Index is a linear combination of stock prices with known weights. In case index is equally weighted, the weights are fixed. Beta measures stock sensitivity to index - by how much stock moves when index moves by 1%. So we regress one component of that linear combination onto the linear combination itself. Shouldn`t the corresponding regression coefficient be equal to the stock weight in the index ?
My own explanation why this might not be the case:
- When regressing single stock on index we do not take into account for confounding effects - correlation of this single stock with other stocks. Hence, if we orthogonalize a single stock returns to all other stock returns and then run regression of residuals on index returns than beta should be equal to index weight ?
- Index is weighted based on prices while beta is calculated on returns. So if we create a returns based index with equal weights and perform orthogonalization as in 1) then beta should be equal to index weight then ?