Geodesic
On random processes as an implicit solution of equations
Kybernetika, Tome 53 (2017) no. 6, pp. 985-991
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Random processes with convenient properties are often employed to model observed data, particularly, coming from economy and finance. We will focus our interest in random processes given implicitly as a solution of a functional equation. For example, random processes AR, ARMA, ARCH, GARCH are belonging in this wide class. Their common feature can be expressed by requirement that stated random process together with incoming innovations must fulfill a functional equation. Functional dependence is linear for AR, ARMA. We consider a general functional dependence, but, existence of a forward and a backward equivalent rewritings of the given functional equation is required. We present a concept of solution construction giving uniqueness of assigned solution. We introduce a class of implicit models where forward and backward equivalent rewritings exist. Illustrative examples are included.
Random processes with convenient properties are often employed to model observed data, particularly, coming from economy and finance. We will focus our interest in random processes given implicitly as a solution of a functional equation. For example, random processes AR, ARMA, ARCH, GARCH are belonging in this wide class. Their common feature can be expressed by requirement that stated random process together with incoming innovations must fulfill a functional equation. Functional dependence is linear for AR, ARMA. We consider a general functional dependence, but, existence of a forward and a backward equivalent rewritings of the given functional equation is required. We present a concept of solution construction giving uniqueness of assigned solution. We introduce a class of implicit models where forward and backward equivalent rewritings exist. Illustrative examples are included.
DOI : 10.14736/kyb-2017-6-0985
Classification : 62M10, 91B70
Keywords: econometric models; ARMA process; implicit definition 
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Lachout, Petr. On random processes as an implicit solution of equations. Kybernetika, Tome 53 (2017) no. 6, pp. 985-991. doi: 10.14736/kyb-2017-6-0985

[1] Box, G. E. P., Jenkins, G. M.: Time-series Analysis Forecasting and Control. Holden-Day, San Francisco 1976. | MR

[2] Brockwell, P. J., Davis, R. A.: Time-series: Theory and Methods. Springer-Verlag, New York 1987. | DOI | MR

[3] Durbin, J., Koopman, S. J.: Time Series Analysis by State Space Models. Oxford University Press, Oxford 2001. | MR

[4] Hommes, C.: Financial markets as nonlinear adaptive evolutionary systems. Quantitative Finance 1 (2001), 1, 149-167. | DOI | MR

[5] Lachout, P.: On functional definition of time-series models. In: Proc. 32th International Conference on Mathematical Methods in Economics, Olomouc (J. Talašová, J. Stoklasa and T. Talášek, eds.), Palacký University, Olomouc 2014, pp. 560-565.

[6] Lachout, P.: Discussion on implicit econometric models. In: Proc. 34th International Conference on Mathematical Methods in Economics, (A. Kocourek and M. Vavroušek, eds.), Technical University of Liberec 2016, pp. 501-505.

[7] Liao, H.-E., Sethares, W. A.: Suboptimal identification of nonlinear ARMA models using an orthogonality approach. IEEE Trans. Circuits and Systems 42 (1995), 1, 14-22. | DOI | MR

[8] Liu, J., Susko, K.: On strict stationarity and ergodicity of a non-linear ARMA model. J. Appl. Probab. 29 (1992), 2, 363-373. | DOI | MR

[9] Masry, E., Tjøstheim, D.: Nonparametric estimation and identification of nonlinear ARCH time-series. Econometr. Theory 11 (1995), 1, 258-289. | DOI | MR

[10] Priestley, M. B.: Non-Linear and Non-Stationary time-series. Academic Press, London 1988. | MR

[11] Shumway, R. H., Stoffer, D. S.: Time Series Analysis and Its Applications - With R Examples. EZ Green Edition, 2015. | MR

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