### Limit theory for high frequency sampled MCARMA models

Vicky Fasen

#### Abstract

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.

#### Article information

Source
Adv. in Appl. Probab., Volume 46, Number 3 (2014), 846-877.

Dates
First available in Project Euclid: 29 August 2014

https://projecteuclid.org/euclid.aap/1409319563

Digital Object Identifier
doi:10.1239/aap/1409319563

Mathematical Reviews number (MathSciNet)
MR3254345

Zentralblatt MATH identifier
06347587

#### Citation

Fasen, Vicky. Limit theory for high frequency sampled MCARMA models. Adv. in Appl. Probab. 46 (2014), no. 3, 846--877. doi:10.1239/aap/1409319563. https://projecteuclid.org/euclid.aap/1409319563

#### References

• Andresen, A., Benth, F. E., Koekebakker, S. and Zakamulin, V. (2014). The CARMA interest rate model. Internat. J. Theoret. Appl. Finance 17, 1450008.
• Bergstrom, A. R. (1990). Continuous Time Econometric Modelling. Oxford University Press.
• 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.
• Brockwell, P. J. (2001). Lévy-driven CARMA processes. Ann. Inst. Statist. Math. 53, 113–124.
• Brockwell, P. J. (2009). Lévy-driven continuous-time ARMA processes. In Handbook of Financial Time Series, Springer, Berlin, pp. 457–480.
• 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.
• Comte, F. (1999). Discrete and continuous time cointegration. J. Econometrics 88, 207–226.
• Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.
• Davis, R., Marengo, J. and Resnick, S. (1985). Extremal properties of a class of multivariate moving averages. Bull. Inst. Internat. Statist. 51, 185–192.
• Doob, J. L. (1944). The elementary Gaussian processes. Ann. Math. Statist. 15, 229–282.
• Engle, R. F. and Granger, C. W. J. (1987). Co-integration and error correction: representation, estimation, and testing. Econometrica 55, 251–276.
• Fasen, V. (2013). Statistical estimation of multivariate Ornstein–Uhlenbeck processes and applications to co-integration. J. Econometrics 172, 325–337.
• Fasen, V. (2013). Time series regression on integrated continuous-time processes with heavy and light tails. Econometric Theory 29, 28–67.
• 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.
• 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.
• 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.
• Garnier, H. and Wang, L. (eds) (2008). Identification of Continuous-Time Models from Sampled Data. Springer, London.
• Gut, A. (1992). Complete convergence of arrays. Period. Math. Hungar. 25, 51–75.
• Hult, H. and Lindskog, F. (2007). Extremal behavior of stochastic integrals driven by regularly varying Lévy processes. Ann. Prob. 35, 309–339.
• Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.
• Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press.
• Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
• Kessler, M. and Rahbek, A. (2001). Asymptotic likelihood based inference for co-integrated homogenous Gaussian diffusions. Scand. J. Statist. 28, 455–470.
• 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.
• Marquardt, T. and Stelzer, R. (2007). Multivariate CARMA processes. Stoch. Process. Appl. 117, 96–120.
• Meerschaert, M. M. and Scheffler, H.-P. (2000). Moving averages of random vectors with regularly varying tails. J. Time Ser. Anal. 21, 297–328.
• Meerschaert, M. M. and Scheffler, H.-P. (2001). Limit Distributions for Sums of Independent Random Vectors. John Wiley, New York.
• 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.
• Paulauskas, V. and Rachev, S. T. (1998). Cointegrated processes with infinite variance innovations. Ann. Appl. Prob. 8, 775–792.
• Phillips, P. C. B. (1991). Error correction and long-run equilibrium in continuous time. Econometrica 59, 967–980.
• Phillips, P. C. B. and Solo, V. (1992). Asymptotics for linear processes. Ann. Statist. 20, 971–1001.
• Pratt, J. W. (1960). On interchanging limits and integrals. Ann. Math. Statist. 31, 74–77.
• Rajput, B. S. and Rosiński, J. (1989). Spectral representation of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451–487.
• Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. Appl. Prob. 18, 66–138.
• Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
• Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.
• Rva$\check{\mbox{c}}$eva, E. L. (1962). On domains of attraction of multi-dimensional distributions. Select. Transl. Math. Statist. Prob. 2, 183–205.
• Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
• 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.
• Stockmarr, A. and Jacobsen, M. (1994). Gaussian diffusions and autoregressive processes: weak convergence and statistical inference. Scand. J. Statist. 21, 403–419.
• Todorov, V. (2009). Estimation of continuous-time stochastic volatility models with jumps using high-frequency data. J. Econometrics 148, 131–148.