Multiple cyclical fractional structures in financial time series

Abstract

This article analyses multiple cyclical structures in financial time series. In particular, we focus on the monthly structure of the Nasdaq, the Dow–Jones and the S&P stock market indices. The three series are modelled as long-memory processes with poles in the spectrum at multiple frequencies, including the long-run or zero frequency.

Acknowledgement

Luis A. Gil-Alana gratefully acknowledges financial support from the Ministerio de Ciencia y Tecnologia (ECO2008-03035 ECON Y FINANZAS, Spain).

Notes

¹Lobato and Robinson (1998) proposed a semiparametric approach for testing this type of model, and Dalla and Hidalgo (2005) suggested a parametric test where the unbounded frequency in the spectrum is assumed to be unknown.

²Note that, if j ₁ = 1, then (1 − 2cos w _{j 1} L + L ²) ^{d ₁} = (1 − 2L + L ²) ^{d ₁} = (1 − L)^{2d ₁} .

³Note that Robinson's (1994) method is based on the Whittle function, which is an approximation of the likelihood function.