Bilinear Time Series Models Research Articles

Bayes Inference for the Spatial Bilinear Time Series Model with Application to Epidemic Data

Spatial time series data can be viewed as a set of time series simultaneously collected at a number of spatial locations. This paper studies Bayesian inferences in a spatial time bilinear model with a Gibbs sampling algorithm to overcome problems in the numerical analysis techniques of a spatial time series model. For illustration, the data set of mumps cases reported from the Korea Center for Disease Control and Prevention monthly over the years 2001~2009 are selected for analysis.

Generalized and subset integrated autoregressive bilinear time series models with applications

Generalized integrated autoregressive bilinear model which is capable of achieving stationary for all non linear series is proposed and compared with subset generalized integrated autoregressive bilinear model using the residual variance to see which perform better. The parameters of the proposed models are estimated using Newton-Raphson iterative method and Marquardt algorithm and the statistical properties of the derived estimates were investigated. An algorithm was proposed to eliminate redundant parameters from the full order generalized integrated autoregressive bilinear models. To determine the order of the models, Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) were adopted. Generalized integrated autoregressive bilinear models are fitted to Wolfer sunspot numbers and stationary conditions are satisfied. Generalized integrated autoregressive bilinear model performed better than subset generalized integrated autoregressive bilinear model.

Covariance analysis of the squares of the purely diagonal bilinear time series models

The covariance structure among other properties of the square of the purely diagonal bilinear time series model is obtained. The time series properties of these squares are compared with those of the linear moving average time series model. It was discovered that the square of a linear moving average process is also identified as a moving average process whereas, while the nonlinear purely diagonal bilinear process is identified as a linear moving average process, its square is identified as an autoregressive moving average process.

On the Estimation and Performance of One-dimensional Autoregressive Integrated Moving Average Bilinear Time Series Models

On the Estimation and Performance of One-dimensional Autoregressive Integrated Moving Average Bilinear Time Series Models

On an independent and identically distributed mixture bilinear time-series model

A class of nonlinear time-series models in which the underlying process follows a finite mixture of bilinear representations is proposed. The mixture feature appears in the conditional distribution of the process which is given as a finite mixture of distributions evaluated at the normed innovations of diagonal bilinear specifications. This class is aimed at capturing special characteristics exhibited by many observed time series such as tail heaviness, multimodality, asymmetry and change in regime. Some probabilistic properties of the proposed model, namely strict and second-order stationarity, geometric ergodicity, covariance structure, existence of higher order moments, tail behaviour and invertibility, are first studied. Parameter estimation is then performed through the EM algorithm, performance of which is shown via simulation experiments. Applications to some real-time-series data are proposed and through which it is shown how neglecting the mixture framework in a bilinear representation results in a loss in adequacy.

Detection of Outliers and Patches in Bilinear Time Series Models

We propose a Gibbs sampling algorithm to detect additive outliers and patches of outliers in bilinear time series models based on Bayesian view. We first derive the conditional posterior distributions, and then use the results of first Gibbs run to start the second adaptive Gibbs sampling. It is shown that our procedure could reduce possible effects on masking and swamping. At last, some simulations are performed to demonstrate the efficacy of detection and estimation by Monte Carlo methods.

Subsetting and identification of optimal models in one-dimensional bilinear time series modeling

To date, significant efforts have been made to study the theory of bilinear time series models, especially simple bilinear models. Much less efforts, however, have been made to identify optimal models. Focused on optimal model identification, this study attempts to fill this gap. Full and subset one-dimensional bilinear models are proposed and shown to be robust in achieving stationarity for all non-linear series. The parameters of the models are estimated using robust nonlinear least-square method and Newton-Raphson iterative method, and statistical properties of the derived estimates are investigated. An algorithm is proposed to eliminate redundant parameters from full order bilinear models.

On Stationarity and Ergodicity of the Bilinear Model with Applications to GARCH Models

We establish sufficient conditions for the bilinear time series model to be strictly stationary and ergodic in terms of its associated Lyapunov exponent. In two special cases, we verify that the conditions are also necessary. We then use these results to give necessary and sufficient conditions for stationarity of specific GARCH models which can be written as a bilinear model, including linear GARCH, Power GARCH, EGARCH amongst others. These results generalise the ones found in Bougerol and Picard (1992b), Duan (1997), and Nelson (1990). In many cases, the conditions are weaker than the ones found elsewhere in the literature.

On stationarity and ergodicity of the bilinear model with applications to GARCH models

Abstract.We establish sufficient conditions for the bilinear time‐series model to be strictly stationary and ergodic in terms of its associated Lyapunov exponent. In two special cases, we verify that the conditions are also necessary. We then use these results to give necessary and sufficient conditions for stationarity of specific generalized autoregressive conditionally heteroskedastic (GARCH) models which can be written as a bilinear model, including linear GARCH, Power GARCH, EGARCH among others. These results generalize the ones found in the studies of, among others, Bougerol and Picard [Journal of Econometrics 52 (1992) 115], Duan [Journal of Econometrics 79 (1997) 97] and Nelson [Econometric Theory 6 (1990) 318]. In many cases, the conditions are weaker than the ones found elsewhere in the literature.

Yule–Walker type estimators in periodic bilinear models: strong consistency and asymptotic normality

This paper studies the covariance structure and the asymptotic properties of Yule–Walker (YW) type estimators for a bilinear time series model with periodically time-varying coefficients. We give necessary and sufficient conditions ensuring the existence of moments up to eighth order. Expressions of second and third order joint moments, as well as the limiting covariance matrix of the sample moments are given. Strong consistency and asymptotic normality of the YW estimator as well as hypotheses testing via Wald’s procedure are derived. We use a residual bootstrap version to construct bootstrap estimators of the YW estimates. Some simulation results will demonstrate the large sample behavior of the bootstrap procedure.

Estimation Of The Parameters Of The Bilinear Seasonal Arima Time Series Model

The parameters of the seasonal bilinear time series model were estimated analytically. This was followed by some numerical illustrations. The closeness of the estimated values to the true values attested to the adequacy of the least square method of minimizing error and the Newton-Raphson iterative procedure. Keywords: Seasonal Bilinear Time Series model, Least Square Method, Newton-Raphson iterative procedure. Global Journal of Mathematical Sciences Vol. 7 (1) 2008: pp. 1-4

Testing coefficients of AR and bilinear time series models by a graphical approach

AR and bilinear time series models are expressed as time series chain graphical models, based on which, it is shown that the coefficients of AR and bilinear models are the conditional correlation coefficients conditioned on the other components of the time series. Then a graphically based procedure is proposed to test the significance of the coefficients of AR and bilinear time series. Simulations show that our procedure performs well both in sizes and powers.

Forecasting performance of a seasonal bilinear arima time series model

Forecasting performance of a seasonal bilinear arima time series model

Initial and final estimates of the Bilinear seasonal time series model

A particular class of non-linear models which has been found to be useful in many fields is the bilinear models. A special class of it is discussed in this paper. In getting the estimates of the parameters of this model special attention was paid to the problem of having good initial estimates as it is proposed that with good initial values of the parameters the estimates obtaining by the Newton-Raphson iterative technique usually not only converge but also are good estimates. In this paper we examined the initial and final estimates of the bilinear seasonal time series model. The Box-Jenkins linear convergence process, the Newton-Raphson iterative procedure, the Fortran Progran and the MINITAB software package were all employed in achieving both the initial estimates and the final estimates of the bilinear seasonal time series model studied. The results showed considerable closeness between the initial estimates and the final estimates for both simulations (n - 100 and n - 500). This confirmed that the initial estimates are good enough. The implication of this is that in estimations of this nature efforts should be made using the right procedures to achieve good initial estimates so that the final estimates could be achieved quickly after few iterations. JONAMP Vol. 11 2007: pp. 615-620

Note on integer-valued bilinear time series models

This note reconsiders the nonnegative integer-valued bilinear processes introduced by Doukhan et al. [Doukhan, P., Latour, A., Oraichi, D., 2006. A simple integer-valued bilinear time series model. Adv. Appl. Prob. 38, 559–578]. Using a hidden Markov argument, we extend their result of the existence of a stationary solution for the INBL(1, 0, 1, 1) process to the class of superdiagonal INBL ( p , q , m , n ) models. Our approach also yields improved parameter restrictions for several moment conditions compared to the ones in [Doukhan, P., Latour, A., Oraichi, D., 2006. A simple integer-valued bilinear time series model. Adv. Appl. Prob. 38, 559–578].

Hellinger distance estimation of general bilinear time series models

In the present paper, minimum Hellinger distance estimates for parameters of a bilinear time series model are presented. The probabilistic properties such as stationarity, existence of moments of the stationary distribution and strong mixing property of the model are well known (see for instance [J. Liu, A note on causality and invertibility of a general bilinear time series model, Adv. Appl. Probab. 22 (1990) 247–250; J. Liu, P.J. Brockwell, On the general bilinear time series model, J. Appl. Probab. 25 (1988) 553–564; D.T. Pham, The mixing property of bilinear and generalised random coefficients autoregressive models, Stoch. Process Appl. 23 (1986) 291–300]). We establish, under some mild conditions, the consistency and the asymptotic normality of the minimum Hellinger distance estimates of the parameters of the model.

Evolutionary transfer functions of bilinear processes with time-varying coefficients

The aim of this paper is to construct the time-varying transfer function systems for bilinear time series models with time-varying coefficient. Recursive formulae are given for time varying symmetrized transfer functions.

A simple integer-valued bilinear time series model

In this paper, we extend the integer-valued model class to give a nonnegative integer-valued bilinear process, denoted by INBL(p,q,m,n), similar to the real-valued bilinear model. We demonstrate the existence of this strictly stationary process and give an existence condition for it. The estimation problem is discussed in the context of a particular simple case. The method of moments is applied and the asymptotic joint distribution of the estimators is given: it turns out to be a normal distribution. We present numerical examples and applications of the model to real time series data on meningitis and Escherichia coli infections.

A simple integer-valued bilinear time series model

In this paper, we extend the integer-valued model class to give a nonnegative integer-valued bilinear process, denoted by INBL(p,q,m,n), similar to the real-valued bilinear model. We demonstrate the existence of this strictly stationary process and give an existence condition for it. The estimation problem is discussed in the context of a particular simple case. The method of moments is applied and the asymptotic joint distribution of the estimators is given: it turns out to be a normal distribution. We present numerical examples and applications of the model to real time series data on meningitis and Escherichia coli infections.

On the non-negative first-order exponential bilinear time series model

In this paper the bilinear model BL(1,0,1,1) driven by exponential distributed innovations is studied in some detail. Conditions under which the model is strictly stationary as well as some properties of the stationary distribution are discussed. Moreover, parameter estimation is also addressed.