Time-varying forecasts by variational approximation of sequential Bayesian inference

Abstract

Methods developed for making time-varying forecasts in economic and financial analysis include (a) equal-weighted moving-window (or rolling) regression, (b) time-weighted (e.g. exponentially weighted) regression, (c) the Kalman filter (KF) and (d) adaptive Kalman filters. This paper developed a new method based on variational approximation of sequential Bayesian inference (VASB). Concepts and notions of the sequential Bayesian analysis and the variational approximation of an intractable posterior are simple and straightforward. Our VASB algorithm is not complicated and is easy to code. For a regression on multiple time-series, the regression coefficients, standard errors, prediction and residual error are time-varying and are estimated jointly at every time step. For a single time-series (e.g. price returns of an asset), its mean and variance are time-varying and are predicted jointly at every time step. The VASB algorithm performs better than the rolling and time-weighted statistics or regressions and the KF in terms of higher predictive power and stronger robustness. Derivations of the VASB algorithm are presented in the appendices.

Keywords:

• Time-varying forecasts
• Time-varying regression
• Time-varying variance
• State space modelling of time-series
• Sequential Bayesian inference
• Variational Bayes
• Kalman filter
• Bayesian filtering

JEL Classification:

• C11
• C22
• C32
• C53

Acknowledgement

The authors truly appreciate the reviewers for their critiques and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.