Adaptive Student's t-distribution with method of moments moving estimator for nonstationary time series
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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 such bias, we will focus on recently proposed agnostic philosophy of moving estimator: in time t finding parameters optimizing e.g. F_t=∑_τ<t (1-η)^t-τln(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 E[|x-μ|^p] evolving for one or multiple powers p∈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)∼ |x|^-ν-1 tail shape, probability of extreme events - which might turn out catastrophic, destabilizing the market.
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