zbMATH — the first resource for mathematics

Quasi-maximum likelihood estimation for cointegrated continuous-time linear state space models observed at low frequencies. (English) Zbl 1434.62095
Summary: In this paper, we investigate quasi-maximum likelihood (QML) estimation for the parameters of a cointegrated solution of a continuous-time linear state space model observed at discrete time points. The class of cointegrated solutions of continuous-time linear state space models is equivalent to the class of cointegrated continuous-time ARMA (MCARMA) processes. As a start, some pseudo-innovations are constructed to be able to define a QML-function. Moreover, the parameter vector is divided appropriately in long-run and short-run parameters using a representation for cointegrated solutions of continuous-time linear state space models as a sum of a Lévy process plus a stationary solution of a linear state space model. Then, we establish the consistency of our estimator in three steps. First, we show the consistency for the QML estimator of the long-run parameters. In the next step, we calculate its consistency rate. Finally, we use these results to prove the consistency for the QML estimator of the short-run parameters. After all, we derive the limiting distributions of the estimators. The long-run parameters are asymptotically mixed normally distributed, whereas the short-run parameters are asymptotically normally distributed. The performance of the QML estimator is demonstrated by a simulation study.

62H12 Estimation in multivariate analysis
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F05 Central limit and other weak theorems
Full Text: DOI Euclid
[1] Andresen, A., Benth, F. E., Koekebakker, S. and Zakamulin, V. (2014). The CARMA interest rate model., Int. J. Theor. Appl. Finance 17 1-27. · Zbl 1290.91170
[2] Aoki, M. (1990)., State Space Modeling of Time Series, 2nd ed. Springer, Berlin.
[3] Barndorff-Nielsen, O. E. (1997). Normal inverse Gaussian distributions and stochastic volatility modelling., Scand. J. Stat. 24 1-13. · Zbl 0934.62109
[4] Bauer, D. and Wagner, M. (2002). Asymptotic Properties of Pseudo Maximum Likelihood Estimates for Multiple Frequency I(1) Processes Diskussionsschriften, Universitaet Bern, Departement, Volkswirtschaft.
[5] Bauer, D. and Wagner, M. (2002). Estimating cointegrated systems using subspace algorithms., J. Econometrics 111 47-84. · Zbl 1031.62069
[6] Benth, F. E., Klüppelberg, C., Müller, G. and Vos, L. (2014). Futures pricing in electricity markets based on stable CARMA spot models., Energ. Econ. 44 392-406.
[7] Bergstrom, A. R. (1990)., Continuous Time Econometric Modelling. Oxford University Press, Oxford.
[8] Bergstrom, A. R. (1997). Gaussian estimation of mixed-order continuous-time dynamic models with unobservable stochastic trends from mixed stock and flow data., Econometric Theory 13 467-505.
[9] Bernstein, D. S. (2009)., Matrix Mathematics: Theory, Facts, and Formulas, 2nd ed. Princeton Univ. Press, Princeton. · Zbl 1183.15001
[10] Blevins, J. R. (2017). Identifying restrictions for finite parameter continuous time models with discrete time data., Econometric Theory 33 739-754. · Zbl 1442.62731
[11] Boubacar Maïnassara, Y. and Francq, C. (2011). Estimating structural VARMA models with uncorrelated but non-independent error terms., J. Multivar. Anal. 102 496-505. · Zbl 1207.62168
[12] Bradley, R. C. (2007)., Introduction to Strong Mixing Conditions 1. Kendrick, Heber City. · Zbl 1133.60001
[13] Brockwell, P. J. and Davis, R. A. (1998)., Time Series: Theory and Methods, 2nd ed. Springer Ser. Statist. Springer, New York. · Zbl 1169.62074
[14] Chambers, A. J. and McCrorie, J. R. (2007). Frequency domain estimation of temporally aggregated Gaussian cointegrated systems., J. Econometrics 136 1-29. · Zbl 1418.62307
[15] Chambers, M. J., McCrorie, J. R. and Thornton, M. A. (2018). Continuous time modelling based on an exact discrete time representation. In, Continuous Time Modelling in the Behavioral and Related Sciences (K. van Montford, J. Oud and M. Voelkle, eds.) Springer, New York. 317-357.
[16] Chambers, M. J. and Thornton, M. A. (2012). Discrete time representation of continuous time ARMA processes., Econometric Theory 28 219-238. · Zbl 1234.62118
[17] Chen, F., Agüero, J. C., Gilson, M., Garnier, H. and Liu, T. (2017). EM-based identification of continuous-time ARMA Models from irregularly sampled data., Automatica 77 293-301. · Zbl 1355.93189
[18] Fasen-Hartmann, V. and Scholz, M. (2019). Cointegrated continuous-time linear state-space and MCARMA models., Stochastics to appear.
[19] Fasen-Hartmann, V. and Scholz, M. (2019). A canonical form for a cointegrated MCARMA process. In, preparation.
[20] Fasen, V. (2013). Time series regresion on integrated continuous-time processeswith heavy and light tails., Econometric Theory 29 28-67. · Zbl 1261.91039
[21] Fasen, V. (2014). Limit theory for high frequency sampled MCARMA models., Adv. Appl. Probab. 46 846-877. · Zbl 1429.62394
[22] Garnier, H. and Wang, L., eds. (2008)., Identification of Continuous-time Models from Sampled Data. Advances in Industrial Control. Springer, London.
[23] Guidorzi, R. (1975). Canonical structures in the identification of multivariable systems., Automatica 11 361-374. · Zbl 0309.93012
[24] Hannan, E. J. and Deistler, M. (2012)., The Statistical Theory of Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia. · Zbl 1239.01122
[25] Hansen, L. P. and Sargent, T. J. (1983). The dimensionality of the aliasing problem in models with rational spectral densities., Econometrica 51 377-387. · Zbl 0516.93063
[26] Harvey, A. C. and Stock, J. H. (1985). The estimation of higher-order continuous time autoregressive models., Econometric Theory 1 97-112.
[27] Harvey, A. C. and Stock, J. H. (1985). Continuous time autoregressive models with common stochastic trends., J. Econom. Dynam. Control. 12 365-384. · Zbl 0647.62104
[28] Harvey, A. C. and Stock, J. H. (1989). Estimating integrated higher-order continuous time autoregressions with an application to money-income causality., J. Econometrics 42 319-336. · Zbl 0692.62099
[29] Ibragimov, I. A. (1962). Some limit theorems for stationary processes., Theory Probab. Appl. 7 349-382. · Zbl 0119.14204
[30] Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models., Econometrica 59 1551-1580. · Zbl 0755.62087
[31] Johansen, S. (1995)., Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Advanced Texts in Econometrics. Oxford Univ. Press, Oxford. · Zbl 0928.62069
[32] Kessler, M. and Rahbek, A. (2001). Asymptotic likelihood based inference for co-integrated homogenous Gaussian diffusions., Scand. J. Statist. 28 455-470. · Zbl 0981.62069
[33] Kessler, M. and Rahbek, A. (2004). Identification and inference for multivariate cointegrated and ergodic Gaussian diffusions., Stat. Inference Stoch. Process. 7 137-151. · Zbl 1056.62092
[34] Krengel, U. (1985)., Ergodic Theorems: With a Supplement. De Gruyter, Berlin. · Zbl 0575.28009
[35] Lütkepohl, H. (2005)., New Introduction to Multiple Time Series Analysis. Springer, Berlin. · Zbl 1072.62075
[36] Lütkepohl, H. and Claessen, H. (1997). Analysis of cointegrated VARMA processes., J. Econometrics 80 223-239. · Zbl 0915.62096
[37] Lütkepohl, H. and Poskitt, D. S. (1996). Specification of echelon-form VARMA models., J. Bus. Econom. Statist. 14 69-79.
[38] Marquardt, T. and Stelzer, R. (2007). Multivariate CARMA processes., Stochastic Process. Appl. 117 96-120. · Zbl 1115.62087
[39] Masuda, H. (2004). On multidimensional Ornstein-Uhlenbeck processes driven by a general Lévy process., Bernoulli 10 97-120. · Zbl 1048.60060
[40] McCrorie, J. R. (2003). The problem of aliasing in identifying finite parameter continuous time stochastic models., Acta Applicandae Mathematicae 79 9-16. · Zbl 1030.62071
[41] McCrorie, J. R. (2009). Estimating continuous-time models on the basis of discrete data via an exact discrete analog., Econometric Theory 25 1120-1137. · Zbl 1253.62095
[42] Phillips, P. C. B. (1973). The problem of identification in finite parameter continuous time models., J. Econometrics 1 351-362. · Zbl 0282.93053
[43] Phillips, P. C. B. (1991). Error correction and long-run equilibrium in continuous time., Econometrica 59 967-980. · Zbl 0725.62101
[44] Reinsel, G. C. (1997)., Elements of Multivariate Time Series Analysis, 2nd ed. Springer-Verlag, New York. · Zbl 0873.62086
[45] Saikkonen, P. (1992). Estimation and testing of cointegrated systems by an autoregressive approximation., Econometric Theory 8 1-27.
[46] Saikkonen, P. (1993). Continuous weak convergence and stochastic equicontinuity results for integrated processes with an application to the estimation of a regression model., Econometric Theory 9 155-188.
[47] Saikkonen, P. (1995). Problems with the asymptotic theory of maximum likelihood estimation in integrated and cointegrated systems., Econometric Theory 11 888-911.
[48] Sato, K.-I. (1999)., Lévy Processes and Infinitely Divisible Distributions. Cambridge Stud. Adv. Math. 68. Cambridge Univ. Press, Cambridge.
[49] Schlemm, E. (2011). Estimation of Continuous-Time ARMA Models and Random Matrices with Dependent Entries, Dissertation, Technische Universität, München.
[50] Schlemm, E. and Stelzer, R. (2012). Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes., Bernoulli 18 46-63. · Zbl 1248.60039
[51] Schlemm, E. and Stelzer, R. (2012). Quasi maximum likelihood estimation for strongly mixing state space models and multivariate Lévy-driven CARMA processes., Electron. J. Stat. 6 2185-2234. · Zbl 1295.62020
[52] Scholz, M. (2016). Estimation of Cointegrated Multivariate Continuous-time Autoregressive Moving Average Processes, Dissertation, Karlsruher Institute of Technology (KIT), Karlsruhe.
[53] Sinha, N. K. and Rao, G. P. (1991)., Identification of Continuous-Time Systems: Methodology and Computer Implementation. Springer, Dordrecht. · Zbl 0770.93021
[54] Sinha, N. K. and Rao, G. P. (1991). Continuous-time models and approaches. In, Identification of Continuous-Time Systems: Methodology and Computer Implementation (N. K. Sinha and G. P. Rao, eds.) 1-15. Springer, Dordrecht.
[55] Stockmarr, A. and Jacobsen, M. (1994). Gaussian diffusions and autoregressive processes: weak convergence and statistical inference., Scand. J. Statist. 21 403-429. · Zbl 0814.62048
[56] Thornton, M. A. and Chambers, M. J. (2016). The exact discretisation of CARMA models with applications in finance., J. Emp. Finance 38 739-761.
[57] Thornton, M. A. and Chambers, M. J. (2017). Continuous time ARMA processes: discrete time representation and likelihood evaluation., J. Econom. Dynam. Control 79 48-65. · Zbl 1401.91506
[58] Todorov, V. (2009). Estimation of continuous-time stochastic volatility models with jumps using high-frequency data., J. Econometrics 148 131-148. · Zbl 1429.62480
[59] Yap, S. F. and Reinsel, G. C. (1995). Estimation and testing for unit roots in a partially nonstationary vector autoregressive moving average model., J. Amer. Statist. Assoc. 90 253-267. · Zbl 0818.62079
[60] Zadrozny, P. (1988). Gaussian likelihood of continuous-time ARMAX models when data are stocks and flows at different frequencies., Econometric Theory 4 108-124.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.