ALTERNATIVE ESTIMATORS AND UNIT ROOT TESTS FOR THE AUTOREGRESSIVE PROCESS

Abstract. We compare several estimators for the second-order autoregressive process and compare the associated tests for a unit root. Monte Carlo results are reported for the ordinary least squares estimator, the simple symmetric least squares estimator and the weighted symmetric least squares estimator. The weighted symmetric least squares estimator of the autoregressive parameters generally has smaller mean square error than that of the ordinary least squares estimator, particularly when one root is close to one in absolute value. For the second-order model with known zero intercept, the one-sided ordinary least squares test for a unit root is more powerful than the symmetric tests. For the model with an estimated intercept, the one-sided weighted symmetric least squares test is the most powerful test.     

SOME ASYMPTOTIC PROPERTIES OF THE SAMPLE COVARIANCES OF GAUSSIAN AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. The paper deals with the asymptotic variances of the sample covariances of autoregressive moving average processes. Using state-space representations and some matrix Lyapunov equation theory, closed-form expressions are derived for the asymptotic variances of the sample covariances and for the Cramer-Rao bounds on the process covariances. The main results obtained from these expressions are as follows: For ARMA ( p, q ) processes with p ≥ q , the sample covariance of order n is asymptotically efficient if and only if 0 ≤ n ≤ p – q .
For ARMA ( p, q ) processes with p < q , none of the sample covariances is asymptotically efficient.     

ASYMPTOTIC PROPERTIES OF THE SAMPLE AUTOCORRELATIONS AND PARTIAL AUTOCORRELATIONS OF A MULTIPLICATIVE ARIMA PROCESS

Abstract. We shall investigate the asymptotic behaviour of the sample autocorrelations and partial autocorrelations of a multiplicative ARIMA process and derive their limiting distributions. Some simulations are presented to illustrate the results obtained.     

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 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.     

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.     

PROPERTIES OF PREDICTORS FOR MULTIVARIATE AUTOREGRESSIVE MODELS WITH ESTIMATED PARAMETERS

Abstract. The k -dimensional p th-order autoregressive processes { Y _t} that are either stationary or have one unstable or explosive root are considered. The properties of the s -periods-ahead predictor Ŷ _n+s, obtained by replacing the unknown parameters in the expression for the best linear predictor YT_n+s by their least-squares estimators, is shown to be asymptotically equivalent to the optimal predictor except in the explosive case. An expression for the mean squared error of Ŷ _n+s is derived through terms of order n ^-1 for normal stationary processes when the parameters are estimated from the realization to be predicted. In addition, small-sample properties of Ŷ _n+s are investigated.     

SOME PROPERTIES OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN THE SIMULTANEOUS SWITCHING AUTOREGRESSIVE MODEL

Abstract. The simultaneous switching autoregressive (SSAR) model proposed by Kunitomo and Sato (A non-linearity in economic time series and disequilibrium econometric models. In Theory and Application of Mathematical Statistics (ed. A. Takemura). Tokyo:University of Tokyo Press (in Japanese), 1994; Asymmetry in economic time series and simultaneous switching autoregressive model. Struct. Change Econ. Dyn. , forthcoming (1994).) is a Markovian non-linear time series model. We investigate the finite sample as well as the asymptotic properties of the least squares estimator and the maximum likelihood (ML) estimator. Due to a specific simultaneity involved in the SSAR model, the least squares estimator is badly biased. However, the ML estimator under the assumption of Gaussian disturbances gives reasonable estimates.     

ASYMPTOTIC EFFICIENCY OF THE SAMPLE COVARIANCES IN A GAUSSIAN STATIONARY PROCESS

Abstract. This paper deals with the asymptotic efficiency of the sample autocovariances of a Gaussian stationary process. The asymptotic variance of the sample autocovariances and the Cramer–Rao bound are expressed as the integrals of the spectral density and its derivative. We say that the sample autocovariances are asymptotically efficient if the asymptotic variance and the Cramer–Rao bound are identical. In terms of the spectral density we give a necessary and sufficient condition that they are asymptotically efficient. This condition is easy to check for various spectra.     

ASYMPTOTIC PROPERTIES OF SOME PRELIMINARY ESTIMATORS FOR AUTOREGRESSIVE MOVING AVERAGE TIME SERIES MODELS

Abstract. Some simple preliminary estimators for the coefficients of mixed autoregressive moving average time series models are considered. As the first step the estimators require the fitting of a long autoregression to the data. The first two methods of the paper are non-iterative and generally inefficient. The estimators are Yule-Walker type modifications of the least squares estimators of the coefficients in auxiliary linear regression models derived, respectively, for the coefficients of the long autoregression and for the coefficients of the corresponding long moving average approximation of the model. Both of these estimators are shown to be strongly consistent and their asymptotic distributions are derived. The asymptotic distributions are used in studying the loss in efficiency and in constructing the third estimator of the paper which is an asymptotically efficient two-step estimator. A numerical illustration of the third estimator with real data is given.     

CONSISTENT ESTIMATION OF THE ASYMPTOTIC COVARIANCE STRUCTURE OF MULTIVARIATE SERIAL CORRELATIONS

Abstract. A method is proposed for estimating, in a consistent way, the asymptotic covariance structure of serial correlations for a multivariate second-order stationary process. To obtain a consistent estimator of this structure, which is also of the non-negative definite type, results relative to the scalar case are generalized. The method consists in weighting appropriately the elements of the sample autocorrelation matrices in a generalization of Bartlett's formula so that the estimator converges in probability. Several useful applications of the results of the paper are mentioned.     

THIRD ORDER ASYMPTOTIC PROPERTIES OF BLUE AND LSE FOR A REGRESSION MODEL WITH ARMA RESIDUAL

Abstract. In this note, we shall investigate third-order asymptotic properties of BLUE and LSE for a regression model with ARMA residual. In the first place we shall evaluate the asymptotic mean square errors of BLUE and LSE up to third order. For appropriate regression variables (constant or harmonic functions), the asymptotic mean square error of LSE coincides with that of BLUE up to second order. Then we shall evaluate the difference of the asymptotic mean square errors of LSE and BLUE at third order. Secondly we shall show that BLUE is third-order asymptotically efficient in the sense of the highest probability concentration around the true value in the third-order Edgeworth expansion.     

DISTRIBUTIONS OF LEAST SQUARES ESTIMATORS OF AUTOREGRESSIVE PARAMETERS FOR A PROCESS WITH COMPLEX ROOTS ON THE UNIT CIRCLE

Abstract. Asymptotic distributions of the autoregressive parameters in the AR(2) model are derived, when the characteristic polynomial has a pair of complex roots on the unit circle. Percentage points are tabulated based on simulations from the asymptotic formulae. The usefulness of the asymptotic results in finite sample situations is investigated by a Monte Carlo study, and an illustrative example is given.     

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.     

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.     

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 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.     

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.     

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.     

A UNIFIED APPROACH TO CONFIDENCE BOUNDS FOR THE AUTOREGRESSIVE SPECTRAL ESTIMATOR

Abstract. Confidence bounds for the spectral density of a stationary time series are derived. A unified method begins with the autoregressive spectral estimate and produces both confidence intervals at single frequencies chosen a priori and a simultaneous confidence band for multiple a posteriori comparisons. The crux is optimization of a quadratic form subject to the constraint imposed by the F -statistic. An approximate method that may produce tighter bounds is described. The former methods are demonstrated on the Waldmeier annual mean sunspot numbers.