TIME-REVERSIBILITY, IDENTIFIABILITY AND INDEPENDENCE OF INNOVATIONS FOR STATIONARY TIME SERIES

Abstract. Weiss ( J. Appl. Prob. 12 (1975) 831–36) has shown that for causal autoregressive moving-average (ARMA) models with independent and identically distributed (i.i.d.) noise, time-reversibility is essentially unique to Gaussian processes. This result extends to quite general linear processes and the extension can be used to deduce that a non-Gaussian fractionally integrated ARMA process has at most one representation as a moving average of i.i.d. random variables with finite variance. In the proof of this uniqueness result, we use a time-reversibility argument to show that the innovations sequence (one-step prediction residuals) of an ARMA process driven by i.i.d. non-Gaussian noise is typically not independent, a result of interest in deconvolution problems. Further, we consider the case of an ARMA process to which independent noise is added. Using a time-reversibility argument we show that the innovations of the ARMA process with added independent noise are independent if and only if both the driving noise of the process and the added noise are Gaussian.     

Residual Empirical Processes and Weighted Sums for Time‐Varying Processes with Applications to Testing for Homoscedasticity

In the context of heteroscedastic time‐varying autoregressive (AR)‐process we study the estimation of the error/innovation distributions. Our study reveals that the non‐parametric estimation of the AR parameter functions has a negligible asymptotic effect on the estimation of the empirical distribution of the residuals even though the AR parameter functions are estimated non‐parametrically. The derivation of these results involves the study of both function‐indexed sequential residual empirical processes and weighted sum processes. Exponential inequalities and weak convergence results are derived. As an application of our results we discuss testing for the constancy of the variance function, which in special cases corresponds to testing for stationarity.     

BOOTSTRAPPING STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. In this paper we develop an asymptotic theory for application of the bootstrap to stationary stochastic processes of autoregressive moving-average (ARMA) type, with known order ( p, q ). We give a proof of the asymptotic validity of the bootstrap proposal applied to M estimators for the unknown parameter vector of the process. For this purpose we derive an asymptotic expansion for M estimators in ARMA models and construct an estimate for the unknown distribution function of the residuals which in principle are not observable. A small simulation study is also included.     

Change point detection in copula ARMA–GARCH Models

In this article, we consider the problem of testing for a copula parameter change based on the cusum test. We first handle this issue in i.i.d. samples and extend it to semiparametric copula ARMA‐GARCH models. We construct the cusum test based on pseudo maximum likelihood estimation of the copula parameter and derive its limiting null distribution. Simulation results are reported for illustration.     

Mixed Portmanteau Tests for Time‐Series Models

Abstract. This paper obtains the joint limiting distribution of residuals and squared residuals of a general time‐series model. Based on this, we propose a mixed portmanteau statistic for testing the adequacy of fitted time‐series models. In some cases, it is shown that this statistic can be simply approximated by the sum of well‐known portmanteau statistics. The finite‐sample performance of the new test is compared with those of well‐known tests through simulations.     

A UNIFIED APPROACH TO THE STUDY OF SUMS, PRODUCTS, TIME-AGGREGATION AND OTHER FUNCTIONS OF ARMA PROCESSES

Abstract. Conditions under which sums, products and time-aggregation of ARMA processes follow ARMA models are derived from a single theorem. This characterizes these processes in terms of difference equations satisfied by their autocovariance function. From this we obtain necessary and sufficient conditions for a function of a Gaussian ARMA process and the product of two possibly dependent Gaussian ARMA processes to be ARMA. We show that the sum and product of two ARMA processes related by a Box and Jenkins transfer function model belong to the ARMA family.     

PARAMETER ESTIMATION FOR PERIODIC ARMA MODELS

Abstract. In time series analysis of data sequences, the estimation of the parameters of an identified autoregressive moving-average (ARMA) model is a well-known and straightforward exercise. However, if the parameters of the model are periodic (i.e. a periodic ARMA (PARMA) model) then the estimation process becomes more difficult. This paper describes an on-line parameter estimation technique, based on methods from automatic control, which is demonstrated to provide consistent estimates of PARMA model parameters.     

A GOODNESS-OF-FIT TEST FOR AUTOREGRESSIVE MOVING-AVERAGE MODELS BASED ON THE STANDARDIZED SAMPLE SPECTRAL DISTRIBUTION OF THE RESIDUALS

Abstract. A stochastic process derived from the standardized sample spectral density of the residuals of a causal and invertible ARMA( p, q ) model is introduced to construct a goodness-of-fit procedure. The test statistics considered have a proper limiting distribution which is free of unknown parameters and which, unlike some well-known goodness-of-fit statistics based on the residuals, does not depend on the sample size.     

AUTOREGRESSIVE MOVING AVERAGE PROCESSES WITH NON-NORMAL RESIDUALS

Abstract. ARMA processes with non-normal residuals have applications in surface metrology and have recently been shown by Nelson and Granger (1979) to occur in modelling economic time series. In this paper we obtain the theoretical relationship between the skewness and kurtosis of an ARMA process and the corresponding parameters of its generating noise series and consider some of the implications of these results. Simulation methods for any ARMA process with given skewness and kurtosis, using Johnson transformations are briefly discussed.     

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.     

Testing for parameter stability in nonlinear autoregressive models

In this article we develop testing procedures for the detection of structural changes in nonlinear autoregressive processes. For the detection procedure, we model the regression function by a single layer feedforward neural network. We show that CUSUM‐type tests based on cumulative sums of estimated residuals, that have been intensively studied for linear regression, can be extended to this case. The limit distribution under the null hypothesis is obtained, which is needed to construct asymptotic tests. For a large class of alternatives, it is shown that the tests have asymptotic power one. In this case, we obtain a consistent change‐point estimator which is related to the test statistics. Power and size are further investigated in a small simulation study with a particular emphasis on situations where the model is misspecified, i.e. the data is not generated by a neural network but some other regression function. As illustration, an application on the Nile data set as well as S&P log‐returns is given.     

PERIODIC CORRELATION IN STRATOSPHERIC OZONE DATA

Abstract. A 50-year time series of monthly stratospheric ozone readings from Arosa, Switzerland, is analyzed. The time series exhibits the properties of a periodically correlated (PC) random sequence with annual periodicities. Spectral properties of PC random sequences are reviewed and a test to detect periodic correlation is presented. An autoregressive moving-average (ARMA) model with periodically varying coefficients (PARMA) is fitted to the data in two stages. First, a periodic autoregressive model is fitted to the data. This fit yields residuals that are stationary but non-white. Next, a stationary ARMA model is fitted to the residuals and the two models are combined to produce a larger model for the data. The combined model is shown to be a PARMA model and yields residuals that have the correlation properties of white noise.     

An Extended Portmanteau Test for VARMA Models With Mixing Nonlinear Constraints

Abstract. The portmanteau test is a widely used diagnostic tool for univariate and multivariate time‐series models. Its asymptotic distribution is known for the unconstrained vector autoregressive moving‐average (VARMA) case and for VAR models with constraints on the autoregressive coefficients. In this article, we give conditions under which the test can be applied to constrained VARMA models. Unfortunately, it cannot generally be applied to models with constraints that simultaneously affect the ARMA polynomial coefficients and the covariance matrix of the innovations (mixing constraints). This happens in latent‐variable models such as dynamic factor models (DFM). In addition, when there are constraints on the covariance matrix it seems convenient to check the goodness of fit using the zero‐lag residual covariances. We propose an extended portmanteau test that not only checks the autocorrelations of the residuals but also whether their covariance matrix is consistent with the constraints. We prove that the statistic is asymptotically distributed as a chi‐square for ARMA models under the assumption that the innovations have Gaussian‐like fourth‐order moments. We also show that the test is appropriate for the DFM, Peña–Box model and factor‐structural vector autoregression (FSVAR).     

ESTIMATION OF MULTIVARIATE TIME SERIES

Abstract. The algorithm proposed here is a multivariate generalization of a procedure discussed by Pearlman (1980) for calculating the exact likelihood of a univariate ARMA model. Ansley and Kohn (1983) have shown how the Kalman filter can be used to calculate the exact likelihood function when not all the observations are known. In Shea (1983) it is shown that this algorithm is much quicker than that of Ansley and Kohn (1983) for all ARMA models except an ARMA (2, 1) and a couple of low-order AR processes and therefore when we have no missing observations this algorithm should be used instead. The Fortran subroutine G13DCF in the NAG (1987) Library fits a vector ARMA model using an adaptation of this algorithm. Experience in the use of this routine suggests that having reasonably good initial estimates of the ARMA parameter matrices, and in particular the residual error covariance matrix, can not only substantially reduce the computing time but more important improve the convergence properties of the minimization procedure. We therefore propose a method of calculating initial estimates of the ARMA parameters which involves using a generalization of the concept of inverse cross covariances from the univariate to the multivariate case. Finally theory is put into practice with the fitting of a bivariate model to a couple of real-life time series.     

PARAMETER IDENTIFICATION IN ARMA PROCESSES IN THE PRESENCE OF REGULAR BUT INCOMPLETE SAMPLING

Abstract. We discuss the parameter identification of multivariate AR (1) models and of univariate ARMA (2,1) and AR (2) models if the variables in the model are observed every m th period where m is some integer greater than unity. The results indicate that the models will often not be globally identified even if they are locally identified and that the likelihood function can have a large number of local maxima.     

ON THE UNBIASEDNESS PROPERTY OF AIC FOR EXACT OR APPROXIMATING LINEAR STOCHASTIC TIME SERIES MODELS

Abstract. A rigorous analysis is given of the asymptotic bias of the log maximum likelihood as an estimate of the expected log likelihood of the maximum likelihood model, when a linear model, such as an invertible, gaussian ARMA ( p, q ) model, with or without parameter constraints, is fit to stationary, possibly non-gaussian observations. It is assumed that these data arise from a model whose spectral density function either (i) coincides with that of a member of the class of models being fit, or, that failing, (ii) can be well-approximated by invertible ARMA ( p, q ) model spectral density functions in the class, whose ARMA coefficients are parameterized separately from the innovations variance. Our analysis shows that, for the purpose of comparing maximum likelihood models from different model classes, Akaike's AIC is asymptotically unbiased, in case (i), under gaussian or separate parametrization assumptions, but is not necessarily unbiased otherwise. In case (ii), its asymptotic bias is shown to be of the order of a number less than unity raised to the power max { p, q } and so is negligible if max { p, q } is not too small. These results extend and complete the somewhat heuristic analysis given by Ogata (1980) for exact or approximating autoregressive models.     

Change-Point Detection in Binomial Thinning Processes,with Applications in Epidemiology

Abstract

Most of the classical change-point detection schemes are designed for the sequences of independent and identically distributed (i.i.d.) random variables. In this article, motivated by the outbreak of 2009 H1N1 pandemic influenza, we develop change-point detection procedures for the susceptible–infected–recovered (SIR) epidemic model, where a change-point in the infection rate parameter signifies either the beginning or the end of an epidemic trend.

The considered model falls into a general class of binomial thinning processes, which is a Markov chain. The cumulative sum (CUSUM) change-point detection procedure is developed for this class, and its performance is evaluated. Apparently, the CUSUM stopping rule is no longer optimal for this non-i.i.d. case. It can be improved by introducing a non constant adaptive threshold. The resulting modified scheme attains a shorter mean delay and at the same time a longer expected time of a false alarm, given that a false alarm eventually occurs.

Proposed detection procedures are applied to the 2001–2012 influenza data published by the Centers for Disease Control and Prevention.     

Parametric and Semi‐Parametric Efficient Tests for Parameter Instability

This article examines asymptotically point optimal tests for parameter instability in realistic circumstances when little information about the unstable parameter process and error distribution is available. We first show that, under a correctly specified error distribution, if the unstable parameter processes converge weakly to a Wiener process, then any asymptotic optimal tests for structural breaks and time‐varying parameters are asymptotically equivalent. Our finding is then extended to a semi‐parametric set‐up in which the error distribution is treated as an unknown infinite‐dimensional nuisance parameter. We find that semi‐parametric tests can be adaptive without further restrictive conditions on the error distribution.     

ON THE INVERTIBILITY OF MULTIVARIATE LINEAR PROCESSES

Abstract. It is shown that a multivariate linear stationary process whose coefficients are absolutely summable is invertible if and only if its spectral density is regular everywhere. This general characterization of invertibility is applied later to the case of a linear process having an autoregressive moving-average (ARMA) representation. Under the usual assumptions, it is deduced that a process Y described by an ARMA(φ, TH) model is invertible if and only if the polynomial detTH( z ) has no roots on the unit circle. Given an invertible process Y which has an ARMA representation, it is finally shown that the process YT , where YT , =ε _i =0^l S _i Y _t-i , is invertible if and only if the matrix S ( z ) =ε _{i =0}^l S _i z ⁱ is of full rank for all z of modulus 1. It follows, in particular, that any subprocess of an invertible ARMA process is also invertible.     

ASYMPTOTIC INFERENCE FOR NON-INVERTIBLE MOVING-AVERAGE TIME SERIES

Abstract. This paper is concerned with statistical inference of nonstationary and non-invertible autoregressive moving-average (ARMA) processes. It makes use of the fact that a derived process of an ARMA( p, q ) model follows an AR( q ) model with an autoregressive (AR) operator equivalent to the moving-average (MA) part of the original ARMA model. Asymptotic distributions of least squares estimates of MA parameters based on a constructed derived process are obtained as corresponding analogs of a nonstationary AR process. Extensions to the nearly non-invertible models are considered and the limiting distributions are obtained as functionals of stochastic integrals of Brownian motions and Ornstein-Uhlenbeck processes. For application, a two-stage procedure is proposed for testing unit roots in the MA polynomial. Examples are given to illustrate the application.