FREQUENCY-DOMAIN ESTIMATION OF BILINEAR TIME SERIES MODELS

Abstract. Two frequency-domain methods of estimation of the parameters of linear time series models–one based on maximum likelihood, called the 'Whittle criterion', and the other based on least squares, called the 'Taniguchi criterion'–are discussed in this paper. A heuristic justification for their use in models such as bilinear models is given. The estimation theory and associated asymptotic theory of these methods are numerically illustrated for the bilinear model BL( p ,0, p , 1). For that purpose, an approach based on the calculus of Kronecker product matrices is used to obtain the derivatives of the spectral density function of the state-space form of the model.     

ON THE ERGODICITY OF BILINEAR TIME SERIES MODELS

Abstract. Existence, strict stationarity and ergodicity of Bilinear Time Series Models for a given input White Noise and parameter values are studied in detail in this paper. The use of ergodicity in the estimation of parameters is also hinted at in this article.     

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.     

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.     

A TEST FOR LINEARITY OF STATIONARY TIME SERIES

Abstract. A standard assumption that is often made in time series analysis is that the series conforms to a linear model. The object of this paper is to describe statistical tests for testing this assumption. The tests are constructed from the bispectral density function, and depend on the application of Hotelling T ². These tests are illustrated with two real time series and four simulated time series. Some guidelines about the choice of the parameters are also included.     

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.     

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.     

SUBSET THRESHOLD AUTOREGRESSION WITH APPLICATIONS

Abstract. In this paper we define subset threshold autoregressive models and suggest a simple algorithm for fitting them. The suggested algorithm is applied to simulated as well as real data. Two well-known time series are studied:the Canadian lynx data and Wolf's sunspot numbers. The fitted models are compared with the threshold models of Tong and Lim and with the subset bilinear models of Gabr and Subba Rao. Some statistical properties of these models are also studied. An examination of forecasting performance is included.     

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.     

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.     

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.     

YULE-WALKER TYPE DIFFERENCE EQUATIONS FOR HIGHER-ORDER MOMENTS AND CUMULANTS FOR BILINEAR TIME SERIES MODELS

Abstract. In this paper we obtain difference equations for the third- and fourth-order lagged moments and cumulants when the time series {X_t} satisfies a bilinear model and is stationary up to fourth order. These equations are similar to the well-known Yule-Walker equations which are available for linear time series models.     

ON THE AUTOCORRELATION STRUCTURE AND IDENTIFICATION OF SOME BILINEAR TIME SERIES

Abstract. For the bilinear time series X _t =β X _t-k e _t-l + e _v , k ≥ l , formulas for the first k -1 autocorrelations of X ² _t are obtained. These results fill in a gap in Granger and Andersen (1978). Simulation experiments are used to study the applicability of theoretical results and to investigate some more general situations. It is found that if ß is not too small, k and l may be identified using the autocorrelations of X ² _t . Application to more general situations is also briefly discussed.     

A SIMPLE CONDITION FOR THE EXISTENCE OF SOME STATIONARY BILINEAR TIME SERIES

Abstract. A sufficient condition is derived for the existence of a strictly stationary solution of some bilinear time series which may have infinite variance innovations. This condition is equivalent to the condition that a polynomial of degree r has no zeros within the unit circle. In the special case when the innovations have finite variance, the computational effort involved in checking this condition is significantly reduced compared with checking the stationarity conditions given by Bhaskara Rao et al. and Liu and Brockwell which requires a knowledge of the maximum eigenvalue in the absolute value of an r ² x r ² matrix.     

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.     

A NOTE ON USING THRESHOLD AUTOREGRESSIVE MODELS FOR MULTI-STEP-AHEAD PREDICTION OF CYCLICAL DATA

Abstract. We give a brief account of how the class of threshold autoregressive time series models may be used to make short, medium and long range predictions of cyclical data.     

THE RECURSIVE FITTING OF SUBSET VARX MODELS

Abstract. A vector time series model of the form A(L)y(t) + B(L)x(t) =ε(t) is known as a vector autoregressive model with exogenous variables (VARX model) and involves a regressand vector y(t) and a regressor vector x(t). This paper provides a method for the recursive fitting of subset VARX models. It suggests the use of ascending recursions in conjunction with an order selection criterion to choose an 'optimum' subset VARX model.     

TIME SERIES RESIDUALS WITH APPLICATION TO PROBABILITY DENSITY ESTIMATION

Abstract. A linear stationary and invertible process y _t models the second-order properties of T observations on a discrete time series, up to finitely many unknown parameters θ. Two estimators of the residuals or innovations ɛ _t of y _t are presented, based on a θ estimator which is root- T consistent with respect to a wide class of ɛ _t distributions, such as a Gaussian estimator. One sets unobserved y _t equal to their mean, the other treats y _t as a circulant and may be best computed via two passes of the fast Fourier transform. The convergence of both estimators to ɛ _t is investigated. We apply the estimated ɛ _t to estimate the probability density function of ɛ _t . Kernel density estimators are shown to converge uniformly in probability to the true density. A new sub-class of linear time series models is motivated.     

A COMPARATIVE STUDY OF VARIOUS UNIVARIATE TIME SERIES MODELS FOR CANADIAN LYNX DATA

Abstract. This paper describes the results of a simulation study aimed at comparing the various time series models, both linear and non-linear, fitted to the annual Canadian lynx trappings for the years 1821–1934.     

A CONDITIONAL LEAST SQUARES APPROACH TO BILINEAR TIME SERIES ESTIMATION

Abstract. In this paper a conditional least squares (CLS) procedure for estimating bilinear time series models is introduced. This method is applied to a special superdiagonal bilinear model which includes the classical linear autoregressive moving-average model as a particular case and it is proven that the limiting distribution of the CLS estimates is Gaussian and that the law of the iterated logarithm holds.