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 THIRD-ORDER MOMENT STRUCTURE AND BISPECTRAL ANALYSIS OF SOME BILINEAR TIME SERIES

For the bilinear time series model X_t=βX_t-k^e_t-t+^e_t, k > l, k = l and k < l formulae for the third-order theoretical moments and an expression for the bispectral density function are obtained. These results can be used to distinguish between bilinear models and white noise and, in general, linear models. Furthermore, they give an indication of the type combination (k, l) in the above model. The modulus of the bispectral density function of the above bilinear time series model for different combinations of (k, l) and values of β are computed and the properties are studied.     

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.     

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 EXISTENCE OF A GENERAL MULTIPLE BILINEAR TIME SERIES

Abstract. A sufficient condition is derived for the existence of a strictly stationary solution of the general multiple bilinear time series equations (without assuming subdiagonality). The condition is shown to reduce to the condition of Stensholt and Tjostheim in the special case which they consider. Under this condition a solution is constructed which is shown to be casual in the sense we define, strictly stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, i.e. the multiple autoregressive moving-average (ARMA) model. the condition given here reduces to the well-known sufficient condition for the existence of a casual stationary solution.     

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.     

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 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.     

EMPIRICAL IDENTIFICATION OF MULTIPLE TIME SERIES

Abstract. In the univariate case the problem of empirical identification consists in determining the order parameters p , d and q of ARIMA ( p, d, q ) processes. In this paper we introduce some new techniques for handling the corresponding problem for a multiple time series X ( t ) with the main emphasis on AR and MA models. Types of joint nonstationarity (or rather almost nonstationarity) are defined and a method of analyzing such structures based on the ordered eigenvalues of the function C ( t ) = K ( t ) K ^-1(0) is discussed, where K ( t ) is the covariance function of X ( t ). It is proposed that the further identification procedure should be based on two X ² statistics and on the estimated trace and eigenvalues of C ( t ), the matrix correlation function p ( t ) and the matrix partial correlation function P ( t ). The suitability for identification purposes of each of these functions is examined in terms of such properties as scale-invariance, existence of normalized eigenvalues and standard errors. The methods introduced are illustrated on a 5-dimensional economic time series first studied by Quenouille and on a 4-dimensional smulated MA series.     

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.     

A NOTE ON NON-STATIONARITY AND CANONICAL ANALYSIS OF MULTIPLE TIME SERIES MODELS

Abstract. Box and Tiao (1977) established the correspondence between non-stationary roots and canonical correlations of an AR(1) process. In this paper, we give an alternative, more direct, proof of the correspondence and extend a special case of that result to AR( p ) processes. The usefulness of these results for multiple time series modelling is also briefly discussed.     

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.     

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.     

ON SOME AMBIGUITIES ASSOCIATED WITH THE FITTING OF ARMA MODELS TO TIME SERIES

Abstract. Examples are presented illustrating some ambiguities associated with the application of ARMA models to problems of signal extraction, multistep-ahead forecasting, spectrum approximation and linear quadratic control. Except in the signal extraction example, the ambiguities arise either from lack of sufficient autocovariance data to completely determine the process, or, often relatedly, from the approximate nature of the models used.     

STATIONARITY AND CENTRAL LIMIT THEOREM ASSOCIATED WITH BILINEAR TIME SERIES MODELS

Abstract. Consider the general bilinear times series model where {X_t; t= 0, L1, …} is a p-variate process, C (p x (s+ 1)), A (p x p). B _t(p x p) (1 ≤j≤q) are arbitrary matrices of constants, εT=[εt,…εt-q+1] and {εt; t=0, ±1, …} is a strictly stationary ergodic sequence of random variables. We investigate a set of minimal regularity conditions (on C, A, B _j and {ε_t}) under which we can establish the existence and causality of X _t and the asymptotic normality of the sample mean derived from { X _t}.     

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.     

LINEAR INTERPOLATORS AND THE INVERSE CORRELATION FUNCTION OF NON-STATIONARY TIME SERIES

Abstract. The inverse correlation function of a stationary time series was introduced by Cleveland (The inverse autocorrelations of a time series and their applications. Technometrics 14 (1972), 277–93). In this paper inverse correlations are defined for non-stationary time series {x_t, integer t} such that y_t= (1 —B^s)^dx_t is second-order stationary. The linear interpolator and the inverse process of {x_t} are also defined:their weights are shown to be time invariant and proportional to the inverse correlations. The interpolation method for the estimation of the inverse correlation function of a stationary time series is extended to the non-stationary series {x_t} and the asymptotic properties of the estimates are found to be similar to those in the stationary case.     

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.     

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.     

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.