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Stochastic difference equations with non-integral differences

Published online by Cambridge University Press:  01 July 2016

P. M. Robinson
P. M. Robinson*
Affiliation:
Harvard University
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Abstract

As an alternative to conventional discrete time models for stochastic processes that fluctuate within the sampling interval, we propose difference equations containing non-integral lags. We discuss the problems of stability, identification and estimation, for which an approximate model is needed. Least squares applied to an approximate Fourier-transformed model yields estimators of the coefficients that are consistent with respect to the true model under some conditions. The conditions are weak when the model contains predetermined variables that obey an “aliasing condition”; estimators of the lags as well as coefficients can then be found that are consistent, efficient and satisfy a central limit theorem. Optimal estimators for stochastic difference-differential equations are also available.

Keywords

NON-INTEGRAL DIFFERENCE EQUATIONSDIFFERENCE-DIFFERENTIAL EQUATIONSFOURIER METHODSSTATIONARY PROCESSESLIMIT THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 3 , September 1974 , pp. 524 - 545
Copyright
Copyright © Applied Probability Trust 1974 

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