# Cross Sectional vs. Time-Series Risk Premia Estimate

Consider the single factor model in time series form, e.g:

$$r_t^i = \alpha_i + \beta_i f_t +\epsilon^i_t \tag{1}$$ Here $$i$$ is not an exponent but a superscript, e.g. it represents the return on security $$i$$.

Assuming we have a single factor model, which then states that: $$\mathbb{E}[r^i] =\alpha_i + \lambda\beta_i \tag{2}$$ e.g. the average return for security $$i$$ is given by a constant times its exposure to the common factor $$f$$ (whatever it may be). As such, there seem to be two ways of estimating $$\lambda$$. The first is the rather simple/possibly too simple way - take equation $$(1)$$ and take empirical time-series averages so that: $$\frac{1}{T}\sum_{t=1}^T r^i_t = \alpha_i + \beta_i \frac{1}{T}\sum_{t=1}^T f_t + \underbrace{\frac{1}{T}\sum_{t=1}^T \epsilon^i_{t}}_{\text{Assumed to be zero}}$$ so that the estimate of the risk premium is:$$\hat{\lambda} = \frac{1}{T}\sum_{t=1}^T f_t$$ For example, in the case of CAPM - this would just be the average return of the benchmark/index.

The other method is the two stage, so-called Fama-Macbeth regression, which would estimate the $$\beta_i$$ for each stock through $$N$$ different time-series OLS regressions, and then for each estimated coefficient, run $$T$$ cross-sectional regressions to estimate $$\lambda_t$$ at each time $$t$$. It would the consider: $$\hat{\lambda} = \frac{1}{T} \sum_{t=1}^T\lambda_t$$

My Question: What exactly is the difference between the results obtained in each of these procedures? I understand methodologically they are quite different, but I am wondering how their estimates of risk premia would differ in practice and why.

No one is better at explaining asset pricing than John Cochrane. He explains it in detail in his brilliant textbook. The following videos from his video course are pure gold

The notes for Week 5b Notes on empirical methods of his course Business 35150 Advanced Investments are also fantastic.

A few notes though

• Time series approaches only apply to traded assets. You can't use time series regression to find the price of consumption risk (because consumption growth isn't a traded return).

• Fama-MacBeth is a variant of the cross-sectional approach (think of constant betas)

• Using the time series approach, you run a regression to perfectly price the the risk-free asset and the factor. The risk premium is the mean of factor. The TS intercept is the cross-sectional error (= alpha).

• Using the cross-sectional (or FM) approach, you try to minimise pricing errors across all assets (and accept that risk-free rate and factor aren't perfectly priced).

• Importantly, if your asset pricing model is well-specified, both approahes (TS and CS) give you broadly similar results.

• Thank you Kevin - your explanation is quite clear. I have been reading through Prof. Cochrane's text although I have only made it through the portions on the theory of the SDF so far. Off topic but I also got to listen to him speak while a student at UofC and thoroughly enjoyed it Aug 1, 2022 at 18:19
• @rubikscube09 That's pretty cool. I also attended one of his talks and it was absolutely brilliant. Note that there are many videos, interviews and postcasts available with him online. It's always a great pleasure to listen to him Aug 2, 2022 at 13:35