SOME DOUBLY STOCHASTIC TIME SERIES MODELS

Abstract. We consider time series models obtained by replacing the parameters of autoregressive models by stochastic processes. Special attention is given to the problem of finding conditions for stationarity and to the problem of forecasting. For the first problem we are only able to obtain solutions in special cases, and the emphasis is on techniques rather than obtaining the most general results in each case. For the second problem more complete results are obtained by exploiting similarities with discrete time (nonlinear) filtering theory. The methods introduced are illustrated on two standard examples, one of state space type and one where the parameter process is a Markov chain.     

RANDOM COEFFICIENT AUTOREGRESSIVE PROCESSES:A MARKOV CHAIN ANALYSIS OF STATIONARITY AND FINITENESS OF MOMENTS

Abstract. Simple yet practically efficient conditions for the ergodicity of a Markov chain on a general state space have recently been developed. We illustrate their application to non-linear time series models and, in particular, to random coefficient autoregressive models.
As well as ensuring the existence of a unique stationary distribution, geometric rates of convergence to stationarity are ensured. Moreover, sufficient conditions for the existence and convergence of moments can be determined by a closely related method. The latter conditions, in particular, are new.     

REGRESSION MODELS FOR NON-STATIONARY CATEGORICAL TIME SERIES

Abstract. Categorical time series often exhibit non-stationary behaviour, due to the influence of exogenous variables. A parsimonious and flexible class of models is proposed for the statistical analysis of such data. These models are extensions of regression models for stochastically independent observations. Statistical inference can be based on asymptotic properties of the maximum likelihood estimator and of test statistics for linear hypotheses. Weak conditions assuring these properties are stated. Some tests which are of special interest in the time series situation are treated in more detail, for example tests of stationarity or independence of parallel time series.     

First-order rounded integer-valued autoregressive (RINAR(1)) process

Abstract.  We introduce a new class of autoregressive models for integer-valued time series using the rounding operator. Compared with classical INAR models based on the thinning operator, the new models have several advantages: simple innovation structure, autoregressive coefficients with arbitrary signs, possible negative values for time series and possible negative values for the autocorrelation function. Focused on the first-order RINAR(1) model, we give conditions for its ergodicity and stationarity. For parameter estimation, a least squares estimator is introduced and we prove its consistency under suitable identifiability condition. Simulation experiments as well as analysis of real data sets are carried out to attest the model performance.     

On stationarity and ergodicity of the bilinear model with applications to GARCH models

We establish sufficient conditions for the bilinear time‐series model to be strictly stationary and ergodic in terms of its associated Lyapunov exponent. In two special cases, we verify that the conditions are also necessary. We then use these results to give necessary and sufficient conditions for stationarity of specific generalized autoregressive conditionally heteroskedastic (GARCH) models which can be written as a bilinear model, including linear GARCH, Power GARCH, EGARCH among others. These results generalize the ones found in the studies of, among others, Bougerol and Picard [Journal of Econometrics 52 (1992) 115], Duan [Journal of Econometrics 79 (1997) 97] and Nelson [Econometric Theory 6 (1990) 318]. In many cases, the conditions are weaker than the ones found elsewhere in the literature.     

Combining Cumulative Sum Change‐Point Detection Tests for Assessing the Stationarity of Univariate Time Series

We derive tests of stationarity for univariate time series by combining change‐point tests sensitive to changes in the contemporary distribution with tests sensitive to changes in the serial dependence. The proposed approach relies on a general procedure for combining dependent tests based on resampling. After proving the asymptotic validity of the combining procedure under the conjunction of null hypotheses and investigating its consistency, we study rank‐based tests of stationarity by combining cumulative sum change‐point tests based on the contemporary empirical distribution function and on the empirical autocopula at a given lag. Extensions based on tests solely focusing on second‐order characteristics are proposed next. The finite‐sample behaviors of all the derived statistical procedures for assessing stationarity are investigated in large‐scale Monte Carlo experiments, and illustrations on two real datasets are provided. Extensions to multi‐variate time series are briefly discussed as well.     

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.     

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.     

A negative binomial integer‐valued GARCH model

This article discusses the modelling of integer‐valued time series with overdispersion and potential extreme observations. For the problem, a negative binomial INGARCH model, a generalization of the Poisson INGARCH model, is proposed and stationarity conditions are given as well as the autocorrelation function. For estimation, we present three approaches with the focus on the maximum likelihood approach. Some results from numerical studies are presented and indicate that the proposed methodology performs better than the Poisson and double Poisson model‐based methods.     

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     

Continuous manufacturing: Is the process mean stationary?

The statistical framework to systematically detect mean stationarity in the context of continuous manufacturing is described in this article. The methods presented in this article use econometric and financial time‐series analysis concepts in the form of unit‐root and stationarity hypothesis tests. The tests under discussion are the augmented Dickey‐Fuller, Philips‐Perron, Leybourne‐McCabe, and Kwiatkowski‐Phillips‐Schmidt‐Shin. These hypothesis tests are evaluated on data generated by a focused‐beam reflectance measurement sensor implemented on‐line in a continuous plug‐flow crystallizer. This contribution has shown that the hypothesis tests can be used to detect steady‐state conditions on‐line in a plug‐flow crystallizer. Furthermore, this econometric framework can be used as a mean stationarity “certificate” of collected samples to document that the process was mean stationary during the sampling. The statistical framework described in this article can be applied to any continuously operated unit operation or sensor measurement. © 2018 American Institute of Chemical Engineers AIChE J, 64: 2426–2437, 2018     

Time Dependence and Moments of a Family of Time-Varying Parameter Garch in Mean Models

Abstract.  In this paper we consider the time series dependence, stationarity, and higher moments issues of a family of first-order conditionally heteroskedastic in mean models with a possibly time-varying mean parameter. The interest in these models lies in the fact that economic theory and physics often require the connection between the first and second conditional moments of time series. Our results reveal important properties of these models, which are consistent with stylized facts in financial and turbulence data sets. They can also be employed for model identification, estimation, and testing.     

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.     

A Nonparametric Model for Stationary Time Series

Stationary processes are a natural choice as statistical models for time series data, owing to their good estimating properties. In practice, however, alternative models are often proposed that sacrifice stationarity in favour of the greater modelling flexibility required by many real‐life applications. We present a family of time‐homogeneous Markov processes with nonparametric stationary densities, which retain the desirable statistical properties for inference, while achieving substantial modelling flexibility, matching those achievable with certain non‐stationary models. A latent extension of the model enables exact inference through a trans‐dimensional Markov chain Monte Carlo method. Numerical illustrations are presented.     

On a Mixture GARCH Time‐Series Model

Abstract. Recently, there has been a lot of interest in modelling real data with a heavy‐tailed distribution. A popular candidate is the so‐called generalized autoregressive conditional heteroscedastic (GARCH) model. Unfortunately, the tails of GARCH models are not thick enough in some applications. In this paper, we propose a mixture generalized autoregressive conditional heteroscedastic (MGARCH) model. The stationarity conditions and the tail behaviour of the MGARCH model are studied. It is shown that MGARCH models have tails thicker than those of the associated GARCH models. Therefore, the MGARCH models are more capable of capturing the heavy‐tailed features in real data. Some real examples illustrate the results.     

Integrating process and controller models for the design of self-optimizing control

The steady-state behavior of an existing plant depends on the independent input variables, process equipment and process controllers. This paper presents a method for formulating models that represent the effects of controllers when they are included within a steady-state process flowsheet. The method replaces the controller equations with the equivalent stationarity conditions representing the relationship between the controlled variables and the implemented manipulated variables at steady state. The method is demonstrated for the centralized multivariable Dynamic Matrix Control algorithm applied to two processes, binary distillation and gasoline blending. The integrated process and control system simulation is used to design controllers that improve the profitability of processes without extensive real-time calculations; this is sometimes termed self-optimizing control. For both processes, controllers were designed that yielded higher profit than standard control methods and that approached the highest possible profit achieved by frequent real-time optimization.     

SPECTRAL RADIUS,KRONECKER PRODUCTS AND STATIONARITY

Abstract. We provide a stochastic proof of the inequality ρ(A?A+B?B) ≥ρ(A?A), where ρ(M) denotes the spectral radius of any square matrix M, i.e. max{|eigenvalues| of M}, and M?N denotes the Kronecker product of any two matrices M and N. The inequality is then used to show that stationarity of the bilinear model will imply stationarity of the linear part, i.e. the linear ARMA model for r= 1 and q= 1. Furthermore, it is shown that stationarity of the subdiagonal model, i.e. the bilinear model with b_ij=0 for i< j, again implies stationarity of its linear part, provided that the stationarity condition given by Bhaskara Rao and his colleagues is met. Interestingly, the conclusion that stationarity of the subdiagonal models, implies that the linear component models cannot be extended to the general non-subdiagonal bilinear models. The last observation is demonstrated via a simple example with p=m= 1, r= 0 and q= 2.     

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 SIMPLE CONDITION FOR THE EXISTENCE OF SOME STATIONARY BILINEAR TIME SERIES

Abstract. A sufficient condition is derived for the existence of a strictly stationary solution of some bilinear time series which may have infinite variance innovations. This condition is equivalent to the condition that a polynomial of degree r has no zeros within the unit circle. In the special case when the innovations have finite variance, the computational effort involved in checking this condition is significantly reduced compared with checking the stationarity conditions given by Bhaskara Rao et al. and Liu and Brockwell which requires a knowledge of the maximum eigenvalue in the absolute value of an r ² x r ² matrix.