If beta is additive i.e. ${\beta}_P =\sum w_i \beta_i$, shouldn't the two methods below yield the same number?
Method 1: Estimate beta for each asset in the portfolio. Then ${\beta}_P =\sum w_i \beta_i$
Method 2: Estimate portfolio returns $r_P =\sum w_i r_i$. Then estimate beta.
The two results though close are not identical. Why is that? Is there an implicit assumption wrt the error terms in the regressions (i.e. uncorrelated, zero mean etc.)?