Stable Identification of Linear Autoregressive Model with Exogenous Variables on the Basis of the Generalized Least Absolute Deviation Method

Least Absolute Deviations (LAD) method is a method alternative to the Ordinary Least Squares OLS method. It allows to obtain robust errors in case of violation of OLS assumptions. We present two types of LAD: Weighted LAD method and Generalized LAD method. The established interrelation of methods made it possible to reduce the problem of determining the GLAD estimates to an iterative procedure with WLAD estimates. The latter is calculated by solving the corresponding linear programming problem. The sufficient condition imposed on the loss function is found to ensure the stability of the GLAD estimators of the autoregressive models coefficients under emission conditions. It ensures the stability of GLAD-estimates of autoregressive models in terms of outliers. Special features of the GLAD method application for the construction of the regression equation and autoregressive equation without exogenous variables are considered early. This paper is devoted to extension of the previously discussed methods to the problem of estimating the parameters of autoregressive models with exogenous variables.

Keywords

algorithm; autoregressive model; linear programming; parameter identification.

References

1. Mudrov V.I., Kushko V.L. Metody obrabotki izmerenii: Kvazpravdopodobnye otcenki. [Methods for Processing Measurements: Quasi-Like Estimates]. Moscow, LEVAND, 2014. (in Russian)
2. Gurin L.S. [On the Consistency of Estimates of the Method of Least Squares]. Matematicheskoe obespechenie kosmicheskikh eksperimentov [Mathematical Support of Cosmic Experiments], Moscow, Nauka, 1978, pp. 69-81. (in Russian)
3. Tyrsin A.N. The Method of Choosing the Best Distribution Law of a Continuous Random Variable on the Basis of the Inverse Mapping. Bulletin of South Ural State University. Series: Mathematics. Mechanics. Physics, 2017, vol. 9, no. 1, pp. 31-38. (in Russian) DOI: 10.14529/mmph170104
4. Shestakov A.L., Keller A.V., Sviridyuk G.A. The Theory of Optimal Measurements. Journal of Computational and Engineering Mathematics, 2014, vol. 1, no. 1, pp. 3-16.
5. Huber P., Ronchetti E.M. Robust Statistics. New Jersey, Wiley, 2009. DOI: 10.1002/9780470434697
6. Pan J., Wang H., Qiwei Y. Weighted Least Absolute Deviations Estimation for Arma Models with Infinite Variance. Econometric Theory, 2007, vol. 23, pp. 852-879. DOI: 10.1017/S0266466607070363
7. Panyukov A.V. Scalability of Algorithms for Arithmetic Operations in Radix Notation. Reliable Computing, 2015, vol. 19, no. 4, pp. 417-434. DOI: 10.1134/S0005117912020063
8. Panyukov A.V., Gorbik V.V. Using Massively Parallel Computations for Absolutely Precise Solution of the Linear Programming Problems. Automation and Remote Control, 2012, vol. 73, no. 2, pp. 276-290. DOI: 10.1134/S0005117912020063
9. Panyukov A.V, Tyrsin A.N. Stable Parametric Identification of Vibratory Diagnostics Objects. Journal of Vibroengineering, 2008, vol. 10, no. 2, pp. 142-146.
10. Tyrsin A.N. Robust Construction of Regression Models Based on the Generalized Least Absolute Deviations Method. Journal of Mathematical Sciences, 2006, vol. 139, no. 3, pp. 6634-6642. DOI: 10.1007/s10958-006-0380-7