THE ESTIMATION AND APPLICATION OF LONG MEMORY TIME SERIES MODELS

Abstract. The definitions of fractional Gaussian noise and integrated (or fractionally differenced) series are generalized, and it is shown that the two concepts are equivalent. A new estimator of the long memory parameter in these models is proposed, based on the simple linear regression of the log periodogram on a deterministic regressor. The estimator is the ordinary least squares estimator of the slope parameter in this regression, formed using only the lowest frequency ordinates of the log periodogram. Its asymptotic distribution is derived, from which it is evident that the conventional interpretation of these least squares statistics is justified in large samples. Using synthetic data the asymptotic theory proves to be reliable in samples of 50 observations or more. For three postwar monthly economic time series, the estimated integrated series model provides more reliable out-of-sample forecasts than do more conventional procedures.     

DETERMINING THE DEGREE OF DIFFERENCING FOR TIME SERIES VIA THE LOG SPECTRUM

Abstract. While many time series require differencing before a model may be fitted it has been shown that 'overdifferencing' may result in a fitted model with poor long term forecasting properties. This may present real problems when the degree of differencing which is appropriate is fractional. We show that the log spectrum is a natural quantity to consider when attempting to determine the degree of differencing required and outline the distribution theory required. The ideas are shown to extend to the seasonal case and can be used to assess whether seasonal differencing is appropriate.     

AN INTRODUCTION TO LONG-MEMORY TIME SERIES MODELS AND FRACTIONAL DIFFERENCING

Abstract. The idea of fractional differencing is introduced in terms of the infinite filter that corresponds to the expansion of (1- B ) ^d . When the filter is applied to white noise, a class of time series is generated with distinctive properties, particularly in the very low frequencies and provides potentially useful long-memory forecasting properties. Such models are shown to possibly arise from aggregation of independent components. Generation and estimation of these models are considered and applications on generated and real data presented.     

TESTING THE ORDER OF DIFFERENCING IN TIME SERIES REGRESSION

Abstract. In this paper we develop a test procedure for detecting overdifferencing or a moving-average unit root in time series regression models with Gaussian autoregressive moving-average errors. In addition to an intercept term the regressors consist of stable or asymptotically stationary variables and non-stationary trending variables generated by an integrated process of order 1. The test of the paper is based on the theory of locally best invariant unbiased tests. Its limiting distribution is derived under the null hypothesis and found to be non-standard but free of unknown nuisance parameters. Asymptotic critical values, which depend on the number of integrated regressors, are obtained by simulation. A limited simulation study is carried out to illustrate the finite sample properties of the test.     

EXPECTATION-MAXIMIZATION ALGORITHMS AND THE ESTIMATION OF TIME SERIES MODELS IN THE PRESENCE OF OUTLIERS

Abstract. The expectation-maximization algorithm is reviewed briefly. The algorithm is applied to time series situations where outliers may be present. An approximation of the algorithm is considered to reduce the computational complexity. Examples are given to illustrate the application of this algorithm.     

THE ESTIMATION AND PREDICTION OF SUBSET BILINEAR TIME SERIES MODELS WITH APPLICATIONS

Abstract. In this paper we define subset bilinear time series models, and then describe an algorithm for the estimation of these models. It is also pointed out that for this class of non-linear time series models, it is possible to obtain optimal several step predictors. The estimation technique of these models is illustrated with respect to three time series, and the optimal several steps ahead forecasts of these time series models are calculated. A comparison of these forecasts is made with the forecasts obtained by the best linear autoregressive and threshold autoregressive models. The residuals obtained from the models are tested for independence and Gaussianity using higher order moments.     

ON THE STRENGTH OF DEPENDENCE OF A TIME SERIES GENERATED BY A CHAOTIC MAP

Abstract. A stochastic sequence generated by a chaotic map has extremely strong dependence in a structural sense, in that any data value may be represented exactly as a known deterministic function of any one of its antecedents. However, the range of dependence of the time series may be very short in a statistical sense - in fact, all its lagged correlations could be zero. In the present paper we study the implications of this property for two of the statistical techniques which weak dependence is often invoked to justify - asymptotic methods based on the central limit theorem, and the bootstrap. It is shown that in the case of the logistic map, the validity of these techniques depends critically on the value of the parameter governing the map. Very small alterations to the parameter value can produce dramatic changes in the strength of dependence, thereby altering the validity of even elementary statistical procedures based on asymptotic normality or resampling.     

RATE OPTIMAL SEMIPARAMETRIC ESTIMATION OF THE MEMORY PARAMETER OF THE GAUSSIAN TIME SERIES WITH LONG-RANGE DEPENDENCE

There exist several estimators of the memory parameter in long- memory time series models with the spectrum specified only locally near zero frequency. In this paper we give an asymptotic lower bound for the minimax risk of any estimator of the memory parameter as a function of the degree of local smoothness of the spectral density at zero. The lower bound allows one to evaluate and compare different estimators by their asymptotic behaviour, and to claim the rate optimality for any estimator attaining the bound. A log-periodogram regression estimator, analysed by Robinson (Log-periodogram regression of time series with long range dependence. Ann. Stat. 23 (1995), 1048--72), is then shown to attain the lower bound, and is thus rate optimal.     

SOME ASPECTS OF THE PERFORMANCE OF DIAGNOSTIC CHECKS IN BIVARIATE TIME SERIES MODELS

Abstract. This paper is concerned with the use of score, or Lagrangian multiplier and portmanteau tests of fitted model adequacy in vector autoregressive-moving average processes. The relation between these alternative diagnostic checking devices is discussed from an asymptotic theoretic standpoint. Some finite sample properties of the tests are investigated in the context of bivariate models using Monte Carlo methods. Asymptotic theory is used to help determine the simulation design and also proves useful in appraising the experimental outcomes. The results provide evidence on the likely relative performance of the two procedures in practice and suggest that the score test is to be preferred.     

A STUDY OF THE APPLICATION OF STATE-DEPENDENT MODELS IN NON-LINEAR TIME SERIES ANALYSIS

Abstract. The theory of state-dependent models was developed by Priestley (1980), and a few simple applications were given in Priestley (1981). In this paper, an extensive study of the application of state-dependent models to a wide variety of non-linear time series data is carried out. Both real and simulated data are used in the study, and the problems encountered are highlighted. The method is demonstrated to be successful in practice in many cases, and suggestions for the further development of the algorithm are also given.     

ON THE UNBIASEDNESS PROPERTY OF AIC FOR EXACT OR APPROXIMATING LINEAR STOCHASTIC TIME SERIES MODELS

Abstract. A rigorous analysis is given of the asymptotic bias of the log maximum likelihood as an estimate of the expected log likelihood of the maximum likelihood model, when a linear model, such as an invertible, gaussian ARMA ( p, q ) model, with or without parameter constraints, is fit to stationary, possibly non-gaussian observations. It is assumed that these data arise from a model whose spectral density function either (i) coincides with that of a member of the class of models being fit, or, that failing, (ii) can be well-approximated by invertible ARMA ( p, q ) model spectral density functions in the class, whose ARMA coefficients are parameterized separately from the innovations variance. Our analysis shows that, for the purpose of comparing maximum likelihood models from different model classes, Akaike's AIC is asymptotically unbiased, in case (i), under gaussian or separate parametrization assumptions, but is not necessarily unbiased otherwise. In case (ii), its asymptotic bias is shown to be of the order of a number less than unity raised to the power max { p, q } and so is negligible if max { p, q } is not too small. These results extend and complete the somewhat heuristic analysis given by Ogata (1980) for exact or approximating autoregressive models.