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.     

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.     

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.     

SPECTRAL ESTIMATION AND DECONVOLUTION FOR A LINEAR TIME SERIES MODEL

Abstract. A vector linear time series model is observed as the sum of a convolution of an unknown signal and an additive noise process. The main objective is the estimation or deconvolution of the signal when the spectra of the signal and noise processes are unknown. We prove the strong consistency of a class of nonparametric spectral estimators derived by maximizing a particular Gaussian likelihood function. We also study the mean square convergence of the finite-sample deconvolution estimators as a function of the sample length T , the filter length M and the spectral bandwidth B_T = L_T/T .     

ESTIMATION IN LONG-MEMORY TIME SERIES MODEL

This study deals with the parameter estimation in long-memory time series models. An unbiased and consistent estimator is proposed. The proposed estimator is based on a least-squares method in the frequency domain, and it is computationally simple. Also, the Cramer–Rao lower bound is derived. The mean-square error of the proposed estimator is order of O(1/N), where N is the number of samples. The accuracy of the estimates is verified using synthetic long-memory time series data.     

INFERENCE FOR A SPECIAL BILINEAR TIME‐SERIES MODEL

It is well known that estimating bilinear models is quite challenging. Many different ideas have been proposed to solve this problem. However, there is not a simple way to do inference even for its simple cases. This article proposes a generalized autoregressive conditional heteroskedasticity‐type maximum likelihood estimator for estimating the unknown parameters for a special bilinear model. It is shown that the proposed estimator is consistent and asymptotically normal under only finite fourth moment of errors.     

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.     

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.     

A FIRST-ORDER TIME CONSTANT ESTIMATION FOR NONLINEAR DIFFUSION PROBLEMS

A Laplace transform–based procedure was proposed to calculate the effective time constant for a class of nonlinear diffusion problems. The governing mathematical representation was first estimated with a linear model by omitting the nonlinear term. The solution to this problem was later introduced into the original equation, which was solved with Laplace transforms, resulting in a first-order approximation of the real system's behavior. A time constant was calculated using frequency-domain expressions. Two case studies were considered to illustrate the methodology. As the rate of heat supplied to a rod is raised, the speed at which the temperature reached an equilibrium value decreased. Increasing the maximum velocity in reaction-diffusion transport by a factor of three lowered the time constant by only 1.7%. The applications of this method range from biosensor dynamics to process control.     

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.     

SPECTRAL DENSITY ESTIMATION VIA NONLINEAR WAVELET METHODS FOR STATIONARY NON-GAUSSIAN TIME SERIES

Abstract. In the present paper we consider nonlinear wavelet estimators of the spectral density f of a zero mean, not necessarily Gaussian, stochastic process, which is stationary in the wide sense. It is known in the case of Gaussian regression that these estimators outperform traditional linear methods if the degree of smoothness of the regression function varies considerably over the interval of interest. Such methods are based on a nonlinear treatment of empirical coefficients that arise from an orthonormal series expansion according to a wavelet basis.
The main goal of this paper is to transfer these methods to spectral density estimation. This is done by showing the asymptotic normality of certain empirical coefficients based on the tapered periodogram. Using these results we can show the risk equivalence to the Gaussian case for monotone estimators based on such empirical coefficients. The resulting estimator of f keeps all interesting properties such as high spatial adaptivity that are already known for wavelet estimators in the case of Gaussian regression.
It turns out that appropriately tuned versions of this estimator attain the optimal uniform rate of convergence of their L ₂ risk in a wide variety of Besov smoothness classes, including classes where linear estimators (kernel, spline) are not able to attain this rate. Some simulations indicate the usefulness of the new method in cases of high spatial inhomogeneity.     

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.     

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}.     

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.     

ASYMPTOTIC EXPANSIONS FOR THE DISTRIBUTION OF AN ESTIMATOR IN THE FIRST-ORDER AUTOREGRESSIVE PROCESS

Abstract. In this paper, two asymptotic expansions for the distribution for an estimator of the parameter in a first-order autoregressive process are derived, according to two situations. Some well known estimators are special cases of the estimator discussed here. The series expansions are carried to terms of order T ^-1.     

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.     

DIFFERENCE EQUATIONS FOR HIGHER-ORDER MOMENTS AND CUMULANTS FOR THE BILINEAR TIME SERIES MODEL BL(p, 0, p, 1)

Abstract. In this paper we obtain difference equations for higher-order moments and cumulants for the time series { X _t} satisfying the bilinear model BL( p , 0, p , 1). These equations are similar to the well-known Yule-Walker equations for the autoregressive moving-average model which are in terms of the second-order covariances. Thus they can be used for the tentative identification of bilinear models. Another application of these equations is in the computation of preliminary estimates of the parameters of the model. We shall illustrate this by means of simulations for two simple examples of bilinear 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.     

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.     

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.