The same book-to-price experiment produces two different numbers, and they answer different
questions. A time-series regression asks whether a portfolio has an intercept after its factor
exposures are controlled. A cross-sectional regression asks whether the average return lines up with
those exposures, period by period. The two-pass narrative is already on the shelf. This note is the researcher's
split: which coefficient is an alpha, which is a premium, and how an error in the first-pass beta
leaks into the second pass.
Learning objectives. By the end of this note you should be able to:
- assign beta estimation to the time series and the price of risk to the cross section;
- state the Fama–MacBeth hypotheses on residual risk, nonlinearity, and the price of risk, including the fair-game check on lagged coefficients;
- explain why an errors-in-variables bias can manufacture a premium on residual variance, and how portfolio grouping and Shanken's adjustment bound that bias;
- read a case in which beta's cross-sectional coefficient dies and then returns once the price of risk is allowed to move with a yield spread.
In this lesson
- Two regressions, two questions
- How an estimated beta contaminates the premium
- The month-by-month premium
- When the premium is allowed to move
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
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