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I have attempted to use Bayesian Model Averaging (BMA) as method to give guidance in the decision to include/omit factors in the linear factor stock return model:

$$r^i = X \beta^{i} + \epsilon^{i} \;\;\mathrm{for} \; i = 1,2,...,m $$

where $r^i$ is a (T x 1) vector, $X$ is an (N x k) matrix, $\beta^{i}$ is a (k x 1) vector, and $\epsilon^{i}$ is $N(0, \sigma_{i}^{2})$

I assumed a g-prior for $\beta^{i}$ and a diffuse prior for $\sigma_{i}^{2}$. After deriving the posterior probability of each model, I get the highest-probability model (i.e. the set of best predictors).

I then move on to testing these selected predictors using the Cross-Sectional Approach, also known as Fama-MacBeth procedure. This test reveals whether the selected predictors have a significant premium or not.

The odd thing is that only one of the BMA selected predictors has a statistically significant premium.

I am confused as to how to reconcile the results of BMA and Fama-MacBeth.

Any help would be greatly appreciated. Thanks!

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The short answer is that there is no particular reason to be surprised at what you are seeing.

The use of p-values to decide which variables to include in a model is rather outdated, and most modern (say, post 1990) model selection tools give very different results from what you would get just from looking at which predictors are "statistically significant". For a relatively nuanced and modern view of model selection, see Burnham and Anderson (2004) .

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