[Paper Review] Fast Times, Slow Times: Timescale Separation in Financial Timeseries Data

Jan Rosenzweig|arXiv (Cornell University)|Jan 16, 2026

Financial Risk and Volatility Modeling0 citations

TL;DR

The paper presents a framework to separate slow and fast components in financial time series via variance and tail stationarity criteria, formulated as generalized eigenvalue problems (tICA variants), and demonstrates robustness of slow components out-of-sample across asset classes.

ABSTRACT

Financial time series exhibit multiscale behavior, with interaction between multiple processes operating on different timescales. This paper introduces a method for separating these processes using variance and tail stationarity criteria, framed as generalized eigenvalue problems. The approach allows for the identification of slow and fast components in asset returns and prices, with applications to parameter drift, mean reversion, and tail risk management. Empirical examples using currencies, equity ETFs and treasury yields illustrate the practical utility of the method.

Motivation & Objective

• Motivate the need to identify multiscale processes in financial time series and their interactions across timescales.
• Propose a framework for timescale separation based on variance and tail stationarity criteria.
• Derive a generalized eigenvalue problem for variance timescales (tICA) and extend to tail timescales (higher-order moments).
• Show empirical demonstrations across currencies, factor ETFs, and US Treasuries to illustrate practical utility.

Proposed method

• Define X_t as an n-dimensional process (prices or returns) and seek weight vectors w_i that minimize drift of variance for unit variance.
• Formulate a generalized eigenvalue problem E[(dX_t)^T X_t + X_t^T dX_t] w = β E[X_t^T X_t] w and discretize to C(t,T) w = λ C(t,0) w, with λ = (1+β)/(2T).
• Interpret eigenvectors as directions of fastest/slowest decay in non-stationarity; eigenvalues λ_i as autocorrelations related to decay times t_i = -2T/(λ_i-1).
• Extend to tail timescales by minimizing drift of higher moments, yielding nonlinear equations solved via fixed-point iteration (FastICA) with updates w ← E[X_t d(w^T X_t)^{2k-1}] − (2k−1) E[(w^T X_t)^{2k−2}] w and normalization steps.
• Provide computationally tractable methods (linear tICA k=1 and nonlinear tICA k=4) and iterative schemes for extracting slow components.

(a) Linear tICA, $k=1$

Experimental results

Research questions

• RQ1 Can financial time series be decomposed into processes operating on different timescales (fast vs. slow)?
• RQ2 Can we estimate the relaxation timescales of these slow/fast components from data?
• RQ3 How can we assess stationarity and tail-stability in a practical, data-driven way for multiscale finance data?
• RQ4 Do linear (variance-based) and nonlinear (tail-based) tICA decompositions yield robust, interpretable slow components across assets and time periods?

Key findings

• The method separates slow and fast components of asset returns and prices using variance and tail criteria framed as generalized eigenvalue problems (tICA-like).
• The linear tICA (k=1) and nonlinear tICA (k=4) decompositions yield slow components with robust out-of-sample time scales across currencies, factor ETFs, and Treasuries.
• Slow components show persistence of their timescales out-of-sample, even as higher-frequency dynamics re-enter during events, indicating a degree of robustness of the slow modes.
• Orthogonality of extracted components decays out-of-sample, indicating limitations in preserving orthogonality beyond the training window.
• Higher tail emphasis (larger k) does not materially change results beyond k=4, suggesting two practical regimes: linear (volatility-driven) and nonlinear (tail-driven).

(b) Nonlinear tICA, $k=4$