Adaptive stable distribution and Hurst exponent by method of moments moving estimator for nonstationary time series

Nonstationarity of real-life time series requires model adaptation. In classical approaches like ARMA-ARCH there is assumed some arbitrarily chosen dependence type. To avoid their bias, we will focus on novel more agnostic approach: moving estimator, which estimates parameters separately for every time t: optimizing Ft=sumτ<t (1-η)t-τ ln(ρθ (xτ)) local log-likelihood with exponentially weakening weights of the old values. In practice such moving estimates can be found by EMA (exponential moving average) of some parameters, like mp=E[|x-μ|p] absolute central moments, updated by mp,t+1 = mp,t + η (|xt-μt|p-mp,t). We will focus here on its applications for alpha-Stable distribution, which also influences Hurst exponent, hence can be used for its adaptive estimation. Its application will be shown on financial data as DJIA time series - beside standard estimation of evolution of center μ and scale parameter σ, there is also estimated evolution of α parameter allowing to continuously evaluate market stability - tails having ρ(x) ~ 1/|x|α+1 behavior, controlling probability of potentially dangerous extreme events.