COMPARISON OF CRITERIA FOR ESTIMATING THE ORDER OF A VECTOR AUTOREGRESSIVE PROCESS

Abstract. Various criteria for estimating the order of a vector autoregressive process are compared in a simulation study. For the considered processes Schwarz's BIC criterion chooses the correct autoregressive order most often and leads to the smallest mean squared forecasting error in samples of the size usually available in practice.     

ESTIMATION OF RANDOM COEFFICIENT AUTOREGRESSIVE PROCESS: AN EMPIRICAL BAYES APPROACH

Abstract. The problem of estimating panel autoregressive time series is considered. The autoregressive parameters vary over independent realizations from an unknown distribution. An empirical Bayes procedure is suggested to estimate the parameters using information from all realizations.     

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.     

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.     

ESTIMATION OF THE MOVING-AVERAGE REPRESENTATION OF A STATIONARY PROCESS BY AUTOREGRESSIVE MODEL FITTING

Abstract. The Hannan-Rissanen procedure for recursive order determination of an autoregressive moving-average process provides 'non-parametric' estimators of the coefficients b ( u ), say, of the moving-average representation of a stationary process by auto-regressive model fitting, and also that of the cross-covariances, c ( u ), between the process and its linear innovations. An alternative 'autoregressive' estimator of the b ( u ) is obtained by inverting the autoregressive transfer function. Some uses of these estimators are discussed, and their asymptotic distributions are derived by requiring that the order k of the fitted autoregression approaches infinity simultaneously with the length T of the observed time series. The question of bias in estimating the parameters is also examined.     

MAXIMUM LIKELIHOOD ESTIMATION FOR AUTOREGRESSIVE PROCESSES DISTURBED BY A MOVING AVERAGE

Abstract. Maximum likelihood estimation for stationary autoregressive processes when the signal is subject to a moving-average sampling error is discussed. A modified maximum likelihood estimator is proposed. An algorithm for computing derivatives of the modified likelihood is suggested. Maximum likelihood estimators of the parameter vector are shown to be strongly consistent and to have a multivariate normal limiting distribution. A Monte Carlo simulation shows that the modified maximum likelihood estimator performs better than other available estimators. US current labour force data are analysed as an example.     

ORDER DETERMINATION OF MULTIVARIATE AUTOREGRESSIVE TIME SERIES WITH UNIT ROOTS

Abstract. This paper deals with order determination of multivariate time series where roots of the characteristic equation are allowed to be equal to one. For estimation of parameters in such processes, least squares were used. For a familiar class of order determination criteria it is shown that results on weak consistency valid in the stationary case can be generalized to processes with unit roots in the characteristic equation. Then a discussion of the possibility of underestimating the order for finite samples is given for a particular model, and it is indicated that nonstationarity of this type decreases the probability of underestimating the order. Finally some Monte Carlo simulation results are given to supplement the theoretical results.     

ASYMPTOTIC EXPANSIONS FOR THE DISTRIBUTION OF AN ESTIMATOR IN THE FIRST-ORDER AUTOREGRESSIVE PROCESS

Abstract. In this paper, two asymptotic expansions for the distribution for an estimator of the parameter in a first-order autoregressive process are derived, according to two situations. Some well known estimators are special cases of the estimator discussed here. The series expansions are carried to terms of order T ^-1.     

ON THE CHOICE OF THE ORDER OF AUTOREGRESSIVE MODELS: A RANKING AND SELECTION APPROACH

Abstract. The problem of estimating the order of autoregressive models is considered from the point of view of ranking and selection procedures. This approach offers a formulation to many problems more realistic than that of classical hypothesis testing or of criteria based on estimation theory (e.g., AIC). In the method considered here, sampling variations are taken into account and the experimenter is also allowed to incorporate any a priori knowledge of the true order (e.g., lower bound as well as upper bound).     

THE CRITERION AUTOREGRESSIVE TRANSFER FUNCTION OF PARZEN

Abstract. A mathematical derivation of the Criterion Autoregressive Transfer Function (CAT) of Parzen (1974) is given and a generalization of this criterion is introduced. The asymptotic distribution of the orders selected by CAT, this generalized criterion, and a new version of CAT introduced by Parzen (1977a) are derived. The joint finite sample behaviour of these three criteria and of the FPE_α,-criterion of Bhansali and Downham (1977) is examined by means of a simulation study.     

ESTIMATION OF COEFFICIENTS OF TIME SERIES REGRESSION WITH A NONSTATIONARY ERROR PROCESS

Abstract. We treat a problem of estimating unknown coefficients of a time series regression when the variance of the error changes with time, i.e. when a process which the error term obeys is nonstationary. First, we show the weak consistency of the ordinary least squares estimator for the coefficients of a polynomial regression under some assumptions on the covariance structure of the error process. Next, we propose a nonparametric method for estimating the variance of the error process and a weighted least squares estimator of the regression coefficients, which is constructed by using the estimator of the variance. We investigate statistical properties of our proposed estimator in the following way. We consider the prediction of a future value of a linear trend by using our proposed estimator and evaluate its prediction error. By simulation studies, we compare the prediction error of the predictor constructed by using our proposed estimator with the prediction errors obtained for other estimators including the ordinary least squares estimator when the variance of the error process increases with time and the sample sizes are small. As a result, our proposed estimator seems to be reasonable.     

A FORMULA FOR THE INVERSE AUTOCORRELATION FUNCTION OF AN AUTOREGRESSIVE PROCESS

Abstract. The determination of the inverse autocorrelation function of a weakly stationary autoregressive process using the autocorrelation function is considered. Usually this is carried out either by using frequency domain methods or by solving first the parameters of the process and then using them. In this paper we give a simple formula by which the inverse autocorrelation function can be determined directly from the autocorrelation function.     

ON THE PROBABILITY OF ERROR WHEN USING A GENERAL AKAIKE-TYPE CRITERION TO ESTIMATE AUTOREGRESSION ORDER

Abstract. A general Akaike-type procedure is studied where the additive penalty is proportional to the autoregression order but the constant of proportionality has a general value γ . For the procedure to be weakly consistent it is necessary and sufficient that γ →0 and nγ ∝ as n ∝, where n denotes the sample size. If n _1-ε γ ∝ for some ε>0 then the probability of erring when estimating the autoregression order converges to zero at a rate n ^{- c} , for all c >0, as n →∝. However, several procedures suggested for practical use have n ^1-ε γ →0 for each ε>0; in particular, they have γ = n ^-1 log n or γ = n ^-1 log log n . To elucidate the properties of error probabilities in these circumstances we study the case of an AR(1) process. It is shown that in this case the probability of underestimating order is usually substantially less then the chance of overestimation, unless the autoregressive constant is particularly small (in fact, of size 2 γ 1/2).     

ESTIMATION OF THE COEFFICIENTS OF A MULTIVARIATE LINEAR FILTER USING THE INNOVATIONS ALGORITHM

It is shown under mild conditions that the estimators of the coefficient matrices obtained by applying the innovations algorithm to the sample covariances of observations of the multivariate linear time series X _t = ∑^∞ _{j =0}ψ _i Z _t , t = 0, ±1, ±2, . . ., are consistent. The asymptotic distribution of the estimators is found to have a very simple form which generalizes the corresponding univariate result of Brockwell and Davis (Simple consistent estimation of the coefficients of a linear filter. In Stochastic Processes and Their Applications . Amsterdam: North- Holland, pp. 47--59). The asymptotic distribution of the corresponding estimator of the spectral density matrix is also derived. Some simulation results are presented to illustrate the small-sample behaviour of the estimators.     

A NOTE ON SOME STATISTICS USEFUL IN IDENTIFYING THE ORDER OF AUTOREGRESSIVE MOVING AVERAGE MODEL

Abstract. Several recursive relations concerning some statistics useful in identifying the order of autoregressive moving average are derived and the asymptotic behaviour of these statistics are studied.     

ESTIMATION OF THE ORDER OF A MOVING AVERAGE MODEL FROM AUTOREGRESSIVE AND WINDOW ESTIMATES OF THE INVERSE CORRELATION FUNCTION

Abstract. The autoregressive and window estimates of the inverse correlation function are used for estimating the order of a finite moving average process by using criteria similar to the FPEα criterion of Bhansali and Downham (1977). The asymptotic distribution of the estimates is derived. Their finite sample behaviour is examined by means of a simulation study.     

ESTIMATION OF THE PARAMETERS OF AN EAR(p) PROCESS

Abstract. Gaver and Lewis and Lawrance and Lewis have described an autoregressive process of order p , EAR( p ), which is such that the marginal distribution of the observations follows an exponential distribution. There is now a rich class of exponential and related distributions time series models. Such models are of importance in queuing and network processes, for example. The properties of these and related models have been well explored, but so far little work has been done toward the important problem of estimation. We attempt here to address this question for the EAR( p ) models. Because of inherent discontinuities in some of the relevant underlying distributions, the standard theory cannot be applied. However, by utilizing a general theory developed by Klimko and Nelson, conditional least-squares estimators are derived. Further, it is shown that these estimators are strongly consistent and asymptotically normally distributed. Small-sample properties are investigated. The results suggest that these estimators are to be preferred compared with those suggested by Lawrance and Lewis.     

LEAST SQUARES ESTIMATES AND ORDER DETERMINATION PROCEDURES FOR AUTOREGRESSIVE PROCESSES WITH A TIME DEPENDENT VARIANCE

Abstract. We study nonstationary autoregressive processes, where the variance of the generating white noise process is allowed to depend on time. It is shown that ordinary least squares estimates are strongly consistent and with a proper scaling factor asymptotically normal, but, as can be expected, they are not efficient. Furthermore, AIC type order determination criteria, used as if the underlying process is stationary, are consistent, whereas identification of order in terms of the partial autocorrelation function may lead one astray.     

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.