Volatility model selection for extremes of financial time series

Although both widely used in the financial industry, there is quite often very little justification why GARCH or stochastic volatility is preferred over the other in practice. Most of the relevant literature focuses on the comparison of the fit of various volatility models to a particular data set, which sometimes may be inconclusive due to the statistical similarities of both processes. With an ever growing interest among the financial industry in the risk of extreme price movements, it is natural to consider the selection between both models from an extreme value perspective. By studying the dependence structure of the extreme values of a given series, we are able to clearly distinguish GARCH and stochastic volatility models and to test statistically which one better captures the observed tail behaviour. We illustrate the performance of the method using some stock market returns and find that different volatility models may give a better fit to the upper or lower tails.     

Volatility estimation from short time series of stock prices

We consider estimation of the historical volatility of stock prices. It is assumed that the stock prices are represented as time series formed as samples of the solution of a stochastic differential equation with random and time-varying parameters; these parameters are not observable directly and have unknown evolution law. The price samples are available with limited frequency only. In this setting, the estimation has to be based on short time series, and the estimation error can be significant. We suggest some supplements to the existing nonparametric methods of volatility estimation. Two modifications of the standard summation formula for the volatility are derived. In addition, a linear transformation eliminating the appreciation rate and preserving the volatility is suggested.     

Volatility forecasting of financial time series using wavelet based exponential generalized autoregressive conditional heteroscedasticity model

Abstract

An improved forecasting model by merging two different computational models in predicting future volatility was proposed. The model integrates wavelet and EGARCH model where the pre-processing activity based on wavelet transform is performed with de-noising technique to eliminate noise in observed signal. The denoised signal is then feed into EGARCH model to forecast the volatility. The predictive capability of the proposed model is compared with the existing EGARCH model. The results show that the hybrid model has increased the accuracy of forecasting future volatility.     

Selection of regressors in econometrics: parametric and nonparametric methods selection of regressors in econometrics

The present paper addresses the selection-of-regressors issue into a general discrimination framework. We show how this framework is useful in unifying various procedures for selecting regressors and helpful in understanding the different strategies underlying these procedures. We review selection of regressors in linear, nonlinear and nonparametric regression models. In each case we successively consider model selection criteria and hypothesis testing procedures.     

Selection of regressors in econometrics: parametric and nonparametric methods selection of regressors in econometrics

The present paper addresses the selection-of-regressors issue into a general discrimination framework. We show how this framework is useful in unifying various procedures for selecting regressors and helpful in understanding the different strategies underlying these procedures. We review selection of regressors in linear, nonlinear and nonparametric regression models. In each case we successively consider model selection criteria and hypothesis testing procedures.     

Generalized Tobit models: diagnostics and application in econometrics

The standard Tobit model is constructed under the assumption of a normal distribution and has been widely applied in econometrics. Atypical/extreme data have a harmful effect on the maximum likelihood estimates of the standard Tobit model parameters. Then, we need to count with diagnostic tools to evaluate the effect of extreme data. If they are detected, we must have available a Tobit model that is robust to this type of data. The family of elliptically contoured distributions has the Laplace, logistic, normal and Student-t cases as some of its members. This family has been largely used for providing generalizations of models based on the normal distribution, with excellent practical results. In particular, because the Student-t distribution has an additional parameter, we can adjust the kurtosis of the data, providing robust estimates against extreme data. We propose a methodology based on a generalization of the standard Tobit model with errors following elliptical distributions. Diagnostics in the Tobit model with elliptical errors are developed. We derive residuals and global/local influence methods considering several perturbation schemes. This is important because different diagnostic methods can detect different atypical data. We implement the proposed methodology in an R package. We illustrate the methodology with real-world econometrical data by using the R package, which shows its potential applications. The Tobit model based on the Student-t distribution with a small quantity of degrees of freedom displays an excellent performance reducing the influence of extreme cases in the maximum likelihood estimates in the application presented. It provides new empirical evidence on the capabilities of the Student-t distribution for accommodation of atypical data.     

Polynomial trends in time series

The J^* test which was previously proposed by the present author for the detection of a trend in a time series does not depend on any quantitative assumptions, but in the case of a polynomial trend it depends on its degree; if this degree is too high, the test cannot be applied. The author finds a bound of the significance level at which the test can be applied when the sample size, as well as a bound of the degree of the trend, are given. Asymptotic results are used only when we trust the asymptotic distribution of J^* under the null hypothesis.     

Assessing influence in time series

This paper studies influential observations on the spectrum of a stationary stochastic process. We introduce a leave-one-out procedure in spectral density estimation to identify influential points. A simulated envelope is proposed to assess the magnitude of influence when the data follow an autoregressive integrated moving average model. Practical illustrations are discussed in two examples.     

Missing observations in time series

A method based on forecasting techniques is proposed to estimate missing observations in time series. Using mean squares, this method is compared to the minimum mean square estimate.     

Developments and new dimensions in econometrics

Summary This special issue of the Journal of the German Statistical Society presents 14 papers with surveys on the development and new topics in econometrics. The articles aim to demonstrate how German econometricians see the discipline from their specific view. They briefly describe the main strands and emphasize some recent methods.     

Tests of specification in econometrics

This survey of recent developments in testing for misspecification of econometric models reviews procedures based on a method due to Hausman. Particular attention is given to alternative forms of the test, its relationship to classical test procedures, and its role in pre-test estimation.     

Cluster of time series models: an example

We show that the various times series models, reported in the literature, for the Canadian lynx data form interesting clusters.     

Tests of specification in econometrics

This survey of recent developments in testing for misspecification of econometric models reviews procedures based on a method due to Hausman. Particular attention is given to alternative forms of the test, its relationship to classical test procedures, and its role in pre-test estimation.     

Self-affine time series: measures of weak and strong persistence

In this paper, we examine self-affine time series and their persistence. Time series are defined to be self-affine if their power-spectral density scales as a power of their frequency. Persistence can be classified in terms of range, short or long range, and in terms of strength, weak or strong. Self-affine time series are scale-invariant, thus they always exhibit long-range persistence. Synthetic self-affine time series are generated using the Fourier power-spectral method. We generate fractional Gaussian noises (fGns), −1β1, where β is the power-spectral exponent. These are summed to give fractional Brownian motions (fBms), 1β3, where the series are self-affine fractals with fractal dimension 1D2; β=2 is a Brownian motion. With β>1, the time series are non-stationary and moments of the time series depend upon its length; with β<1 the time series are stationary. We define self-affine time series with β>1 to have strong persistence and with β<1 to have weak persistence. We use a variety of techniques to quantify the strength of persistence of synthetic self-affine time series with −3β5. These techniques are effective in the following ranges: (1) semivariograms, 1β3, (2) rescaled-range (R/S) analyses, −1β1, (3) Fourier spectral techniques, all values of β, and (4) wavelet variance analyses, all values of β. Wavelet variance analyses lack many of the inherent problems that are found in Fourier power-spectral analysis.