Computation of the Asymptotic Covariance Matrix of the Generalized Exponential Autoregressive Ozaki Model for the Least Squares Method

Abstract

High-order mathematical models and their theoretical properties remain the subject of active research over the past decades. They are playing an important role in solving economic, financial, engineering and medical problems. One of the most common examples is the generalized exponential autoregressive Ozaki model. The asymptotic covariance matrix of the generalized Ozaki model was computed for the least squares estimation by its expansion into the Taylor series. The paper compares the speed, at which the model separate implementations tend to its asymptotic behavior for several distributions of the updating process, i.e., normal (Gaussian), contaminated normal with various combinations in the contamination frequency and magnitude parameters, Student, Laplace and logistic. Scientific novelty of this work lies in direct determination of the asymptotic covariance matrix of the generalized Ozaki model. The practical novelty is the possibility of using the tabular results in comparing its implementations to decide on introducing the Ozaki model or any other model in the engineering calculations

Please cite this article in English as:

Goryainov V.B., Masyagin M.M. Computation of the asymptotic covariance matrix of the generalized exponential autoregressive Ozaki model for the least squares method. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2024, no. 3 (114), pp. 24--44 (in Russ.). EDN: PPICLB

References

[1] Chan E.P. Machine trading. Deploying computer algorithms to conquer the markets. Wiley, 2017.

[2] Naik N., Mohan B.R. Stock price volatility estimation using regime switching technique empirical study on the Indian stock market. Mathematics, 2021, vol. 9, iss. 14, pp. 1595--1608. DOI: https://doi.org/10.3390/math9141595

[3] Hsu B., Sherina V., McCall M.N. Auto-regressive modeling and diagnostics for qPCR amplification. Bioinformatics, 2020, vol. 36, no. 22-23, pp. 5386--5391. DOI: https://doi.org/10.1101/665596

[4] Mohamed H.S., Cordeiro G.M., Yousuf H.M. The synthetic autoregressive model for the insurance claims payment data: modeling and future prediction. Statistics Optimization & Information Computing, 2022, vol. 11, pp. 524--533.

[5] Lapin V.A., Yerzhanov S.E., Aidakhov Y.S. Statistical modeling of a seismic isolation object under random seismic exposure. J. Phys.: Conf. Ser., 2019, vol. 1425, art. 012006. DOI: https://doi.org/10.1088/1742-6596/1425/1/012006

[6] Ozaki T. Non-linear phenomena and time series models. Invited paper 43 Session of the International Statistical Institute. Buenos Aires, 1981.

[7] Ozaki T. The statistical analysis of perturbed limit cycle processes using nonlinear time series models. J. Time Ser. Anal., 1982, vol. 3, iss. 1, pp. 29--41. DOI: https://doi.org/10.1111/j.1467-9892.1982.tb00328.x

[8] Tong H. Non-linear time series. Oxford Univ. Press, 1990.

[9] Terasvirta T. Specification, estimation, and evaluation of smooth transition autoregressive models. J. Am. Stat. Assoc., 1994, vol. 89, iss. 425, pp. 208--218. DOI: https://doi.org/10.1080/01621459.1994.10476462

[10] Chan K.S., Tong H. On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations. Adv. Appl. Probab., 1985, vol. 17, iss. 3, pp. 666--668. DOI: https://doi.org/10.2307/1427125

[11] White H. Asymptotic theory for econometricians. Academic Press, 2014.

[12] Hausler E., Luschgy H. Stable convergence and stable limit theorems. In: Probability Theory and Stochastic Modelling, vol. 74. Cham, Springer, 2015. DOI: https://doi.org/10.1007/978-3-319-18329-9

[13] Goryainov V.B. Least-modules estimates for spatial autoregression coefficients. J. Comput. Syst. Sci. Int., 2011, vol. 50, no. 4, pp. 565--572. DOI: https://doi.org/10.1134/S1064230711040101

[14] Goryainov A.V., Goryainova E.R. Comparison of efficiency of estimates by the methods of least absolute deviations and least squares in the autoregression model with random coefficient. Autom. Remote Control, 2016, vol. 77, no. 9, pp. 1579--1588. DOI: https://doi.org/10.1134/S000511791609006X

[15] Andersen P.K., Gill R.D. Cox’s regression model for counting processes: a large sample study. Ann. Statist., 1982, vol. 10, iss. 4, pp. 1100--1120. DOI: https://doi.org/10.1214/aos/1176345976

[16] Shiryaev A.N. Veroyatnost-1 [Probability]. Moscow, MTSNMO Publ., 2011.

[17] Billingsley P. Convergence of probability measures. Wiley, 1999.

[18] Magnus J.R., Neudecker H. Matrix differential calculus with applications in statistics and econometrics. Wiley, 2019.

[19] Broyden C.G. The convergence of a class of double-rank minimization algorithms. IMA J. Appl. Maths, 1970, vol. 6, iss. 1, pp. 76--90. DOI: https://doi.org/10.1093/imamat/6.1.76

[20] Fletcher R. A new approach to variable metric algorithms. Comput. J., 1970, vol. 13, iss. 3, pp. 317--322. DOI: https://doi.org/10.1093/comjnl/13.3.317

[21] Goldfarb D. A family of variable-metric methods derived by variational means. Math. Comp., 1970, vol. 24, no. 109, pp. 23--26. DOI: https://doi.org/10.1090/S0025-5718-1970-0258249-6

[22] Shanno D.F. Conditioning of quasi-Newton methods for function minimization. Math. Comp., 1970, vol. 24, no. 111, pp. 647--656. DOI: https://doi.org/10.2307/2004840