Assume I have
n signals, which I would like to linearly weight and combine to form an aggregate signal. Two possible ways of doing this based on historical data are:
- Regress the
nsignals on historical returns. Use the betas estimated from the regression as the weights on the signals.
- Estimate a covariance matrix of the signals based on historical signal values. Estimate future returns (mean) of the signals based off of the historical returns of the signals. Perform a mean-variance optimization and obtain optimal weights of signals
I understand the technical differences between these two approaches (different objective functions) but I'm trying to grasp (in practice) what is a better approach, as empirically they come up with similar answers.
In both cases, you are using historical signal performance to estimate future signal performance (e.g. momentum of the signal returns). In both cases, you can constrain the optimization to achieve different objectives (restrict weights to be positive, sum to 1 etc.). The linear regression objective - squared loss - is somewhat similar to the mean variance objective in that you are maximizing return. Not quite sure if the minimizing variance part is as obvious
It seems as though the mean variance approach is a little more flexible, you can detangle the expected returns from the covariance estimates (or even having no estimates of expected returns - e.g. minimum variance). Wondering if anyone has any insight