ASYMPTOTIC EXPANSIONS FOR THE DISTRIBUTIONS OF SERIAL CORRELATIONS

Abstract. Let X ₁, …, X _n be a random sample from a population with a distribution function F and let E ( X ₁) = 0, E ( X ₁²) < ∞. Let r ₁=Σ _{t =1} ^{n -1} X _t X _{t +1}/Σ _{t =1} ^{n -1}( X _t ²+ X _{t +1}²). We derive a proper Edgeworth type expansion for the sampling distribution of r ₁ under the assumption that F is a mixture of Gaussian distributions of one of two given types. The result can easily be extended to the sampling distributions of serial correlations of arbitrary lag s .     

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.     

APPROXIMATE DISTRIBUTION OF PARAMETER ESTIMATORS FOR FIRST-ORDER AUTOREGRESSIVE MODELS

Abstract. Formulae for the exact bias and mean square error for the least squares for forward-backward least squares estimators are obtained based on the explicit expressions for the moment-generating and characteristic functions of quadratic form in the first-order autoregressive process. Asymptotic expressions for their cumulants and the maximum likelihood estimator are given. Approximations of the distributions of the above estimators are proposed based on the Ornstein-Ulenbeck process. A simple computational procedure for the exact distribution is developed, and some numerical comparisons are given which support the overall good accuracy of the approximation and confirm that the maximum likelihood estimator performs better than the others.     

OPTIMALITY OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN FIRST-ORDER AUTOREGRESSIVE PROCESSES

Abstract. Let ρρ* be the maximum likelihood estimator (MLE) of the parameter ρ in the first-order autoregressive process with normal errors. The problem of optimality in the sense of weighted squared error is considered rather than moments of asymptotic distributions. Many unbiased estimators can be constructed, but the two-dimensional sufficient statistic is incomplete. It is shown that dEρ* - ρ→ 0 uniformly in ρ, and that dE (ρ*)/ d ρ→ 1 for all |ρ| > 1. For |ρ| > 1, it is known that the asymptotic distribution of {I(ρ)}¹²⁽ρ* - ρ), where I (ρ) is the Fisher information, is Cauchy. It follows that the Cramèr-Rao inequality will not yield useful results for investigating the limit of exact efficiency of any asymptotically unbiased estimator, including the MLE. For all |ρ| < 1, ρ* is asymptotically optimal in the sense of minimizing expected weighted squared error. In addition, for |ρ| < 1, ρ* minimizes the variance asymptotically.     

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.     

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.     

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.     

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.     

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.     

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.     

THE ASYMPTOTIC PROPERTIES OF THE SAMPLE AUTOCORRELATIONS FOR A MULTIPLE AUTOREGRESSIVE PROCESS WITH ONE UNIT ROOT

Abstract. In this paper the large sample behaviour of the sample autocorrelation matrix R _n( h ), ( h being the lag, n the sample size), of a multivariate autoregressive time series with one of its characteristic roots equal to unity and the rest of the roots lying inside the unit circle is studied. It is shown that R _n( h ) converges almost surely to a constant matrix. Further, the asymptotic distribution of R _n( h ) is characterized as that of a random matrix which is a function of jointly normal random variables.     

ESTIMATION OF COEFFICIENTS OF AN AUTOREGRESSIVE PROCESS BY USING A HIGHER ORDER MOMENT

Abstract. When we use the estimators, obtained by solving Yule-Walker equations, of the coefficients of an autoregressive process, we cannot discriminate X _t and Y _t where all the solutions of the associated polynomial equation of X _t are less than 1 in the absolute value and, at least, one of the solutions of that of Y _t is greater than 1 in the absolute value. To discriminate between X _t and Y _t Rosenblatt proposed a method. We propose another method by using a higher order moment.     

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.     

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.     

VALIDITY OF EDGEWORTH EXPANSIONS FOR STATISTICS OF TIME SERIES

Abstract. In this paper, we discuss the validity of the multivariate Edgeworth expansion of distribution functions of statistics which need not be standardized sums of independent and identically distributed vectors. We apply this result to statistics of time series. In particular, we shall give the asymptotic expansion of the distribution of the maximum likelihood estimator of a parameter of a circular autoregresive moving average process.     

HIGHER-ORDER ASYMPTOTIC PROPERTIES OF A WEIGHTED ESTIMATOR FOR GAUSSIAN ARMA PROCESSES

Abstract. Let { X _t } be a Gaussian ARMA process with spectral density f _θ(Λ), where θ is an unknown parameter. To estimate θ, we propose an estimator θC_w of the Bayes type. Since our standpoint in this paper is different from Bayes's original approach, we call it a weighted estimator. We then investigate various higher-order asymptotic properties of θC_w. It is shown that θC_w is second-order asymptotically efficient in the class of second-order median unbiased estimators. Furthermore, if we confine our discussions to an appropriate class D of estimators, we can show that θC_w is third-order asymptotically efficient in D . We also investigate the Edgeworth expansion of a transformation of θC_w. We can then give the transformation of θC_w which makes the second-order part of the Edgeworth expansion vanish. Finally we consider the problem of testing a simple hypothesis H:θ=θ_o against the alternative A:θ#θ_o. For this problem we propose a class of tests δ_A which are based on the weighted estimator. We derive the X ² type asymptotic expansion of the distribution of S (ζδ_A) under the sequence of alternatives A n :θ=θ_o+ε n ^1/2, ε > 0. We can then compare the local powers of various tests on the basis of their asymptotic expansions.     

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.     

THE STABILITY OF THE AR(1) PROCESS WITH AN AR(1) COEFFICIENT

Abstract. In this paper we consider a simple time varying coefficient ARMA process:the AR (1) process with an AR (1) coefficient. A basic requirement of the process is that the output has finite variance, and we derive a condition on the parameters for this to be satisfied. The analysis is complicated by the interaction between the equations for the data and the varying coefficient.     

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.     

THIRD-ORDER ASYMPTOTIC PROPERTIES OF ESTIMATORS IN GAUSSIAN ARMA PROCESSES WITH UNKNOWN MEAN

Abstract. This paper deals with the third-order asymptotic theory for Gaussian autoregressive moving-average (ARMA) processes with unknown mean μ. We are interested in the estimation of ρ = ( α₁…, α_p, β₁…, β_q ), where α ₁…, α_ρ and β ₁…, β_q are the coefficients of the autoregressive part and the moving-average part, respectively. First, we investigate the third-order asymptotic optimality of the bias adjusted maximum likelihood estimator (MLE) of ρ in the presence of the nuisance parameters μ and ² (innovation variance). Next, for a Gaussian AR(1μ μ, ²), we propose a mean corrected estimator α_c1c2 of the autoregressive coefficient. We make a comparison between the bias adjusted estimator α_c1c2* and the bias adjusted MLE, in terms of their probabilities of concentration around the true value, or equivalently, in terms of their mean squared errors. Finally some numerical studies are provided in order to verify the third-order asymptotic theory.