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Estimating and Testing Stochastic Volatility Models using Realized Measures Valentina Corradi and Walter DistasoEstimating and Testing Stochastic Volatility Models using Realized Measures ∗
, 2004
"... This paper proposes a procedure to test for the correct specification of the functional form of the volatility process, within the class of eigenfunction stochastic volatility models (Meddahi, 2001). The procedure is based on the comparison of the moments of realized volatility measures with the cor ..."
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This paper proposes a procedure to test for the correct specification of the functional form of the volatility process, within the class of eigenfunction stochastic volatility models (Meddahi, 2001). The procedure is based on the comparison of the moments of realized volatility measures with the corresponding ones of integrated volatility implied by the model under the null hypothesis. We first provide primitive conditions on the measurement error associated with the realized measure, which allow to construct asymptotically valid specification tests. Then we establish regularity conditions under which realized volatility, bipower variation (BarndorffNielsen & Shephard, 2004d), and modified subsampled realized volatility (Zhang, Mykland & Aït Sahalia, 2003), satisfy the given primitive assumptions. Finally, we provide an empirical illustration based on three stock from the Dow Jones Industrial Average.
A Martingale Approach for Testing Diffusion Models Based on Infinitesimal Operator
, 2009
"... I develop an omnibus specification test for diffusion models based on the infinitesimal operator instead of the already extensively used transition density. The infinitesimal operatorbased identification of the diffusion process is equivalent to a "martingale hypothesis" for the new proces ..."
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I develop an omnibus specification test for diffusion models based on the infinitesimal operator instead of the already extensively used transition density. The infinitesimal operatorbased identification of the diffusion process is equivalent to a "martingale hypothesis" for the new processes transformed from the original diffusion process. The transformation is via the celebrated "martingale problems". My test procedure is to check the "martingale hypothesis" via a multivariate generalized spectral derivative based approach which enjoys many good properties. The infinitesimal operator of the diffusion process enjoys the nice property of being a closedform expression of drift and diffusion terms. This makes my test procedure capable of checking both univariate and multivariate diffusion models and particularly powerful and convenient for the multivariate case. In contrast checking the multivariate diffusion models is very difficult by transition densitybased methods because transition density does not have a closedform in general. Moreover, different transformed martingale processes contain different separate information about the drift and diffusion terms and their interactions. This motivates us to suggest a separate inferencebased test procedure to explore the sources when rejection of a parametric form happens. Finally, simulation studies are presented and possible future researches using the infinitesimal operatorbased martingale characterization are discussed.
Analysis of Nonstationary Stochastic Processes with Application to the Fluctuations in the Oil Price
"... We describe a method for analyzing a nonstationary stochastic process x(t), and utilize it to study the fluctuations in the oil price. Evidence is presented that the fluctuations in the returns y(t), defined as, y(t) = ln{x(t+1)/x(t)}, where x(t) is the datum at time t, constitute a Markov process, ..."
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We describe a method for analyzing a nonstationary stochastic process x(t), and utilize it to study the fluctuations in the oil price. Evidence is presented that the fluctuations in the returns y(t), defined as, y(t) = ln{x(t+1)/x(t)}, where x(t) is the datum at time t, constitute a Markov process, characterized by a Markov time scale tM. We compute the coefficients of the KramersMoyal expansion for the probability distribution function P (y, ty0, t0), and show that P (y, t, y0, t0) satisfies a FokkerPlanck equation, which is equivalent to a Langevin equation for y(t). The Langevin equation provides quantitative predictions for the oil price over Markov time scale tM. Also studied is the average frequency of positiveslope crossings, ν+α = P (yi> α, yi−1 < α), for the returns y(t), where T (α) = 1/ν α is the average waiting time for observing y(t) = α again. The method described is applicable to a wide variety of nonstationary stochastic processes which, unlike many of the previous methods, does not require the data to have any scaling feature. 1 PACS numbers(s): 05.10.Gg, 05.40.a, 05.45.Tp
NonRigid Motion Behaviour Learning: A Spectral and Graphical Approach
, 2007
"... In this thesis graph spectral methods and kernel methods are combined together for the tasks of rigid and nonrigid feature correspondence matching and consistent labelling. The thesis is divided into five chapters. In Chapter 1 we give a brief introduction and an outline of the thesis. In Chapter ..."
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In this thesis graph spectral methods and kernel methods are combined together for the tasks of rigid and nonrigid feature correspondence matching and consistent labelling. The thesis is divided into five chapters. In Chapter 1 we give a brief introduction and an outline of the thesis. In Chapter 2 we review the main techniques in the literature which are related to the developments in this thesis. Topics covered include data representation, the data classification methods, spectral graph matching, and the kernel methods. Chapter 3 aims at developing a new feature correspondence matching algorithm for rigid and articulated motion. We focus on the point features extracted from consecutive image frames. Specifically, a graph structure is used to represent the datasets, and spectral graph theory is used for the correspondence localization. The novelty is that a kernel viewpoint is adopted in constructing the proximity matrix, and a consistent labelling process is incorporated into the matching process when the objects under investigation undergoes articulated