*Oxford Bulletin of Economics and Statistics*ha da poco messo on line un mio articolo sulla tendenza dei beta (e degli alfa) di portafoglio a cambiare nel corso del tempo. La versione working paper dello scritto può essere scaricata da qui. Di seguito un estratto dall'introduzione.

According to the Capital Asset Pricing Model (CAPM), differences between expected asset returns reflect differences in their exposure to systematic risk. Sensitivities to market risk are commonly measured by the slope coefficients of OLS regressions of asset returns on the market return. The success of this procedure rests on the key assumption that true parameters are time-invariant. However, there are important reasons, both theoretical and practical, which instead call for a conditional specification in order to account for parameter uncertainty and time variation. In this paper, I construct and estimate a parsimonious one-factor model with time-varying intercepts (alphas), slopes (betas) and idiosyncratic risk, all endogenous with respect to the uncertainty surrounding their expected values. The market-risk sensitivities that I obtain reveal superior predictive ability for portfolio returns relative to constant and rolling-window OLS estimates. I also evaluate their medium-term fluctuations: alphas and betas of portfolios characterised by different book-to-market values (BE/ME) and market capitalization (size) evolve according to different cyclical patterns.

Investors' uncertainty (Knight, 1921) affects asset returns in several ways. Volatility and heteroskedasticity of fundamentals may influence the investors' ability to identify the distribution of asset payoffs. Investors' forecasts of key quantities like market betas are, plausibly, the outcome of some complex learning process that reflects uncertainty. Puzzles and anomalies are indeed pervasive in empirical asset pricing. At least in part, they might be the result of some key parameters being in fact uncertain and subject to learning effects. For example, Bekaert et al. (2009) study how heteroskedasticity of fundamentals and stochastic risk aversion impact on various asset prices and returns. Pástor and Veronesi (2009) argue that market betas of innovative firms are likely to increase during technological revolutions. These and other arguments suggest that unconditional measures of portfolio betas might mismeasure asset risk. Changes in the structure of the economy and in financial markets make reasonable to model risk sensitivities as potentially time-varying quantities, particularly over long samples and at lower frequencies.

The model estimated in this paper features time-varying alphas and betas that are determined endogenously with respect to uncertainty. The latter is measured by the conditional error variance of the optimal forecast of alphas and betas. I assume that investors can infer the risk loadings from available information, and optimally update them as new information becomes available. Accordingly, I design a model with changes in perceived risks due to effects, such as shifts in the quantity of market risk, which are unobserved by the econometrician and might be learning-induced. I then estimate a conditional one-factor relationship using the Kalman filter, and extract monthly alpha and beta time series. There are various advantages of this methodology over existing alternatives. First, its simplicity, as well as its ability to adapt to assets' or portfolios' actual loadings on market risk in a way that constant-coefficient, and widely employed fixed-window OLS regressions, simply do not permit. Second, unlike some recent contributions, the joint estimates of each period's conditional alphas and betas share the returns' frequency, without further assumptions on their period-to-period variation. Estimated parameters fluctuate significantly over time. However, alphas denote much less and betas much more volatility than in previous studies. Finally, I study whether market-risk sensitivities evolve according to some cyclical pattern. I find clear-cut evidence that their relationship with the macroeconomy depends crucially on portfolio characteristics, such as size and book-to-market. I apply the methodology to 1926-2007 monthly returns of U.S. equity portfolios sorted by size and BE/ME, capturing rich temporal and cross-sectional dynamics for alphas, betas and pricing uncertainty. The time-varying, Kalman-filter based (TVK, henceforth) parameters that I obtain depend only on portfolio and market returns, and appear to be precisely estimated. To assess their predictive accuracy against conventional rolling OLS betas, I perform an out-of-sample simulation over a hedging strategy alternatively based on these measures. A large literature attributes the risk premia associated with size and BE/ME effects to the link between those characteristics and fluctuations in aggregate consumption or wealth. I further examine the association of TVK betas with key state variables and macroeconomic indicators, obtaining fresh evidence on the evolution of market-risk sensitivities at business-cycle frequency.

The remainder of the paper and its main findings are as follows. Next Section sets out the literature background. In Section 3 I test for the presence of shifts in time-invariant CAPM coefficients, detecting multiple structural breaks. Section 4 presents a model for risk loadings that accounts for the learning problem of investors under uncertainty. Section 5 reports estimates of time-varying alphas, betas and pricing uncertainty, and tests their relative information content for market returns. TVK betas are more tightly estimated and have superior predictive ability for actual portfolio returns than those obtained via the conventional rolling-window approach. In addition, the evolution of TVK parameters is rich and differentiated across portfolios sorted on the basis of book-to-market ratios and market capitalization. Section 6 evaluates the association of time-varying betas with business-cycle indicators, at monthly and quarterly frequencies. The betas of high BE/ME stocks broadly turn out to move pro-cyclically, whereas those of low BE/ME stocks reveal an opposite, though weaker tendency. Large-cap portfolios have betas strongly correlated with state variables, which also anticipate future output developments. Finally, investment growth, much more than consumption growth, helps forecast the betas of high BE/ME and small firms portfolios. These results lend support to recent production-based asset-pricing models. Section 7 concludes.

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