Can I use coefficients of one set of regressions as dependent variable in a new regression?

Let’s take CAPM model in finance as an example. I have stock prices of $N$ different companies. First, I do regression below and obtain betas of each company $i$.

$$Price=a+beta_i*Index$$

Now I am curious about what company characteristics affect $beta$. Can I then use $beta$s from the first stage as the new dependent variable and regress against other variables? Why I can and why not? Thanks.