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Variational sums and power variation: a unifying approach to model selection and estimation in semimartingale models

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  • Woerner Jeannette H. C.

Abstract

In the framework of general semimartingale models we provide limit theorems for variational sums including the p-th power variation, i.e. the sum of p-th absolute powers of increments of a process. This gives new insight in the use of quadratic and realised power variation as an estimate for the integrated volatility in finance. It also provides a criterion to decide from high frequency data, whether a jump component should be included in the model. Furthermore, results on the asymptotic behaviour of integrals with respect to Lévy processes, estimates for integrals with respect to Lévy measures and non-parametric estimation for Lévy processes will be derived and viewed in the framework of variational sums.

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  • Woerner Jeannette H. C., 2003. "Variational sums and power variation: a unifying approach to model selection and estimation in semimartingale models," Statistics & Risk Modeling, De Gruyter, vol. 21(1/2003), pages 47-68, January.
    • Handle: RePEc:bpj:strimo:v:21:y:2003:i:1/2003:p:47-68:n:6
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    Cited by:

    1. Todorov, Viktor, 2013. "Power variation from second order differences for pure jump semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2829-2850.
    2. Tim Bollerslev & Viktor Todorov, 2011. "Estimation of Jump Tails," Econometrica, Econometric Society, vol. 79(6), pages 1727-1783, November.
    3. Diop, Assane & Jacod, Jean & Todorov, Viktor, 2013. "Central Limit Theorems for approximate quadratic variations of pure jump Itô semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 839-886.
    4. Vetter, Mathias, 2014. "Inference on the Lévy measure in case of noisy observations," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 125-133.
    5. Ole E. Barndorff-Nielsen & José Manuel Corcuera & Mark Podolskij, 2009. "Multipower Variation for Brownian Semistationary Processes," CREATES Research Papers 2009-21, Department of Economics and Business Economics, Aarhus University.
    6. Ghysels, Eric & Santa-Clara, Pedro & Valkanov, Rossen, 2006. "Predicting volatility: getting the most out of return data sampled at different frequencies," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 59-95.
    7. Barndorff-Nielsen, Ole E. & Corcuera, José Manuel & Podolskij, Mark, 2009. "Power variation for Gaussian processes with stationary increments," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1845-1865, June.
    8. Liu, Guangying & Zhang, Xinsheng, 2011. "Power variation of fractional integral processes with jumps," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 962-972, August.
    9. Figueroa-López, José E. & Houdré, Christian, 2009. "Small-time expansions for the transition distributions of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3862-3889, November.
    10. Figueroa-López, José E., 2008. "Small-time moment asymptotics for Lévy processes," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3355-3365, December.

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