Factor Analysis of Multivariate Stochastic Volatility Model

Abstract

Modeling the time-varying covariance structures of high-dimensional variables is critical across diverse scientific and industrial applications; however, existing approaches exhibit notable limitations in either modeling flexibility or inferential efficiency. For instance, change-point modeling fails to account for the continuous time-varying nature of covariance structures, while GARCH and stochastic volatility models suffer from over-parameterization and the risk of overfitting. To address these challenges, we propose a Bayesian factor modeling framework designed to enable simultaneous inference of both the covariance structure of a high-dimensional time series and its time-varying dynamics. The associated Expectation-Maximization (EM) algorithm not only features an exact, closed-form update for the M-step but also is easily generalizable to more complex settings, such as spatiotemporal multivariate factor analysis. We validate our method through simulation studies and real-data experiments using climate and financial datasets.

Figures (21)

• Figure 1: A graphical depiction of the proposed Bayesian factor model.
• Figure 2: Locations of the 50 US airports (blue dots) over 8 time windows. A pair of airports is connected by a solid red line if their cosine measure exceeds 5/6 and by a dashed red line if it exceeds 1/2 but less than 5/6.
• Figure 3: The first spatial factor loadings over locations and time.
• Figure 4: (A) Sums of the predictive log-likelihoods (i.e., that evaluated by the true log-daily returns) of various models. (B): log-likelihoods of the true log-daily returns of the RHeFM minus those of the EWMA covariance model with $\alpha = 0.979$ over dates. (D): log-likelihoods of the true log-daily returns of the RHeFM minus those of the GHeFM over dates. (C) and (E): Histograms of the respective log-likelihood differences in the middle panels. Among 128 test business days, the RHeFM beats the GHeFM and EWMA with $\alpha=0.979$ for 103 days each.
• Figure E.1: Some simulated datasets with $Q=1$ under the time-varying volatility. Black dots and gray regions are the simulated datasets and true 95% confidence bands, respectively. Red curves are the estimated 95% confidence bands by the simplified model in Section \ref{['sec4_5']} and the criterion to determine the bandwidth in Appendix \ref{['secD']}.