Testing Whether an Asset Pricing Model Needs a Correlated Factor
Summary
The document explains a test for whether a factor such as the size factor in a three-factor equity model adds pricing power beyond other, correlated factors. Simply checking whether the factor’s average return is small can mislead: removing it may change the estimated exposures to the remaining factors. The proposed approach first regresses the candidate factor on the other factors and examines the intercept of the residual component. If that intercept is zero, the document argues, the remaining factors can price the candidate factor, so dropping it need not change the model’s pricing of the test assets.
The algebraic intuition is to replace the candidate factor with an orthogonalized version, separating its shared variation from the part unique to it. The shared component can be absorbed into the other factors’ loadings; the unique component is what may justify retaining the candidate. The document presents this as an explanation of Cochrane’s model comparison argument and raises, without resolving, whether the same logic applies to characteristics rather than traded factors. It supplies no new empirical test or general proof across model settings.
Key ideas
- A factor’s small average return alone does not show that it can be removed from a pricing model.
- Removing a correlated factor can change estimated loadings on the remaining factors.
- Regressing the candidate factor on the others isolates the component unique to that factor.
- The unique component’s intercept is used to assess whether the remaining factors can price it.
- The discussion leaves open how this reasoning extends from factors to firm characteristics.
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# Testing one asset pricing model against another a la Cochrane: why this works
# Testing one asset pricing model against another a la Cochrane: why this works
I am reading section section 14.6 of John Cochrane's lectures notes for the course Business 35150 Advanced Investments. On p. 239-240, he discusses testing one asset pricing model against another. I have quite some trouble following his arguments. Here is the essence:
> Example. FF3F. $$ E(R^{ei}) = \alpha_i + b_i\lambda_{rmrf} + h_i\lambda_{hml} + s_i\lambda_{smb} \tag{i} $$ Do we really need the size factor? Or can we write $$ E(R^{ei}) = \alpha_i + b_i\lambda_{rmrf} + h_i\lambda_{hml} \tag{ii} $$ and do as well? ($\alpha$ will rise, but will they rise “much”?)
> A common misconception: Measure $\lambda_{smb} = E(smb)$. If $\lambda_{smb} = 0$ (and “small”) we can drop it. Why is this wrong? Because if you drop $smb$ from the regression, $b_i$ and $h_i$ also change!
> <...>
> Solution: (a) First run a regression of $smb_t$ on $rmrf_t$ and $hml_t$ and take the residual, $$ smb_t = \alpha_{smb} + b_s rmrf_t + h_s hml_t + \varepsilon_t \tag{iii} $$ Now, we can drop $smb$ from the three factor model if and only $\alpha_{smb}$ is zero. Intuitively, if the other assets are enough to price $smb$, then they are enough to price anything that $smb$ prices. (b) “Drop smb” means the 25 portfolio alphas are the same with or without $smb$ (c) *Equivalently, we are forming an “orthogonalized factor” $$ smb_t^* = \alpha_{smb} + \varepsilon_t = smb_t − b_s rmrf_f − h_s hml_t $$ This is a version of $smb$ purged of its correlation with $rmrf$ and $hml$. Now it is OK to drop $smb$ if $E(smb^{*})$ is zero, because the $b$ and $h$ are not affected if you drop $smb^*$ (d) *Why does this work? Think about rewriting the original model in terms of $smb^{*}$, \begin{align*} R_t^{ei} &= \alpha_i + b_i rmrf_t + h_i hml_t + s_i smb_t + \varepsilon_t^i \\ &= \alpha_i + (b_i+s_i b_s) rmrf_t + (h_i+s_i h_s) hml_t + s_i (smb_t - b_s rmrf_t - h_s hml_t) + \varepsilon_t^i \\ &= \alpha_i + (b_i+s_i b_s) rmrf_t + (h_i+s_i h_s) hml_t + s_i smb_t^* + \varepsilon_t^i \end{align*} The other factors would now get the betas that were assigned to $smb$ merely because $smb$ was correlated with the other factors. This part of the $smb$ premium can be captured by the other factors, we don’t need $smb$ to do it. The only part that we need $smb$ for is the last part. Thus average returns can be explained without $smb$ if and only if $E(smb_t^{*}) = 0$.
Part 4. is unclear to me. I do not get why $\text{(iii)}$ is the relevant regression to run and $\alpha_{smb}$ in it the relevant coefficient to test. (Attempting to show that this approach fails, I provide a counterexample here.) I think I need a formal proof in addition to the intuition. I guess once I see the proof, Cochrane's intuition will become more intuitive to me, too. Could you help me understand this?
Also, does this apply as is if the factor we consider kicking out of the model is actually a characteristic, not a factor (such as the size of the firm rather than the firm's sensitivity to the $smb$ factor)?
Update: Another source that discusses this is chapter 13 of Cochrane "Asset Pricing" (2005), especially sections 13.4 and 13.6.
#### References
- Cochrane, J. (2005). Asset Pricing: Revised Edition. Princeton University Press.
- Cochrane, J. H. (2014). Week 5 Empirical methods notes. Business 35150 Advanced Investments, 225-247.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.