Investigating Chaotic Behavior in Economic Series: The Delay Time in the Grassberger–procaccia Algorithm

Abstract

In this paper we try to bring forward evidence on the practical application of the Grassberger–Procaccia algorithm, in particular on the determination of the "delay time" parameter. For this purpose, we analyze the results obtained from applying the main methods proposed to calculate this delay time for series simulated from well-known chaotic systems. As the most relevant result we conclude that, in general, all the methods display inadequate behavior, except for that based on the previous filtering of the series according to singular value decomposition. In a second stage we apply the same study to three financial series, with the results appearing to confirm the advantage of this method.

A Generalized BDS Statistic

“…As was commented earlier (Aparicio et al, 1999;Kanz and Schreiber, 2004) the choice of delay time is crucial when estimating certain dynamical properties, such as the correlation dimension, up to the point that an unfortunate delay time choice can yield misleading results.…”

“…This is particularly interesting when the practical method of reconstruction has to overcome the objections which arise from a insufficiently conditioned time series (relative to Takens' theorem, like for example when dealing with finite and noise data, as is usual in economics) and therefore it seems crucial to select delay time also. Moreover, recent research (Aparicio et al, 1999) shows that the choice of delay time is crucial when estimating the correlation dimension (based on the correlation integral), up to the point that an unfortunate delay time choice yields misleading results concerning the dimension of well known attractors. As Kantz and Schreiber (2004) indicate, in spite of the proper choice of delay time being quite important in applications, the relevant mathematical framework has not been convincingly studied.…”

Prediction and Volatility of Black Market Currencies: Evidence From Renminbi and Rial Exchange Rates

“…In the exercises of prediction, we usually use a scalar mapping instead of the mapping in (2), that is,…”

“…Aparicio, et al [2] calculated the correlation dimensions for the daily Mark/USD exchange rate, the federal funds rate, and the daily NY Stock Market Composite Index, and concluded that the dynamics of the federal funds rate may be chaotic. Bajo-Rubio et.…”

Does Chaos Matter in Financial Time Series Analysis?

“…Therefore, the logic of the random walk is that if the flow of information is immediately reflected in stock prices, then tomorrow's price change will reflect only tomorrow's news and will be independent of the price changes today. Consequently, stock returns are considered Independent and Identically Distributed (IID) random variables that move as a random walk (Aparicio et al, 1999;Gilmore, 1993;Kyrtsou et al, 2004;Muckley, 2004;Sewell et al, 1996;Varson and Doran 1995). Nevertheless, the restricted assumptions and the presence of irregularities in financial time series have led to criticise the validity of EMH theory.…”