This paper performs a thorough statistical examination of the time-series properties of the daily market volatility index (VIX) from the Chicago Board Options Exchange (CBOE). The motivation lies on the widespread consensus that the VIX is a barometer to the overall market sentiment as to what concerns investors’ risk appetite. Our preliminary analysis suggests that the VIX index displays long-range dependence. This is well line with the strong empirical evidence in the literature supporting long memory in both options-implied and realized variances. We thus resort to both parametric and semiparametric heterogeneous autoregressive (HAR) processes for modeling and forecasting purposes. Our main findings are as follows. First, we confirm the evidence in the literature that there is a strong negative relationship between the VIX index and the S&P 500 index return as well as a positive contemporaneous link with the volume of the S&P 500 index. Second, we find that the VIX index tends to decline as the long-run oil price increases. This is not entirely surprising given the high demand from oil in the last years as well as the recent trend of shorting energy prices in the hedge fund industry. Third, the term spread has no long-run impact in the VIX index despite of the positive contemporaneous link. Fourth, there is some weak evidence that increases in the value of the US dollar tend to move down options-implied market volatility. Finally, we cannot reject the linearity of the above relationships, neither in sample nor out of sample. As for the latter, we actually show that it is pretty hard to beat the pure HAR process because of the very persistent nature of the VIX index. It is not impossible, though. We set out a semiparametric HAR-type model that performs very well across different forecasting horizons by using the above explanatory variables in a quite efficient manner.

Publicado no  Journal of Banking and Finance, 40, 1-10, 2014

This paper shows that bagging can improve the forecast accuracy of time series models for realized volatility. We consider 23 stocks from the Dow Jones Industrial Average over the sample period 1995 to 2005 and employ two different forecast models, a log-linear specification in the spirit of the heterogeneous autoregressive model and a nonlinear specification with logistic transitions. Both forecast model types benefit from bagging, in particular in the 1990s part of our sample. The log-linear specification shows larger improvements than the nonlinear model. Bagging the log-linear model yields the highest forecast accuracy on our sample.

Publicado no Econometric Reviews, 29, 571-593. 

In this paper we propose a flexible model to capture nonlinearities and long-range dependence in time series dynamics. The new model is a multiple regime smooth transition extension of the Heterogenous Autoregressive (HAR) model, which is specifically designed to model the behaviour of the volatility inherent in financial time series. The model is able to describe simultaneously long memory, as well as sign and size asymmetries. A sequence of tests is developed to determine the number of regimes, and an estimation and testing procedure is presented. Monte Carlo simulations evaluate the finite-sample properties of the proposed tests and estimation procedures. We apply the model to several Dow Jones Industrial Average index stocks using transaction level data from the Trades and Quotes database that covers ten years of data. We find strong support for long memory and both sign and size asymmetries. Furthermore, the new model, when combined with the linear HAR model, is viable and flexible for purposes of forecasting volatility

Publicado no Journal of Econometrics, v. 147, 2008

publicado em Journal of  Mathematical Economics, v. 44, n. 5-6, abril 2008

publicado em Journal of Public Economics v. 93, n. 1-2, p. 280-295, 2009

In this paper, we propose a class of logarithmic ACD-type models that accommodates overdispersion, intermittent dynamics, multiple regimes, and asymmetries in financial durations. In particular, our functional coefficient logarithmic autoregressive conditional duration (FC-LACD) model relies on a smooth-transition autoregressive specification. The motivation lies on the fact that the latter yields a universal approximation if one lets the number of regimes grows without bound. After establishing sufficient conditions for strict stationarity, we address model identifiability as well as the asymptotic properties of the quasi-maximum likelihood (QML) estimator for the FC-LACD model with a fixed number of regimes. In addition, we also discuss how to consistently estimate a semiparametric variant of the FC-LACD model that takes the number of regimes to infinity. An empirical illustration indicates that our functional coefficient model is flexible enough to model IBM price durations.

A ser publicado em Econometric Reviews

The goal of this paper is twofold. First, using five of the most actively traded stocks in the Brazilian financial market, this paper shows that the normality assumption commonly used in the risk management area to describe the distributions of returns standardized by volatilities is not compatible with volatilities estimated by EWMA or GARCH models. In sharp contrast, when the information contained in high frequency data is used to construct the realized volatility measures, we attain the normality of the standardized returns, giving promise of improvements in Value-at-Risk statistics. We also describe the distributions of volatilities of the Brazilian stocks, showing that they are nearly lognormal. Second, we estimate a simple model of the log of realized volatilities that differs from the ones in other studies. The main difference is that we do not find evidence of long memory. The estimated model is compared with commonly used alternatives in out-of-sample forecasting experiment.

Publicado na Revista Brasileira de Finanças, Volume 4, p.321-343, 2006

Does volatility reflect a continuous reaction to past shocks or do changes in the markets induce shifts in the volatility dynamics? In this paper, we provide empirical evidence that cumulated price variations convey meaningful information about multiple regimes in the realized volatility of stocks, where large falls (rises) in prices are linked to persistent regimes of high (low) variance in stock returns. Incorporating past cumulated daily returns as an explanatory variable in a flexible and systematic nonlinear framework, we estimate that falls of different magnitudes over less than two months are associated with volatility levels 20% and 60% higher than the average of periods with stable or rising prices. We show that this effect accounts for large empirical values of long memory parameter estimates. Finally, we show that, while introducing more realistic dynamics for volatility, the model is able to overall improve or at least retain out-of-sample performance in forecasting when compared to standard methods. Most importantly, the model is more robust to periods of financial crises, when it attains significantly better forecasts.

Publicado em International Journal of Forecasting, v.25, p. 304-327, 2009

This article reviews the exciting and rapidly expanding literature on realized volatility. After presenting a general univariate framework for estimating realized volatilities, a simple discrete time model is presented in order to motivate the main results. A continuous time specification provides the theoretical foundation for the main results in this literature. Cases with and without microstructure noise are considered, and it is shown how microstructure noise can cause severe problems in terms of consistent estimation of the daily realized volatility. Independent and dependent noise processes are examined. The most important methods for providing consistent estimators are presented, and a critical exposition of different techniques is given. The finite sample properties are discussed in comparison with their asymptotic properties. A multivariate model is presented to discuss estimation of the realized covariances. Various issues relating to modelling and forecasting realized volatilities are considered. The main empirical findings using univariate and multivariate methods are summarized.

Publicado na Econometric Reviews, v. 27, jan-junho 2008