Nonlinear Time Series Models Research Articles

Kalman filter-based modelling and forecasting of stochastic volatility with threshold

We propose a parametric nonlinear time-series model, namely the Autoregressive-Stochastic volatility with threshold (AR-SVT) model with mean equation for forecasting level and volatility. Methodology for estimation of parameters of this model is developed by first obtaining recursive Kalman filter time-update equation and then employing the unrestricted quasi-maximum likelihood method. Furthermore, optimal one-step and two-step-ahead out-of-sample forecasts formulae along with forecast error variances are derived analytically by recursive use of conditional expectation and variance. As an illustration, volatile all-India monthly spices export during the period January 2006 to January 2012 is considered. Entire data analysis is carried out using EViews and matrix laboratory (MATLAB) software packages. The AR-SVT model is fitted and interval forecasts for 10 hold-out data points are obtained. Superiority of this model for describing and forecasting over other competing models for volatility, namely AR-Generalized autoregressive conditional heteroscedastic, AR-Exponential GARCH, AR-Threshold GARCH, and AR-Stochastic volatility models is shown for the data under consideration. Finally, for the AR-SVT model, optimal out-of-sample forecasts along with forecasts of one-step-ahead variances are obtained.

Exploiting the interpretability and forecasting ability of the RBF-AR model for nonlinear time series

In this paper, we explore the radial basis function network-based state-dependent autoregressive (RBF-AR) model by modelling and forecasting an ecological time series: the famous Canadian lynx data. The interpretability of the state-dependent coefficients of the RBF-AR model is studied. It is found that the RBF-AR model can account for the phenomena of phase and density dependencies in the Canadian lynx cycle. The post-sample forecasting performance of one-step and two-step ahead predictors of the RBF-AR model is compared with that of other competitive time-series models including various parametric and non-parametric models. The results show the usefulness of the RBF-AR model in this ecological time-series modelling.

Adaptive particle allocation in iterated sequential Monte Carlo via approximating meta-models

Sequential Monte Carlo (SMC) filters (also known as particle filters) are widely used in the analysis of non-linear and non-Gaussian time series models in diverse application areas such as engineering, finance, and epidemiology. When a time series contains an observation that is very unlikely given the previous observations, evaluation of its conditional log likelihood by SMC can suffer from high variance. The presence of one or more such observations can result in poor Monte Carlo estimate of the overall likelihood. In this article, we develop a novel strategy of particle allocation for off-line iterated SMC based filters, in order to reduce the overall variance of the likelihood estimate to enable efficient computation. The complications arising from the intractability of the actual SMC variance is handled via an approximating meta-model, in which we model the SMC errors in the evaluation of conditional log likelihood of the observations as an autoregressive process. We demonstrate numerical results on both simulated and real data sets where adaptive particle allocation results in 54 % lower overall variance over the naive equal allocation of particles at all time points in simulations and 53 % lower variance on a real time series model of epidemic malaria transmission. The approximating model approach presented in this article is novel in the context of SMC and offers a computationally attractive procedure for practical analysis of a broad class of time series models.

Fitting EXPAR Models Through the Extended Kalman Filter

Exponential autoregressive (EXPAR) family of parametric nonlinear time-series models, which is a discrete-time approximation of continuous-time nonlinear stochastic dynamical system, is considered. A heartening feature of this model is that it is capable of describing those data sets that depict cyclical variations. The estimation procedure for EXPAR models is developed using extended Kalman filter (EKF). Through simulation studies, it is shown that EKF is very efficient for fitting EXPAR models. Formulae for optimal one-step and two-step ahead out-of-sample forecasts are derived analytically by recursive use of conditional expectation. Conditions for the existence of limit cycle behaviour for EXPAR models are also established. Superiority of EKF method vis-a-vis Genetic algorithms (GA) method for fitting EXPAR models is shown through simulation studies. As an illustration, EXPAR models are employed for modelling and forecasting Oil sardine, Mackerel and Bombay duck time-series landings data in India. It is shown that all the three fitted models exhibit the desirable feature of existence of limit cycle behaviour. It is concluded that the EXPAR model performs better than ARIMA methodology for both modelling and forecasting purposes for the data sets under consideration.

Algorithm Analysis on Internal Automatic Calibration of Category Theory Based on Nonlinear Time Series Model

Category reflects the basic concept of nature and universal connection, and is the logical form of human rational thinking. In this paper, through the rich practical experience in the field of English teaching, on the basic of prototype category theory to study of English grammar teaching methods, the author attempts to explore the scientific and rational English grammar teaching concept and strategy.

Double Generalized Threshold Models with constraint on the dispersion by the mean

Generalized Threshold Model (GTM) is a non-linear time series model which generalizes the Threshold Autoregressive Model (TAR) to implement the idea of the Generalized Linear Model under the threshold time series framework. However, the dispersion parameter is usually assumed as constant in the context of Generalized Linear Model which does not hold in general. In this paper, the GTM is extended to a Double Generalized Threshold Model (DGTM) where the dispersion parameter, defined as the expected deviance of the individual response about its mean, varies throughout the entire sample. The variation of the dispersion parameter can be predicted by another threshold type generalized linear model, which is interlinked with the threshold model for the mean and can be estimated simultaneously.

The CUSUM Test for Detecting Structural Changes in Strong Mixing Processes

Strong mixing property holds for a broad class of linear and nonlinear time series models such as Auto-Regressive Moving Average Processes and Generalized Auto-Regressive Conditional Heteroscedasticity Processes models. In this article, we study correlation structure of strong mixing sequences, and some asymptotic properties are presented. We also present a new method for detecting change point in correlation structure of strong mixing sequences, and present a nonparametric sequential analysis for detecting changes named cumulative sum test statistic for this. Asymptotic consistency of this test statistics is shown. This method is applied to simulated data of some linear and nonlinear models and power of the test is evaluated. For linear models, it is shown that this method has a better performance in comparison to Berkes et al. (2009).

Research on a Kind of Deformation Prediction Model of Wavelet Neural Network

Combining with the localized analysis capability of the wavelet and the learning capability of the neural network, this article investigates the wavelet neural network nonlinear time series model and the wavelet neural network architecture integrated model which combines the affine transformation and the rotation transformation ,moreover, based on wavelet neural network algorithm that substitutes the Morlet wavelet function for the Sigmoid function, researchers analyze the application of Wavelet Neural Network in settlement deformation of high-speed rail. The example this paper discussed shows that the wavelet neural network has good results in the deformation prediction, especially in large complicated engineering.

The Importance of Nonlinearity in Reproducing Business Cycle Features

This paper considers the ability of simulated data from linear and nonlinear time-series models to reproduce features in U.S. real GDP data related to business cycle phases. We focus our analysis on a number of linear ARIMA models and nonlinear Markov-switching models. To determine the timing of business cycle phases for the simulated data, we present a model-free algorithm that is more successful than previous methods at matching NBER dates and associated features in the postwar data. We find that both linear and Markov-switching models are able to reproduce business cycle features such as the average growth rate in recessions, the average length of recessions, and the total number of recessions. However, we find that Markov-switching models are better than linear models at reproducing the variability of growth rates in different business cycle phases. Furthermore, certain Markov-switching specifications are able to reproduce high-growth recoveries following recessions and a strong correlation between the severity of a recession and the strength of the subsequent recovery. Thus, we conclude that nonlinearity is important in reproducing business cycle features.

Influence of News in Moscow and New York on Returns and Risks on Baltic State Stock Indices

The impact of news of the Moscow and New York stock market exchanges on the returns and volatilities of the Baltic state stock market indices is studied using daily return data for the period of 2000-2005. A nonlinear time series model that accounts for asymmetries in the conditional mean and variance functions is used for the empirical work. News from New York have stronger effect on returns in Tallinn, than news from Moscow. High risk shocks in New York have a strong impact on volatility in Tallinn, whereas volatility of Vilnius is more influenced by high risk shocks from Moscow. Riga seems to be autonomous to news arriving from abroad.

Bayesian Analysis of the Functional-Coefficient Autoregressive Heteroscedastic Model

In this paper, we propose a new model called the functional-coefficient autoregressive heteroscedastic (FARCH) model for nonlinear time series. The FARCH model extends the existing functional-coefficient autoregressive models and double-threshold autoregressive heteroscedastic models by providing a flexible framework for the detection of nonlinear features for both the conditional mean and conditional variance. We propose a Bayesian approach, along with the Bayesian P-splines technique and Markov chain Monte Carlo algorithm, to estimate the functional coefficients and unknown parameters of the model. We also conduct model comparison via the Bayes factor. The performance of the proposed methodology is evaluated via a simulation study. A real data set derived from the daily S&P 500 Composite Index is used to illustrate the methodology.

Specifying Asymmetric STAR models with Linear and Nonlinear GARCH Innovations: Monte Carlo Approach

Economic and finance time series are mostly asymmetric and are expected to be modeled using asymmetrical nonlinear time series models. Smooth Transition Autoregressive (STAR) models: Logistic (LSTAR) and Exponential (ESTAR) are known to be asymmetric and symmetric respectively, and these have gained their popularity in empirical econometric modelling. Estimating only the LSTAR model for economic and finance data may give models that are not well diagnosed. Under non-normal and heteroscedastic innovations, the residuals of these models are estimated using Generalized Autoregressive Conditionally Heteroscedastic (GARCH) models with variants which include linear and nonlinear forms. The small sample properties of these STAR-GARCH variants are yet to be established but these properties are investigated using Monte Carlo (MC) simulation approach. An MC investigation was conducted to investigate the performance of selections of STAR-GARCH models by the classical nonlinear selection approaches. The ARCH(1) and GARCH(1,1) were the linear GARCH while the Logistic Smooth Transition-ARCH (LST-ARCH(1,1)), Logistic Smooth Transition-GARCH (LST-GARCH(1,1)) and Asymmetric Nonlinear Smooth Transition-GARCH (ANST-GARCH(1,1)) were the nonlinear GARCH. The nonlinearity parameter in the variance equations and Autoregressive (AR) parameters were varied along with different sample sizes. With the assumption of normality, the results showed that the selection of LSTAR models were actually affected by the structure of the innovations and this improved as sample size increased. Misspecification tests actually showed that these models cannot be misrepresented in the real sense.

Procedures for outlier detection in circular time series models

It is known that the occurrence of outliers in linear or non-linear time series models may have adverse effects on the modelling and statistical inference of the data. Consequently, extensive research has been conducted on developing outlier detection procedures so that outliers may be properly managed. However, no work has been done on the problem of outliers in circular time series data. This problem is the focus of this paper. The main objective is to develop novel numerical and graphical procedures for detecting these outliers in circular time series data.A number of circular time series models have been proposed including the circular autoregressive model. We extend the iterative outlier detection procedure which has been successfully used in linear time series models to the circular autoregressive model. The proposed procedure shows a good performance when investigated via simulation for the circular autoregressive model of order one. At the same time, several statistical techniques have been used to detect the change of preferred trend in time series data using SLIME and CUSUM plots. While the methods fail to indicate directly the outliers in circular time series data, we use the ideas employed to develop three novel graphical procedures for identifying the outliers. For illustration, we apply the procedures to a particular set of wind direction data. An agreement between the results of the graphical and iterative detection procedures is observed. These procedures could be very useful in improving the modelling and inferential processes for circular time series data.

DECISION SUPPORT SYSTEM ARCHITECTURE FOR FORECASTING OF NONSTATIONARY FINANCIAL PROCESSES AND CORRESPONDING RISKS

A concept of a decision support system (DSS) for modeling and forecasting of economic and financial processes is proposed as well as its software implementation. The main functions of the DSS are in modeling and short-term forecasting of nonstationary nonlinear time series in various areas of human activities. On the basis of the proposed concept the software system was developed that possesses a set of the following useful features such as constructing of high quality forecasting mathematical models using time series statistical data; testing automatically quality of models and the forecasts based on them using appropriate sets of statistical criteria; combining various computational procedures for improving forecasts estimates; easy modification and expanding of existing system functionality. The software has been created with the use of cross platform instrumentation Qt and programming languages C++ and QML. Thanks to the open system architecture the functional possibilities of the system created can be expanded substantially at the expense of new model types, extra computational techniques and analytical instruments, possibilities for graphical representation of results as well as the means for interaction with user.

Editor's Special Invited Paper: Sequential Estimation for Time Series Models

This article revisits sequential estimation of the autoregressive parameter β in a first-order autoregressive (AR(1)) model and construction of a sequential confidence region for a parameter vector θ in a first-order threshold autoregressive (TAR(1)) model. To resolve a theoretical conjecture raised in Sriram (1986), we provide a comprehensive numerical study that strongly suggests that the regret in using a sequential estimator of β can be significantly negative for many heavy-tailed error distributions and even for normal errors. Secondly, to investigate yet another conjecture about the limiting distribution of a sequential pivotal quantity for θ in a TAR(1) model, we conduct an extensive numerical study that strongly suggests that the sequential confidence region has much better coverage probability than that of a fixed sample counterpart, regardless of whether the θ values are inside or on or near the boundary of the ergodic region of the series. These highlight the usefulness of sequential sampling methods in fitting linear and nonlinear time series models.

Learning causal graphs of nonlinear structural vector autoregressive model using information theory criteria

Detection and clarification of cause-effect relationships among variables is an important problem in time series analysis. Traditional causality inference methods have a salient limitation that the model must be linear and with Gaussian noise. Although additive model regression can effectively infer the nonlinear causal relationships of additive nonlinear time series, it suffers from the limitation that contemporaneous causal relationships of variables must be linear and not always valid to test conditional independence relations. This paper provides a nonparametric method that employs both mutual information and conditional mutual information to identify causal structure of a class of nonlinear time series models, which extends the additive nonlinear times series to nonlinear structural vector autoregressive models. An algorithm is developed to learn the contemporaneous and the lagged causal relationships of variables. Simulations demonstrate the effectiveness of the proposed method.

Consistency of kernel density estimators for causal processes

Using the blocking techniques and m-dependent methods, the asymptotic behavior of kernel density estimators for a class of stationary processes, which includes some nonlinear time series models, is investigated. First, the pointwise and uniformly weak convergence rates of the deviation of kernel density estimator with respect to its mean (and the true density function) are derived. Secondly, the corresponding strong convergence rates are investigated. It is showed, under mild conditions on the kernel functions and bandwidths, that the optimal rates for the i.i.d. density models are also optimal for these processes.

Stochastic equicontinuity in nonlinear time series models

In this paper I provide simple and easily verifiable conditions under which a strong form of stochastic equicontinuity holds in a wide variety of modern time series models. In contrast to most results currently available in the literature, my methods avoid mixing conditions. I discuss several applications in detail.

Generalized duration models and optimal estimation using estimating functions

This article introduces a class of generalized duration models and shows that the autoregressive conditional duration (ACD) models and stochastic conditional duration (SCD) models discussed in the literature are special cases. The martingale estimating functions approach, which provides a convenient framework for deriving optimal inference for nonlinear time series models, is described. It is shown that when the first two conditional moments are functions of the same parameter, and information about higher order conditional moments of the observed duration process become available, combined estimating functions are optimal and are more informative than component estimating functions. The combined estimating functions approach is illustrated on three classes of generalized duration models, viz., multiplicative random coefficient ACD models, random coefficient models with ACD errors, and log-SCD models. Recursive estimation of model parameters based on combined estimating functions provides a mechanism for fast estimation in the general case, and is illustrated using simulated data sets.

Bayesian inference for nonlinear structural time series models

We consider efficient methods for likelihood inference applied to structural models. In particular, we introduce a particle filter method which concentrates upon disturbances in the Markov state of the approximating solution to the structural model. A particular feature of such models is that the conditional distribution of interest for the disturbances is often multimodal. We provide a fast and effective method for approximating such distributions. We estimate a neoclassical growth model using this approach. An asset pricing model with persistent habits is also considered. The methodology we employ allows many fewer particles to be used than alternative procedures for a given precision.