint and that the possibility of transforming the scale ought to be considered as a further option for improving the quality of seasonal adjustment.     

NONPARAMETRIC ESTIMATORS FOR TIME SERIES

Abstract. Kernel multivariate probability density and regression estimators are applied to a univariate strictly stationary time series X _r We consider estimators of the joint probability density of X _t at different t -values, of conditional probability densities, and of the conditional expectation of functionals of X _v given past behaviour. The methods seem of particular relevance in light of recent interest in non-Gaussian time series models. Under a strong mixing condition multivariate central limit theorems for estimators at distinct points are established, the asymptotic distributions being of the same nature as those which would derive from independent multivariate observations.     

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.     

Asymptotics of Quantiles and Rank Scores in Nonlinear Time Series

This paper extends the concept of regression and autoregression quantiles and rank scores to a very general nonlinear time series model. The asymptotic linearizations of these nonlinear quantiles are then used to obtain the limiting distributions of a class of L-estimators of the parameters. In particular, the limiting distributions of the least absolute deviation estimator and trimmed estimators are obtained. These estimators turn out to be asymptotically more efficient than the widely used conditional least squares estimator for heavy-tailed error distributions. The results are applicable to linear and nonlinear regression and autoregressive models including self-exciting threshold autoregressive models with known threshold.     

Detecting misspecifications in autoregressive conditional duration models and non‐negative time‐series processes

We develop a general theory to test correct specification of multiplicative error models of non‐negative time‐series processes, which include the popular autoregressive conditional duration (ACD) models. Both linear and nonlinear conditional expectation models are covered, and standardized innovations can have time‐varying conditional dispersion and higher‐order conditional moments of unknown form. No specific estimation method is required, and the tests have a convenient null asymptotic N(0,1) distribution. To reduce the impact of parameter estimation uncertainty in finite samples, we adopt Wooldridge's (1990a) device to our context and justify its validity. Simulation studies show that in the context of testing ACD models, finite sample correction gives better sizes in finite samples and are robust to parameter estimation uncertainty. And, it is important to take into account time‐varying conditional dispersion and higher‐order conditional moments in standardized innovations; failure to do so can cause strong overrejection of a correctly specified ACD model. The proposed tests have reasonable power against a variety of popular linear and nonlinear ACD alternatives.     

DYNAMIC STATE-SPACE MODELS

Abstract. In many cases, multiple time series can be viewed as realizations of the same underlying process and such data usually accumulate in time. The historic time-series data provide important information for our current prediction. In this paper, we extend the traditional state-space model to a general dynamic scheme, in which estimation and prediction across time series and within a time series are handled by a unified O ( N ) sequential procedure. Under this framework, the information from historic data serves as the prior for the current time series and the estimation and prediction of a time series can incorporate the information from other time series as well as its own history. The solution is to construct sequentially a new state-space model for the next time series conditional on the past time series. Because we achieve the general dynamic estimation and prediction through constructing new conditional state-space models, existing estimation procedures for state-space models can be adapted into this framework with minimal modifications. An application to infant growth curves is used as illustration.     

Periodic autoregressive model identification using genetic algorithms

Periodic autoregressive (PAR) models extend the classical autoregressive models by allowing the parameters to vary with seasons. Selecting PAR time‐series models can be computationally expensive, and the results are not always satisfactory. In this article, we propose a new automatic procedure to the model selection problem by using the genetic algorithm. The Bayesian information criterion is used as a tool to identify the order of the PAR model. The success of the proposed procedure is illustrated in a small simulation study, and an application with monthly data is presented.     

A family of multivariate non-gaussian time series models

In this article, we propose a class of multivariate non-Gaussian time series models which include dynamic versions of many well-known distributions and consider their Bayesian analysis. A key feature of our proposed model is its ability to account for correlations across time as well as across series (contemporary) via a common random environment. The proposed modeling approach yields analytically tractable dynamic marginal likelihoods, a property not typically found outside of linear Gaussian time series models. These dynamic marginal likelihoods can be tied back to known static multivariate distributions such as the Lomax, generalized Lomax, and the multivariate Burr distributions. The availability of the marginal likelihoods allows us to develop efficient estimation methods for various settings using Markov chain Monte Carlo as well as sequential Monte Carlo methods. Our approach can be considered to be a multivariate generalization of commonly used univariate non-Gaussian class of state space models. To illustrate our methodology, we use simulated data examples and a real application of multivariate time series for modeling the joint dynamics of stochastic volatility in financial indexes, the VIX and VXN.     

Improved prediction limits for a general class of Gaussian models

In this article we consider the problem of prediction for a general class of Gaussian models, which includes, among others, autoregressive moving average time‐series models, linear Gaussian state space models and Gaussian Markov random fields. Using an idea presented in Sjöstedt‐De Luna and Young (2003) , in the context of spatial statistics, we discuss a method for obtaining prediction limits for a future random variable of interest, taking into account the uncertainty introduced by estimating the unknown parameters. The proposed prediction limits can be viewed as a modification of the estimative prediction limit, with unconditional, and eventually conditional, coverage error of smaller asymptotic order. The modifying term has a quite simple form and it involves the bias and the mean square error of the plug‐in estimators for the conditional expectation and the conditional variance of the future observation. Applications of the results to Gaussian time‐series models are presented.