Adaptive Student's t-distribution with method of moments moving estimator for nonstationary time series

The real life time series are usually nonstationary, bringing a difficult question of model adaptation. Classical approaches like ARMA-ARCH assume arbitrary type of dependence. To avoid their bias, we will focus on recently proposed agnostic philosophy of moving estimator: in time t finding parameters optimizing e.g. Ft=sumτ<t (1-η)t-τ ln(ρθ (xτ)) moving log-likelihood, evolving in time. It allows for example to estimate parameters using inexpensive exponential moving averages (EMA), like absolute central moments mp=E[|x-μ|p] evolving for one or multiple powers p∈ℝ^+ using mp,t+1 = mp,t + η (|xt-μt|p-mp,t). Application of such general adaptive methods of moments will be presented on Student's t-distribution, popular especially in economical applications, here applied to log-returns of DJIA companies. While standard ARMA-ARCH approaches provide evolution of μ and σ, here we also get evolution of nu describing ρ(x)~ |x|-nu-1 tail shape, probability of extreme events - which might turn out catastrophic, destabilizing the market.