Bilinear Time Series Models Research Articles

On the sieve M-estimation for a special bilinear time series model with time-functional variance noises

. The bilinear model has been frequently used in fields like control theory and economics to model seismic data. In this article, time-functional variance (TFV) noises are embedded into a specific bilinear model. We propose a generalized autoregressive conditional heteroskedasticity-type maximum likelihood estimator (GMLE) with a sieve method and then provide various inferential techniques based on this GMLE. It is shown that under the finite fourth moment of errors, the GMLE is consistent and asymptotically normally distributed. A simulation study and analysis of real data are additionally carried out to evaluate GMLE’s finite sample performance.

Comparative Analysis of Bilinear Time Series Models with Time-Varying and Symmetric GARCH Coefficients: Estimation and Simulation

This paper makes a significant contribution by focusing on estimating the coefficients of a sample of non-linear time series, a subject well-established in the statistical literature, using bilinear time series. Specifically, this study delves into a subset of bilinear models where Generalized Autoregressive Conditional Heteroscedastic (GARCH) models serve as the white noise component. The methodology involves applying the Klimko–Nilsen theorem, which plays a crucial role in extracting the asymptotic behavior of the estimators. In this context, the Generalized Autoregressive Conditional Heteroscedastic model of order (1,1) noted that the GARCH (1,1) model is defined as the white noise for the coefficients of the example models. Notably, this GARCH model satisfies the condition of having time-varying coefficients. This study meticulously outlines the essential stationarity conditions required for these models. The estimation of coefficients is accomplished by applying the least squares method. One of the key contributions lies in utilizing the fundamental theorem of Klimko and Nilsen, to prove the asymptotic behavior of the estimators, particularly how they vary with changes in the sample size. This paper illuminates the impact of estimators and their approximations based on varying sample sizes. Extending our study to include the estimation of bilinear models alongside GARCH and GARCH symmetric coefficients adds depth to our analysis and provides valuable insights into modeling financial time series data. Furthermore, this study sheds light on the influence of the GARCH white noise trace on the estimation of model coefficients. The results establish a clear connection between the model characteristics and the nature of the white noise, contributing to a more profound understanding of the relationship between these elements.

Novel Insights into Estimation of Bilinear Time Series Models with Exponential and Symmetric Coefficients

This paper focuses on the estimation and simulation of a specific subset of bilinear time series models characterized by dynamic exponential coefficients. Employing an exponential framework, we delve into the implications of the exponential function for our estimation process. Our primary aim is to estimate the coefficients of the proposed model using exponential coefficients derived from time-varying parameters. Through this investigation, our goal is to shed light on the asymptotic behaviors of the estimators and scrutinize their existence and probabilistic traits, drawing upon the foundational theorem established by Klimko and Nilsen. The least squares approach is pivotal in both estimating coefficients and analyzing estimator behavior. Moreover, we present a practical application to underscore the real-world implications of our research. By offering concrete examples of applications and simulations, we endeavor to provide readers with a comprehensive understanding of the implications of our work within the realm of time series analysis, specifically focusing on bilinear models and time-varying exponential coefficients. This multifaceted approach underscores the potential impact and practical relevance of our findings, contributing to the advancement of the field of time series analysis. To discern the symmetry characteristics of the model, we estimate it using coefficients that sum to zero and conduct a brief comparative analysis of two bilinear models.

A special integer-valued bilinear time series model with applications

A special integer-valued bilinear time series model with applications

Integer-valued Bilinear Model with Dependent Counting Series

The present work proposes a new stationary integer-valued bilinear time series model with dependent counting series. The model will enable one to tackle the presence of some correlation between underlying events. The various probabilistic and statistical properties of the model are discussed, unknown parameters are estimated by several methods. Moreover, the performance of the estimation methods is illustrated through a simulation study and an empirical application to two data sets.

Integer-valued bilinear time series model with signed generalized power series thinning operator

ABSTRACT We propose an integer-valued bilinear model with signed generalized power series thinning operator which provides more flexibility in modelling negative integer-valued time series together with negative correlation. Strict stationarity and some properties of the process are determined, unknown parameters are estimated by several methods and the behaviour of the estimators is described through the numerical results. Finally, the analysis of three real data sets has been presented for illustrative purposes.

On the Performance of the Mixed Seasonal Autoregressive Integrated Moving Average One-Dimensional Bilinear Time series model

On the Performance of the Mixed Seasonal Autoregressive Integrated Moving Average One-Dimensional Bilinear Time series model

An integer-valued bilinear time series model via two random operators

ABSTRACTThis paper presents a new stationary integer-valued bilinear time series model of the first order by mixing the thinning and Pegram operators. Some statistical properties of the model are obtained, involving the conditional moments, autocovariance and spectral density function. Estimation of the model parameters is discussed using the Yule-Walker and conditional least squares methods with a simulation study for evaluating the performance of those estimators. Applicability of the process is investigated using a practical count data set with comparing the model to a competitive bilinear model using some marginal distributions of innovations. Issue of forecasting data is discussed under the proposed model.

Model‐based forecast adjustment: With an illustration to inflation

AbstractThis paper introduces the idea of adjusting forecasts from a linear time series model where the adjustment relies on the assumption that this linear model is an approximation of a nonlinear time series model. This way of creating forecasts could be convenient when inference for a nonlinear model is impossible, complicated or unreliable in small samples. The size of the forecast adjustment can be based on the estimation results for the linear model and on other data properties such as the first few moments or autocorrelations.An illustration is given for a first‐order diagonal bilinear time series model, which in certain properties can be approximated by a linear ARMA(1, 1) model. For this case, the forecast adjustment is easy to derive, which is convenient as the particular bilinear model is indeed cumbersome to analyze in practice.An application to a range of inflation series for low‐income countries shows that such adjustment can lead to some improved forecasts, although the gain is small for this particular bilinear time series model.

Identification of stable elementary bilinear time-series model

Abstract The paper presents new approach to estimation of the coefficients of an elementary bilinear time series model (EB). Until now, a lot of authors have considered different identifiability conditions for EB models which implicated different identifiability ranges for the model coefficient. However, all of these ranges have a common feature namely they are significantly narrower than the stability range of the EB model. This paper proposes a simple but efficient solution which makes an estimation of the EB model coefficient possible within its entire stability range.

Vector bilinear autoregressive time series model and its superiority over its linear autoregressive counterpart

In this research, a vector bilinear autoregressive time series model was proposed and used to model three revenue series ( X 1 , X 2 , X 3 ) . The “orders” of the three series were identified on the basis of the distribution of autocorrelation and partial autocorrelation functions and were used to construct the vector bilinear models. The estimates obtained from the bilinear fits were compared graphically with those obtained from fitting linear (autoregressive) models. Residual variance and Box-Ljung Q statistic comparisons were also made. The result showed that vector bilinear autoregressive (BIVAR) models provide better estimates than the long embraced linear models. Keywords: Linear time series, Autoregressive process, Autocorrelation function, Partial autocorrelation function, Vector time series and bilinear vector process

Periodic integer-valued bilinear time series model

ABSTRACTThis paper deals with the study of some probabilistic and statistical properties of a periodic integer-valued diagonal bilinear model. The existence of a periodically strict stationary integer-valued process is shown. Sufficient conditions for the periodically stationary, both in the first and second orders, are established. The closed-forms of the mean and the second moment are obtained. The closed-form of the periodic autocovariance function is established. The Yule–Walker estimations of the underlying parameters are obtained. A simulation study is provided.

On periodic time-varying bilinear processes: structure and asymptotic inference

This paper is devoted to the bilinear time series models with periodic-varying coefficients $$\left( { PBL}\right) $$ . So, firstly conditions ensuring the existence of periodic stationary solutions of the $${ PBL}$$ and the existence of higher-order moments of such solutions are given. A distribution free approach to the parameter estimation of $${ PBL}$$ is presented. The proposed method relies on minimum distance estimator based on the first and second order empirical moments of the observed process. Consistency and asymptotic normality of the estimator are discussed. Examples and Monte Carlo simulation results illustrate the practical relevancy of our general theoretical results are presented.

The LASSO Method for Bilinear Time Series Models

In this article we propose a method called GLLS for the fitting of bilinear time series models. The GLLS procedure is the combination of the LASSO method, the generalized cross-validation method, the least angle regression method, and the stepwise regression method. Compared with the traditional methods such as the repeated residual method and the genetic algorithm, GLLS has the advantage of shrinking the coefficients of the models and saving the computational time. The Monte Carlo simulation studies and a real data example are reported to assess the performance of the proposed GLLS method.

Imputation of Missing Values for Pure Bilinear Time Series Models with Normally Distributed Innovations

In this study, estimates of missing values for bilinear time series models with normally distributed innovations were derived by minimizing the h-steps-ahead dispersion error. For comparison purposes, missing value estimates based on artificial neural network (ANN) and exponential smoothing (EXP) techniques were also obtained. Simulated data was used in the study. 100 samples of size 500 each were generated for different pure bilinear time series models using the R-statistical software. In each sample, artificial missing observations were created at data positions 48, 293 and 496 and estimated using these methods. The performance criteria used to ascertain the efficiency of these estimates were the mean absolute deviation (MAD) and mean squared error (MSE). The study found that optimal linear estimates were the most efficient estimates for estimating missing values. Further, the optimal linear estimates were equivalent to one step-ahead forecast of the missing value. The study recommends OLE estimates for estimating missing values for pure bilinear time series data with normally distributed innovations.

Consistency of quasi-maximum likelihood estimator for Markov-switching bilinear time series models

In this paper, we consider the class of Markov switching bilinear processes (MS–BL) that offer remarkably rich dynamics and may be considered as an alternative to model non Gaussian data which exhibit structural changes. In these models, the parameters are allowed to depend upon a latent time-homogeneous Markov chain with finite state space. Analysis based on models with time-varying coefficients has long suffered from the lack of an asymptotic theory except in very restrictive cases. So, first, some basic issues concerning this class of models including sufficient conditions ensuring the existence of stationarity (in strict sense) and ergodic solutions are given. Second, we illustrate the fundamental problems linked with MS–BL models, i.e., parameters estimation by considering a maximum likelihood (ML) approach. So, we provide the detail on the asymptotic properties of ML, in particular, we discuss conditions for its strong consistency.

Imputation of Missing Values for BL (P,0,P,P) Models with Normally Distributed Innovations

This study derived estimates of missing values for bilinear time series models BL (p, 0, p, p) with normally distributed innovations by minimizing the h-steps-ahead dispersion error. For comparison purposes, missing value estimates based on artificial neural network (ANN) and exponential smoothing (EXP) techniques were also obtained. Simulated data was used in the study. 100 samples of size 500 each were generated for bilinear time series models BL (1, 0, 1, 1) using the R-statistical software. In each sample, artificial missing observations were created at data positions 48, 293 and 496 and estimated using these methods. The performance criteria used to ascertain the efficiency of these estimates were the mean absolute deviation (MAD) and mean squared error (MSE). The study found that optimal linear estimates were the most efficient estimates for estimating missing values for BL (p, 0, p, p). The study recommends OLE estimates for estimating missing values for bilinear time series data with normally distributed innovations.

Forecasting In One-Dimensional And Generalized Integrated Autoregressive Bilinear Time Series Models

In this paper, forecast of one-dimensional integrated autoregressive bilinear is compared with forecast of generalized integrated autoregressive bilinear model. We describe the method for estimation of these models and the forecast. It is also pointed out that for this class of non-linear time series models; it is possible to obtain optimal forecast. The estimation technique is illustrated with respect to a time series, and the optimal forecast of these time series are calculated. A comparison of these forecasts is made using the two models under study. The mean square error for forecast in generalized integrated autoregressive bilinear model is smaller than the mean square error for forecast in one-dimensional integrated autoregressive bilinear model. Though the two models are adequate for forecast when compared with the real series but forecast with generalized integrated autoregressive bilinear model is more adequate.Keywords: Optimal Forecast, Non-Linear Time Series Models, Bilinear Models, Estimation Technique, Mean Square Error.

패널 중선형 시계열 모형의 동질성 검정

패널 시계열자료 분석에서 모수축약의 원칙에 충실하기 위해서 동질성 검정을 수행한다. 본 논문에서는 독립적인 중선형 시계열 패널 자료의 동질성 검정을 수행하기 위하여 먼저 중선형 시계열 모형의 정상성 조건을 구하고 최우추정량과 동질성 검정통계량과 극한 분포를 이끌어내며, 실증분석으로 우리나라 8도의 Mumps 패널자료를 이용해 8개 지역의 발병 추이에 대한 동질성 검정을 수행한다. The acceptance of the test of the homogeneity for panel time series models allows for the pooling of the series to achieve parsimony. In this paper, we introduce a panel bilinear time series model as well as derive the stationary condition and the limiting distribution of the test statistic of the homogeneity test for the model. For the applications study, we use Korea Mumps data from January 2001 to December 2008. Finally, we perform test of homogeneity for the panel data with 8 independent bilinear time series.

Synthetic detection of change point and outliers in bilinear time series models

This paper proposes a procedure of synthetic detection for the location of a change point and outliers in bilinear time series models with a change after an unknown time point. Based on Bayesian framework, we first derive the conditional posterior distribution of the change point and from that distribution estimate the position of the change point. Then we use these results to detect the outliers in the time series before and after that change point via Gibbs sampler algorithm. Our simulation studies show that the proposed procedure is effective.