Temporal Aggregation of Volatility Models

No abstract.

Testing Normality: A GMM Approach

In this paper, we consider testing marginal normal distributional assumptions. More precisely, we propose tests based on moment conditions implied by normality. These moment conditions are known as the Stein (1972) equations. They coincide with the first class of moment conditions derived by Hansen and Scheinkman (1995) when the random variable of interest is a scalar diffusion. Among other examples, Stein equation implies that the mean of Hermite polynomials is zero. The GMM approach we adopted is well suited for two reasons. It allows us to study in detail the parameter uncertainty problem, i.e., when the tests depend on unknown parameters that have to be estimated. In particular, we characterize the moment conditions that are robust against parameter uncertainty and show that Hermite polynomials are special examples. This is the main contribution of the paper. The second reason for using GMM is that our tests are also valid for time series. In this case, we adopt a Heteroskedastic-Autocorrelation-Consistent approach to estimate the weighting matrix when the dependence of the data is unspecified. We also make a theoretical comparison of our tests with Jarque and Bera (1980) and OPG regression tests of Davidson and MacKinnon (1993). Finite sample properties of our tests are derived through a comprehensive Monte Carlo study. Finally, three applications to GARCH and realized volatility models are presented. Dans cet article, nous testons des hypothèses de normalité marginale. Plus précisément, nous proposons des tests fondés sur des conditions de moments connues sous le nom d?équations de Stein. Ces conditions coïncident avec la première classe de conditions de moments obtenues par Hansen et Scheinkman (1995) quand la variable d?intérêt est une diffusion. L?équation de Stein implique, par exemple, que l?espérance de chaque polynôme de Hermite est nulle. L?approche GMM est utile pour deux raisons. Elle nous permet de tenir compte du problème d?incertitude des paramètres préalablement estimés. En particulier, nous caractérisons les conditions de moments qui sont robustes à ce problème et montrons que c?est le cas des polynômes de Hermite. C?est la principale contribution de l?article. Le second avantage de l?approche GMM est que nos tests sont aussi valides pour des séries temporelles. Dans ce cas, nous adoptons une approche HAC (Heteroskedastic-Autocorrelation-Consistent) pour estimer la matrice de poids qui intervient dans la statistique de test quand la forme sérielle des données n?est pas spécifiée. Nous comparons nos tests de manière théorique avec les tests de Jarque et Bera (1981) et les tests dits OPG de Davidson et MacKinnon (1993). Les propriétés de petits échantillons de nos tests sont étudiées par simulation. Finalement, nous appliquons nos tests à trois exemples de modèles de volatilité GARCH et volatilité réalisée.Normality, Stein-Hansen-Scheinkman equation, GMM, Hermite polynomials, parameter uncertainty, HAC, OPG regression, Normalité, équation de Stein-Hansen-Scheinkman, GMM, polynômes de Hermite, incertitude des paramètres, HAC, régression OPG

Volatility forecasting when the noise variance Is time-varying

This paper explores the volatility forecasting implications of a model in which the friction in high-frequency prices is related to the true underlying volatility. The contribution of this paper is to propose a framework under which the realized variance may improve volatility forecasting if the noise variance is related to the true return volatility. The realized variance is defined as the sum of the squared intraday returns. When based on high-frequency returns, the realized variance would be non-informative for the true volatility under the standard framework. In this new setting, we revisit the results of Andersen et al. (2011) and quantify the predictive ability of several measures of integrated variance. Importantly, the time-varying aspect of the noise variance implies that the forecast of the integrated variance is different from the forecast of a realized measure. We characterize this difference, which is time-varying, and propose a feasible bias correction. We assess the usefulness of our approach for realistic models, then study the empirical implication of our method when dealing with forecasting integrated variance or trading options. The empirical results for Alcoa stock show several improvements resulting from the assumption of time-varying noise variance

A distributional approach to realized volatility

This paper proposes new measures of the integrated variance, measures which use highfrequency bid-ask spreads and quoted depths. The traditional approach assumes that the mid-quote is a good measure of frictionless price. However, the recent high-frequency econometric literature takes the mid-quote as a noisy measure of the frictionless price and proposes new and robust estimators of the integrated variance. This paper forgoes the common assumption of an additive friction term, and demonstrates how the quoted depth may be used in the construction of refined realized volatility measures under the assumption that the true frictionless price lies between the bid and the ask. More specifically, we make assumptions about the conditional distribution of the frictionless price given the available information, including quotes and depths. This distributional assumption leads to new measures of the integrated variance that explicitly incorporate the depths. We then empirically compare the new measures with the robust ones when dealing with forecasting integrated variance or trading options. We show that, in several cases, the new measures dominate the traditional measures

Testing Normality : A GMM Approach

In this paper, we consider testing marginal normal distributional assumptions. More precisely, we propose tests based on moment conditions implied by normality. These moment conditions are known as the Stein (1972) equations. They coincide with the first class of moment conditions derived by Hansen and Scheinkman (1995) when the random variable of interest is a scalar diffusion. Among other examples, Stein equation implies that the mean of Hermite polynomials is zero. The GMM approach we adopted is well suited for two reasons. It allows us to study in detail the parameter uncertainty problem, i.e., when the tests depend on unknown parameters that have to be estimated. In particular, we characterize the moment conditions that are robust against parameter uncertainty and show that Hermite polynomials are special examples. This is the main contribution of the paper. The second reason for using GMM is that our tests are also valid for time series. In this case, we adopt a Heteroskedastic-Autocorrelation-Consistent approach to estimate the weighting matrix when the dependence of the data is unspecified. We also make a theoretical comparison of our tests with Jarque and Bera (1980) and OPG regression tests of Davidson and MacKinnon (1993). Finite sample properties of our tests are derived through a comprehensive Monte Carlo study. Finally, three applications to GARCH and realized volatility models are presented