# Calibrating an Ornstein Uhlenbeck process on residuals of regression

I am trying a basic statistical arbitrage strategy as follows:

1. Perform PCA on a log return series of a basket of stocks
2. Regress returns against top principal components identified
3. Calculate the residuals of regression for each stock
4. Fit a OU process on the residuals

To fit an OU process, calculated the sum of residuals for each stock and regressed them on the lagged sum of residuals. However sometimes the intercept and slope are negative.

How do I calibrate this to an OU process when intercept or slope is negative?

You work in discrete time so you should not fit an OU-process but simply an AR(1) process which is its analogon in discrete time. Look here to see why this is true.

Calibrating the AR(1) boils down to do a regression on your residuals.

• Probably suuuuper late but... given that AR(1) is the discrete time analogue, why isn't there a way to use the AR(1) fit results to get reasonable OU parameters? – Lagerbaer Jan 6 '17 at 17:49
• Look at the link in the answer. I think it is reasonable to do this.. – Ric Jan 6 '17 at 19:31
• I don't know. Just slapping the $\Delta t$-factor onto the other factors doesn't seem to do the trick. – Lagerbaer Jan 6 '17 at 19:56
• Try to follow the formulas in the link. Keep in mind on which time scale you observe the discretized process (Delta t) and then the relationship just holds. – Ric Jan 6 '17 at 20:15