I am doing orthogonal regression. My X matrix consists of returns on a broad market index, value index, growth index, a few sectors,.....(my Y is the returns on an equity fund)
I am regressing the Y on the (1st two) principal components of X (this is to avoid the problem of multicollinearity in X). I then back out the betas for the original X variables by the matrix multiplication of the eigenvector matrix and the betas for the principal components. All good so far
I wanted to make sure everything was right so decided as a test to regress the returns of the broad market index on X (and remember that the broad market index returns are actually in X). I would expect the beta for the broad market index to be very near 1 but when I do that orthogonal regression it is not - it is similar in value to all the other betas (around 0.15).
This surely does not make sense right? When I do plain old regression of the returns of the market index on X (which would suffer from multicollinearity given the high correlation amongst the X variables, right?), the beta estimate is exactly 1 (and the betas for the other factors are very small in comparison), but when I use orthogonal regression the beta is 0.15.
Is the small beta for the market index factor not a concern?