On the question of parameters reconstruction accuracy for linear dynamic models with discrete time

Abstract

Modern methods of analysis of social and economic processes suggest the application of economic and mathematical models among which dynamic models with discrete time are widely spread. The accuracy of these models identification significantly impacts the analysis results validity and decisions made on these results basis. The opportunity and conditions to get guaranteed estimations of reconstruction accuracy of parameters of linear dynamic models with discrete time are discussed in the article. The study is based on the observation results of a trajectory including random errors with different nature (measurement errors, model specification errors, external disturbances). Particular attention is paid to an opportunity of obtaining guaranteed estimates of accuracy of an approximate parameters reconstruction of linear systems with discrete time and random but finite memory. An overview of known approaches and techniques for construction of parameter estimates of vector autoregression models is presented in the study. All options of estimates of construction methods are based on relatively strong theories about the nature of random errors and disturbances (strict exogenity, orthogonality, non-correlatedness, normal distribution, etc.) in the framework of an econometric method. In this case, only interval estimates for unknown parameters and confident intervals corresponding to a given significance level can be constructed. The works that potentially estimate system parameter reconstruction accuracy have been revealed among the results referring to the research of the given task from the view point of the theory of linear difference systems with discrete time. However, we have revealed that all studies in this field are characterized by one concept that suggests the estimates to be made during complex computational task. This fact significantly complicates the derivation of evident guaranteed estimates of accuracy expressed in the frameworks of general restrictions with regard to observation errors. An original mathematically substantiated approach to the task at minimum of assumptions regarding observation errors is the novelty of the study. Corresponding theory about parameters reconstruction accuracy estimation has been suggested and proven in the frameworks of the approach. An algorithm and illustrating case study have also been described in the framework of the approach. The method and algorithm allow us solving the task of parameter reconstruction of dynamic models on the basis of observation results over the simulation process with an accuracy that exceeds of a commonly used procedure of the least square method and its generalizations. However, the significant distinction from the well-known results is that in the terms of the proven theory the accuracy of parameter estimates is guaranteed in a strictly deterministic sense. Further studies in this field will consider the application of the developed tools for obtaining guaranteed estimates to the analysis of dynamic models with discrete and continuous time (hybrid models) including in the system both difference equations with discrete time and equations with continuous time in a form of autonomous functional differential equations. This task solution will allow us to achieve new results in the field of consequence effect identification during simulation of real social and economic processes.

Keywords

economic dynamic models, identification problems, guaranteed estimates, difference equations, systems with discrete time, social and economic process simulation.

Acknowledgements

The study was financially supported by the Ministry of Education and Science of the Russian Federation (Contract No. 02.G25.31.0039) and the Russian Foundation for Basic Research (Project No. 18-01-00332).

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