A stock beta model describes the correlation between a stock and the general market and whether it amplifies () or
dampens () market volatility. A dynamic factor model for stock beta describes how the betas evolve with time. A stock beta varies over the long run due to strategy, industry prospects, the composition of the market overall. The pandemic shock of 2020 highlights how a significant event can significantly change market betas: discretionary sector beta increased sharply, staples sector beta decreased sharply.
The observation equation describes how a sector's returns are a linear function of the market returns include a return differential the stock risk multiple . The stock multiple is time varying, evolving as a random walk according to the observation equation. This construct is a dynamic linear regression model where the fitted regression coefficients is time varying, a Kalman Filter and Smoother is used to fit the unobserved beta, where the innovations are drawn from the covariance matrix W.
References:
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Campbell, John Y, 1996. Understanding Risk and Return, Journal of Political Economy, University of Chicago Press, vol. 104(2), pages 298-345, April.
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Adrian, Tobias & Franzoni, Francesco, 2009.
Learning about beta: Time-varying factor loadings, expected returns, and the conditional CAPM, Journal of Empirical Finance, Elsevier, vol. 16(4), pages 537-556, September.