The scale-dependent market trend: Empirical evidences using the lagged DFA method

In this paper we make an empirical research and test the efficiency of 44 important market indexes in multiple scales. A modified method based on the lagged detrended fluctuation analysis is utilized to maximize the information of long-term correlations from the non-zero lags and keep the margin of errors small when measuring the local Hurst exponent. Our empirical result illustrates that a common pattern can be found in the majority of the measured market indexes which tend to be persistent (with the local Hurst exponent >0.5) in the small time scale, whereas it displays significant anti-persistent characteristics in large time scales. Moreover, not only the stock markets but also the foreign exchange markets share this pattern. Considering that the exchange markets are only weakly synchronized with the economic cycles, it can be concluded that the economic cycles can cause anti-persistence in the large time scale but there are also other factors at work. The empirical result supports the view that financial markets are multi-fractal and it indicates that deviations from efficiency and the type of model to describe the trend of market price are dependent on the forecasting horizon.