Time-Series and Cross-Sectional Tests of Factor Pricing Models

Summary

The document lays out two ways to examine a factor model such as the CAPM. In a time-series approach, each asset’s returns are regressed on a factor, and the model implies that each asset-specific intercept should be zero. In a cross-sectional approach, estimated factor loadings are used to explain assets’ expected returns, with an intercept or residual representing a pricing error. The author asks whether these two quantities are theoretically the same and how they relate to Jensen’s alpha.

The document is framed as a question and supplies no answer, empirical results, or testing procedure. It usefully identifies a distinction: a time-series regression intercept and a cross-sectional pricing residual arise from different regressions, even though both can be discussed as alphas. It also raises the implication of a correctly specified model, under which pricing errors are expected to vanish. Further assumptions and estimation details would be needed to establish precise relationships in a particular test.

Key ideas

• A time-series factor regression estimates an intercept separately for each asset.
• A cross-sectional regression relates expected returns to estimated factor loadings.
• The cross-sectional residual is interpreted as a pricing error.
• The document raises, but does not resolve, how these two alpha concepts relate to Jensen’s alpha.
• A correctly specified factor model implies zero pricing errors in theory.

Tags

# Alpha - Time Series vs Cross Section Approach


# Alpha - Time Series vs Cross Section Approach












I am currently reading Cochranes book on asset pricing. However, I get confused about one thing. He says that one could test a factor model (I will use the CAPM, just as he does), via a time series approach (a) and a cross section approach (b) (and the Fama McBeth procedure which I do not want to discuss here, though).

(a) Let $t$ denote the period, then estimate $R_t = \alpha + \beta f_t + \epsilon_t$ for every asset $i$. The respective factor model, in this case the CAPM, implies that each $\alpha$ should be zero. One can test for this by use of certain tests.

(b) Take the $\beta$ of each asset $i$ from (a). Then estimate $E_T(R_i)= \beta_i \lambda + \alpha_i$. He says that in this case $\alpha_i$ denotes the cross sectional regression residuals and the pricing errors.

My questions are: Does the $\alpha$ in (a) equal the $\alpha$ in (b), at least from a theoretical perspective, i.e. intercept in (a) is equal to the residual in (b)? How do the $\alpha$ values relate to Jensens Alpha? From my understanding, Jensens Alpha is the difference between expected return and the predicted return. This clearly corresponds to the residual in (b), i.e. the $\alpha$ in (b). If this connection is in fact true, do factor models predict Jensens Alpha to be zero by construction?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.