Volume 8, Issue 2 - Linear and Nonlinear Dynamics in Time Series Estella Bee Dagum and Tommaso Proietti, Editors

This paper extends the forward search technique to the analysis of structural time series data. It provides a series of powerful new forward plots that use information from the whole sample to display the effect of each observation on a wide variety of aspects of the fitted model and shows how the forward search, free from masking and swamping problems, can detect the main underlying features of the series under study (masked multiple outliers, level shifts or transitory changes). The effectiveness of the suggested approach is shown through the analysis of real and simulated data.

In this paper we suggest an extension of the forward search methodology to GARCH models which are often used for forecasting stock market volatility. It is frequently found that estimated residuals from GARCH models have excess kurtosis, even when one allows for conditional t-distributed errors. Some papers have appeared on outlier detection in GARCH models but the proposed methods are iterative and may suffer from masking effects. The forward search is a method for determining the effect of outliers on fitted parameters and for detecting also masked outliers. In the case of GARCH models outliers are strictly related to extreme observations which are responsible for the well-known volatility clustering of financial returns. It is possible, through the forward search, to visualize the effect on estimated parameters of patches of extremal observations.

Several methods have been proposed for identifying clusters of extreme values leading to estimators of the extremal index; the latter represents,in the limit, the mean-size of each cluster of thresholds exceedances. The detection of clusters of extremes is relevant for the class of processes commonly used in financial econometrics, such as GARCH processes. The paper illustrates a novel approach to the above identification that exploits additional knowledge of the trajectory of the process around extreme events, and compares it to traditional approaches, using simulation from a GARCH process. We assess the relative performance of estimators in terms of bias, mean square error and distributional properties.

In this paper, we describe and compare two simulated Maximum Likelihood estimation methods for a basic stochastic volatility model. For both methods, the likelihood function is estimated using importance sampling techniques. Based on a Monte Carlo study, we assess which method is more effective. Further, we validate the two methods using diagnostic importance sampling test procedures. Stochastic volatility models with Gaussian and Student-t distributed disturbances are considered.

In this paper we present a stochastic volatility model assuming that the return shock has a Skew-GED distribution. This allows a parsimonious yet flexible treatment of asymmetry and heavy tails in the conditional distribution of returns. The Skew-GED distribution nests both the GED, the Skew-normal and the normal densities as special cases so that specification tests are easily performed. Inference is conducted under a Bayesian framework using Markov Chain MonteCarlo methods for computing the posterior distributions of the parameters. More precisely, our Gibbs-MH updating scheme makes use of the Delayed Rejection Metropolis-Hastings methodology as proposed by Tierney and Mira (1999), and of Adaptive-Rejection Metropolis sampling. We apply this methodology to a data set of daily and weekly exchange rates. Our results suggest that daily returns are mostly symmetric with fat-tailed distributions while weekly returns exhibit both significant asymmetry and fat tails.

GARCH-type models have been analyzed assuming various nongaussian distributions of errors. In general, the asymmetric generalized Student-t random variable seems to be the distribution which better captures the nonnormality features of financial data. However, a drawback of this distribution is represented by the technical dificulties due to the evaluation of moments, especially in the case of fractional degrees of freedom. In this paper we propose to model high frequency time series returns using GARCH-type models with a generalized secant hyperbolic (GSH) distribution. The main advantage of the GSH distribution over the Student-t distribution is that all the moments are finite for each value of the shape parameter. The distribution is symmetric with respect to the mean, but we show that it is still possible to obtain the density in a closed form introducing a skewness parameter according to the method proposed by Fernandez and Steel. We use a Monte Carlo experiment to validate this distribution in the context of GARCH models with maximum likelihood estimates of parameters. Finally, we show an application to log returns of a stock index.

The instantaneous volatility of the price process is analyzed through the intraday financial durations between price changes. Previous research has traditionally dealt with parametric models without reaching a satisfactory level of adequacy. In this study, it is shown that by using a mixture of two exponential distributions a highly satisfactory fit can be obtained. The presence on financial markets of traders with different information sets makes reasonable the mixture assumption.

State space models provide a useful stochastic description for dynamic phenomena, based on unobserved or latent variables. When the model rests on linear and Gaussian assumptions there exists a well-known iterative procedure, called the Kalman filter, which gives analytic updating recursion for the filtering, the prediction and the smoothing distributions. However, this is rare and a state space model does not usually admit such a filter. For this reason, instead of looking for analytic solutions, a number of papers aim to define alternative procedures, giving numerical or approximate solutions. This paper concerns a particular class of models based on the assumption that the mixed process, obtained by alternating states and observations, is a Markov process. The main features of this class of models, proposed for stochastic volatility description by Barndorff-Nielsen (1997), are emphasized. In this framework, some new non-linear Gaussian state space models, computationally tractable and of potential interest for applications, may be defined.

In order to develop statistical tests for the Lyapunov exponents of deterministic dynamical systems, we develop bootstrap tests based on empirical likelihood for percentiles and expectiles of strictly stationary processes. The percentiles and expectiles are estimated in terms of asymmetric least deviations and asymmetric least squares methods. Asymptotic distributional properties of the estimators are established.

Chaos theory offers to time series analysis new perspectives as well as concepts and ideas that have a through contribution to statistics. On the other hand, statistical methodology has shown to play a crucial role for the comprehension of nonlinear and chaotic phenomena. One peculiar feature of chaotic systems is sensitivity to initial conditions, which is responsible of the unpredictability we experience in such phenomena. One of the most popular quantity that measures this property is the maximum Lyapunov characteristic exponent (MLCE). In this paper we discuss from a statistical perspective the problems arising in estimating both the MLCE and its generalizations in time series, issues that have recently deserved attention in the community of time series analysts. We also present a method based on resampling in order to assign confidence interval to the estimates of the MLCE. It is shown that in addition to answering the question of the presence of chaos, these methods give relevant contributions to the characterization of many other aspects of nonlinear time series.

In the present paper we consider the general class of first-order nonlinear models. The main contributions concern primerly a generalization of the conditions for geometric ergodicity presented in Ferrante et al. (2003). The obtained result is then applied to two classes of first-order nonlinear models not previously addressed. Secondly we apply to general firstorder nonlinear models some recently developed conditions for the existence of the invariant measure of a Markov process. For this class of nonlinear models we also prove that the usual drift-condition for geometric ergodicity for Markov chains still holds even in the presence of an alternative assumption than T-continuity.

We describe an algorithm for the construction of optimum experimental designs for the parameters in a regression model when the errors have a correlation structure. Our example is drawn from chemical kinetics, so that the model is nonlinear. Our algorithm has been implemented to be used when the model consists of a set of differential equations for which only numerical solutions ar available. However, the algorithm can also be used for standard regression models when the errors are correlated. The paper concludes with some discussion of outstanding issues in optimum design with correlated errors.

Practical aspects of likelihood-based inference and forecasting of series with long memory are considered, based on the arfima(p; d; q) model with deterministic regressors. Sampling characteristics of approximate and exact first-order asymptotic methods are compared. The analysis is extended using modified profile likelihood analysis, which is a higher-order asymptotic method suggested by Cox and Reid (1987). The relevance of the differences between the methods is investigated for models and forecasts of monthly core consumer price inflation in the US and quarterly overall consumer price inflation in the UK.

In situations where groups of economic time series are likely to have seasonal dynamics in common, use of seasonal information across series in estimating their seasonal factors should improve the seasonal factor estimates for individual series. Such information sharing can also be the basis for consistent seasonal adjustment of combinations of series. It also provides information about dominant seasonal patterns in component series. A methodology based on principal component analysis, developed earlier by the author with Eric Bartelsman, is applied to series of price indexes and production indexes from EU countries. Evidence of common structure is revealed. The implications of common estimation of seasonal factors are explored.

The paper introduces the class of seasonal specific structural time series models, according to which each season follows specific dynamics, but is also tied to the others by a common random effect. Seasonal specific models are dynamic variance components models that account for some kind of periodic behaviour, such as periodic heteroscedasticity, and are also tailored to deal with situations such that one or a group of seasons behave differently. Trends and non periodic features can still be extracted and their nature is discussed. Multivariate extensions entertain the case when cointegration pertains only to groups of seasons. It is finally shown that a circular correlation pattern for the idiosyncratic disturbances yields a periodic component that is isomorphic to a trigonometric seasonal com- ponent.

Following Henderson (1916) who developed a smoothing measure as a function of the weight system of a linear filter, Dagum and Luati (2002a) proposed a set of local statistical measures of bias, variance and mean square error which are intrinsic to the smoother and, thus, independent of the data to which they will be applied on. Theoretical measures of local fitting (LMSE) and smoothing (S) are calculated on the basis of the weight systems of the following nonparametric function estimators, Loess of degree 1 and 2, the cubic smoothing spline, the Gaussian kernel, and the 13-term Henderson filter.Our aim is to evaluate the extent to which these local or weight-based measures of fitting and smoothing can be used to obtain a priori general information on the global (data-based) goodness of fit and smoothness when such filters are applied to real time series. A priori knowledge of a smoother fit and smoothness performances when applied to real data is relevant, among others, for current economic analysis, the main interest of which is the detection of true turning points.For each function estimator, we calculate global measures of fitting (MSE) and smoothing (Q) using two large samples of real and simulated series characterized by different degrees of variability.The results show that the theoretical (weight-based) local smoothing measures are always in agreement with the global empirical ones. Similarly, the local (weight-based) mean square error, analyzed in terms of bias and variance composition, provides sound a priori information on the global goodness of fit given by the symmetric filters of each nonparametric estimator. For the asymmetric filters, the above analysis must be done taking into consideration also the impact of phase shifts which can be inferred from the smoothing measures.