Robust Identification of an Exponential Autoregressive Model

Authors: Goryainov A.V., Goryainov V.B., Khing W.M. Published: 08.08.2020
Published in issue: #4(91)/2020  
DOI: 10.18698/1812-3368-2020-4-42-57

Category: Mathematics and Mechanics | Chapter: Computational Mathematics  
Keywords: exponential autoregression, robust estimate, consistency, asymptotic normality, asymptotic relative efficiency

One of the most common nonlinear time series (random processes with discrete time) models is the exponential autoregressive model. In particular, it describes such nonlinear effects as limit cycles, resonant jumps, and dependence of the oscillation frequency on amplitude. When identifying this model, the problem arises of estimating its parameters --- the coefficients of the corresponding autoregressive equation. The most common methods for estimating the parameters of an exponential model are the least squares method and the least absolute deviation method. Both of these methods have a number of disadvantages, to eliminate which the paper proposes an estimation method based on the robust Huber approach. The obtained estimates occupy an intermediate position between the least squares and least absolute deviation estimates. It is assumed that the stochastic sequence is described by the autoregressive equation of the first order, is stationary and ergodic, and the probability distribution of the innovations process of the model is unknown. Unbiased, consistency and asymptotic normality of the proposed estimate are established by computer simulation. Its asymptotic variance was found, which allows to obtain an explicit expression for the relative efficiency of the proposed estimate with respect to the least squares estimate and the least absolute deviation estimate and to calculate this efficiency for the most widespread probability distributions of the innovations sequence of the equation of the autoregressive model


[1] Ozaki T. Time series modeling of neuroscience data. CRC Press, 2012.

[2] Olugbode M., El-Masry A., Pointon J. Exchange rate and interest rate exposure of UK industries using first-order autoregressive exponential GARCH-in-mean (EGARCH-M) approach. Manch. Sch., 2014, vol. 82, iss. 4, pp. 409--464. DOI: https://doi.org/10.1111/manc.12029

[3] Gurung B. An exponential autoregressive (EXPAR) model for the forecasting of all India annual rainfall. Mausam, 2015, vol. 66, no. 4, pp. 847--849.

[4] Ghosh H., Gurung B., Gupta P. Fitting EXPAR models through the extended Kalman filter. Sankhya B, 2015, vol. 77, no. 1, pp. 27--44. DOI: https://doi.org/10.1007/s13571-014-0085-8

[5] Goryainov A.V., Goryainov V.B. M-Estimates of autoregression with random coefficients. Autom. Remote Control, 2018, vol. 79, pp. 1409--1421. DOI: https://doi.org/10.1134/S0005117918080040

[6] Goryainov V.B., Goryainova E.R. Comparative analysis of robust and classical methods for estimating the parameters of a threshold autoregression equation. Autom. Remote Control, 2019, vol. 80, no. 4, pp. 666--675. DOI: https://doi.org/10.1134/S0005117919040052

[7] Chan K.S., Tong H. On the use of deterministic Lyapunov function for the ergodicity of stochastic difference equations. Adv. Appl. Probab., 1985, vol. 17, iss. 3, pp. 666--678. DOI: https://doi.org/10.2307/1427125

[8] Tong H. Threshold models in time series analysis 30 years on. Stat. Interface, 2011, vol. 4, no. 2, pp. 107--118. DOI: https://dx.doi.org/10.4310/SII.2011.v4.n2.a1

[9] Huber P., Ronchetti E.M. Robust statistics. Wiley, 2009.

[10] Rhinehart R.R. Nonlinear regression modeling for engineering applications: modeling, model validation, and enabling design of experiments. Wiley, 2016.

[11] White H. Asymptotic theory for econometricians. Academic Press, 2000.

[12] Goryainov V.B. M-estimates of the spatial autoregression coefficients. Autom. Remote Control, 2012, vol. 73, no. 8, pp. 1371--1379. DOI: https://doi.org/10.1134/S0005117912080103

[13] Shiryaev A.N. Veroyatnost’ [Probability]. Moscow, Nauka Publ., 2011.

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

[15] Magnus J.R., Neudecker H. Matrix differential calculus with applications in statistics and econometrics. Wiley, 1999.

[16] Hampel F.R., Ronchetti E.M., Rousseeuw P.J., et al. Robust statistics. Wiley, 2005.

[17] Goryainov V.B., Goryainova E.R. The influence of anomalous observations on the least squares estimate of the parameter of the autoregressive equation with random coefficient. Herald of the Bauman Moscow State Technical University. Series Natural Sciences, 2016, no. 2 (65), pp. 16--24 (in Russ.). DOI: http://doi.org/10.18698/1812-3368-2016-2-16-24