What quantities (means, betas) must be constant over time for the GRS test to be valid?

Question

I am interested in testing the CAPM using the GRS test. Consider $N$ assets observed for $T$ time periods. Using the notation of Cochrane "Asset Pricing" (2005), the GRS test amounts to running $N$ time series regressions of the form $$ R^{ei}_t=\alpha_i+\beta_i f_t+\varepsilon^i_t \tag{12.1} $$ and testing the joint hypothesis $H_0\colon \alpha_1=\dots=\alpha_N=0$. The $\alpha$s are treated as pricing errors, so they better be zero if the CAPM is an adequate model.

What quantities must be constant over time for the GRS test to be valid?
I suppose $\beta_i$s should be constant, but what about $E(R_t^{ei})$ and $E(f_t)$; can these be time-varying?

Answer 1

Here is my attempt. I do not find it thorough enough to convince myself, but at least it is a start.

I think the OLS estimator of $\beta_i$ used in the GRS test implies that all three of $\beta_i$, $E(R_t^{ei})$ and $E(f_t)$ are (implicitly) assumed to be constant over time. After all, $$ \hat\beta_i^{OLS}=\frac{ \frac{1}{T-1}\sum_{t=1}^T (R^{ei}_t-\bar{R}^{ei})(f_t-\bar{f}) }{ \frac{1}{T-1}\sum_{t=1}^T (f_t-\bar{f})^2 } $$ where we use $\bar{R}^{ei}$ as the empirical counterpart to $E(R_t^{ei})$ for all $t$ and $\bar{f}$ as the empirical counterpart to $E(f_t)$ for all $t$. There is no accounting (empirically) for the possibility that $E(R_t^{ei})$ or $E(f_t)$ are time varying.