Nonlinear Time Series Models Research Articles

A Volatility Model for Financial Time Series in the Generalized Pearson Setting

Our objective is to review the existing literature on modeling financial asset return distributions and propose additional models and techniques that provide a better fit for a given financial asset return series such as global stock indices, industry segments and foreign exchange. One possible way is to adjust the model within the whole family of parameters, or only within the family of parameters, which make economic sense. The ideal model specification is the one that can handle structural change and time dependence in conditional mean, variance, skewness, and kurtosis. This may have more economic appeal than assuming fundamental nonnormality. We postulate that the simultaneous estimation of time-varying first-four moments using a flexible family of probability distributions such as the Pearson type distributions might provide a better explanation of risks, and hence, robust design of portfolio allocation systems beyond the traditional first-two moments framework. In a study of fractual structures in exchange rates, Richards (2000) finds in a simulation experiment that the best performing model among non-linear time series models is a GARCH, in which a generalized error distribution was modified to allow for wider tails.In this research, we will combine an autoregressive conditional heteroskedasticity model with an asymmetric information structure due to Daniel B. Nelson (1991) with a flexible family of distributions, developed by Karl Pearson (1895). Within this framework, a nonlinear parametric model with time-varying higher moments is proposed. We, hereby, attempt to extend the time-varying conditional variance nature of traditional ARCH/GARCH-type models to include either time-varying skewness or kurtosis for it might improve our understanding of risks and risk premia seen in financial markets. To this end, we will explore the possible ill behavior of standardized residuals in many types of the time-varying conditional variance models and show that these residuals lead us to the modeling of time-varying skewness and kurtosis. We will fit the Pearson distribution directly to sample data by calculating the second, third and fourth central moments of the observed values and using the definitions of skewness and kurtosis. However, observed values of the third and fourth moments could be sensitive to outliers. This would question the validity of the equation, which takes advantage of the time-varying properties of kurtosis. Moreover, the sensitivity of higher moments’ estimates to a small number of extreme returns also means that ex-post returns may have quite different properties from ex-ante returns. We are also planning to incorporate the concepts of L-moments, introduced by Hosking (1990). L-moments are defined to be the expected values of linear expectations of the order statistics. They are less sensitive to outliers than ordinary moments, and often provide a better identification of the parent distribution that generates a particular data sample. One might use L-moments to fit a particular distribution to data by equating the first few sample and population L-moments, analogously to the method of moments. The resulting estimators of parameters and quantiles are sometimes more accurate than the maximum-likelihood estimates, in case of generalized extreme value distributions, Hosking et al. (1985), and some instances of the generalized Pareto distribution, Hosking and Wallis (1987). If the mean of the distribution exists, then all the higher-order L-moments do as well (see “Theorem 1” in Hosking (1996)). Thus, L-moments can describe fat-tailed distributions whose variance or higher-order regular moments may be infinite. Moreover, for standard errors of sample L-moments to be finite, the largest moment that needs to be finite is the second moment; i.e., variance. We believe the use of L-moments for a leptokurtic distribution such as the Pearson type-VII might be of some value due to the success of L-moments with other heavy-tailed distributions.

Stability analysis of RBF network-based state-dependent autoregressive model for nonlinear time series

Varying-coefficient models have attracted great attention in nonlinear time series analysis recently. In this paper, we consider a semi-parametric functional-coefficient autoregressive model, called the radial basis function network-based state-dependent autoregressive (RBF-AR) model. The stability conditions and existing conditions of limit cycle of the RBF-AR model are discussed. An efficient structured parameter estimation method and the modified multi-fold cross-validation criterion are applied to identify the RBF-AR model. Application of the RBF-AR model to the famous Canadian lynx data is presented. The forecasting capability of the RBF-AR model is compared to those of other competing time series models, which shows that the RBF-AR model is as good as or better than other models for the postsample forecasts.

Investigating asymptotic properties of vector nonlinear time series models

Analyses and simulations of vector nonlinear time series typically run into weeks or even months because the methods used are computationally intensive. Statisticians have been known to base empirical results on a relatively small number of simulation replications, sacrificing precision and reliability of results in the interest of time and productivity. The simulations are amenable for parallelization. However, parallel computing technology has not yet been widely used in this specific research area. This paper proposes an approach to the parallelization of statistical simulation codes to address the challenge of long running times. Requiring minimal code revision, this approach takes advantage of recent advances in dynamic loop scheduling to achieve high performance on general-purpose clusters, even with the presence of unpredictable load imbalance factors. Preliminary results of applying this approach in the simulation of normal white noise and threshold autoregressive model obtains efficiencies in the range 95%–98% on 8–64 processors. Furthermore, previously unobserved properties of the statistical procedures for the models are uncovered by the simulation.

Strategic Business Cycle Management and Merger and Acquisition Patterns

Strategic business cycle management (BCM) involves exploiting movements of the economy to gain competitive advantage over rivals. BCM is applicable to mergers and acquisitions (M&A) which occur in cyclical, non-periodic waves. Acquiring targets at a wave’s beginning is profitable, making waves important opportunities for value creation. Econometricians have attempted to model waves, with little consensus. Using a model-selection approach, we compare five linear and nonlinear time series models for U.S. M&A data from 1895-2008. Our results suggest that a seasonal autoregressive moving average (SARIMA) is both well-specified and parsimonious. We discuss how our study extends the prescriptive BCM research agenda.

Boosting techniques for nonlinear time series models

Many of the popular nonlinear time series models require a priori the choice of parametric functions which are assumed to be appropriate in specific applications. This approach is mainly used in financial applications, when sufficient knowledge is available about the nonlinear structure between the covariates and the response. One principal strategy to investigate a broader class on nonlinear time series is the Nonlinear Additive AutoRegressive (NAAR) model. The NAAR model estimates the lags of a time series as flexible functions in order to detect non-monotone relationships between current and past observations. We consider linear and additive models for identifying nonlinear relationships. A componentwise boosting algorithm is applied for simultaneous model fitting, variable selection, and model choice. Thus, with the application of boosting for fitting potentially nonlinear models we address the major issues in time series modelling: lag selection and nonlinearity. By means of simulation we compare boosting to alternative nonparametric methods. Boosting shows a strong overall performance in terms of precise estimations of highly nonlinear lag functions. The forecasting potential of boosting is examined on the German industrial production (IP); to improve the model’s forecasting quality we include additional exogenous variables. Thus we address the second major aspect in this paper which concerns the issue of high dimensionality in models. Allowing additional inputs in the model extends the NAAR model to a broader class of models, namely the NAARX model. We show that boosting can cope with large models which have many covariates compared to the number of observations.

Autocopulas: Investigating the Interdependence Structure of Stationary Time Series

Here we present a novel approach to the description of the lagged interdependence structure of stationary time series. The idea is to extend the use of copulas to the lagged (one-dimensional) series, to the analogy of the autocorrelation function. The use of such autocopulas can reveal the specifics of the lagged interdependence in a much finer way. However, the lagged interdependence is resulted from the dynamics, governing the series, therefore the known and popular copula models have little to do with that type of interdependence. True though, it seems rather cumbersome to calculate the exact form of the autocopula even for the simplest nonlinear time series models, so we confine ourselves here to an empirical and simulation based approach. The advantage of using autocopulas lays in the fact that they represent nonlinear dependencies as well, and make it possible e.g. to study the interdependence of high (or low) values of the series separately. The presented methods are capable to check whether autocopulas of an observed process can be distinguished significantly from the autocopulas a of given time series model. The proposed approach is based on the Kendall’s transform which reduces the multivariate problem to one dimension. After illustrating the use of our approach in detecting conditional heteroscedasticity in the AR-ARCH vs. AR case, we apply the proposed methods to investigate the lagged interdependence of river flow time series with particular focus on model choice based on the synchronized appearance of high values.

Multi–regime models for nonlinear nonstationary time series

Nonlinear nonstationary models for time series are considered, where the series is generated from an autoregressive equation whose coefficients change both according to time and the delayed values of the series itself, switching between several regimes. The transition from one regime to the next one may be discontinuous (self-exciting threshold model), smooth (smooth transition model) or continuous linear (piecewise linear threshold model). A genetic algorithm for identifying and estimating such models is proposed, and its behavior is evaluated through a simulation study and application to temperature data and a financial index.

Asset allocation under threshold autoregressive models

We discuss the asset allocation problem in the important class of parametric non-linear time series models called the threshold autoregressive model in (J. Roy. Statist. Soc. Ser. A 1977; 140:34–35; Patten Recognition and Signal Processing. Sijthoff and Noordhoff: Netherlands, 1978; and J. Roy. Statist. Soc. Ser. B 1980; 42:245–292). We consider two specific forms, one self-exciting (i.e. the SETAR model) and the other smooth (i.e. the STAR) model developed by Chan and Tong (J. Time Ser. Anal. 1986; 7:179–190). The problem of maximizing the expected utility of wealth over a planning horizon is considered using a discrete-time dynamic programming approach. This optimization approach is flexible enough to deal with the optimal asset allocation problem under a general stochastic dynamical system, which includes the SETAR model and the STAR model as particular cases. Numerical studies are conducted to demonstrate the practical implementation of the proposed model. We also investigate the impacts of non-linearity in the SETAR and STAR models on the optimal portfolio strategies. Copyright © 2011 John Wiley & Sons, Ltd.

Variance decompositions of nonlinear time series using stochastic simulation and sensitivity analysis

In this paper, A variance decomposition approach to quantify the effects of endogenous and exogenous variables for nonlinear time series models is developed. This decomposition is taken temporally with respect to the source of variation. The methodology uses Monte Carlo methods to affect the variance decomposition using the ANOVA-like procedures proposed in Archer et al. (J. Stat. Comput. Simul. 58:99---120, 1997), Sobol' (Math. Model. 2:112---118, 1990). The results of this paper can be used in investment problems, biomathematics and control theory, where nonlinear time series with multiple inputs are encountered.

Forecasting the Polish Zloty with Non-Linear Models

The literature on exchange rate forecasting is vast. Many researchers have tested whether implications of theoretical economic models or the use of advanced econometric techniques can help explain future movements in exchange rates. The results of the empirical studies for major world currencies show that forecasts from a naive random walk tend to be comparable or even better than forecasts from more sophisticated models. In the case of the Polish zloty, the discussion in the literature on exchange rate forecasting is scarce. This article fills this gap by testing whether non-linear time series models are able to generate forecasts for the nominal exchange rate of the Polish zloty that are more accurate than forecasts from a random walk. Our results confirm the main findings from the literature, namely that it is difficult to outperform a naive random walk in exchange rate forecasting contest.

Modeling threshold conditional heteroscedasticity with regime-dependent skewness and kurtosis

Construction of nonlinear time series models with a flexible probabilistic structure is an important challenge for statisticians. Applications of such a time series model include ecology, economics and finance. In this paper we consider a threshold model for all the first four conditional moments of a time series. The nonlinear structure in the conditional mean is specified by a threshold autoregression and that of the conditional variance by a threshold generalized autoregressive conditional heteroscedastic (GARCH) model. There are many options for the conditional innovation density in the modeling of the skewness and kurtosis such as the Gram–Charlier (GC) density and the skewed- t density. The Gram–Charlier (GC) density allows explicit modeling of the skewness and kurtosis parameters and therefore is the main focus of this paper. However, its performance is compared with that of Hansen’s skewed- t distribution in the data analysis section of the paper. The regime-dependent feature for the first four conditional moments allows more flexibility in modeling and provides better insights into the structure of a time series. A Lagrange multiplier (LM) test is developed for testing for the presence of threshold structure. The test statistic is similar to the classical tests for the presence of a threshold structure but allowing for a more general regime-dependent structure. The new model and the LM test are illustrated using the Dow Jones Industrial Average, the Hong Kong Hang Seng Index and the Yen/US exchange rate.

Forecasting Performance of Nonlinear Models for Intraday Stock Returns

ABSTRACTWe studied the predictability of intraday stock market returns using both linear and nonlinear time series models. For the S&P 500 index we compared simple autoregressive and random walk linear models with a range of nonlinear models, including smooth transition, Markov switching, artificial neural network, nonparametric kernel regression and support vector machine models for horizons of 5, 10, 20, 30 and 60 minutes. The empirical results indicate that nonlinear models outperformed linear models on the basis of both statistical and economic criteria. Specifically, although return serial correlation receded by around 10 minutes, return predictability still persisted for up to 60 minutes according to nonlinear models, even though profitability decreases as time elapses. More flexible nonlinear models such as support vector machines and artificial neural network did not clearly outperform other nonlinear models. Copyright © 2011 John Wiley & Sons, Ltd.

Single-index coefficient models for nonlinear time series

The single-index coefficient model, where the coefficients are functions of an index of a covariate vector, is a powerful tool for modelling nonlinearity in multivariate estimation. By reducing the covariate vector to an index which is usually a linear combination of covariates, the single-index coefficient model overcomes the well-known phenomenon of ‘curse-of-dimensionality’. We estimate the univariate varying coefficients with penalised splines (PS). An iterative data-driven algorithm is developed, adaptively selecting the index. The algorithm is based on the observation that given an estimated index, the varying-coefficient model using PS is essentially a linear ridge regression with spline bases. Our experiments show that the proposed algorithm gives rapid convergence. We also establish large sample properties assuming fixed number of knots. The usual jointly stationary assumption for dependent data is relaxed with weaker size requirements for either φ-mixing or α-mixing. Finally, we present an application to a gross national product data set and a simulated example.

Infulence of news from Moscow and New York on returns and risks of Baltic States’ stock markets

The impact of news from the Moscow and New York stock exchanges on the daily returns and volatilities of Baltic stock market indices is studied. A nonlinear time series model that accounts for asymmetries in conditional mean and variance functions is used for the empirical work. News from New York has stronger effects on returns in Tallinn than news from Moscow. High-risk shocks in New York have a stronger impact on volatility in Tallinn, whereas volatility in Vilnius is more influenced by high-risk shocks from Moscow. Riga seems not to be affected by news arriving from abroad.

Macro-grammatical evolution for nonlinear time series modeling—a case study of reservoir inflow forecasting

Streamflow forecasting is significantly important for planning and operating water resource systems. However, streamflow formation is a highly nonlinear, time varying, spatially distributed process and difficult to forecast. This paper proposes a nonlinear model which incorporates improved real-coded grammatical evolution (GE) with a genetic algorithm (GA) to predict the ten-day inflow of the De-Chi Reservoir in central Taiwan. The GE is a recently developed evolutionary-programming algorithm used to express complex relationships among long-term nonlinear time series. The algorithm discovers significant input variables and combines them to form mathematical equations automatically. Utilizing GA with GE optimizes an appropriate type of function and its associated coefficients. To enhance searching efficiency and genetic diversity during GA optimization, the macro-evolutionary algorithm (MA) is processed as a selection operator. The results using an example of theoretical nonlinear time series problems indicate that the proposed GEMA yields an efficient optimal solution. GEMA has the advantages of its ability to learn relationships hidden in data and express them automatically in a mathematical manner. When applied to a real world case study, the fittest equation generated through GEMA used only a single input variable in a reasonable nonlinear form. The predicting accuracies of GEMA were better than those of the traditional linear regression (LR) model and as good as those of the back-propagation neural network (BPNN). In addition, the predicting of ten-day reservoir inflows reveals the effectives of GEMA, and standardization is beneficial to model for seasonal time series.

An Alternative Bayesian Approach to Structural Breaks in Time Series Models

We propose a new approach to deal with structural breaks in time series models. The key contribution is an alternative dynamic stochastic specification for the model parameters which describes potential breaks. After a break new parameter values are generated from a so-called baseline prior distribution. Modeling boils down to the choice of a parametric likelihood specification and a baseline prior with the proper support for the parameters. The approach accounts in a natural way for potential out-of-sample breaks where the number of breaks is stochastic. Posterior inference involves simple computations that are less demanding than existing methods. The approach is illustrated on nonlinear discrete time series models and models with restrictions on the parameter space.

Two Distributed-State Models For Generating High-Dimensional Time Series

In this paper we develop a class of nonlinear generative models for high-dimensional time series. We first propose a model based on the restricted Boltzmann machine (RBM) that uses an undirected model with binary latent variables and real-valued variables. The latent and visible variables at each time step receive directed connections from the visible variables at the last few time-steps. This conditional RBM (CRBM) makes on-line inference efficient and allows us to use a simple approximate learning procedure. We demonstrate the power of our approach by synthesizing various sequences from a model trained on motion capture data and by performing on-line filling in of data lost during capture. We extend the CRBM in a way that preserves its most important computational properties and introduces multiplicative three-way interactions that allow the effective interaction weight between two variables to be modulated by the dynamic state of a third variable. We introduce a factoring of the implied three-way weight tensor to permit a more compact parameterization. The resulting model can capture diverse styles of motion with a single set of parameters, and the three-way interactions greatly improve its ability to blend motion styles or to transition smoothly among them. Videos and source code can be found at http://www.cs.nyu.edu/~gwtaylor/publications/jmlr2011.

Nonparametric model validations for hidden Markov models with applications in financial econometrics

We address the nonparametric model validation problem for hidden Markov models with partially observable variables and hidden states. We achieve this goal by constructing a nonparametric simultaneous confidence envelope for transition density function of the observable variables and checking whether the parametric density estimate is contained within such an envelope. Our specification test procedure is motivated by a functional connection between the transition density of the observable variables and the Markov transition kernel of the hidden states. Our approach is applicable for continuous-time diffusion models, stochastic volatility models, nonlinear time series models, and models with market microstructure noise.

Structure and asymptotic theory for nonlinear models with GARCH errors

Abstract Nonlinear time series models, especially those with regime-switching and/or conditionally heteroskedastic errors, have become increasingly popular in the economics and finance literature. However, much of the research has concentrated on the empirical applications of various models, with little theoretical or statistical analysis associated with the structure of the processes or the associated asymptotic theory. In this paper, we derive sufficient conditions for strict stationarity and ergodicity of three different specifications of the first-order smooth transition autoregressions with heteroskedastic errors. This is essential, among other reasons, to establish the conditions under which the traditional LM linearity tests based on Taylor expansions are valid. We also provide sufficient conditions for consistency and asymptotic normality of the Quasi-Maximum Likelihood Estimator for a general nonlinear conditional mean model with first-order GARCH errors.

Chaotic time series method combined with particle swarm optimization and trend adjustment for electricity demand forecasting

Electricity demand forecasting plays an important role in electric power systems planning. In this paper, nonlinear time series modeling technique is applied to analyze electricity demand. Firstly, the phase space, which describes the evolution of the behavior of a nonlinear system, is reconstructed using the delay embedding theorem. Secondly, the largest Lyapunov exponent forecasting method (LLEF) is employed to make a prediction of the chaotic time series. In order to overcome the limitation of LLEF, a weighted largest Lyapunov exponent forecasting method (WLLEF) is proposed to improve the prediction accuracy. The particle swarm optimization algorithm (PSO) is used to determine the optimal weight parameters of WLLEF. The trend adjustment technique is used to take into account the seasonal effects in the data set for improving the forecasting precision of WLLEF. A simulation is performed using a data set that was collected from the grid of New South Wales, Australia during May 14–18, 2007. The results show that chaotic characteristics obviously exist in electricity demand series and the proposed prediction model can effectively predict the electricity demand. The mean absolute relative error of the new prediction model is 2.48%, which is lower than the forecasting errors of existing methods.