# Does including an additional pricing factor necessarily reduce the pricing errors?

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.

1. 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”?)

Does including an additional pricing factor necessarily reduce the pricing errors $$\alpha$$? Is that an algebraic fact? Does it hold on the level of the population (or the data generating process), sample or both?

### References

Let me share some thoughts. They are not formulated in complete detail nor are they rigorously proven, but I hope the intuition is correct.

### At the level of the population / data generating process

Suppose the expected returns $$E(R^{ei})$$ and factor sensitivities $$\beta_{1,i}$$, $$\beta_{2,i}$$ are known, and our goal is to come up with a good asset pricing model. For starters, consider a one-factor model that explicitly acknowledges its imperfection via incorporating nonzero pricing errors $$\alpha_i^{(iii)}$$. Let the factor be denoted $$X_1$$ and the corresponding risk premium $$\lambda_1$$. The model is $$E(R^{ei}) = \alpha_i^{(iii)} + \beta_{1,i} \lambda_1. \tag{iii}$$ Without knowing the true values $$(\alpha_i^{(iii)},\lambda_1)$$, we can choose a pair $$(\hat\alpha_i^{(iii)},\hat\lambda_1)$$ that minimizes some norm of $$\hat\alpha_i^{(iii)}$$: $$||\hat\alpha_i^{(iii)}||$$.

Now consider including an additional factor $$X_2$$ with a risk premium $$\lambda_2$$ into the model. The extended model becomes $$E(R^{ei}) = \alpha_i^{(iv)} + \beta_{1,i} \lambda_1 + \beta_{2,i} \lambda_2. \tag{iv}$$ Without knowing the true values $$(\alpha_i^{(iv)},\lambda_1,\lambda_2)$$, it is always possible to choose a set of values $$(\tilde\alpha_i^{(iv)},\tilde\lambda_1,\tilde\lambda_2)$$ so that $$||\tilde\alpha_i^{(iv)}||\leq||\hat\alpha_i^{(iii)}||$$. (In the worst case, use $$\tilde\alpha_i^{(iv)}=\hat\alpha_i^{(iii)},\tilde\lambda_1=\hat\lambda_1,\tilde\lambda_2=0$$ to achieve equality.)

Now suppose the risk premia $$\lambda_1$$ and $$\lambda_2$$ are known. The the only thing that is uknown are the pricing errors, but we can obtain them immediately from $$(\text{iii})$$ and $$(\text{iv})$$. The pricing errors being determined that way, there is no algebraic guarantee that $$||\tilde\alpha_i^{(iv)}||\leq||\hat\alpha_i^{(iii)}||$$ anymore.

### At the level of the sample

The values that we assumed to be known in population will usually have to be replaced by estimates (unless we have some of the values implied by some theory). E.g. if we assume some parameters stay constant over time, we could estimate them from time-series observations of the variables. Or if we build a model that determines how these values evolve over time, we could still estimate them from a time-series sample. If we do not impose any restrictions on $$\lambda_1$$ and $$\lambda_2$$, the sample counterpart of $$||\tilde\alpha_i^{(iv)}||\leq||\hat\alpha_i^{(iii)}||$$ should still hold. If we do impose restrictions on $$\lambda_1$$ and $$\lambda_2$$*, it is possible that the sample counterpart of $$||\tilde\alpha_i^{(iv)}||\leq||\hat\alpha_i^{(iii)}||$$ fails to hold.

*E.g. if $$X_j$$ is an excess return and we assume parameter constancy over time, we can estimate $$\lambda_j$$ by the sample mean of $$X_j$$.