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.     

ON THE LAGRANGE MULTIPLIER TEST FOR AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. This paper examines the score or Lagrange multiplier statistic for testing the adequacy of a fitted autoregressive moving-average model and gives a simple closed-form expression for this test statistic. Some singularities arising as the order of the alternative model is increased are examined.     

A GENERALIZED LEAST-SQUARES APPROACH FOR ESTIMATION OF AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. In this paper we present a generalized least-squares approach for estimating autoregressive moving-average (ARMA) models. Simulation results based on different model structures with varying numbers of observations are used to contrast the performance of our procedure with that of maximum likelihood estimates. Existing software packages can be utilized to derive these estimates.     

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.     

ESTIMATION OF AUTOREGRESSIVE MOVING-AVERAGE ORDER GIVEN AN INFINITE NUMBER OF MODELS AND APPROXIMATION OF SPECTRAL DENSITIES

Abstract. A modification of the minimum Akaike information criterion (AIC) procedure (and of related procedures like the Bayesian information criterion (BIC)) for order estimation in autoregressive moving-average (ARMA) models is introduced. This procedure has the advantage that consistency for the order estimators obtained via this procedure can be established without restricting attention to only a finite number of models. The behaviour of these newly introduced order estimators is also analysed for the case when the data-generating process is not an ARMA process (transfer function/spectral density approximation). Furthermore, the behaviour of the order estimators obtained via minimization of BIC (or of related criteria) is investigated for a non-ARMA data-generating process.     

A NOTE ON SQUARE ROOT FILTERING FOR VECTOR AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. A simplified version of the square root Kalman filter is obtained for a vector autoregressive moving-average (VARMA) model. The algorithm is computationally more efficient that the standard square root algorithm and its output can be used to compute the likelihood of a VARMA model accurately.     

A LINEAR ESTIMATION PROCEDURE FOR THE PARAMETERS OF AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. A linear estimation procedure for the parameters of autoregressive moving-average processes is proposed. The basic idea is to write the spectrum for the moving-average part as a linear function of a properly selected set of parameters and to use Chiu's weighted least-squares procedure to reduce the problem to a weighted linear least-squares problem. The proposed procedure finds estimates by solving systems of linear equations and does not need optimization programs. An one-step estimate is also suggested. It is shown that the estimates are asymptotically equal to the commonly used 'approximate' maximum likelihood estimate described in the paper. For Gaussian processes, the estimates obtained by the proposed procedures are asymptotically efficient.     

LARGE SAMPLE ANALYSIS OF AUTOREGRESSIVE MOVING-AVERAGE MODELS WITH ERRORS IN VARIABLES

Abstract. We consider estimation of parameters of an unobservable ARMA(p, q) process {U_t; t= 1,2,…} based on a set of n observables, X₁, …, X_n, where X_t=U_t, +ε_t, 1 ≤t≤n, it being assumed that {ε_t} is independent of {U_t}. We examine the asymptotic properties of these ARMA estimators under a set of weak regularity conditions on {ε_t}.     

YULE-WALKER ESTIMATES FOR CONTINUOUS-TIME AUTOREGRESSIVE MODELS

Abstract. I consider continuous-time autoregressive processes of order p and develop estimators of the model parameters based on Yule-Walker type equations. For continuously recorded data, it is shown that these estimators are least squares estimators and have the same asymptotic distribution as maximum likelihood estimators.
In practice, though, data can only be observed discretely. For discrete data, I consider approximations to the continuous-time estimators. It is shown that some of these discrete-time estimators are asymptotically biased. Alternative estimators based on the autocovariance function are suggested. These are asymptotically unbiased and are a fast alternative to the maximum likelihood estimators described by Jones. They may also be used as starting values for maximum likelihood estimation.     

ESTIMATION OF AUTOREGRESSIVE PARAMETERS AND ORDER SELECTION FOR ARMA MODELS

Abstract. A quick algorithm for obtaining estimates of autoregressive parameters for autoregressive moving-average model is presented. The algorithm is recursive in the orders, and can be used for model selection by providing a criterion and a two-way table of certain partial covariances. Consistency and asymptotic normality of the estimates are shown.     

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.     

ESTIMATION OF AUTOREGRESSIVE MOVING-AVERAGE MODELS VIA HIGH-ORDER AUTOREGRESSIVE APPROXIMATIONS

Abstract. In this paper the problem of estimating autoregressive moving-average (ARMA) models is dealt with by first estimating a high-order autoregressive (AR) approximation and then using the AR estimate to form the ARMA estimate. We show how to obtain an efficient ARMA estimate by allowing the order of the AR estimate to tend to infinity as the number of observations tends to infinity. This approach is closely related to the work of Durbin. By transforming the approach into the frequency domain, we can view it as an L ²-norm model approximation of the relative error of the spectral factors. It can also be seen as replacing the periodogram estimate in the Whittle approach by a high-order AR spectral density estimate. Since L ²-norm approximation is a difficult task, we replace it by a modification of a recent model approximation technique called balanced model reduction. By an example, we show that this technique gives almost efficient ARMA estimates without the use of numerical optimization routines.     

FAST LINEAR ESTIMATION METHODS FOR VECTOR AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. Three linear methods for estimating parameter values of vector auto-regressive moving-average (VARMA) models which are in general at least an order of magnitude faster than maximum likelihood estimation are developed in this paper. Simulation results for different model structures with varying numbers of component series and observations suggest that the accuracy of these procedures is in most cases comparable with maximum likelihood estimation. Procedures for estimating parameter standard error are also discussed and used for identification of nonzero elements in the VARMA polynomial structures. These methods can also be used to establish the order of the VARMA structure. We note, however, that the primary purpose of these estimates is to generate initial estimates for the nonzero parameters in order to reduce subsequent computational time of more efficient estimation procedures such as exact maximum likelihood.     

THE ESTIMATION OF PARAMETERS FOR AUTOREGRESSIVE MOVING AVERAGE MODELS

Abstract. This paper provides some new and improved versions of an earlier procedure for the estimation of parameters for autoregressive moving average models suggested by the author (1979). Some numerical examples of the application of the procedure are also given.     

ON VECTOR AUTOCORRELATIONS AND GENERALIZED SECOND-ORDER FUNCTIONS FOR TIME SERIES

Abstract. Some structural properties of certain vector generalizations of second-order functions of a stationary stochastic process based on determinantial functions of autocovariances are discussed. In particular, a generalized autocovariance function which retains all properties of the ordinary autocovariance function is considered and the linear dependence structure of certain scalar stochastic processes associated with this function is investigated. Properties of the normalized function are discussed and a duality property is found, according to which this function also generalizes in a natural way the ordinary partial autocorrelation function of stochastic processes.     

ON THE SELECTION OF SUBSET AUTOREGRESSIVE TIME SERIES MODELS

Abstract. The estimation of subset autoregressive time series models has been a difficult problem because of the large number of possible alternative models involved. However, with the advent of model selection criteria based on the maximum likelihood, subset model fitting has become feasible. Using an efficient technique for evaluating the residual variance of all possible subset models, a method is proposed for the fitting of subset autoregressive models. The application of the method is illustrated by means of real and simulated data.     

ESTIMATION FOR NON-LINEAR TIME SERIES MODELS USING ESTIMATING EQUATIONS

Godambe's (1985) theorem on optimal estimating equations for stochastic processes is applied to non-linear time series estimation problems. Examples are considered from the usual classes of non-linear time series models. A recursive estimation procedure based on optimal estimating equations is provided. It is also shown that pre-filtered estimates can be used to obtain the optimal estimate from a non-linear state-space model.     

A FREQUENCY DOMAIN APPROACH TO LAGRANGE MULTIPLIER TEST FOR AUTOREGRESSIVE MOVING AVERAGE MODELS

Abstract. The paper derives a goodness of fit test for autoregressive moving average models using the frequency domain approximation to the log likelihood and the Lagrange multiplier approach. The test statistic is based on the sample autocovariances and can be quickly computed through a recursive procedure.     

ROBUST FITTING OF INARCH MODELS

We discuss robust M‐estimation of INARCH models for count time series. These models assume the observation at each point in time to follow a Poisson distribution conditionally on the past, with the conditional mean being a linear function of previous observations. This simple linear structure allows us to transfer M‐estimators for autoregressive models to this situation, with some simplifications being possible because the conditional variance given the past equals the conditional mean. We investigate the performance of the resulting generalized M‐estimators using simulations. The usefulness of the proposed methods is illustrated by real data examples.