Non-linear correlations and dependence

Non identical variables

The previous chapter presented a statistical analysis of financial data that implicitly assumes homogenous time series. For example, when we constructed the empirical distributions of price returns ηk, we pooled together all the data, thereby assuming that P1(η1), P2(η2), …, PN(ηN) are all identical.

It turns out that the distributions of the elementary random variables P1(η1), P2(η2), …, PN(ηN) are often not all identical. This is the case, for example, when the variance of the random process itself depends upon time – in financial markets, it is a wellknown fact that the daily volatility is time dependent, taking rather high levels in periods of uncertainty, and reverting back to lower values in calmer periods. This phenomena is called heteroskedasticity. For example, the volatility of the bond market has been very high during 1994, and decreased in later years. Similarly, the stock markets since the beginning of the century have witnessed periods of high volatility separated by rather quiet periods (Fig. 7.1). Another interesting analogy is turbulent flows which are well known to be intermittent: periods of relative tranquility (laminar flow), where the velocity field is smooth, are interrupted by intense ‘bursts’ of activity.