Abstract
In this paper, we propose criteria for the existence of memory of power-law type (PLT) memory in economic processes. We give the criterion of existence of power-law long-range dependence in time by using the analogy with the concept of the long-range alpha-interaction. We also suggest the criterion of existence of PLT memory for frequency domain by using the concept of non-integer dimensions. For an economic process, for which it is known that an endogenous variable depends on an exogenous variable, the proposed criteria make it possible to identify the presence of the PLT memory. The suggested criteria are illustrated in various examples. The use of the proposed criteria allows apply the fractional calculus to construct dynamic models of economic processes. These criteria can be also used to identify the linear integro-differential operators that can be considered as fractional derivatives and integrals of non-integer orders.
Highlights
In models of economic processes, variables of two types are used
This dependence is due to the fact that people, which participate in the economic process, can remember the previous changes of exogenous variable X(t) and the impact of these changes on the endogenous variable Y(t)
For with it is known that an endogenous variable Y(t) depends on exogenous variables X(t), there is a multi-parametric memory of power-law type if there are at least two of the limits (23)–(24) and lim ω→0+
Summary
In models of economic processes, variables of two types are used. The first type includes exogenous variables that are external to the considered model. The long-range time dependencies have been empirically observed in economics [1,2,3] For these dependences the correlations between values of variables decay to zero slower than it can be expected from independent variables or variables from classical Markov and autoregressive moving average models [1,2,3,4,5,6,7]. Note that the fractional differencing and integrating, which are usually used in economics [4,5,6], are the Grunwald–Letnikov fractional differences, which have been proposed more than hundred fifty years ago [8,9] This fact shows the importance of fractional calculus for modeling processes with memory. It shows the necessity of criteria for the existence of PLT memory
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