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 $F_t={\sum }_{\tau <t}(1-\eta {)}^{t-\tau }\ln ({\rho }_{\theta }(x_{\tau }))$ 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 $m_p=E[|x-\mu {|}^p]$ absolute central moments, updated by $m_{p,t+1}=m_{p,t}+\eta (|x_t-{\mu }_t{|}^p-m_{p,t})$. 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 $\mu$ and scale parameter $\sigma$, there is also estimated evolution of $\alpha$ parameter allowing to continuously evaluate market stability - tails having $\rho (x)\sim 1/|x{|}^{\alpha +1}$ behavior, controlling probability of potentially dangerous extreme events.

5 pages, 7 figures. arXiv admin note: text overlap with arXiv:2304.03069