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Limit Theory for High Frequency Sampled MCARMA Models

Part of: Inference from stochastic processes Limit theorems Mathematical economics

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: [email protected]
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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 

References

Andresen, A., Benth, F. E., Koekebakker, S. and Zakamulin, V. (2014). “The CARMA interest rate model. Internat. J. Theoret. Appl. Finance 17, 1450008.CrossRefGoogle Scholar
Bergstrom, A. R. (1990). “Continuous Time Econometric Modelling.” Oxford University Press.Google Scholar
Beveridge, S. and Nelson, C. R. (1981). “A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘business cycle’.” J. Monetary Econom. 7, 151–174.Google Scholar
Brockwell, P. J. (2001). “Lévy-driven CARMA processes.” Ann. Inst. Statist. Math. 53, 113–124.CrossRefGoogle Scholar
Brockwell, P. J. (2009). “Lévy-driven continuous-time ARMA processes.“In Handbook of Financial Time Series, Springer, Berlin, pp. 457–480.Google Scholar
Brockwell, P. J., Ferrazzano, V. and Klüppelberg, C. (2013). “High-frequency sampling and kernel estimation for continuous-time moving average processes.” J. Time Ser. Anal. 34, 385–404.Google Scholar
Comte, F. (1999). “Discrete and continuous time cointegration.” J. Econometrics 88, 207–226.CrossRefGoogle Scholar
Davis, R. and Resnick, S. (1985). “Limit theory for moving averages of random variables with regularly varying tail probabilities.” Ann. Prob. 13, 179–195.Google Scholar
Davis, R., Marengo, J. and Resnick, S. (1985). “Extremal properties of a class of multivariate moving averages.” Bull. Inst. Internat. Statist. 51, 185–192.Google Scholar
Doob, J. L. (1944). “The elementary Gaussian processes.” Ann. Math. Statist. 15, 229–282.Google Scholar
Engle, R. F. and Granger, C. W. J. (1987). “Co-integration and error correction: representation, estimation, and testing.” Econometrica 55, 251–276.CrossRefGoogle Scholar
Fasen, V. (2013). “Statistical estimation of multivariate Ornstein–Uhlenbeck processes and applications to co-integration.” J. Econometrics 172, 325–337.Google Scholar
Fasen, V. (2013). “Time series regression on integrated continuous-time processes with heavy and light tails.” Econometric Theory 29, 28–67.Google Scholar
Fasen, V. and Fuchs, F. (2013). “On the limit behavior of the periodogram of high-frequency sampled stable CARMA processes.” Stoch. Process. Appl. 123, 229–273.Google Scholar
Fasen, V. and Fuchs, F. (2013). “Spectral estimates for high-frequency sampled continuous-time autoregressive moving average processes.” J. Time Series Anal. 34, 532–551.Google Scholar
García, I., Klüppelberg, C. and Müller, G. (2011). “Estimation of stable CARMA models with an application to electricity spot prices.” Statist. Modelling 11, 447–470.Google Scholar
Garnier, H. and Wang, L. (eds) (2008). “Identification of Continuous-Time Models from Sampled Data. “Springer, London.Google Scholar
Gut, A. (1992). “Complete convergence of arrays.” Period. Math. Hungar. 25, 51–75.Google Scholar
Hult, H. and Lindskog, F. (2007). “Extremal behavior of stochastic integrals driven by regularly varying Lévy processes.” Ann. Prob. 35, 309–339.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). “Limit Theorems for Stochastic Processes, 2nd edn. “Springer, Berlin.CrossRefGoogle Scholar
Johansen, S. (1995). “Likelihood-Based Inference in Cointegrated Vector Autoregressive Models.” Oxford University Press.Google Scholar
Kallenberg, O. (1997). “Foundations of Modern Probability. “Springer, New York.Google Scholar
Kessler, M. and Rahbek, A. (2001). “Asymptotic likelihood based inference for co-integrated homogenous Gaussian diffusions.” Scand. J. Statist. 28, 455–470.Google Scholar
Larsson, E. K., Mossberg, M. and Söderström, T. (2006). “An overview of important practical aspects of continuous-time ARMA system identification.” Circuits Systems Signal Process. 25, 17–46.Google Scholar
Marquardt, T. and Stelzer, R. (2007). “Multivariate CARMA processes.” Stoch. Process. Appl. 117, 96–120.CrossRefGoogle Scholar
Meerschaert, M. M. and Scheffler, H.-P. (2000). “Moving averages of random vectors with regularly varying tails.” J. Time Ser. Anal. 21, 297–328.CrossRefGoogle Scholar
Meerschaert, M. M. and Scheffler, H.-P. (2001). “Limit Distributions for Sums of Independent Random Vectors.” John Wiley, New York.Google Scholar
Moser, M. and Stelzer, R. (2011). “Tail behavior of multivariate Lévy-driven mixed moving average processes and supOU stochastic volatility models.” Adv. Appl. Prob. 43, 1109–1135.Google Scholar
Paulauskas, V. and Rachev, S. T. (1998). “Cointegrated processes with infinite variance innovations.” Ann. Appl. Prob. 8, 775–792.Google Scholar
Phillips, P. C. B. (1991). “Error correction and long-run equilibrium in continuous time.” Econometrica 59, 967–980.Google Scholar
Phillips, P. C. B. and Solo, V. (1992). “Asymptotics for linear processes.” Ann. Statist. 20, 971–1001.Google Scholar
Pratt, J. W. (1960). “On interchanging limits and integrals.” Ann. Math. Statist. 31, 74–77.Google Scholar
Rajput, B. S. and Rosiński, J. (1989). “Spectral representation of infinitely divisible processes.” Prob. Theory Relat. Fields 82, 451–487.Google Scholar
Resnick, S. I. (1986). “Point processes, regular variation and weak convergence.” Adv. Appl. Prob. 18, 66–138.Google Scholar
Resnick, S. I. (1987). “Extreme Values, Regular Variation, and Point Processes. “Springer, New York.Google Scholar
Resnick, S. I. (2007). “Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. “Springer, New York.Google Scholar
Rvačeva, E. L. (1962). “On domains of attraction of multi-dimensional distributions.” Select. Transl. Math. Statist. Prob. 2, 183–205.Google Scholar
Sato, K.-I. (1999). “Lévy Processes and Infinitely Divisible Distributions.” Cambridge University Press.Google Scholar
Schlemm, E. and Stelzer, R. (2012). “Multivariate CARMA processes, continous-time state space models and complete regularity of the innovations of the sampled processes.” Bernoulli 18, 46–63.Google Scholar
Stockmarr, A. and Jacobsen, M. (1994). “Gaussian diffusions and autoregressive processes: weak convergence and statistical inference.” Scand. J. Statist. 21, 403–419.Google Scholar
Todorov, V. (2009). “Estimation of continuous-time stochastic volatility models with Jumps using high-frequency data.” J. Econometrics 148, 131–148.Google Scholar