Adaptive Student's t-distribution with method of moments moving estimator for nonstationary time series

TL;DR

The paper tackles nonstationary time series by replacing static parameter estimates with moving estimators that adapt $\theta_t=(\mu_t,\sigma_t,\nu_t)$ over time using a time-local objective $F_t$ and past data. It develops a method-of-moments based adaptation using absolute central moments and EMA updates to estimate $\sigma$ and $\nu$, plus log-absolute-moment and asymmetric extensions, all within the Student's $t$ framework. Empirical evaluation on 1900–2007 DJIA daily log-returns (and 2008–2018 stock components) demonstrates meaningful evolution of $\mu_t$, $\sigma_t$, and $\nu_t$, with adaptive $\sigma$ improving log-likelihood and $\nu$ tracking tail risk; incorporating asymmetry yields additional improvement. The work offers a flexible online, model-agnostic approach to nonstationary heavy-tailed time series and motivates further enhancements to $\nu$ estimation, skewed variants, and multivariate extensions such as online regression and HCR.

Abstract

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.

Figures (10)

• Figure 1: Mathematica code used for moving estimation of all $\theta=(\mu,\sigma,\nu)$ Student's t-distributions parameters (using $M_{\nu p}=E[|(x-\mu)/\sigma|^p]$ moment formula (\ref{['mom']})), and results of its application to 107 years of daily log-returns of DJIA (Dow Jones Industrial Average) time series. The parameters were manually tuned for this case to maximize log-likelihood: mean $\ln(\rho_t(x_t))$ showed at the bottom. We can see interesting evolution through this century which might be worth a deeper investigation, like $\approx 5$ year period cyclic behavior of the center $\mu$, huge $\approx 25\times$ change of width $\sigma$, and a few nearly Gaussian $\nu\to \infty$ periods mostly during 1967-1983. While $\mu$ describes the general up/down trend, $\sigma$ is close to volatility, additional $\nu$ complements it with kind of stability - probability of potentially catastrophic extreme events.
• Figure 2: Log-likelihoods (mean $\ln(\rho_t(x_t))$) evaluations for log-returns of 107 years DJIA time series, and 10 years for 29 individual companies. In horizontal axis there is $1/\nu$ Student's t-distribution degrees of freedom (from Gauss to Cauchy distributions), for static parameters (orange), and adaptive $\sigma$ scale parameter (blue, using $p=1$ power and $\eta_2=0.05$ learning rate), all for $\mu=0$ center. We can see adaptation has allowed for less heavy tails (larger $\nu$ in maximum). There are also shown analogously the best from $\sigma$ adaptation for $\rho(x)\sim \exp(-|x|^\kappa)$ Exponential Power Distribution in the previous article adaptive (gray). Red line shows evaluation of $\sigma$ adaptation by standard GARCH(1,1) model - which is comparable with $\nu=\infty$ Gaussian case, but usually slightly worse.
• Figure 3: Probability distribution function (PDF, asymptotically $\sim |x|^{-1-\nu}$) and cumulative distribution function (CDF) for Student's t-distribution with fixed center $\mu=0$ and scale parameter $\sigma=1$, but various shape parameter $\nu$. We get Gaussian distribution for $\nu\to\infty$, Cauchy distribution for $\nu=1$, and can also cover different types of heavy tails and bodies of distribution.
• Figure 4: Top: error dependence for choice of power $p$ in $\sigma$ estimation as $\hat{\sigma} = \sqrt[\leftroot{5}p]{T^{-1} \sum_t |x_i-\hat{\mu}|^p}/M_{\nu p}$. We can see that for Gaussian distribution $\nu\to\infty$ we should choose $p=2$ as in standard variance estimation, but to improve prediction should reduce this $p$ for lower $\nu$ to $p\approx \nu/6$. Bottom: monotonous functions for $\nu$ estimation for various choices of 2 powers $p_1,p_2$.
• Figure 5: The actual and expected numbers of events $|X-\mu|>k\sigma$: for $k=1,\ldots,10$, complete time series of 29349 values 1900-2007 (top) and restricted to 4012 values 1967-1983 (bottom). The marked green second column are numbers of values in the data, on the right there are expected numbers of events (length $\times$ probability) for Student's t-distribution for various $\nu$. In the top table we see large numbers of extreme events, after using adaptive $\sigma$ close to $\nu\in(3,5)$ Student's t-distribution. In contrast, the 1967-1983 range, suggested by $\nu$ evolution in Fig. \ref{['djiastud']}, has much lower $\nu\sim 10$ probability of extreme events - suggesting more stable market. Fig. \ref{['nu']} shows more detailed $\nu$ evolutions, what might be helpful with localizing, understand the crucial mechanisms, and maybe exploiting them to make the market more stable.