Time-Series and Fama–MacBeth Estimates of Risk Premia

# Cross Sectional vs. Time-Series Risk Premia Estimate


# 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.

## Answer by Kevin (score 4, accepted)

https://quant.stackexchange.com/a/71795

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

- 2.2 Time-Series and GRS

- 2.3 Cross-Sectional Regressions

- 2.4 Comments

- 2.5 Fama-MacBeth Regressions

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.