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Limit Theory for High Frequency Sampled MCARMA Models

Part of: Mathematical economics Inference from stochastic processes Limit theorems

Published online by Cambridge University Press:  22 February 2016

Vicky Fasen
Vicky Fasen*
Affiliation:
ETH Zürich
*
∗ Postal address: Institute of Stochastics, Karlsruhe Institute of Technology, Kaiserstrasse 89, 76133 Karlsruhe, Germany. Email address: vicky.fasen@kit.edu
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Abstract

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We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {hn, 2hn,…, nhn}, where hn ↓ 0 and nhn → ∞ as n → ∞, or at a constant time grid where hn = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.

Keywords

Central limit theoremcointegrationdomain of attractionhigh-frequency datamultivariate CARMA processregular variationOrnstein-Uhlenbeck processsample variance

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc. 60F05: Central limit and other weak theorems
Secondary: 91B84: Economic time series analysis
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 3 , September 2014 , pp. 846 - 877
Copyright
© Applied Probability Trust 

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