Multiscale analysis of economic time series by scale-dependent Lyapunov exponent

Economic time series usually exhibit complex behavior such as nonlinearity, fractal long-memory, and non-stationarity. Recently, considerable efforts have been made to detect chaos and fractal long-memory in finance. While evidence supporting fractal scaling in finance has been accumulating, it is now generally thought that financial time series may not be modeled by chaos or noisy chaos, since the estimated Lyapunov exponent (LE) is negative. A negative LE amounts to a negative Kolmogorov entropy, and thus implies simple regular dynamics of the economy. This is at odds with the general observation that the economy is highly complicated due to nonlinear and stochastic interactions among component systems and hierarchical regulations in the world economy. To resolve this dilemma, and to provide an effective means of characterizing fractal long-memory properties in non-stationary economic time series, we employ a multiscale complexity measure, the scale-dependent Lyapunov exponent (SDLE), to characterize economic time series. SDLE cannot only unambiguously distinguish low-dimensional chaos from noise, but also detect high-dimensional and intermittent chaos, as well as effectively deal with non-stationarity. With SDLE, we are able to show that the reported negative LE may correspond to large-scale convergence, but not imply the absence of small-scale divergence or noisy chaos in the world economy. Using US foreign exchange rate data as examples, we further show how SDLE can readily characterize fractal, persistent or anti-persistent long-range correlations in economic time series.