Fractional integration and data frequency

Abstract

This paper examines the robustness of fractional integration estimates to different data frequencies. We show by means of Monte Carlo experiments that if the number of differences is an integer value (e.g. 0 or 1), there is no distortion when data are collected at wider intervals; however, if it is a fractional value, the distortion increases as the number of periods between the observations increases, which results in lower orders of integration than those of the true DGP. An empirical application using the S&P 500 index is also carried out.

Keywords:

• fractional integration
• data frequency
• stock market

JEL Classification :

• C15

Acknowledgements

L.A. Gil-Alana gratefully acknowledges financial support from the Ministerio de Ciencia y Tecnologia (SEJ2005-07657, Spain).

Notes

Throughout this paper we assume that x _t =0 for t=0, which is a standard assumption in applied work using fractional integration techniques (see the type I and II definitions of fractional integration, e.g. in Gil-Alana and Hualde 16).

That means that if the test is defined against local alternatives of the form: , with δ≠0, the limit distribution is normal, with unit variance and mean that cannot be exceeded in absolute value by any rival regular statistic under Gaussianity of u _t .

Other recent developments of fractional integration in semiparametric contexts and based on the Whittle function have been proposed in Velasco 24 and Phillips and Shimotsu 25 26 but these methods require additional user-chosen parameters, which may be very sensitive to the estimation of d.

We generate Gaussian series using the routines GASDEV and RAN3 of Press et al. 32.

Some methods to calculate the optimal bandwidth numbers in semiparametric contexts have been examined in Delgado and Robinson 33 and Robinson and Henry 34. However, in the case of the Whittle estimator of Robinson 22 23, the use of optimal values has not been theoretically justified. Other authors, such as Lobato and Savin 35 use an interval of values for m, but we have preferred to report the results for the whole range of values of m.