| Peer-Reviewed

Detection and Estimation of Change Point in Volatility Function of Foreign Exchange Rate Returns

Received: 13 January 2023     Accepted: 22 February 2023     Published: 6 April 2023
Views:       Downloads:
Abstract

This work aims at detection and estimation of a change point in conditional variance function of a Nonparametric Auto-Regressive Conditional Heteroscedastic model. The conditional mean and conditional variance functions are not specified a priori but estimated using Nadaraya Watson kernel. This is because inferences based on nonparametric approaches are robust against misspecification of the conditional mean function and the conditional variance function of returns. The squared residuals obtained after estimating the regression function of the returns are used in estimating the conditional variance function. Further, the squared residuals are used in developing a test statistic for unknown abrupt change point in volatility of the exchange rate returns. The test statistic takes into consideration the conditional heteroskedasticity of the disturbances, dependence of the returns, heterogeneity and fourth moment of returns. This does not require prior knowledge of the marginal or the conditional densities of the returns as opposed to maximum likelihood estimation methods. The estimator for change point is considered as the augmented maximum of the test statistic. The consistency of the estimator is stated as a theorem. The asymptotic distribution associated with the test for unknown break points is the Bessel process distribution. The Bessel process distributions have no known simple closed-form expression for the distribution function which makes it difficult to compute exact p-values. Also, the Bessel process distributions depend on two parameters which makes it hard to tabulate the critical values hence one needs to simulate them. After simulating the critical values, hypothesis testing is done in the presence and absence of a change point in volatility of a simulated time series and the test is shown to reject the null hypothesis in the presence of a change point at alpha level of significance. Further, the test fails to reject the null hypothesis in the absence of a change point at alpha level of significance. An application to United States Dollar/Kenya Shilling historical exchange rates returns is made from 1st January 2010 to 27th November 2020 where the sample size n = 2839 is done. Through binary segmentation method, three change points are detected, estimated and accounted for. A significant improvement in describing a time series is expected if a point in time for volatility change has been detected and estimated.

Published in International Journal of Data Science and Analysis (Volume 9, Issue 1)
DOI 10.11648/j.ijdsa.20230901.11
Page(s) 1-12
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

Nonparametric, Kernel, Volatility

References
[1] B. Chen and Y. Hong, “Detecting for smooth structural changes in garch models,” Econometric Theory, vol. 32, no. 3, pp. 740-791, 2016.
[2] T. Bollerslev, R. F. Engle, and D. B. Nelson, “Arch models,” Handbook of econometrics, vol. 4, pp. 2959- 3038, 1994.
[3] C. Inclan and G. C. Tiao, “Use of cumulative sums of squares for retrospective detection of changes of varinace,” Journal of the American Statistical Association, vol. 89, no. 427, pp. 913-923, 1994.
[4] G. Chen, Y. K. Choi, and Y. Zhou, “Nonparametric estimation of structural change points in volatility models for time series,” Journal of Econometrics, vol. 126, no. 1, pp. 79-114, 2005.
[5] G. J. Ross, “Modelling financial volatility in the presence of abrupt changes,” Physica A: Statistical Mechanics and its Applications, vol. 392, no. 2, pp. 350-360, 2013.
[6] S. Aminikhanghahi and D. J. Cook, “A survey of methods for time series change point detection,” Knowledge and information systems, vol. 51, no. 2, pp. 339-367, 2017.
[7] C. Gerstenberger, “Robust wilcoxon-type estimation of change-point location under shortrange dependence,” Journal of Time Series Analysis, vol. 39, no. 1, pp. 90- 104, 2018.
[8] J. Chen and A. K. Gupta, Parametric statistical change point analysis: with applications to genetics, medicine, and finance. Springer Science & Business Media, 2011.
[9] E. Brodsky and B. S. Darkhovsky, Nonparametric methods in change point problems, vol. 243. Springer Science & Business Media, 2013.
[10] S. Lee, O. Na, and S. Na, “On the cusum of squares test for variance change in nonstationary and nonparametric time series models,” Annals of the Institute of Statistical Mathematics, vol. 55, no. 3, pp. 467-485, 2003.
[11] A. Sansó, J. Carrion, and V. Aragó, “Testing for changes in the unconditional variance of financial time series,” Revista de Econom´ıa Financiera, 2004, vol. 4, p. 32-52, 2020.
[12] I. Berkes, E. Gombay, L. Horváth, and P. Kokoszka, “Sequential change-point detection in garch (p, q) models,” Econometric theory, vol. 20, no. 6, pp. 1140- 1167, 2004.
[13] A. Koubková, Sequential change-point analysis. PhD thesis, 2006.
[14] C. Perna and M. Sibillo, Mathematical and statistical methods in insurance and finance. Springer, 2008.
[15] M. Rosenblatt, “A central limit theorem and a strong mixing condition,” Proceedings of the National Academy of Sciences of the United States of America, vol. 42, no. 1, p. 43, 1956.
[16] E. A. Nadaraya, “On estimating regression,” Theory of Probability & Its Applications, vol. 9, no. 1, pp. 141- 142, 1964.
[17] G. S. Watson, “Smooth regression analysis,” Sankhya: The Indian Journal of Statistics, Series A, pp. 359-372, 1964.
[18] J. Fan and Q. Yao, “Efficient estimation of conditional variance functions in stochastic regression,” Biometrika, vol. 85, no. 3, pp. 645-660, 1998.
[19] H. Oodaira and K.-i. Yoshihara, “Functional central limit theorems for strictly stationary processes satisfyinc the strong mixing condition,” in Kodai Mathematical Seminar Reports, vol. 24, pp. 259-269, Department of Mathematics, Tokyo Institute of Technology, 1972.
[20] P. Hall and C. Hyde, Martingale limit theory and its application. Academic Press, 1980.
[21] A. W. Van der Vaart, Asymptotic statistics, vol. 3. Cambridge university press, 2000.
[22] S. Schwaar, Asymptotics for change point tests and change point estimators. PhD thesis, 2017.
[23] M. Csorgo and L. Horváth, Limit theorems in change- point analysis. John Wiley & Sons Chichester, 1997.
Cite This Article
  • APA Style

    Josephine Njeri Ngure, Anthony Gichuhi Waititu, Simon Maina Mundia. (2023). Detection and Estimation of Change Point in Volatility Function of Foreign Exchange Rate Returns. International Journal of Data Science and Analysis, 9(1), 1-12. https://doi.org/10.11648/j.ijdsa.20230901.11

    Copy | Download

    ACS Style

    Josephine Njeri Ngure; Anthony Gichuhi Waititu; Simon Maina Mundia. Detection and Estimation of Change Point in Volatility Function of Foreign Exchange Rate Returns. Int. J. Data Sci. Anal. 2023, 9(1), 1-12. doi: 10.11648/j.ijdsa.20230901.11

    Copy | Download

    AMA Style

    Josephine Njeri Ngure, Anthony Gichuhi Waititu, Simon Maina Mundia. Detection and Estimation of Change Point in Volatility Function of Foreign Exchange Rate Returns. Int J Data Sci Anal. 2023;9(1):1-12. doi: 10.11648/j.ijdsa.20230901.11

    Copy | Download

  • @article{10.11648/j.ijdsa.20230901.11,
      author = {Josephine Njeri Ngure and Anthony Gichuhi Waititu and Simon Maina Mundia},
      title = {Detection and Estimation of Change Point in Volatility Function of Foreign Exchange Rate Returns},
      journal = {International Journal of Data Science and Analysis},
      volume = {9},
      number = {1},
      pages = {1-12},
      doi = {10.11648/j.ijdsa.20230901.11},
      url = {https://doi.org/10.11648/j.ijdsa.20230901.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijdsa.20230901.11},
      abstract = {This work aims at detection and estimation of a change point in conditional variance function of a Nonparametric Auto-Regressive Conditional Heteroscedastic model. The conditional mean and conditional variance functions are not specified a priori but estimated using Nadaraya Watson kernel. This is because inferences based on nonparametric approaches are robust against misspecification of the conditional mean function and the conditional variance function of returns. The squared residuals obtained after estimating the regression function of the returns are used in estimating the conditional variance function. Further, the squared residuals are used in developing a test statistic for unknown abrupt change point in volatility of the exchange rate returns. The test statistic takes into consideration the conditional heteroskedasticity of the disturbances, dependence of the returns, heterogeneity and fourth moment of returns. This does not require prior knowledge of the marginal or the conditional densities of the returns as opposed to maximum likelihood estimation methods. The estimator for change point is considered as the augmented maximum of the test statistic. The consistency of the estimator is stated as a theorem. The asymptotic distribution associated with the test for unknown break points is the Bessel process distribution. The Bessel process distributions have no known simple closed-form expression for the distribution function which makes it difficult to compute exact p-values. Also, the Bessel process distributions depend on two parameters which makes it hard to tabulate the critical values hence one needs to simulate them. After simulating the critical values, hypothesis testing is done in the presence and absence of a change point in volatility of a simulated time series and the test is shown to reject the null hypothesis in the presence of a change point at alpha level of significance. Further, the test fails to reject the null hypothesis in the absence of a change point at alpha level of significance. An application to United States Dollar/Kenya Shilling historical exchange rates returns is made from 1st January 2010 to 27th November 2020 where the sample size n = 2839 is done. Through binary segmentation method, three change points are detected, estimated and accounted for. A significant improvement in describing a time series is expected if a point in time for volatility change has been detected and estimated.},
     year = {2023}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Detection and Estimation of Change Point in Volatility Function of Foreign Exchange Rate Returns
    AU  - Josephine Njeri Ngure
    AU  - Anthony Gichuhi Waititu
    AU  - Simon Maina Mundia
    Y1  - 2023/04/06
    PY  - 2023
    N1  - https://doi.org/10.11648/j.ijdsa.20230901.11
    DO  - 10.11648/j.ijdsa.20230901.11
    T2  - International Journal of Data Science and Analysis
    JF  - International Journal of Data Science and Analysis
    JO  - International Journal of Data Science and Analysis
    SP  - 1
    EP  - 12
    PB  - Science Publishing Group
    SN  - 2575-1891
    UR  - https://doi.org/10.11648/j.ijdsa.20230901.11
    AB  - This work aims at detection and estimation of a change point in conditional variance function of a Nonparametric Auto-Regressive Conditional Heteroscedastic model. The conditional mean and conditional variance functions are not specified a priori but estimated using Nadaraya Watson kernel. This is because inferences based on nonparametric approaches are robust against misspecification of the conditional mean function and the conditional variance function of returns. The squared residuals obtained after estimating the regression function of the returns are used in estimating the conditional variance function. Further, the squared residuals are used in developing a test statistic for unknown abrupt change point in volatility of the exchange rate returns. The test statistic takes into consideration the conditional heteroskedasticity of the disturbances, dependence of the returns, heterogeneity and fourth moment of returns. This does not require prior knowledge of the marginal or the conditional densities of the returns as opposed to maximum likelihood estimation methods. The estimator for change point is considered as the augmented maximum of the test statistic. The consistency of the estimator is stated as a theorem. The asymptotic distribution associated with the test for unknown break points is the Bessel process distribution. The Bessel process distributions have no known simple closed-form expression for the distribution function which makes it difficult to compute exact p-values. Also, the Bessel process distributions depend on two parameters which makes it hard to tabulate the critical values hence one needs to simulate them. After simulating the critical values, hypothesis testing is done in the presence and absence of a change point in volatility of a simulated time series and the test is shown to reject the null hypothesis in the presence of a change point at alpha level of significance. Further, the test fails to reject the null hypothesis in the absence of a change point at alpha level of significance. An application to United States Dollar/Kenya Shilling historical exchange rates returns is made from 1st January 2010 to 27th November 2020 where the sample size n = 2839 is done. Through binary segmentation method, three change points are detected, estimated and accounted for. A significant improvement in describing a time series is expected if a point in time for volatility change has been detected and estimated.
    VL  - 9
    IS  - 1
    ER  - 

    Copy | Download

Author Information
  • Department of Mathematics, Pan African University Institute for Basic Sciences, Technology and Innovation (PAUSTI), Nairobi, Kenya

  • Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology (JKUAT), Nairobi, Kenya

  • Department of Statistics and Actuarial Sciences, Dedan Kimathi University of Technology (DEKUT), Nyeri, Kenya

  • Sections