Adaptive stable distribution and Hurst exponent by method of moments moving estimator for nonstationary time series
This work addresses the challenge of nonstationarity in time series analysis for fields like finance, though it appears incremental as it builds on existing methods with a novel adaptation approach.
The paper tackles the problem of modeling nonstationary time series by introducing a moving estimator that adaptively estimates parameters like those of the alpha-Stable distribution and Hurst exponent, with application to DJIA financial data showing continuous evaluation of market stability through tail behavior.
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 $F_t=\sum_{τ<t} (1-η)^{t-τ} \ln(ρ_θ(x_τ))$ 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 $m_p=E[|x-μ|^p]$ absolute central moments, updated by $m_{p,t+1} = m_{p,t} + η(|x_t-μ_t|^p-m_{p,t})$. 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 $μ$ and scale parameter $σ$, there is also estimated evolution of $α$ parameter allowing to continuously evaluate market stability - tails having $ρ(x) \sim 1/|x|^{α+1}$ behavior, controlling probability of potentially dangerous extreme events.