Nonlinear Time Series Models Research Articles

Jump-Preserving Varying-Coefficient Models for Nonlinear Time Series

An important and widely used class of semiparametric models is formed by the varying-coefficient models. Although the varying coefficients are traditionally assumed to be smooth functions, the varying-coefficient model is considered here with the coefficient functions containing a finite set of discontinuities. Contrary to the existing nonparametric and varying-coefficient estimation of piecewise smooth functions, the varying-coefficient models are considered here under dependence and are applicable in time series with heteroscedastic and serially correlated errors. Additionally, the conditional error variance is allowed to exhibit discontinuities at a finite set of points too. The (uniform) consistency and asymptotic normality of the proposed estimators are established and the finite-sample performance is tested via a simulation study.

Information criteria for nonlinear time series models

AbstractIn this paper the performance of different information criteria for simultaneous model class and lag order selection is evaluated using simulation studies. We focus on the ability of the criteria to distinguish linear and nonlinear models. In the simulation studies, we consider three different versions of the commonly known criteria AIC, SIC and AICc. In addition, we also assess the performance of WIC and evaluate the impact of the error term variance estimator. Our results confirm the findings of different authors that AIC and AICc favor nonlinear over linear models, whereas weighted versions of WIC and all versions of SIC are able to successfully distinguish linear and nonlinear models. However, the discrimination between different nonlinear model classes is more difficult. Nevertheless, the lag order selection is reliable. In general, information criteria involving the unbiased error term variance estimator overfit less and should be preferred to using the usual ML estimator of the error term variance.

Application of Chaos Theory in the Prediction of Motorised Traffic Flows on Urban Networks

In recent times, urban road networks are faced with severe congestion problems as a result of the accelerating demand for mobility. One of the ways to mitigate the congestion problems on urban traffic road network is by predicting the traffic flow pattern. Accurate prediction of the dynamics of a highly complex system such as traffic flow requires a robust methodology. An approach for predicting Motorised Traffic Flow on Urban Road Networks based on Chaos Theory is presented in this paper. Nonlinear time series modeling techniques were used for the analysis of the traffic flow prediction with emphasis on the technique of computation of the Largest Lyapunov Exponent to aid in the prediction of traffic flow. The study concludes that algorithms based on the computation of the Lyapunov time seem promising as regards facilitating the control of congestion because of the technique’s effectiveness in predicting the dynamics of complex systems especially traffic flow.

On business cycle fluctuations in USA macroeconomic time series

This study employs eighteen USA macroeconomic time series variables to investigate possible existence of asymmetries in business cycle fluctuations in the series. Detection of asymmetric fluctuations in economic activity is important for policymakers since effective monetary policy relies on asymmetric business cycle fluctuations in all the series. The asymmetric deviations from the long-term growth trend in each of the series are modeled using regime switching models and artificial neural networks. The results based on nonlinear switching time series models reveal strong evidence of business cycle asymmetries in most of the series. The results based on in-sample approximations from artificial neural networks show statistically significant evidence of asymmetries in all the series. Similar results are obtained when jackknife out-of-sample approximations from artificial neural networks are used. Thus, the study results show statistically significant evidence of asymmetries in all the series which indicates that business cycle fluctuations in the series are asymmetric, thus alike. Therefore, the impact of monetary policy shocks on the output and the other macroeconomic variables can be anticipated using nonlinear models only. The results on asymmetric business cycle fluctuations in real GDP are in line with recent studies but in sharp contrast with Balke and Fomby (1994).

The structure and sustainability of current account deficit: Turkish evidence from regime switching

ABSTRACTThis study focuses on the current account deficit dynamics and sustainability, using data of the period between 1990:Q1 and 2014:Q2 in the context of Turkish economy. The main findings of the study can be put into two categories. The first category covers energy consumption, openness rate, gross domestic product, exchange rate and investments, which are the most important determinants of the current deficit. The second one asserts that sustainability is weak for Turkish economy; however, it is even weaker during the economic contraction. There is extensive literature about structure and sustainability of the current account deficit. However, many of the studies have analysed the sustainability of the current account deficits without considering the economic conjuncture. For this purpose, the study employs the Markov-switching method which is a non-linear time series model.

Orthogonal Series Estimation in Nonlinear Cointegrating Models with Endogeneity

This paper considers a nonlinear time series model associated with both nonstationarity and endogeneity. The proposed model is then estimated by a nonparametric series method. An asymptotic theory is established in both point–wise and the space metric sense for the estimator. The Monte Carlo simulation results show that the performance of the proposed estimate is numerically satisfactory.

Forecasting in nonlinear univariate time series using penalized splines

In this article we discuss penalized splines for fitting and forecasting univariate nonlinear time series models. While penalized splines have been excessively used in smooth regression, their use in nonlinear time series models is less far developed. This paper focuses on univariate autoregressive processes and discuss different nonlinear (functional) time series models including parsimonious estimation and model selection ideas. Furthermore, in simulations and an application we show how this approach compares to common parametric nonlinear models.

Forecasting exchange rates using multivariate threshold models

AbstractThis paper investigates the ability of a broad range of non-linear time series models to forecast the EUR/USD exchange rate. Using a variant of the well-known Dornbusch (Dornbusch, R. 1976. “Expectations and Exchange Rate Dynamics.”

Estimation of Multiple-Regime Threshold Autoregressive Models With Structural Breaks

The threshold autoregressive (TAR) model is a class of nonlinear time series models that have been widely used in many areas. Due to its nonlinear nature, one major difficulty in fitting a TAR model is the estimation of the thresholds. As a first contribution, this article develops an automatic procedure to estimate the number and values of the thresholds, as well as the corresponding AR order and parameter values in each regime. These parameter estimates are defined as the minimizers of an objective function derived from the minimum description length (MDL) principle. A genetic algorithm (GA) is constructed to efficiently solve the associated minimization problem. The second contribution of this article is the extension of this framework to piecewise TAR modeling; that is, the time series is partitioned into different segments for which each segment can be adequately modeled by a TAR model, while models from adjacent segments are different. For such piecewise TAR modeling, a procedure is developed to estimate the number and locations of the breakpoints, together with all other parameters in each segment. Desirable theoretical results are derived to lend support to the proposed methodology. Simulation experiments and an application to an U.S. GNP data are used to illustrate the empirical performances of the methodology. Supplementary materials for this article are available online.

On the complex dynamics of functional-coefficients nonlinear autoregressive time series models

In this article, a new chaotic functional-coefficient nonlinear autoregressive time series model is formulated. Asymptotic stability of the equilibria of the skeleton is studied. Complex dynamics of the proposed model are investigated by means of the bifurcations, time series diagrams, and phase portraits. The effects of noise intensity on its dynamics and the intermittency phenomenon are also discussed via simulation. Two chaotic indicators, namely, the fractal dimension and the Lyapunov exponent methods are investigated for the model.

First-order random coefficient autoregressive (RCA(1)) model: Joint Whittle estimation and information

Random coefficient autoregressive model, RCA(p), has been discussed widely as a suitable model for nonlinear time series. The conditional least squares and likelihood parameter estimation of RCA(p) model has also been discussed in [3]. The statistical inference of RCA(1) model has been presented in [4] while the conditional least square estimates for nonstationary processes is studied in [7]. The optimal estimation for nonlinear time series using estimating equations has been investigated in [6]. Recently there has been interest in joint prediction based on spectral density of popular nonlinear time series models such as RCA(p) models. Another way of estimating the parameters of the RCA(1) model is to do Whittle's estimation. In this paper the Whittle estimates of the parameters of an RCA(p) model are studied. It is shown that the Whittle information of the autoregressive parameter in an RCA(p) model is larger than the corresponding information in an autoregressive (AR) model.

A Potential Method for Determining Nonlinearity in Wind Data

This paper investigates a basic problem in modeling time series of wind data: whether there exists detectable correlation or nonlinearity in the observed wind time series. At present, a variety of linear and nonlinear time series models have been applied to predict the wind data. The first question that should be answered before building a model, however, is whether the data studied are correlated or carry nonlinearity. It would be futile to model the relationships if the pertaining wind data cannot be distinguished from the white noise. Advanced nonlinear prediction models are also not necessary if there are no nonlinear structures in the data. In this paper, we test by the surrogate data method: 1) whether the differenced wind speed time series (taking the first difference of the time series) is white noise, and 2) the presence of nonlinearity in the original wind speed time series. Nine data sets, including 10 min and hourly wind speed data, are examined. The results show that all of the differenced wind speed time series are correlated, and three out of the nine original wind speed time series satisfy the hypothesis of a linear stochastic generating process. It is concluded that for a specific wind speed time series, the nonlinearity is data-dependent from the perspective of practical time series analysis.

The impact of model risk on capital reserves: a quantitative analysis

This paper analyzes and quantifies the idea of model risk in the environment of internal model building. We define various types of model risk including estimation risk, model risk in distribution and model risk in functional form. By the quantification of these concepts we analyze the impact of the modeling process of an econometric model on the resulting company model. Utilizing real insurance data we specify, estimate and simulate various linear and nonlinear time series models for the inflation rate and examine its impact on pension liabilities under the aspect of model risk. Under consideration of different risk measures it is shown that model risk can differ profoundly due to the specification process of the econometric model resulting in remarkable monetary differences concerning capital reserves. We furthermore propose a specification strategy for univariate time series models and demonstrate that thereby market risk and capital reserves can be reduced distinctively. JEL codes: G12, G18

Forecasting daily river flows using nonlinear time series models

Summary In the hydrology studies, it is well known that the river flows are affected by various factors, and therefore the dynamics in their associated time series are complicated and have nonlinear behaviors. In an empirical study, we investigate the capability of five classes of nonlinear time series models, namely Threshold Autoregressive (TAR), Smooth Transition Autoregressive (STAR), Exponential Autoregressive (EXPAR), Bilinear Model (BL) and Markov Switching Autoregressive (MSAR) to capture the dynamics in the Colorado river discharge time series. Least Squares (LS) and Maximum Likelihood (ML) methods are employed to estimate parameters of the models. For model comparison three criteria, namely loglikelihood, Akaike information criterion (AIC) and Bayesian information criterion (BIC) are calculated. The results show that a self-exciting TAR (SETAR) model performs better than other four competing models. To forecast future river discharge values an iterative method is applied and forecasting confidence intervals are constructed. The out-of-sample 1-day to 5-day ahead forecasting performances of the models based on ten forecast accuracy measures are evaluated. Comparing verification metrics of all models, SETAR model presents the best forecasting performance.

Estimation of structural changes in nonlinear time series models by using particle filters and genetic programming

SummarySeveral works have demonstrated detection of changes of state equations (called structural changes) based on statistical measures but have given no suggestions regarding the functional forms of the state equations after changes. This paper deals with the estimation of structural changes in nonlinear time series models by using particle filters, genetic programming (GP), and its applications. We consider the problems of state estimation from the observed time series that are generated based on nonlinear state equations. It is assumed that structural changes can be detected by some measure of likelihood and that the state equation after changes is modified from its current functional form. Individuals corresponding to functional forms in the GP pool are generated at random, and we apply the crossover operation between the current functional form and the individuals by giving possible multiple functional forms. Then, we have the optimal functional form among the possible functional forms generated by GP from the current form. As an application, we show the estimation of structural change for an artificially generated time series and also discuss the estimation of functional forms for a real economic time series before and after structural changes. Copyright © 2015 John Wiley & Sons, Ltd.

Statistical inference for conditional quantiles in nonlinear time series models

This paper studies the statistical properties of a two-step conditional quantile estimator in nonlinear time series models with unspecified error distribution. The asymptotic distribution of the quasi-maximum likelihood estimators and the filtered empirical percentiles is derived. Three applications of the asymptotic result are considered. First, we construct an interval estimator of the conditional quantile without any distributional assumptions. Second, we develop a specification test for the error distribution. Finally, using the specification test, we propose methods for estimating the tail index of the error distribution that supports the construction of a new estimator for the conditional quantile at the extreme tail. The asymptotic results and their applications are illustrated by simulations and real data analyses in which our methods for analyzing daily and intraday financial return series have been adopted.

Statistical Inference of Conditional Quantiles in Nonlinear Time Series Models

This paper studies the statistical properties of a two-step conditional quantile estimator in nonlinear time series models with unspecified error distribution. The asymptotic distribution of the quasi-maximum likelihood estimators and the filtered empirical percentiles is derived. Three applications of the asymptotic result are considered. First, we construct an interval estimator of the conditional quantile without any distributional assumptions. Second, we develop a specification test for the error distribution. Finally, using the specification test, we propose methods for estimating the tail index of the error distribution that would support the construct of a new high conditional quantile estimator. The asymptotic results and their applications are illustrated by simulations and real data analyses in which our methods for analyzing daily and intraday financial return series have been adopted.

Specification testing in nonstationary time series models

Summary In this paper, we consider a specification testing problem in nonlinear time series models with nonstationary regressors, and we propose using a nonparametric kernel-based test statistic. The null asymptotics for the proposed nonparametric test statistic have been well developed in the existing literature. In this paper, we study the local asymptotics of the test statistic (i.e. the asymptotic properties of the test statistic under a sequence of general nonparametric local alternatives) and show that the asymptotic distribution depends on the asymptotic behaviour of the distance function, which is the local deviation from the parametrically specified model in the null hypothesis. In order to implement the proposed test in practice, we introduce a bootstrap procedure to approximate the critical values of the test statistic and establish a new Edgeworth expansion, which is used to justify the use of such an approximation. Based on the approximate critical values, we develop a bandwidth selection method, which chooses the optimal bandwidth that maximizes the local power of the test while its size is controlled at a given significance level. The local power is defined as the power of the proposed test for a given sequence of local alternatives. Such a bandwidth selection is made feasible by an approximate expression for the local power of the test as a function of the bandwidth. A Monte Carlo simulation study is provided to illustrate the finite sample performance of the proposed test.

Modeling Autoregressive Processes with Moving-Quantiles-Implied Nonlinearity

We introduce and investigate some properties of a class of nonlinear time series models based on the moving sample quantiles in the autoregressive data generating process. We derive a test fit to detect this type of nonlinearity. Using the daily realized volatility data of Standard & Poor’s 500 (S&P 500) and several other indices, we obtained good performance using these models in an out-of-sample forecasting exercise compared with the forecasts obtained based on the usual linear heterogeneous autoregressive and other models of realized volatility.

Threshold Vector Arma Models

In this article, we propose the threshold vector autoregressive moving average model (TVARMA). It is a multivariate nonlinear time series model characterized by two or more regimes that follow a vector ARMA structure and where the switching among them is regulated by a latent variable. The TVARMA model represents a generalization of some nonlinear models proposed in the literature and shows interesting features that are explored. The condition for the strong and weak stationarity of the TVARMA model are presented and the moments up to order two of the process are derived.