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

Abstract

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 F_t=_<t (1-)^t- (_ (x_)) 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 m_p=E[|x-|^p] absolute central moments, updated by m_p,t+1 = m_p,t + (|x_t-_t|^p-m_p,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 and scale parameter , there is also estimated evolution of parameter allowing to continuously evaluate market stability - tails having (x) 1/|x|^+1 behavior, controlling probability of potentially dangerous extreme events.

Tasks

Reproductions