Unveiling Nonlinearities Through Time-Scale Transformations

Dominique Guillaume
Center for Economic Studies, Catholic University of Leuven
ndaab07@econmsmail.kuleuven.ac.be

Olivier Pictet, Urs Müller, and Michel Dacorogna
Olsen & Associates
olivier@olsen.ch

Abstract

In this paper, we show that intra-daily foreign exchange rate returns exhibit even stronger nonlinearities that daily or weekly returns. These nonlinearities result from the intra-daily seasonality and the presence of market participants with different time-horizons. Moreover, we present some evidence that both these nonliearities and the autocorrelation patterns of the volatility can successfully be accounted for by successive time scale transformations.

The recent empirical literature on the structure of the intra-daily foreign exchange (FX) markets has unveiled many features of FX markets not apparent at the daily or weekly frequencies (see Guillaume et al. (1994) for a survey). The purpose of this paper is twofold: first, we want to test whether this finer structure is indeed reflected by stronger nonlinearities at the intra-daily frequencies that at the daily and weekly ones (Hsieh, 1989; Hsieh, 1991; Brock et al., 1992); second, once we describe the sources of these nonlinearities, we want to show how most of them can be accounted for by two successive time scale transformations. The first time scale is an intra-day extension of the business time scale implicitly used at daily intervals by omitting week-end or holidays (Dacorogna et al., 1993b). The second time scale (Dacorogna et al., 1993a) is very similar in spirit to the time deformations introduced in Mandelbrot and Taylor (1967), Clark (1973), Stock (1988), Ghysels and Jasiak (1994). One major difference however is the modelling of the heteroskedasticity by directly fitting the volatility instead of the volume.

The remainder of this paper is divided as follows. In the next section, we present our methodology. The third section summarizes the results. The fourth section concludes the paper.