Adaptive Student's t-distribution with method of moments moving estimator for nonstationary time series
This work addresses nonstationary time series modeling for financial applications, offering an incremental improvement over classical methods by incorporating tail shape evolution.
The paper tackles the problem of modeling nonstationary time series by developing an adaptive method of moments estimator for Student's t-distribution, applied to log-returns of DJIA companies, allowing evolution of tail shape parameters to estimate extreme event probabilities.
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. $F_t=\sum_{τ<t} (1-η)^{t-τ} \ln(ρ_θ(x_τ))$ moving log-likelihood, evolving in time. It allows for example to estimate parameters using inexpensive exponential moving averages (EMA), like absolute central moments $m_p=E[|x-μ|^p]$ evolving for one or multiple powers $p\in\mathbb{R}^+$ using $m_{p,t+1} = m_{p,t} + η(|x_t-μ_t|^p-m_{p,t})$. 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 $ν$ describing $ρ(x)\sim |x|^{-ν-1}$ tail shape, probability of extreme events - which might turn out catastrophic, destabilizing the market.