Nonlinear Time Series Models Research Articles

Discussion of “An analysis of global warming in the Alpine region based of nonlinear nonstationary time series models” by F. Battaglia and M. K. Protopapas

We discuss the scientific contribution of Battaglia and Protopapas’ paper concerning the debate on global warming supported by an extensive analysis of temperature time series in the Alpine region. In the work, Authors use several exploratory and modelling tools for assessing and discriminating the presence of different patterns in the data. We add some general and specific considerations mainly devoted to the modelling stage of their analysis.

An analysis of global warming in the Alpine region based on nonlinear nonstationary time series models

The annual temperatures recorded for the last two centuries in fifteen european stations around the Alps are analyzed. They show a global warming whose growth rate is not however constant in time. An analysis based on linear Arima models does not provide accurate results. Thus, we propose threshold nonlinear nonstationary models based on several regimes both in time and in levels. Such models fit all series satisfactorily, allow a closer description of the temperature changes evolution, and help to discover the essential differences in the behavior of the different stations.

Inference for random coefficient volatility models

Estimating functions have been shown to be convenient to study inference for nonlinear time series models. One such model is the recently proposed Random Coefficient Autoregressive (RCA) model with Generalized Autoregressive Heteroscedasticity (GARCH) errors (Thavaneswaran et al., 2009). We derive the martingale estimating functions for the joint estimation of the conditional mean and variance parameters and we show the information gain relative to conditional least square estimation.

Discussion of “An analysis of global warming in the Alpine region based on nonlinear nonstationary time series models” by F. Battaglia and M. K. Protopapas

The paper by Battaglia and Protopapas (Stat Method Appl 2012) is stimulating. It gives an elegant mathematical generalization of autoregressive models (the nine types). It explains state-of-the-art model fitting techniques (genetic algorithm combined with fitness function and least squares). It is written in a fluent and authoritative manner. Important for having a wider impact: it is accessible to non-statisticians. Finally, it has interesting results on the temperature evolution over the instrumental period (roughly the past 200 years). These merits make this paper an important contribution to applied statistics as well as climatology. As a climate researcher, coming from Physics and having had an affiliation with a statistical institute only as postdoc, I re-analyse here three data series with the aim of providing motivation for model selection and interpreting the results from the climatological perspective.

Analysing nonlinear time series with central subspace

Traditionally, time series analysis involves building an appropriate model and using either parametric or nonparametric methods to make inference about the model parameters. Motivated by recent developments for dimension reduction in time series, an empirical application of sufficient dimension reduction (SDR) to nonlinear time series modelling is shown in this article. Here, we use time series central subspace as a tool for SDR and estimate it using mutual information index. Especially, in order to reduce the computational complexity in time series, we propose an efficient estimation method of minimal dimension and lag using a modified Schwarz–Bayesian criterion, when either of the dimensions and the lags is unknown. Through simulations and real data analysis, the approach presented in this article performs well in autoregression and volatility estimation.

Testing for intercept-scale switch in linear autoregression

Autoregressive models with switching regime are a frequently used class of nonlinear time series models, which are popular in finance, engineering, and other fields. We consider linear switching autoregressions in which the intercept and variance possibly switch simultaneously, while the autoregressive parameters are structural and hence the same in all states, and we propose quasi-likelihood-based tests for a regime switch in this class of models. Our motivation is from financial time series, where one expects states with high volatility and low mean together with states with low volatility and higher mean. We investigate the performance of our tests in a simulation study, and give an application to a series of IBM monthly stock returns. The Canadian Journal of Statistics 40: 427–446; 2012 © 2012 Statistical Society of Canada Les modèles autorégressifs avec changement de régime forment une classe de modèles de séries chronologiques non linéaires utilisés fréquemment et ils sont populaires en finance, génie et autres domaines. Nous considérons les autorégressions linéaires pour lesquelles l'ordonnées à l'origine et la variance peuvent changés simultanément tandis que les paramètres autorégressifs sont structuraux et par conséquent, ils sont les mêmes pour tous les régimes. Nous proposons un test de quasi-vraisemblance sur le changement de régime pour les modèles de cette classe. Notre motivation provient des séries chronologiques financières, où nous pouvons avoir des régimes avec une grande volatilité et une moyenne faible et d'autres avec une faible volatilité et une grande moyenne. La performance de notre test est étudiée à l'aide de simulation et nous l'appliquons aux rendements mensuels des actions d'IBM. La revue canadienne de statistique 40: 427–446; 2012 © 2012 Société statistique du Canada Additional supporting information may be found in the online version of this article. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

Discussion of ‘An analysis of global warming in the Alpine region based on nonlinear nonstationary time series models’ by Battaglia and Protopapas

This discussion focuses on threshold nonstationary–nonlinear time series modelling; it raises various issues to do with identifiability and model complexity. It also gives some background history concerning smooth threshold/transition autoregressive models and hidden Markov switching models.

Misspecification tests based on quantile residuals

Summary We develop quantile residual-based misspecification tests and apply them to non-linear time series models for which conventional residuals are unsuited. We formulate a general framework and use it to obtain computationally simple tests aimed at detecting autocorrelation, conditional heteroscedasticity and non-normality in quantile residuals. These tests are generalizations of similar previous tests based on conventional residuals and the Lagrange Multiplier principle. According to simulations on mixture models our tests are reasonable in size and more powerful than alternatives in the literature. An empirical example on interest rate data illustrates the usefulness of these methods.

Exchange rate pass-through and inflation: A nonlinear time series analysis

This paper investigates the relationship between the exchange rate pass-through (ERPT) and inflation by estimating a nonlinear time series model. Based on a simple theoretical model of ERPT determination, we show that the dynamics of ERPT can be well approximated by a class of smooth transition autoregressive (STAR) models using the past inflation rate as a transition variable. We employ several U-shaped transition functions in the estimation of the time-varying ERPT to US domestic prices. The estimation result suggests that declines in the ERPT during the 1980s and 1990s are associated with lowered inflation.

The marked empirical process to test nonlinear time series against a large class of alternatives when the random vectors are nonstationary and absolutely regular

In this paper, we propose a method for testing absolutely regular and possibly nonstationary nonlinear time-series, with application to general AR-ARCH models. Our test statistic is based on a marked empirical process of residuals which is shown to converge to a Gaussian process with respect to the Skohorod topology. This testing procedure was first introduced by Stute [Nonparametric model checks for regression, Ann. Statist. 25 (1997), pp. 613–641] and then widely developed by Ngatchou-Wandji [Weak convergence of some marked empirical processes: Application to testing heteroscedasticity, J. Nonparametr. Stat. 14 (2002), pp. 325–339; Checking nonlinear heteroscedastic time series models, J. Statist. Plann. Inference 133 (2005), pp. 33–68; Local power of a Cramer-von Mises type test for parametric autoregressive models of order one, Compt. Math. Appl. 56(4) (2008), pp. 918–929] under more general conditions. Applications to general AR-ARCH models are given.

Nonlinear time series modeling and prediction using functional weights wavelet neural network-based state-dependent AR model

This paper presents a Functional Weights Wavelet Neural Network-based state-dependent AR (FWWNN-AR) model with the main objective to address the modeling and prediction problem of nonlinear time series. The FWWNN-AR model is a state-dependent autoregressive (SD-AR) model, which has its coefficients approximated by a set of Functional Weights Wavelet Neural Network (FWWNN). The FWWNN is an enhanced type of wavelet neural network comprising of five layers: input, wavelet, product, output and functional weight layer that computes the weights as function of inputs thus making the weights to vary with the inputs and to share the dynamics with the wavelet compartment. The FWWNN-AR model possesses both the advantages of the state-dependent AR model in the description of nonlinear dynamics using few nodes and of the FWWNN in functional approximation considering mutually the time and frequency spaces. It learns the nonlinear dynamics from three distinct levels: AR level, Wavelet compartment level and functional weights level. A Structured Nonlinear Parameter Optimization Method (SNPOM) is applied to estimate the FWWNN-AR model parameters. This learning approach divides the parameter search space into linear and nonlinear subspaces and centers the search in the nonlinear subspace, but at each iteration in the optimization process, a search in the nonlinear (or linear) subspace is executed on the basis of the estimated values just obtained in linear (or nonlinear) subspace. The search in the nonlinear subspace uses a method similar to the Levemberg–Marquardt Method (LMM), and the search in the linear subspace uses the Least Square Method (LSM). The proposed model is validated by comparing its performances and effectiveness with those achieved by some well known models on both generated and real nonlinear time series.

Ship's tracking control based on nonlinear time series model

For actualization of ship tracking control along a desired path with a constant velocity, a hybrid model is proposed to represent the ship's tracking dynamic behavior. Firstly, a single-input single-output nonlinear time series model is built for characterizing the responses between the ship's heading angle deviation and its rudder angle. To represent nonlinearity of the ship motion, the rolling angle is used as the model index to make the model parameters vary with the ship sailing states considering the yaw-heel-effect. The nonlinear time series model is identified offline by using previously observed real data. Then, a state-space model combined with the relationship between the heading angle deviations and the cross track errors is proposed to represent the tracking dynamic behavior. On the basis of the identified state-space type tracking motion model, a predictive controller is designed to steer the ship sailing forward with the constant velocity along the predefined reference path. The effectiveness of the nonlinear time series model-based method for the tracking control proposed in this paper is demonstrated by simulation studies and actual experiments.

Limit experiments of GARCH

GARCH is one of the most prominent nonlinear time series models, both widely applied and thoroughly studied. Recently, it has been shown that the COGARCH model (which was introduced a few years ago by Kl\"{u}ppelberg, Lindner and Maller) and Nelson's diffusion limit are the only functional continuous-time limits of GARCH in distribution. In contrast to Nelson's diffusion limit, COGARCH reproduces most of the stylized facts of financial time series. Since it has been proven that Nelson's diffusion is not asymptotically equivalent to GARCH in deficiency, in the present paper, we investigate the relation between GARCH and COGARCH in Le Cam's framework of statistical equivalence. We show that GARCH converges generically to COGARCH, even in deficiency, provided that the volatility processes are observed. Hence, from a theoretical point of view, COGARCH can indeed be considered as a continuous-time equivalent to GARCH. Otherwise, when the observations are incomplete, GARCH still has a limiting experiment, which we call MCOGARCH, which is not equivalent, but nevertheless quite similar, to COGARCH. In the COGARCH model, the jump times can be more random than for the MCOGARCH, a fact practitioners may see as an advantage of COGARCH.

The Asymmetric Business Cycle

Abstract The business cycle is a fundamental yet elusive concept in macroeconomics. In this paper, we consider the problem of measuring the business cycle. First, we argue for the output-gap view that the business cycle corresponds to transitory deviations in economic activity away from a permanent, or trend, level. Then we investigate the extent to which a general model-based approach to estimating trend and cycle for the U.S. economy leads to measures of the business cycle that reflect models versus the data. We find empirical support for a nonlinear time series model that produces a business cycle measure with an asymmetric shape across NBER expansion and recession phases. Specifically, this business cycle measure suggests that recessions are periods of relatively large and negative transitory fluctuations in output. However, several close competitors to the nonlinear model produce business cycle measures of widely differing shapes and magnitudes. Given this model-based uncertainty, we construct a mode...

Prediction for noisy nonlinear time series by echo state network based on dual estimation

When using echo state networks (ESNs) to establish a regression model for noisy nonlinear time series, only the output uncertainty was usually concerned in some literature. However, the unconsidered internal states uncertainty is actually important as well. In this study, an improved ESN model with noise addition is proposed, in which the additive noises describe the internal state uncertainty and the output uncertainty. In terms of the parameters determination of this prediction model, a nonlinear/linear dual estimation consisting of a nonlinear Kalman filter and a linear one is proposed to perform the supervised learning. For verifying the effectiveness of the proposed method, the noisy Mackey Glass time series and the generation flow of blast furnace gas (BFG) in steel industry practice are both employed. The experimental results demonstrate that the proposed method is effective and robust for noisy nonlinear time series prediction.

Specification Testing Driven by Orthogonal Series in Nonlinear and Nonstationary Time Series Models

This paper establishes two simple and new specification tests based on the use of an orthogonal series. The paper then establishes an asymptotic theory for each of the proposed tests. The first test is initially proposed for the case where the regression function involved is integrable and the second test is an extended version of the first test for covering a class of non-integrable functions. The finite sample performance of the proposed tests is examined through using several simulated examples. Meanwhile, an application of the second test shows that a second-order polynomial model is more appropriate than a commonly used linear model for modelling the relationship between the United States consumers' consumption expenditure and disposable income over the time period of 1960-2009. Our experience generally shows that the proposed tests are easily implementable and also have stable sizes and good power properties even when the 'distance' between the null hypothesis and a sequence of local alternatives is asymptotically negligible.

G100055 非線形時系列モデルによる振動系の解析および推定について

This technique examined reply analysis in the non-linear oscillation system and estimation. We examined the application to an equivalent non-linear model and the conversion to a non-linear time series model, and the thing that was effective about a smooth nonlinearity characteristic and the discontinuous nonlinearity characteristics such as the coulomb friction showed this article by numerical simulation for non-linear oscillation system.

Pattern recognition using enhanced non-linear time-series models for predicting dynamic real-time decision making environments

The abundance of data and importance of knowledge extraction to foresee the future has made time dependent data analysis an inevitable and challenging task in all areas of science and engineering. High dimensionality and the presence of noise in the non-linear time-series data makes it difficult for the existing clustering algorithms to produce efficient results. Hence, two approaches for time series representation (TSR) techniques by name hybrid dimensionality reduction (HDR) and extended hybrid dimensionality reduction (EHDR) and high low non-overlapping (HLN) clustering algorithm that produces efficient results by controlling noise and reducing the dimensionality optimally are proposed. A comparison of the experimental results on intraday non-linear stock data sets to predict the similarity in their intraday behaviour using K-means clustering algorithm with MINDIST as distance measure using symbolic aggregate approximation (SAX) and HLN using HDR and EHDR has proved that EHDR and HDR TSRs outperforms the other models.

DNA Optimization Threshold Autoregressive Prediction Model and Its Application in Ice Condition Time Series

There are many parameters which are very difficult to calibrate in the threshold autoregressive prediction model for nonlinear time series. The threshold value, autoregressive coefficients, and the delay time are key parameters in the threshold autoregressive prediction model. To improve prediction precision and reduce the uncertainties in the determination of the above parameters, a new DNA (deoxyribonucleic acid) optimization threshold autoregressive prediction model (DNAOTARPM) is proposed by combining threshold autoregressive method and DNA optimization method. The above optimal parameters are selected by minimizing objective function. Real ice condition time series at Bohai are taken to validate the new method. The prediction results indicate that the new method can choose the above optimal parameters in prediction process. Compared with improved genetic algorithm threshold autoregressive prediction model (IGATARPM) and standard genetic algorithm threshold autoregressive prediction model (SGATARPM), DNAOTARPM has higher precision and faster convergence speed for predicting nonlinear ice condition time series.

Robust Spectral Analysis

In this paper I introduce quantile spectral densities that summarize the cyclical behavior of time series across their whole distribution by analyzing periodicities in quantile crossings. This approach can capture systematic changes in the impact of cycles on the distribution of a time series and allows robust spectral estimation and inference in situations where the dependence structure is not accurately captured by the auto-covariance function. I study the statistical properties of quantile spectral estimators in a large class of nonlinear time series models and discuss inference both at fixed and across all frequencies. Monte Carlo experiments illustrate the advantages of quantile spectral analysis over classical methods when standard assumptions are violated.