# If returns are random variables, then how can CAPM be posed as a simple linear regression?

The Capital Asset Pricing model proposes that, $$R_i=R_f+\beta(R_m-R_f)$$

where $$R_i$$ is the return of the i-th asset, $$R_f$$ is the risk-free rate and $$R_m$$ is the Market returns. $$\beta$$ is generally estimated using simple linear regression, but the covariates are random variables which violates the assumptions of regression.

First, the CAPM does not state that. It states that (using your notation) $$\mathbb{E}(R_i)=R_f+\beta(\mathbb{E}(R_m)-R_f).$$ Second, $$\beta:=\frac{ \text{Cov}(R_i,R_m) }{ \text{Var}(R_m) }$$ is well defined when $$R_i$$ and $$R_m$$ are random variables, as long as $$\text{Var}(R_m)$$ exists. Also, having a random covariate is not a violation of regression assumptions in general. There is no single set of regression assumptions. There are different sets for obtaining different properties of different estimators. This is a related thread that goes into more detail.
• Thank you for your answer. This estimator of $\beta$ arises as the least squares solution of $R_{i,t}=\alpha_i+\beta(R_{m,t})+\epsilon_t$. One can indeed forego the assumption of covariates being fixed, but then one would have to pose it as a measurement error model. The second answer in the link you shared mentions that. Commented Apr 3, 2023 at 9:27
• @Whocares, might you be confusing (however slightly) the CAPM with the definition of $\beta$? For estimation of $\beta$, the regression equation in your comment is spot on. It allows estimating $\beta$ and is not problematic from a statistical perspective. For CAPM, it replaces $\mathbb{E}(R_m)$ with $R_{m,t}$, and that indeed introduces measurement error. So use it for estimating $\beta$ directly, but do not use it for implementing the CAPM without taking care of the measurement error. Commented Apr 3, 2023 at 9:36
• @Whocares, we do treat $R_{m,t}$ as a random variable, but that does not introduce measurement error for $\beta$. Why do you think it should contain measurement error? $\beta$ is defined based on $R_i$ and $R_m$, and we measure both of these precisely (no error). Commented Apr 3, 2023 at 9:56
• Since $R_{m,t}$ is a random variable, we can break it down as $R_{m,t}=u+\gamma_t$, where $u$ is a constant and $\gamma_t$ is a random variable with mean 0. So $\epsilon_t$ and $R_{m,t}$ may be correlated. It is in this sense that it carries a measurement error, bias and maybe even consistency of $\hat{\beta}$ is lost. Commented Apr 3, 2023 at 10:12
• @Whocares, no, in the regression equation for $\beta$ the error term is defined to be uncorrelated with $R_{m,t}$. Otherwise the regression coefficient would not be $\beta$, it would be something else. Commented Apr 3, 2023 at 10:20