THE AUTOCORRELATION FUNCTION OF SEASONAL ARMA MODELS

Abstract. This note obtains the theoretical autocorrelation function of an ARMA model with multiplicative seasonality. It is shown that this function can be interpretated as the result of the interaction between the seasonal and regular autocorrelation patterns of the ARMA model. The use of this result makes easier the identification of the structure of the model, is helpful in choosing between a multiplicative or additive seasonal component and leads to a better understanding of the properties of the estimated autocorrelation function of scalar ARMA processes.     

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.     

Least absolute deviation estimation for general autoregressive moving average time‐series models

We study least absolute deviation (LAD) estimation for general autoregressive moving average time‐series models that may be noncausal, noninvertible or both. For ARMA models with Gaussian noise, causality and invertibility are assumed for the parameterization to be identifiable. The assumptions, however, are not required for models with non‐Gaussian noise, and hence are removed in our study. We derive a functional limit theorem for random processes based on an LAD objective function, and establish the consistency and asymptotic normality of the LAD estimator. The performance of the estimator is evaluated via simulation and compared with the asymptotic theory. Application to real data is also provided.     

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.     

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.     

RANDOM AGGREGATION OF UNIVARIATE AND MULTIVARIATE LINEAR PROCESSES

Abstract. The paper is devoted to random aggregation of multivariate autoregressive moving-average (ARMA) processes. We derive second-order characteristics of random aggregate models. We show that random aggregation preserves the ARMA structure. Moreover, we specify a functional relation between the initial model poles and aggregate ones. We then examine the case of univariate ARMA processes. Theorem 4 shows that, if the initial process is ARMA( p, q ), the random aggregate process is an ARMA( p*, q* ) model with p* at most equal to p ; * depends, among other things, on the sampling distribution L . This theorem generalizes the well-known results on the topic of time interval aggregation without overlapping.     

ASYMPTOTIC DISTRIBUTIONS OF LIKELIHOOD RATIOS FOR OVERPARAMETRIZED ARMA PROCESSES

Abstract. This paper is devoted to an extension of a classical problem of statistics to the asymptotic distribution of likelihood ratios. Two main types of likelihood ratios are considered for Gaussian ARMA processes. It is assumed in both cases that the asymptotic Fisher information matrix of estimation is singular in the higher order models. It is proved that the asymptotic distributions of the log likelihood ratios are invariant with respect to the parameters generating the process. A simulation shows that the sample distribution of the log likelihood ratio approaches the asymptotic one. Finally, the likelihood ratio test is proposed for model order reduction.     

High Moment Partial Sum Processes of Residuals in ARMA Models and their Applications

Abstract. In this article, we study high moment partial sum processes based on residuals of a stationary autoregressive moving average (ARMA) model with known or unknown mean parameter. We show that they can be approximated in probability by the analogous processes which are obtained from the i.i.d. errors of the ARMA model. However, if a unknown mean parameter is used, there will be an additional term that depends on model parameters and a mean estimator. When properly normalized, this additional term will vanish. Thus the processes converge weakly to the same Gaussian processes as if the residuals were i.i.d. Applications to change‐point problems and goodness‐of‐fit are considered, in particular, cumulative sum statistics for testing ARMA model structure changes and the Jarque–Bera omnibus statistic for testing normality of the unobservable error distribution of an ARMA model.     

SOME PROPERTIES OF CONDITIONAL QUASI-LIKELIHOOD FUNCTIONS FOR TIME SERIES MODEL FITTING

Abstract. We consider fitting a parametric model to a time series and obtain the maximum likelihood estimates of unknown parameters included in the model by regarding the time series as a Gaussian process satisfying the model. We evaluate the asymptotic value of the conditional quasi-likelihood function when the number of observations tends to infinity. We show what properties of the time series we can find by examining the behaviour of the conditional quasi-likelihood function, even when the time series does not necessarily satisfy the model and is not necessarily Gaussian.     

ON RESULTS OF PORAT CONCERNING ASYMPTOTIC EFFICIENCY OF SAMPLE COVARIANCES OF GAUSSIAN ARMA PROCESSES

Abstract. An alternative derivation is given of results first obtained by Porat (1987) concerning the asymptotic efficiencies of sample autocovariances of a stationary Gaussian ARMA process. This is based on an approximation to the likelihood of these autocovariances.     

ON THE ASYMPTOTIC DISTRIBUTION OF THE GENERALIZED PARTIAL AUTOCORRELATION FUNCTION IN AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. It has been conjectured and illustrated that the estimate of the generalized partial autocorrelation function (GPAC), which has been used for the identification of autoregressive moving-average (ARMA) models, has a thick-tailed asymptotic distribution. The purpose of this paper is to investigate the asymptotic behaviour of the GPAC in detail. It will be shown that the GPAC can be represented as a ratio of two functions, known as the θ function and the Λ function, each of which itself has a useful pattern for ARMA model identification. We shall show the consistencies of the extended Yule-Walker estimates of the three functions and present their asymptotic distributions.     

A NOTE ON THE EMBEDDING OF DISCRETE-TIME ARMA PROCESSES

Abstract. Let {X_n, n= 0, 1, 2,…} be a discrete-time ARMA(p, q) process with q < p whose autoregressive polynomial has r (not necessarily distinct) negative real roots. According to a recent result of He and Wang (On embedding a discrete-parameter ARMA model in a continuous-parameter ARMA model. J. Time Ser. Anal. 10 (1989), 315–23) there exists a continuous-time ARMA (p', q') process {Y(t), t≥0} with q' < p'=p+r such that {Y(n), n= 0, 1, 2,…} has the same autocorrelation function as {X_n}. In this paper we show that this result is false by considering the case when {X_n} is a discrete-time AR(2) process whose autoregressive polynomial has distinct complex conjugate roots. We identify the proper subset of such processes which are embeddable in a continuous-time ARMA(2, 1) process. We show that every discrete-time AR(2) process with distinct complex conjugate roots can be embedded in either a continuous-tie ARMA(2, 1) process or a continuous-time ARMA(4, 2) process, or in some cases both. We derive an expression for the spectral density of the process obtained by sampling a general continuous-time ARMA(p, q) process (with distinct autoregressive roots) at arbitrary equally spaced time points. The expression clearly shows that the sampled process is a discrete-time ARMA (p', q') process with q' < p.     

AN APPLICATION OF THE SCHUR-COHN ALGORITHM TO TIME SERIES ANALYSIS

Abstract. Standard least squares analysis of autoregressive moving-average (ARMA) processes with errors-in-variables entails the construction of a new set of parameters which are functions of the original ARMA parameters, and requires that derivatives of these new parameters of order three or less with respect to the ARMA parameters exist and be bounded. The boundedness of these derivatives in turn depends critically on the nonsingularity of a matrix B which is a function of the ARMA parameters via the new parameters in the model. A particular version of the classical Schur–Cohn algorithm enables us to establish this nonsingularity.     

RECURSIVE COMPUTATION OF THE PARAMETERS OF PERIODIC AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. An algorithm for recursive computation of the parameters of periodic autoregressive moving-average (ARMA) processes is given. It also provides recursions for stationary multivariate ARMA processes. A procedure for simultaneous estimation of the order and the parameters of a periodic ARMA process is outlined.     

RESULTS ON ESTIMATION AND TESTING FOR A UNIT ROOT IN THE NONSTATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODEL

Abstract. We review the limiting distribution theory for Gaussian estimation of the univariate autoregressive moving-average (ARMA) model in the presence of a unit root in the autoregressive (AR) operator, and present the asymptotic distribution of the associated likelihood ratio (LR) test statistic for testing for a unit root in the ARMA model. The finite sample properties of the LR statistic as well as other unit root test procedures for the ARMA model are examined through a limited simulation study. We conclude that, for practical empirical work that relies on standard computations, the LR test procedure generally performs better than other standard procedures in the presence of a substantial moving-average component in the ARMA model.     

ON THE ASYMPTOTIC EFFICIENCY OF ESTIMATORS OF THE PARAMETERS OF AN ARMA PROCESS

Abstract. In this paper we derive a lower bound on the asymptotic covariance matrix of an estimator of the parameters of an autoregressive moving average (ARMA) process when the innovations are not necessarily Gaussian.     

An algorithm for computing the asymptotic fisher information matrix for seasonal SISO models

Abstract.  The paper presents an algorithm for computing the asymptotic Fisher information matrix of a possibly seasonal single-input single-output (SISO) time-series model. That matrix is a block matrix whose elements are basically integrals of rational functions over the oriented unit circle. The procedure makes use of the autocovariance or the cross-covariance function of two autoregressive processes based on the same noise. The algorithm also works when the input variable is omitted, the case of a seasonal ARMA model.     

ON EMBEDDING A DISCRETE-PARAMETER ARMA MODEL IN A CONTINUOUS-PARAMETER ARMA MODEL

Abstract. It is shown that a real-valued discrete-parameter Gaussian ARMA ( p. q ) model with q < p can be embedded in a real-valued continuous-parameter Gaussian ARMA( p', q' ) model with q' < p' . The problem of embedding a real-valued discrete-parameter Gaussian AR( p ) into a real-valued continuous-parameter Gaussian AR( p ) is also discussed.     

Portmanteau tests for ARMA models with infinite variance

Abstract. Autoregressive and moving‐average (ARMA) models with stable Paretian errors are some of the most studied models for time series with infinite variance. Estimation methods for these models have been studied by many researchers but the problem of diagnostic checking of fitted models has not been addressed. In this article, we develop portmanteau tests for checking the randomness of a time series with infinite variance and for ARMA diagnostic checking when the innovations have infinite variance. It is assumed that least squares or an asymptotically equivalent estimation method, such as Gaussian maximum likelihood, is used. It is also assumed that the distribution of the innovations is identically and independently distributed (i.i.d.) stable Paretian. It is seen via simulation that the proposed portmanteau tests do not converge well to the corresponding limiting distributions for practical series length so a Monte Carlo test is suggested. Simulation experiments show that the proposed Monte Carlo test procedure works effectively. Two illustrative applications to actual data are provided to demonstrate that an incorrect conclusion may result if the usual portmanteau test based on the finite variance assumption is used.     

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.