Nonlinear ARMA models with functional MA coefficients

Abstract. In the present article, we propose and study a new class of nonlinear autoregressive moving‐average (ARMA) models, in which each moving‐average (MA) coefficient is enlarged to an arbitrary univariate function. We first provide a sufficient condition for the existence of the stationary solution and further discuss the moment structure. We investigate the estimation method to the proposed models. The global estimates of parameters and local linear estimates of functional coefficients are obtained by using a back‐fitting algorithm. For testing whether the functional coefficients are some specified parametric forms, a bootstrap test approach is provided. The proposed models are illustrated by both simulated and real data examples.     

Some Notes on Mutual Information Between Past and Future

Abstract. We present some new results on the mutual information between past and future for Gaussian stationary sequences. We provide several formulae to calculate this quantity. As a by‐product, we establish the so‐called reflectrum identity that links partial autocorrelation coefficients and cepstrum coefficients. So as to obtain these results, we provide an account of several regularity conditions for Gaussian stationary processes in terms of properties of the associated Toeplitz and Hankel operators. We discuss conditions under which the mutual information is finite. These results lead us to an interesting perspective towards the definition of long‐memory processes. Our result implies that zeros on the unit circle can cause mutual information to be infinite. Examples include fractional autoregressive integrated moving average (ARIMA) models. In addition, we consider a finite sample from a Gaussian stationary sequence. In the expansion of the determinant of its covariance matrix, the Toeplitz matrix, the first and second term are, entropy and mutual information respectively. A form of approximation to the likelihood using entropy and mutual information is presented.     

A Class of Antipersistent Processes

Abstract. We introduce a class of stationary processes characterized by the behaviour of their infinite moving average parameters. We establish the asymptotic behaviour of the covariance function and the behaviour around zero of the spectral density of these processes, showing their antipersistent character. Then, we discuss the existence of an infinite autoregressive representation for this family of processes, and we present some consequences for fractional autoregressive moving average models.     

Selection of weak VARMA models by modified Akaike's information criteria

This article considers the problem of order selection of the vector autoregressive moving‐average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. These models are called weak VARMA by opposition to the standard VARMA models, also called strong VARMA models, in which the error terms are supposed to be i.i.d. We relax the standard independence assumption to extend the range of application of the VARMA models, allowing us to treat linear representations of general nonlinear processes. We propose a modified version of the Akaike information criterion for identifying the orders of weak VARMA models.     

Order Patterns in Time Series

Abstract. Recent use of order patterns in time‐series analysis shows the need for a corresponding theory. We determine probabilities of order patterns in Gaussian and autoregressive moving‐average (ARMA) processes. Two order functions are introduced which characterize a time series in a way similar to autocorrelation. For stationary ergodic processes, all finite‐dimensional distributions are obtained from the one‐dimensional distribution plus the order structure of a typical time series.     

A Note on Non‐Negative Arma Processes

Abstract. Recently, there has been much research on developing models suitable for analysing the volatility of a discrete‐time process. Since the volatility process, like many others, is necessarily non‐negative, there is a need to construct models for stationary processes which are non‐negative with probability one. Such models can be obtained by driving autoregressive moving average (ARMA) processes with non‐negative kernel by non‐negative white noise. This raises the problem of finding simple conditions under which an ARMA process with given coefficients has a non‐negative kernel. In this article, we derive a necessary and sufficient condition. This condition is in terms of the generating function of the ARMA kernel which has a simple form. Moreover, we derive some readily verifiable necessary and sufficient conditions for some ARMA processes to be non‐negative almost surely.     

Asymptotic Properties of Weighted Least Squares Estimation in Weak PARMA Models

The aim of this work is to investigate the asymptotic properties of weighted least squares (WLS) estimation for causal and invertible periodic autoregressive moving average (PARMA) models with uncorrelated but dependent errors. Under mild assumptions, it is shown that the WLS estimators of PARMA models are strongly consistent and asymptotically normal. It extends Thm 3.1 of Basawa and Lund (2001) on least squares estimation of PARMA models with independent errors. It is seen that the asymptotic covariance matrix of the WLS estimators obtained under dependent errors is generally different from that obtained with independent errors. The impact can be dramatic on the standard inference methods based on independent errors when the latter are dependent. Examples and simulation results illustrate the practical relevance of our findings. An application to financial data is also presented.     

A NOTE ON ASYMPTOTIC INFERENCE IN AUTOREGRESSIVE MODELS WITH ROOTS ON THE UNIT CIRCLE

Abstract. An estimation and inference procedure is proposed for parameters of the p th order autoregressive model with roots both on the unit circle and outside the unit circle. The procedure is motivated by the fact that the parameter estimates of the nonstationary part of the model have higher order consistency properties than the parameter estimates of the stationary part. The procedure allows the use of the known asymptotic distributional results of purely nonstationary models and purely stationary models. Only ordinary least squares routines are needed.     

Relative entropy and spectral constraints: some invariance properties of the ARMA class

Abstract. We determine the form of spectral densities of multidimensional scalar processes which minimize a relative entropy under a finite number of general moment‐type constraints. The obtained theoretical results are applied to spectral densities of weakly stationary processes under covariances, inverse covariances and cepstral or impulse response constraints. Invariance properties of the class of autoregressive moving‐average (ARMA) processes are shown to hold under the relative entropy minimization principle for many choices of entropy.     

First-Order Autoregressive Processes with Heterogeneous Persistence

Abstract. We propose a semi-nonparametric method of identification and estimation for Gaussian autoregressive processes with stochastic autoregressive coefficients. The autoregressive coefficient is considered as a latent process with either a moving average or regime switching representation. We develop a consistent estimator of the distribution of the autoregressive coefficient based on nonlinear canonical decomposition of the observed process. The approach is illustrated by simulations.     

A Stationary Spatio-Temporal GARCH Model

We introduce a lagged nearest-neighbour, stationary spatio-temporal generalized autoregressive conditional heteroskedasticity (GARCH) model on an infinite spatial grid that opens for GARCH innovations in a space-time ARMA model. This is illustrated by a real data application to a classical dataset of sea surface temperature anomalies in the Pacific Ocean. The model and its translation invariant neighbourhood system are wrapped around a torus forming a model with finite spatial domain, which we call circular spatio-temporal GARCH. Such a model could be seen as an approximation of the infinite one and simulation experiments show that the circular estimator with a straightforward bias correction performs well on such non-circular data. Since the spatial boundaries are tied together, the well-known boundary issue in spatial statistical modelling is effectively avoided. We derive stationarity conditions for these circular processes and study the spatio-temporal correlation structure through an ARMA representation. We also show that the matrices defined by a vectorized version of the model are block circulants. The maximum quasi-likelihood estimator is presented and we prove its strong consistency and asymptotic normality by generalizing results from univariate GARCH theory.     

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.     

Mean shift testing in correlated data

Several tests for detecting mean shifts at an unknown time in stationary time series have been proposed, including cumulative sum (CUSUM), Gaussian likelihood ratio (LR), maximum of F(F_max) and extreme value statistics. This article reviews these tests, connects them with theoretical results, and compares their finite sample performance via simulation. We propose an adjusted CUSUM statistic which is closely related to the LR test and which links all tests. We find that tests based on CUSUMing estimated one‐step‐ahead prediction residuals from a fitted autoregressive moving average perform well in general and that the LR and F_max tests (which induce substantial computational complexities) offer only a slight increase in power over the adjusted CUSUM test. We also conclude that CUSUM procedures work slightly better when the changepoint time is located near the centre of the data, but the adjusted CUSUM methods are preferable when the changepoint lies closer to the beginning or end of the data record. Finally, an application is presented to demonstrate the importance of the choice of method.     

Subset ARMA Model Identification Using Genetic Algorithms

Subset models are often useful in the analysis of stationary time series. Although subset autoregressive models have received a lot of attention, the same attention has not been given to subset autoregressive moving-average (ARMA) models, as their identification can be computationally cumbersome. In this paper we propose to overcome this disadvantage by employing a genetic algorithm. After encoding each ARMA model as a binary string, the iterative algorithm attempts to mimic the natural evolution of the population of such strings by allowing strings to reproduce, creating new models that compete for survival in the next population. The success of the proposed procedure is illustrated by showing its efficiency in identifying the true model for simulated data. An application to real data is also considered.     

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.     

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.     

Time Deformation,Continuous Euler Processes and Forecasting

Abstract. A continuous Euler model has time‐varying coefficients. Through a logarithmic time transformation, a continuous Euler model can be transformed to a continuous autoregressive (AR) model. By using the continuous Kalman filtering through the Laplace method, this article explores the data application of a continuous Euler process. This time deformation of an Euler process deforms specific time‐variant (non‐stationary) behaviour to time‐invariant (stationary) data on the deformed time scale. With these time‐invariant data on the transformed time scale, one may use traditional tools to conduct parameter estimation and forecasts. The obtained results can then be transformed back to the original time scale. Simulated data and actual data such as bat echolocation and the US residential investment growth are used to demonstrate the usefulness of time deformation in forecasting. The results indicate that fitting a traditional autoregressive moving‐average (ARMA) model on an Euler data set without imposing time transformation leads to forecasts that are out of phase while the forecasts of an Euler model stay mostly in phase.     

Boundary Limit Theory for Functional Local to Unity Regression

This article studies functional local unit root models (FLURs) in which the autoregressive coefficient may vary with time in the vicinity of unity. We extend conventional local to unity (LUR) models by allowing the localizing coefficient to be a function which characterizes departures from unity that may occur within the sample in both stationary and explosive directions. Such models enhance the flexibility of the LUR framework by including break point, trending, and multidirectional departures from unit autoregressive coefficients. We study the behavior of this model as the localizing function diverges, thereby determining the impact on the time series and on inference from the time series as the limits of the domain of definition of the autoregressive coefficient are approached. This boundary limit theory enables us to characterize the asymptotic form of power functions for associated unit root tests against functional alternatives. Both sequential and simultaneous limits (as the sample size and localizing coefficient diverge) are developed. We find that asymptotics for the process, the autoregressive estimate, and its t‐statistic have boundary limit behavior that differs from standard limit theory in both explosive and stationary cases. Some novel features of the boundary limit theory are the presence of a segmented limit process for the time series in the stationary direction and a degenerate process in the explosive direction. These features have material implications for autoregressive estimation and inference which are examined in the article.     

Exact Maximum Likelihood Estimation of an ARMA(1, 1) Model with Incomplete Data

For a first-order autoregressive and first-order moving average model with nonconsecutively observed or missing data, the closed form of the exact likelihood function is obtained, and the exact maximum likelihood estimation of parameters is derived in the stationary case.