arXiv 6 Apr 2023 · Statistics — Methodology
arXiv:2304.03069 · PDF · DOI · OpenAlex · Extracted main text
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. $F_t=\sum_{\tau<t} (1-\eta)^{t-\tau} \ln(\rho_\theta (x_\tau))$ moving log-likelihood, evolving in time. It allows for example to estimate parameters using inexpensive exponential moving averages (EMA), like absolute central moments $m_p=E[|x-\mu|^p]$ evolving for one or multiple powers $p\in\mathbb{R}^+$ using $m_{p,t+1} = m_{p,t} + \eta (|x_t-\mu_t|^p-m_{p,t})$. 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 $\mu$ and $\sigma$, here we also get evolution of $\nu$ describing $\rho(x)\sim |x|^{-\nu-1}$ tail shape, probability of extreme events - which might turn out catastrophic, destabilizing the market.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | B. Basu and V. Pakrashi, “Parameter estimates of alpha-stable distri… (2018) Parameter estimates of alpha-stable distribution and hurst coefficients | 0.737 | 3 | 2 | 100% |
| 2 | J. Duda, “Adaptive exponential power distribution with moving estima… (2020) Adaptive exponential power distribution with moving estimator for nonstationary time series | 0.693 | 5 | 1 | 100% |
| 3 | J. Duda, “Parametric context adaptive laplace distribution for multi… (2018) Exploiting statistical dependencies of time series with hierarchical correlation reconstruction | 0.644 | 2 | 2 | 100% |
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| 6 | J. Duda, “Improving SGD convergence by tracing multiple promising di… (2019) Improving SGD convergence by tracing multiple promising directions and estimating distance to minimum | 0.405 | 1 | 1 | 100% |
| 7 | H. Cardot and D. Degras, “Online principal component analysis in hig… (2018) Online principal component analysis in high dimension: Which algorithm to choose? | 0.405 | 1 | 1 | 100% |
| 8 | T. Bollerslev, “Generalized autoregressive conditional heteroskedast… (1986) Generalized autoregressive conditional heteroskedasticity | 0.405 | 1 | 1 | 100% |
| 9 | N. Johnson and B. Welch, “Applications of the non-central t-distribu… (1940) Applications of the non-central t-distribution | 0.405 | 1 | 1 | 100% |
| 10 | K. E. Bassler, G. H. Gunaratne, and J. L. McCauley, “Markov processe… (2006) Markov processes, hurst exponents, and nonlinear diffusion equations: With application to finance | 0.405 | 1 | 1 | 100% |
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