Nonlinear Time Series Models Research Articles

Break detection in the covariance structure of multivariate time series models

In this paper, we introduce an asymptotic test procedure to assess the stability of volatilities and cross-volatilites of linear and nonlinear multivariate time series models. The test is very flexible as it can be applied, for example, to many of the multivariate GARCH models established in the literature, and also works well in the case of high dimensionality of the underlying data. Since it is nonparametric, the procedure avoids the difficulties associated with parametric model selection, model fitting and parameter estimation. We provide the theoretical foundation for the test and demonstrate its applicability via a simulation study and an analysis of financial data. Extensions to multiple changes and the case of infinite fourth moments are also discussed.

Markov‐Switching Estimation of Spatial Maize Price Transmission Processes between Tanzania and Kenya

We discuss the Markov-switching vector au toregressive (MS-VAR) class of nonlinear time series models that can be used to analyze recurring discrete structural changes in time series. Hamilton's (1989) seminal Markov switching (MS) model of the U.S. business cy cle triggered considerable interest in the MS approach in economics. Most empirical appli cations to date have focused on the business cy cle and financial markets, but we see potential for MS-VAR models in agricultural economics, for example, in price transmission analysis. In the following, we first provide an overview of the MS-VAR framework. We then present an illustrative application to maize price trans mission between Tanzania and Kenya. The article closes with a discussion of strengths, weaknesses, and potential uses of the MS-VAR approach in price transmission analysis.

Granular RBF NN Approach and Statistical Methods Applied to Modelling and Forecasting High Frequency Data

We examine the ARCH-GARCH models for the forecasting of the bond price time series provided by VUB bank and make comparisons the forecast accuracy with the class of RBF neural network models. A limited statistical or computer science theory exists on how to design the architecture of RBF networks for some specific nonlinear time series, which allows for exhaustive study of the underlying dynamics, and determination of their parameters. To illustrate the forecasting performance of these approaches the learning aspects of RBF networks are presented and an application is included. We show a new approach of function estimation for nonlinear time series model by means of a granular neural network based on Gaussian activation function modelled by cloud concept. In a comparative study is shown, that the presented approach is able to model and predict high frequency data with reasonable accuracy and more efficient than statistical methods.

Forecast Comparison of Nonlinear Time Series Models of Us GDP: A Bayesian Approach

This thesis utilizes modern Bayesian tools to evaluate the forecasting performance of two of the most widely used nonlinear time series models of post-war US GDP, the Markov Switching (MS) model and the Self-Exciting threshold autoregressive (SETAR) model. We develop a clear, empirical ground for model selection, forecast comparison and forecast combination in the context of nonlinear economic time series. The use of the Bayesian methodology introduces an immediate framework to deal with nonlinearities. This holds not only in estimation, by treating nuisance parameters as unknown random variables, and in testing, by avoiding the approximation of unknown distributions of likelihood ratio statistics in order to find evidence of nonlinearity, but also in the choice of the best model according to its forecasting potential. To our knowledge, the literature avoids welcoming existing Bayesian methods in forecasting and comparison of nonlinear time series models; all known attempts from a Bayesian perspective are consumed only to the estimation part. Here we propose to use predictive likelihoods, an easy to adopt and to interpret criterion, as an alternative to classical forecast evaluation criterions like the root mean squared forecast error (RMSE). The advantage of our approach is very clear in the nonlinear time series framework that we examine. Preservation of computational resources, efficiency in manipulation of nuisance parameters, and parameter uncertainty are fully incorporated while we improve over individual forecasts by implementing Bayesian Model Averaging.

A partially linearized sigma point filter for latent state estimation in nonlinear time series models

A new technique for the latent state estimation of a wide class of nonlinear time series models is proposed. In particular, we develop a partially linearized sigma point filter in which random samples of possible state values are generated at the prediction step using an exact moment-matching algorithm and then a linear programming based procedure is used in the update step of the state estimation. The effectiveness of the new filtering procedure is assessed via a simulation example that deals with a highly nonlinear, multivariate time series representing an interest rate process.

Trends, time-varying and nonlinear time series models for NGRIP and VOSTOK paleoclimate data

In order to gain further insights into stochastic behaviour of paleoclimate data, including timescales at and below Milankovitch forcing, three specific questions are discussed using δ18O NGRIP and Vostok Deuterium content data. A comparison of ordinary and time-varying coefficients autoregressive (AR) models shows that both data sets are distinguishable from data generated by suitable low-order AR processes in contrast to earlier conclusions. A harmonic regression analysis clearly distinguishing between discrete and continuous spectra detects cycles corresponding to variations of eccentricity, obliquity and precession. Contribution of eccentricity to the total variance in the last 422,766-year Vostok data is close to, while the variance reduction delivered jointly by obliquity and precession is substantially smaller than a previous recent finding. A harmonic regression analysis with time-varying frequencies and amplitudes is also performed. This approach delivers a gain over the constant frequency model at any reasonable significance level. It is demonstrated that variations of frequencies are at least partly due to real variations and not merely to timescale uncertainties. In order to consider nonlinearity in paleoclimate data, threshold autoregressive (TAR) models are applied to time series examined. A bivariate TAR model describing simultaneous NGRIP and Vostok data exhibits three fix points and one limit cycle related to a part of Dansgaard–Oeschger events. The model selected suggests that Greenland has a primary role in the Greenland–Antarctica climate variation relationship.

SUP-TESTS FOR LINEARITY IN A GENERAL NONLINEAR AR(1) MODEL

We consider linearity testing in a general class of nonlinear time series models of order one, involving a nonnegative nuisance parameter that (a) is not identified under the null hypothesis and (b) gives the linear model when equal to zero. This paper studies the asymptotic distribution of the likelihood ratio test and asymptotically equivalent supremum tests. The asymptotic distribution is described as a functional of chi-square processes and is obtained without imposing a positive lower bound for the nuisance parameter. The finite-sample properties of the sup-tests are studied by simulations.

ASYMPTOTIC DISTRIBUTION-FREE DIAGNOSTIC TESTS FOR HETEROSKEDASTIC TIME SERIES MODELS

This article investigates model checks for a class of possibly nonlinear heteroskedastic time series models, including but not restricted to ARMA-GARCH models. We propose omnibus tests based on functionals of certain weighted standardized residual empirical processes. The new tests are asymptotically distribution-free, suitable when the conditioning set is infinite-dimensional, and consistent against a class of Pitman’s local alternatives converging at the parametric raten−1/2, withnthe sample size. A Monte Carlo study shows that the simulated level of the proposed tests is close to the asymptotic level already for moderate sample sizes and that tests have a satisfactory power performance. Finally, we illustrate our methodology with an application to the well-known S&P 500 daily stock index. The paper also contains an asymptotic uniform expansion for weighted residual empirical processes when initial conditions are considered, a result of independent interest.

Multivariate Non-Linear Time Series Modelling of Exposure and Risk in Road Safety Research

Summary A multivariate non-linear time series model for road safety data is presented. The model is applied in a case-study into the development of a yearly time series of numbers of fatal accidents (inside and outside urban areas) and numbers of kilometres driven by motor vehicles in the Netherlands between 1961 and 2000. The model accounts for missing entries in the disaggregated numbers of kilometres driven although the aggregated numbers are observed throughout. We consider a multivariate non-linear time series model for the analysis of these data. The model consists of dynamic unobserved factors for exposure and risk that are related in a non-linear way to the number of fatal accidents. The multivariate dimension of the model is due to its inclusion of multiple time series for inside and outside urban areas. Approximate maximum likelihood methods based on the extended Kalman filter are utilized for the estimation of unknown parameters. The latent factors are estimated by extended smoothing methods. It is concluded that the salient features of the observed time series are captured by the model in a satisfactory way.

Exploring the nature of Pacific climate variability using a “toy” nonlinear stochastic model

The Pacific Decadal Oscillation (PDO) is the leading mode of North-Pacific temperature variability. This climatic oscillation is nominally dominated by regime shifts having wide-ranging impacts. An ongoing point of discussion around the PDO regards whether such regimes genuinely represent multiple, stable, discrete states in a nonlinear dynamical system, or instead simply extended runs of above- or below-normal values in a continuous stochastic memory process. A simple nonlinear Markov time series model, involving a (hybrid) stochastically driven tipping-point process, is considered as a vehicle to explore the fundamental nature of the PDO. This toy model is found to be sufficiently plausible for our purposes, insofar as it reproduces some key observational PDO dynamics: in particular, de facto regime shifts and a 1/f power spectrum. The primary conclusion is that the line between the two competing general views may be blurred, and the red-noise and bi-stability hypotheses might be better considered as end members in a continuum of possible system conceptualizations for the PDO.

Density estimation for nonlinear parametric models with conditional heteroscedasticity

This article studies density and parameter estimation problems for nonlinear parametric models with conditional heteroscedasticity. We propose a simple density estimate that is particularly useful for studying the stationary density of nonlinear time series models. Under a general dependence structure, we establish the root n consistency of the proposed density estimate. For parameter estimation, a Bahadur type representation is obtained for the conditional maximum likelihood estimate. The parameter estimate is shown to be asymptotically efficient in the sense that its limiting variance attains the Cramér–Rao lower bound. The performance of our density estimate is studied by simulations.

Asymptotic Distribution-Free Diagnostic Tests for Heteroskedastic Time Series Models

This article investigates model checks for a class of possibly nonlinear heteroskedastic time series models, including but not restricted to ARMA-GARCH models. We propose omnibus tests based on functionals of certain weighted standardized residual empirical processes. The new tests are asymptotically distribution-free, suitable when the conditioning set is in nite-dimensional, and consistent against a class of Pitman's local alternatives converging at the parametric rate n1=2; with n the sample size. A Monte Carlo study shows that the simulated level of the proposed tests is close to the asymptotic level already for moderate sample sizes and that tests have a satisfactory power performance. Finally, we illustrate our methodology with an application to the well-known S&P 500 daily stock index. The paper also contains an asymptotic uniform expansion for weighted residual empirical processes when initial conditions are considered, a result of independent interest.

Non-parametric tests of real exchange rates in the post-Bretton Woods era

This paper examines the mean-reverting property of real exchange rates. Earlier studies have generally not been able to reject the null hypothesis of a unit-root in real exchange rates, especially for the post-Bretton Woods floating period. The results imply that long-run purchasing power parity does not hold. More recent studies, especially those using panel unit-root tests or nonlinear time series models, have found more favorable results, however. But the results from these recent studies are far from conclusive. Consistent individual country time series evidence that supports long-run purchasing power parity continues to be scarce. In this paper, we test for long memory using (Lo’s in Econometrica 59:1279–1313, 1991) the modified rescaled range test, and the rescaled variance test of Giraitis et al. (J Econ 112:265–294, 2003a). Our testing procedure provides a non-parametric alternative to the parametric tests commonly used in this literature. Our data set consists of monthly observations from April 1973 to April 2001 of the G-6 countries (excluding the US) in the OECD. Using the modified rescaled range test, we find only 2 cases out of 15 where the null hypothesis of a unit-root with short-term dependence could be rejected in favor of the alternative hypothesis of long-term dependence and none using the rescaled variance test. Our results therefore do not provide strong empirical support for the stationarity of real exchange rates.

Single‐Index Additive Vector Autoregressive Time Series Models

Abstract. We study a new class of nonlinear autoregressive models for vector time series, where the current vector depends on single‐indexes defined on the past lags and the effects of different lags have an additive form. A sufficient condition is provided for stationarity of such models. We also study estimation of the proposed model using P‐splines, hypothesis testing, asymptotics, selection of the order of the autoregression and of the smoothing parameters and nonlinear forecasting. We perform simulation experiments to evaluate our model in various settings. We illustrate our methodology on a climate data set and show that our model provides more accurate yearly forecasts of the El Niño phenomenon, the unusual warming of water in the Pacific Ocean.

Conditional inference in linear versus nonlinear models for binary time series

The modelling of discrete such as binary time series, unlike the continuous time series, is not easy. This is due to the fact that there is no unique way to model the correlation structure of the repeated binary data. Some models may also provide a complicated correlation structure with narrow ranges for the correlations. In this paper, we consider a nonlinear dynamic binary time series model that provides a correlation structure which is easy to interpret and the correlations under this model satisfy the full−1 to 1 range. For the estimation of the parameters of this nonlinear model, we use a conditional generalized quasilikelihood (CGQL) approach which provides the same estimates as those of the well-known maximum likelihood approach. Furthermore, we consider a competitive linear dynamic binary time series model and examine the performance of the CGQL approach through a simulation study in estimating the parameters of this linear model. The model mis-specification effects on estimation as well as forecasting are also examined through simulations.

Detection of low-dimensional chaos in buildings energy consumption time series

Abstract In this paper, nonlinear time series modeling techniques are applied to analyze building energy consumption data. The time series were obtained for the benchmark data set Proben 1, and comes from the first energy prediction contest, the Great Building Energy Predictor Shootout I, organized by ASHRAE. The phase space, which describes the evolution of the behavior of a nonlinear system, is reconstructed using the delay embedding theorem suggested by TAKENS. The embedding parameters, e.g. the delay time and the embedding dimension are estimated using the average mutual information (AMI) of the data and the false nearest neighbor (FNN) algorithm, respectively. Nonlinearity was detected, by applying the surrogate data sets method. Numerically estimated non-integral fractal dimension and a positive Lyapunov exponent are not necessarily sufficient indication of chaos; therefore we apply a more stringent criterion, developed by Gao and Zheng, which is based on the logarithmic displacement of time-dependent exponent curves, and show that these data are chaotic. Based on this analysis and proof, we then calculate the correlation dimension of the resulting attractor and the largest Lyapunov exponent. The correlation dimension 3.47 and largest Lyapunov exponent 0.047 are estimated. These results indicate that chaotic characteristics obviously exist in the specific energy consumption data set, and thus techniques based on phase space dynamics can be used to analyze and predict buildings energy use.

Selecting nonlinear time series models using information criteria

Abstract. This article considers the problem of selecting among competing nonlinear time series models by using complexity‐penalized likelihood criteria. An extensive simulation study is undertaken to assess the small‐sample performance of several popular criteria in selecting among nonlinear autoregressive models belonging to some families that have been popular with practitioners.

Falling and Explosive, Dormant and Rising Markets via Multiple-Regime Financial Time Series Models

A multiple-regime threshold nonlinear financial time series model, with a fat-tailed error distribution, is discussed and Bayesian estimation and inference is considered. Further, approximate Bayesian posterior model comparison among competing models with different numbers of regimes is considered: effectively a test for the number of required regimes. An adaptive MCMC sampling scheme is designed, while importance sampling is employed to estimate Bayesian residuals for model diagnostic testing. Our modeling framework provides a parsimonious representation of well-known stylized features of financial time series and facilitates statistical inference in the presence of high or explosive persistence and dynamic conditional volatility. We focus on the three-regime case: the main feature of the model is the capturing of mean and volatility asymmetries in financial markets, while allowing an explosive volatility regime. A simulation study highlights the properties of our MCMC estimators and the accuracy and favourable performance as a model selection tool, compared to a deviance criterion, of the posterior model probability approximation method. An empirical study of eight international oil & gas markets illustrates strong support for the three-regime model over its competitors, in most markets, in terms of model posterior probability and in showing three distinct regime behaviours: falling/explosive, dormant and rising markets.

Seasonality with Trend and Cycle Interactions in Unobserved Components Models

SummaryUnobserved components time series models decompose a time series into a trend, a season, a cycle, an irregular disturbance and possibly other components. These models have been successfully applied to many economic time series. The standard assumption of a linear model, which is often appropriate after a logarithmic transformation of the data, facilitates estimation, testing, forecasting and interpretation. However, in some settings the linear–additive framework may be too restrictive. We formulate a non-linear unobserved components time series model which allows interactions between the trend–cycle component and the seasonal component. The resulting model is cast into a non-linear state space form and estimated by the extended Kalman filter, adapted for models with diffuse initial conditions. We apply our model to UK travel data and US unemployment and production series, and show that it can capture increasing seasonal variation and cycle-dependent seasonal fluctuations.

Adaptive ARFIMA Models of Inflation

Many previous analyses of inflation have used either long memory or nonlinear time series models. This paper suggests a simple adaptive modification of the basic ARFIMA model, which uses a Flexible Fourier Form to allow for a time varying intercept. Simulation evidence suggests the model provides a good representation of various forms of structural breaks and also that the new model can be efficiently estimated by a QMLE approach. We investigate monthly CPI inflation series for the G7 countries and find evidence of stable long memory parameters across regimes and also of significant nonlinear effects. The estimated adaptive ARFIMA models generally have less persistent long memory parameters than previous studies, with the estimated time dependent intercept being an important component. The model is also supplemented with an adaptive FIGARCH component, yielding a double nonlinear long memory model.