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.     

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.     

RECURSIVE GENERALIZED M ESTIMATES FOR AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. Outliers in time series seriously affect conventional parameter estimates. In this paper a robust recursive estimation procedure for the parameters of auto-regressve moving-average models with additive outliers is proposed. Using 'cleaned' residuals from an initial robust fit of an autoregression of high order as input, bounded influence regression is applied recursively. The proposal follows certain ideas of Hannan and Rissanen, who suggested a three-stage procedure for order and parameter estimation in a conventional setting.
A Monte Carlo study is performed to investigate the robustness properties of the proposed class of estimates and to compare them with various other suggestions, including least squares, M estimates, residual autocovariance and truncated residual autocovariance estimates. The results show that the recursive generalized M estimates compare favourably with them. Finally, possible modifications to master even vigourous situations are suggested.     

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}.     

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.     

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.     

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.     

SOME SIMPLE TESTS OF THE MOVING-AVERAGE UNIT ROOT HYPOTHESIS

Abstract. This paper deals with three test statistics for a moving-average (MA) unit root. The spectral test is based on the estimate of the spectral density at frequency zero. The variance difference statistic compares the sample variance of the integrated series with the estimated variance imposing the MA unit root constraint. Furthermore, Tanaka's score type test statistic is modified to improve the power in higher order models. The asymptotic power of the tests is considered and Monte Carlo experiments are performed to investigate the small sample properties of the tests. Finally, the tests are applied to a number of economic time series to determine the degree of integration.     

ASYMPTOTIC MEAN SQUARE PREDICTION ERROR FOR A MULTIVARIATE AUTOREGRESSIVE MODEL WITH RANDOM COEFFICIENTS

In this paper, an expression for the asymptotic mean square error in predicting more than one step ahead from a p-variate autoregressive model with random coefficients is derived. Two cases are investigated: (i) when the parameters are known, and (ii) when the parameters are estimated.     

A NOTE ON THE GENERATION OF INDEPENDENT REALIZATIONS OF A VECTOR AUTOREGRESSIVE MOVING-AVERAGE PROCESS

Abstract. Barone has described a method for generating independent realizations of a vector autoregressive moving-average (ARMA) process which involves recasting the ARMA model in state space form. We discuss a direct method of computing the initial state covariance matrix T ₀ which, unless the number of time series is large, is usually faster than using the doubling algorithm of Anderson and Moore. Our numerical comparisons are particularly valuable because T ₀ must also be computed when calculating the likelihood function. A number of other computational refinements are described. In particular, we advocate the use of Choleski factorizations rather than spectral decompositions. For a pure moving-average process computational savings can be achieved by working directly with the ARMA model rather than with its state space representation.     

ORDER IDENTIFICATION STATISTICS IN STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS:VECTOR AUTOCORRELATIONS AND THE BOOTSTRAP

Abstract. In this paper we consider the vector autocorrelation approach for identifying ARMA ( p, q ) models and use a bootstrap procedure in order to evaluate the distribution of the corresponding sample statistics by means of a resampling scheme for the residuals when p and q are unknown. The asymptotic validity of the bootstrap procedure applied to the vector autocorrelation estimates is established. Some simulations and examples demonstrating the appropriateness of the proposed bootstrap procedure in comparison with large-sample Gaussian approximations are included.     

A METHODOLOGY FOR SELECTING SUBSET AUTOREGRESSIVE TIME SERIES MODELS

Abstract. In time series modelling, subset models are often desirable, especially when the data exhibit some form of periodic behaviour with a range of different natural periods in terms of days, weeks, months and years. Recently, Hokstad proposed a method based on personal judgement for selecting the first tentative model to obtain the best subset autoregressive model. The subjective approach adopted in the Hokstad method is a disadvantage in building up a computer program which could automatically select the appropriate model of a given time series. In this paper, we propose overcoming this disadvantage by employing the inverse autocorrelation function to select the first tentative model. In addition to sets of synthetic data, some well-known real series such as the D, E and F series of Box and Jenkins and the Canadian lynx data are analysed to validate the proposed method. The results indicate that the method can successfully detect the true model for a given time series.     

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 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.     

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.     

IDENTIFYING MULTIVARIATE TIME SERIES MODELS

Abstract. This paper is concerned with how canonical variate analysis can be used to identify the structure of a linear multivariate time series model. The procedure used is based on that of Akaike and Cooper and Wood. A correction and a refinement are made, however. The correction is on the testing statistic and the refinement on the allowed order ( p, q ). Appropriate asymptotic distributions for testing zero canonical correlations are also given.     

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.     

A NOTE ON USING THRESHOLD AUTOREGRESSIVE MODELS FOR MULTI-STEP-AHEAD PREDICTION OF CYCLICAL DATA

Abstract. We give a brief account of how the class of threshold autoregressive time series models may be used to make short, medium and long range predictions of cyclical data.     

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.