Econometric Theory

ARTICLES

REPRESENTATION AND WEAK CONVERGENCE OF STOCHASTIC INTEGRALS WITH FRACTIONAL INTEGRATOR PROCESSES

James Davidsona1 c1 and Nigar Hashimzadea2

a1 University of Exeter

a2 University of Reading

Abstract

This paper considers the asymptotic distribution of the sample covariance of a nonstationary fractionally integrated process with the stationary increments of another such process—possibly itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analyzed in a previous paper (Davidson and Hashimzade, 2008), and the construction derived from moving average representations in the time domain. Depending on the values of the long memory parameters and choice of normalization, the limiting integral is shown to be expressible as the sum of a constant and two Itô-type integrals with respect to distinct Brownian motions. In certain cases the latter terms are of small order relative to the former. The mean is shown to match that of the harmonic representation, where the latter is defined, and satisfies the required integration by parts rule. The advantages of our approach over the harmonic analysis include the facts that our formulas are valid for the full range of the long memory parameters and that they extend to non-Gaussian processes.

Correspondence

c1 Address correspondence to James Davidson, Department of Economics, University of Exeter, Exeter EX4 4PU, United Kingdom; e-mail: james.davidson@exeter.ac.uk.

Footnotes

We thank, among others, Javier Hualde, Philipp Sibbertsen, Toni Stocker, Peter Phillips, Bent Nielsen, editor Robert Taylor, and three anonymous referees for discussion and comments on previous versions of the paper.

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