Predictive Systems: Living with Imperfect Predictors

Lubos Pastor and Robert F. Stambaugh

In this paper, the authors revisit predictive regressions of expected stock returns and replace them with predictive systems.

A predictive regression starts with the conditional expected return relationship of the type:

r(t+1)=mu(t)+u(t+1)

r is the return between time t and t+1

mu is the expected return for the same period

u is the unexpected return and has zero mean expectation at time t

then the expected return is perfectly explained by some variables x(t)

mu(t)=a+b*x(t)

The predictive system assumes that the variables x(t) do not necessarily explain exactly the expected return but are correlated. Furthermore, the authors exploit the prior beliefs that that the unexpected return is negatively correlated with the change in expected return and that predictability of returns is relatively low. They also complement the explanatory variables with their lagged values as well as the lagged returns.

By incorporating useful prior information on the structure of returns, their methodology improves on the traditional predictive regression approach. The priors are economically motivated and the posteriors incorporate potentially complicated priors and functional forms as well as the parameter uncertainties.

The authors use Gibbs sampling to calculate the posterior distributions. They apply their analysis to the period 1952Q1 to 2003Q4 on US stock returns and compare the predictive system with the predictive regression. They find that the priors improve the predictive power of the dividend yield and CAY but weaken the predictive power of the bond yield.

They compare the estimates of expected returns using both methods and find that predictive systems give more stable expected returns but the prior on the correlation between unexpected returns and innovation in expected returns matter greatly.

They proceed with a variance decomposition of expected return. The predictive system is able to outperform the predictive regression because it uses lagged innovations in unexpected returns and predictors. These innovations explain a large part of the return volatility.

Pastor, Lubos and Stambaugh, Robert F., "Predictive Systems: Living With Imperfect Predictors" (December 22, 2006). CRSP Working Paper No. 617 Available at SSRN: http://ssrn.com/abstract=953272

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