This paper measures the degrees of persistence of the daily returns of eight European stock market indices, after their lack of ergodicity and stationarity has been established. The proper identification of the nature of the persistence of financial time series forms a crucial step in deciding what kind of diffusion modeling of such series might provide invariant results. The results indicate that ergodicity and stationarity are very difficult to establish with only daily observations of market indices and thus, various price diffusion models cannot be successfully identified. However, the measured degrees of persistence point to the existence of long-term dependencies or Long Memory (LM), most likely of a non-linear nature. Global Hurst exponents, computed from wavelet multi-resolution analysis using a fractal Brownian motion model, measure the degree of persistence of the data series. The FTSE turns out to be anti-persistent, i.e., an ultra-efficient market with abnormally fast mean-reversion, faster than that of a geometric Brownian motion. The various measurement methodologies reported in the financial literature produce non-unique empirical results. Thus, it is very difficult to obtain definite conclusions regarding the presence or absence of long-term dependence phenomena based on the global Hurst exponents. Most stock markets in Europe appear to be slightly anti-persistent, but more powerful methods, such as the computation of the multifractal spectra of financial time series from intra-day pricing data, may be required to establish this as a definite scientific conclusion. Still, we demonstrate that the visualization of the wavelet resonance coefficients and their power spectra, in the form of localized scalograms and averaged scalegrams, forcefully assist the detection and measurement of several types of persistent market price diffusion.

For a long time financial researchers have struggled with the identification of properly specified econometric and time series models that can capture the dynamic dependencies in financial time series. The most popular models, accounting for lagged observations, are the family of ARIMA models and GARCH models. Some models from these families have become very popular among technical analysts, due to their ability to capture short-term dependence. But these, often linear, models are criticized for not being able to model longterm dependence (also called persistence or Long Memory, or LM) and for requiring (and often presuming) Gaussian distribution characteristics for the innovation residuals. Loretan and Phillips (1994) recognize that distribution characteristics of time series often vary over time. Cochrane (1988) already pointed out several weaknesses of the ARIMA models and suggested a measure for long-term dependence, like unit root integration I(1), because of the presence of approximate common factors in the AR and MA polynomials.