Asset pricing model factor need to be excess return?

In John Cochrane's Asset Pricing book and his video lecture, he states that asset pricing factors need to be excess returns, a traded portfolio. Is there a reason for that? I can't find explanation anywhere. GDP is not tradable, can GDP be a asset pricing factor?

AP factors do not need to be excess returns. In case they are, corresponding prices of risk are conveniently equal to average factor values, since "factors price themselves":

$$E[R_i] = \beta_{i} \cdot \lambda_f, \\ E[f] = 1 \cdot \lambda_f, \\ \Leftrightarrow \\ \lambda_f = E[f],$$

where there is just one factor $$f$$, $$\beta_i$$ is the loading of asset $$i$$ thereon, $$\lambda_f$$ is the price of risk.

On the other hand, what is the price of risk for GDP? You'll need an additional set of equations to determine it.

• Isn't $\beta$ the time series coefficient of the stock return against factor. and the $\lambda_f$ is the second stage cross-sectional regression coefficient of expected return against stocks beta. If factor is return itself, how would it help in this two regressions? If the factor is GDP, I could still run time series regression of stock return against GDP to get beta, then run cross-sectional regression of expected stock return against beta to get risk premium right?
– JOHN
Nov 9, 2018 at 14:18
• @JOHN you might be running into a chicken-and-egg problem. An asset pricing model starts with what I wrote, namely, with the identity that expected returns differ as long as betas do. No first-stage or second-stage or even time-series whatever so far. It is your job then to test the model, estimating the betas, the price of risk and the 'second-stage' alphas. This is hard! But if the factor is an excess return, testing is easy and void of gory econometrics. Nov 9, 2018 at 14:56
• Thanks for reply. Can I understand this as it help us to understand the problem, but the steps to proof the essential question 'higher beta leads to higher return' are the same no matter the pricing factor is return or not?
– JOHN
Nov 9, 2018 at 15:08
• @JOHN the point is, the steps must be different. If the factor is an excess return, its price of risk is defined by its average value, full stop. You then check if time-series betas times this average are aligned with $E[R_i]$. In case the factor is GDP or anything else, you need the two-step regressions etc. Of course, you can pretend an excess return is not an excess return and go for two-step regressions, but this is bad science. Nov 9, 2018 at 16:06
• thanks a lot for the explanation. That makes sense.
– JOHN
Nov 9, 2018 at 17:06