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Fama–MacBeth

Time-varying beta

We want to allow for beta to vary over time.

\[\tilde{r}^i_t = \alpha^i + \boldsymbol{\beta}^{i,\boldsymbol{x}}_t \boldsymbol{x}_t + \epsilon_t^i\]

So far, we have been estimating unconditional \(\beta\)

\[\tilde{r}^i_t = \alpha^i + \boldsymbol{\beta}^{i,x} \boldsymbol{x}_t + \epsilon_t^i\]

Must choose a model for how \(\beta\) changes over time.

• Consider stochastic vol models above.

• Often see estimates of \(\beta_t\) using rolling window of data. 5 years?

• Can use GARCH, other models to capture nonlinear impact.

Fama-Macbeth estimates

The Fama-Macbeth procedure is widely used to deal with time-varying betas. \vv

• Imposes little on the cross-sectional returns.

• Does assume no correlation across time in returns.

• Equivalent to certain GMM specifications under these assumptions.

Fama-Macbeth estimation

1. Estimate \(\beta_t\).

For each security, \(i\), estimate the time-series of \(\boldsymbol{\beta}^i_t\). This could be done for each \(t\) using a rolling window or other methods. (If using a constant \(\boldsymbol{\beta}\) just run the usual time-series regression for each security.)

\[\begin{aligned} \tilde{r}^i_t =& \alpha^i + \boldsymbol{\beta}^{i,\boldsymbol{x}}_t~ \boldsymbol{\beta}_t + \epsilon_t^i \end{aligned}\]

1. Estimate \(\lambda,\upsilon\). (Could include an intercept here, though LFM implies no intercept.)

For each \(t\), estimate a cross-sectional regression to obtain \(\lambda_t\) and estimates of the \(N\) pricing errors, \(\upsilon_t^i\).

\[\tilde{r}^i_t = \boldsymbol{\beta}^{i,\boldsymbol{x}}_t \boldsymbol{\lambda}_t + \upsilon^i_t\]

Illustration of time and cross regressions

Use sample means of the estimates:

\[\hat{\lambda} = \frac{1}{T}\sum_{t=1}^T \lambda_t,\; \; \hat{\upsilon}^i = \frac{1}{T}\sum_{t=1}^T \upsilon^i_t\]

• This allowed flexible model for \(\boldsymbol{\beta}^{i,\boldsymbol{x}}_t\).

• Running \(t\) cross-sectional regressions allowed \(t\) (unrelated) estimates \(\lambda_t\) and \(\upsilon_t\).

Fama-MacBeth standard errors

Get standard errors of the estimates by using Law of Large Numbers for the sample means, \(\hat{\lambda}\) and \(\hat{\upsilon}\).

\[\begin{split}\begin{aligned} s.e.(\hat{\lambda}) =& \frac{1}{\sqrt{T}}\sigma_{\lambda} \\ =& \frac{1}{T}\sqrt{\sum_{t=1}^T\left(\lambda_t - \hat{\lambda}\right)^2} \end{aligned}\end{split}\]

• These standard errors correct for cross-sectional correlation.

• If there is no time-series correlation in the OLS errors, then the Fama-Macbeth standard errors will equal the GMM errors.

Beyond Fama-MacBeth

The Fama-MacBeth, two-pass, regression approach is very popular to incorporate dynamic betas. (Note that there would be no point of using Fama-MacBeth if we are using full-sample time-series betas. This will just give us the usual cross-sectional estimates.)

• It is easy to implement.

• It is (relatively!) easy to understand.

• It gives reasonable estimates of the standard errors.

If we want to calculate more precise standard errors, we could easily use the Generalized Method of Moments (GMM).

• GMM would account for any serial correlation.

• GMM would account for the imprecision of the first-stage (time-series) estimates.

References

• Back, Kerry. Asset Pricing and Portfolio Choice Theory. 2010. Chapter 6.

• Bodie, Kane, and Marcus. Investments. 2011. Chapters 9 and 10.

• Cochrane. Discount Rates. Journal of Finance. August 2011.

• Frazzini, Adrea and Lasse Pedersen. Betting Against Beta. Working Paper. October 2011.