Change detection in the Cox–Ingersoll–Ross model
-
Gyula Pap
Abstract
We propose an offline change detection method for the famous Cox–Ingersoll–Ross model based on a continuous sample. We develop one- and two-sided testing procedures for both drift parameters of the process. The test process is based on estimators that are motivated by the discrete time least-squares estimators, and its asymptotic distribution under the no-change hypothesis is that of a Brownian bridge. We prove the asymptotic weak consistence of the test, and derive the asymptotic properties of the change-point estimator under the alternative hypothesis of change at one point in time.
Funding source: European Social Fund
Award Identifier / Grant number: TÁMOP 4.2.4. A/2-11-1-2012-0001
Funding statement: This research was supported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP 4.2.4. A/2-11-1-2012-0001 “National Excellence Program”.
Acknowledgements
The authors are grateful to Professor Péter Major at the University of Szeged for supplying the basic idea of the proof of Lemma 7.8. We express our gratitude to the anonymous reviewers, whose comments have been most helpful in improving the readability of the paper.
References
[1] Barczy M. and Pap G., Asymptotic properties of maximum likelihood estimators for Heston models based on continuous time observations, Statistics 50 (2016), no. 2, 389–417. 10.1080/02331888.2015.1044991Suche in Google Scholar
[2] Ben Alaya M. and Kebaier A., Parameter estimation for the square-root diffusions: Ergodic and nonergodic cases, Stoch. Models 28 (2012), no. 4, 609–634. 10.1080/15326349.2012.726042Suche in Google Scholar
[3] Ben Alaya M. and Kebaier A., Asymptotic behavior of the maximum likelihood estimator for ergodic and nonergodic square-root diffusions, Stoch. Anal. Appl. 31 (2013), no. 4, 552–573. 10.1080/07362994.2013.798175Suche in Google Scholar
[4] Cox J. C., Ingersoll J. E. and Ross S. A., A theory of the term structure of interest rates, Econometrica 53 (1985), no. 2, 385–407. 10.1142/9789812701022_0005Suche in Google Scholar
[5] Csörgő M. and Horváth L., Limit Theorems in Change-Point Analysis, Wiley, New York, 1977. Suche in Google Scholar
[6] Feller W., Two singular diffusion problems, Ann. of Math. (2) 54 (1951), 173–182. 10.1007/978-3-319-16856-2_9Suche in Google Scholar
[7] Gombay E., Change detection in autoregressive time series, J. Multivariate Anal. 99 (2008), 451–464. 10.1016/j.jmva.2007.01.003Suche in Google Scholar
[8] Guo M. and Härdle W., Adaptive interest rate modelling, Discussion Paper SFB649DP2010-029, Collaborative Research Center 649, Humboldt University, Berlin, 2010. Suche in Google Scholar
[9] Hu Y. and Long H., Parameter estimation for Ornstein–Uhlenbeck processes driven by α-stable Lévy motions, Commun. Stoch. Anal. 1 (2007), no. 2, 175–192. 10.31390/cosa.1.2.01Suche in Google Scholar
[10] Jacod J. and Shiryaev A. N., Limit Theorems for Stochastic Processes, Springer, Berlin, 2003. 10.1007/978-3-662-05265-5Suche in Google Scholar
[11] Jin P., Mandrekar V., Rüdiger B. and Trabelsi C., Positive Harris recurrence of the CIR process and its applications, Commun. Stoch. Anal. 7 (2013), no. 3, 409–424. 10.31390/cosa.7.3.04Suche in Google Scholar
[12] Karatzas I. and Shreve S. E., Brownian Motion and Stochastic Calculus, 2nd ed., Springer, New York, 1991. Suche in Google Scholar
[13] Kokoszka P. and Leipus R., Change-point in the mean of dependent observations, Statist. Probab. Lett. 40 (1998), 385–393. 10.1016/S0167-7152(98)00145-XSuche in Google Scholar
[14] Kokoszka P. and Leipus R., Change-point estimation in ARCH models, Bernoulli 6 (2000), no. 3, 513–539. 10.2307/3318673Suche in Google Scholar
[15] Liptser R. and Shiryaev A., Statistics of Random Processes II. Applications, 2nd ed., Springer, Berlin, 2001. 10.1007/978-3-662-13043-8Suche in Google Scholar
[16] Meyn S. P. and Tweedie R. L., Stability of Markovian processes II: Continuous-time processes and sampled chains, Adv. in Appl. Probab. 25 (1993), 487–517. 10.2307/1427521Suche in Google Scholar
[17] Overbeck L., Estimation for continuous branching processes, Scand. J. Stat. 25 (1998), 111–126. 10.1111/1467-9469.00092Suche in Google Scholar
[18] Overbeck L. and Rydén T., Estimation in the Cox–Ingersoll–Ross model, Econometric Theory 13 (1997), no. 3, 430–461. 10.1017/S0266466600005880Suche in Google Scholar
[19] Pap G. and Szabó T. T., Change detection in INAR(p) processes against various alternative hypotheses, Comm. Statist. Theory Methods 42 (2013), no. 7, 1386–1405. 10.1080/03610926.2012.732181Suche in Google Scholar
[20] Pap G. and Szabó T. T., Change detection in the Cox–Ingersoll–Ross model, preprint 2015, http://arxiv.org/abs/1502.07102. 10.1515/strm-2015-0008Suche in Google Scholar
[21] Schmid W. and Tzotchev D., Statistical surveillance of the parameters of a one-factor Cox–Ingersoll–Ross model, Sequential Anal. 23 (2004), no. 3, 379–412. 10.1081/SQA-200027052Suche in Google Scholar
© 2016 by De Gruyter
