GEOMETRIC ERGODICITY OF A DOUBLY STOCHASTIC TIME SERIES MODEL

Abstract. We demonstrate that a large class of doubly stochastic time series models are geometrically ergodic, and hence admit second-order stationary solutions.
We also establish a version of the strong law of large numbers, the law of the interated logorithm and the central limit theorem for the stochastic processes under consideration.     

ON THE EXISTENCE OF SOME BILINEAR TIME SERIES MODELS

Abstract. The existence of a multivariate strictly stationary stochastic process conforming to a certain bilinear time series model is discussed.     

NONPARAMETRIC ESTIMATORS FOR TIME SERIES

Abstract. Kernel multivariate probability density and regression estimators are applied to a univariate strictly stationary time series X _r We consider estimators of the joint probability density of X _t at different t -values, of conditional probability densities, and of the conditional expectation of functionals of X _v given past behaviour. The methods seem of particular relevance in light of recent interest in non-Gaussian time series models. Under a strong mixing condition multivariate central limit theorems for estimators at distinct points are established, the asymptotic distributions being of the same nature as those which would derive from independent multivariate observations.     

MULTIPLE BILINEAR TIME SERIES MODELS

Abstract. A definition of multiple bilinear time series models is given. Sufficient conditions are obtained for the existence of strictly stationary solutions conforming to the model, and a brief discussion of the first and second order structure is included.     

DIAGNOSTIC CHECKING ARMA TIME SERIES MODELS USING SQUARED-RESIDUAL AUTOCORRELATIONS

Abstract. Squared-residual autocorrelations have been found useful in detecting nonlinear types of statistical dependence in the residuals of fitted autoregressive-moving average (ARMA) models (Granger and Andersen, 1978; Miller, 1979). In this note it is shown that the normalized squared-residual autocorrelations are asymptotically unit multivariate normal. The results of a simulation experiment confirming the small-sample validity of the proposed tests is reported.     

ON THE IDENTIFICATION OF SOME BILINEAR TIME SERIES MODELS

Abstract. In their book on bilinear time series models Granger and Andersen (1978, p. 43) dismiss the use of third order moments for identifying models on the grounds that for some bilinear models they will all be zero and hence are of no use in discriminating between true white noise and some bilinear models. However, in this paper it is shown that some of the third order moments do not vanish for some superdiagonal and diagonal bilinear models and the pattern of non zero moments can be used to discriminate between true white noise and these bilinear models and also between different bilinear models. Simulation experiments are used to study the applicability of theoretical results.     

MARGINALS OF MULTIVARIATE FIRST-ORDER AUTOREGRESSIVE TIME SERIES MODELS

This paper is concerned with the marginal models associated with a given multivariate first-order autoregressive model. A general theory is developed to determine when reductions in the known orders of the marginal models will occur. When the auto-regressive coefficient matrix has repeated eigenvalues, there may be global reductions in the marginal models. Zeros in the eigenvectors and generalized eigenvectors of the auto-regressive coefficient matrix lead to local reductions in the marginal models. The case when the autoregressive parameter matrix has systematic zeros is also investigated.     

A TEST FOR CONDITIONAL HETEROSKEDASTICITY IN TIME SERIES MODELS

Abstract. When testing for conditional heteroskedasticity and nonlinearity, the power of the test in general depends on the functional forms of conditional heteroskedasticity and nonlinearity that are allowed under the alternative hypothesis. We suggest a test for conditional heteroskedasticity and nonlinearity with the nonlinear autoregressive conditional heteroskedasticity model of Higgins and Bera as the alternative. Standard testing procedures are not applicable since our nonlinear autoregressive conditional heteroskedasticity (ARCH) parameter is not identified under the null hypothesis. To resolve this problem, we apply the procedure recently proposed by Davies. Power and size of the suggested test are investigated through simulation, and an empirical application of testing for ARCH in exchange rates is also discussed.     

BAYESIAN THRESHOLD AUTOREGRESSIVE MODELS FOR NONLINEAR TIME SERIES

Abstract. This paper provides a Bayesian approach to statistical inference in the threshold autoregressive model for time series. The exact posterior distribution of the delay and threshold parameters is derived, as is the multi-step-ahead predictive density. The proposed methods are applied to the Wolfe's sunspot and Canadian lynx data sets.     

COMPARATIVE POWER STUDIES FOR GOODNESS OF FIT TESTS OF TIME SERIES MODELS

Abstract. Power studies for three tests of fit, namely the classical test given by Whittle, the widely-known test given by Ljung and Box, and the recent test given by Godolphin, are derived for the invertible time series models considered previously in an empirical study. The results demonstrate that it is possible to improve on the Ljung-Box test by employing more empirically-based alternatives. In particular, the test derived by Godolphin appears to be more powerful than the other tests. The results also suggest that the Whittle and Ljung-Box tests are complementary, so that the combination of both may yield a sensitive test procedure.     

MULTIPLICATIVE EXPONENTIAL MODELS FOR STATIONARY TIME SERIES

Abstract. A class of models for one dimensional time series is presented. The spectrum of such a model is obtained by raising the spectrum of a known parameterized model to an exponent, allowed to attain arbitrary real values. For a moving average model this for example means that the roots of the moving average operator are allowed to have any real order. This method adds a further flexibility to the model which for example allows us to model long memory time series using only a few parameters. The exponent is parameterized in a special way to make the estimation of the parameter determining the exponent asymptotically independent of the estimation of the other model-parameters. The asymptotic distribution of the estimators is derived. The idea is also used for multiplicative models with an exponent for each seasonal factor. In this case the estimators are only approximately independent for a large season length. Finally an application of the model is given using the Beveridge wheat price index.     

AN EXACT TEST FOR A STOCHASTIC COEFFICIENT IN A TIME SERIES REGRESSION MODEL

An exact small-sample test is developed for testing the hypothesis that a regression coefficient is constant against the alternative that it is generated by a random walk process. The test is mean- and scale-invariant and approximates the most powerful invariant test against any specific alternative. It thus outperforms tests previously given in the literature. Computationally efficient algorithms are given to compute the test statistic and its distribution using a modified version of the Kalman filter.     

THRESHOLD TIME SERIES MODELS AS MULTIMODAL DISTRIBUTION JUMP PROCESSES

Abstract. Recent contributions by Tong and others in modelling time series exhibiting threshold points have generally been based on approximating non-linear processes by piecewise linear time series models. In this paper we provide an alternative framework in which to model time series displaying jump behaviour by using a multimodal conditional distribution to capture the jump process. Each subordinate model of the distribution is determined by an autoregressive process, and jump behaviour occurs when the relative heights of the modes of the distribution change whilst the threshold points are identified by the antimodes of the distribution. This class of models is referred to as multipredictor autoregressive time series (MATS).     

FORECASTING TIME SERIES:A COMPARATIVE ANALYSIS OF ALTERNATIVE CLASSES OF TIME SERIES MODELS

Abstract. Performance of the state dependent model developed by Priestley is evaluated relative to that of bilinear and standard linear models using two well-known time series. The results indicate the use of broader classes of time series models beyond the conventional ARMA class is likely to lead to significant reductions in forecasting error. However, there are difficult problems relating to the identification of the order of the model, estimation of the parameters, and determination of the correct nonlinear model.     

SPECTRAL ANALYSIS FOR AMPLITUDE-MODULATED TIME SERIES

Abstract. Let { X ( n )} be a non-observed strictly stationary process, { a ( n )} a sequence independent of { X ( n )} and Y ( n ) = a ( n ) X ( n ) the observed process. This work deals with the estimation of the spectral density function fx ( Λ ) of the process of interest, { X ( n )}, using observations of the modulated process { Y ( n )}. We obtain estimators of fx ( Λ ) for three types of modulating functions:deterministic, random independent and random correlated.     

ON STATIONARITY OF THE SOLUTION OF A DOUBLY STOCHASTIC MODEL

Abstract. Consider the discrete parameter process {X_I} satisfying the doubly stochastic model X_t =ø_tX_t-1+ε_t where {ø} and {ε_t} are also stochastic processes. Necessary and sufficient conditions on {ø} are given for { X₁ } to be a second order process. When {ø_t} is a strictly stationary process, some sufficient conditions in terms of {ø} are given which guarantee the wide sense stationarity of {X_t} . It turns out that for these problems the distribution and dependence structure of the process {log |ø|} play an important role.     

SOME THEORY ON M-SMOOTHING OF TIME SERIES

Abstract. In recent years many robust smoothing procedures for time series have been introduced. Their extreme nonlinearity made them mathematically untractable and their behaviour was mostly analysed by means of Monte Carlo studies. In this paper we develop some mathematical theory of a specific class of nonlinear smoothers. We investigate the asymptotics of so-called M-smoothers and discuss robustness of M-smoothers in some special cases.     

ESTIMATION FOR NON-LINEAR TIME SERIES MODELS USING ESTIMATING EQUATIONS

Godambe's (1985) theorem on optimal estimating equations for stochastic processes is applied to non-linear time series estimation problems. Examples are considered from the usual classes of non-linear time series models. A recursive estimation procedure based on optimal estimating equations is provided. It is also shown that pre-filtered estimates can be used to obtain the optimal estimate from a non-linear state-space model.     

NEAREST-NEIGHBOUR METHODS FOR TIME SERIES ANALYSIS

Abstract. The nearest-neighbour method, because of its intuitively appealing nature and competitive theoretical properties, deserves consideration in time-series applications akin to attention it has received lately in the i.i.d. case. Here it is shown that as a nonparametric regression device, like the kernel method, under the G ₂ mixing assumption, it converges in quadratic mean at the Stone-optimal rate. In the closing sections, our methodology is extended to a broader pattern-recognition context, and applied to hydrologic data.     

ON THE SELECTION OF SUBSET AUTOREGRESSIVE TIME SERIES MODELS

Abstract. The estimation of subset autoregressive time series models has been a difficult problem because of the large number of possible alternative models involved. However, with the advent of model selection criteria based on the maximum likelihood, subset model fitting has become feasible. Using an efficient technique for evaluating the residual variance of all possible subset models, a method is proposed for the fitting of subset autoregressive models. The application of the method is illustrated by means of real and simulated data.