I have a predictor variable (x) and dependent variable (y). Both are monthly rolling annualized returns, which naturally induces significant autocorrelation in x and y. They both also fail to be stationary under ADF tests (is this a natural consequence of their construction of being rolling returns?) In terms of regression, is it more appropriate to 1) difference both x and y first or 2)run the normal regression (y~x) and simply adjust standard errors by using heteroskedasticity and autocorrelation (HAC) consistent covariance matrix estimation methods(such as NeweyWest)? Under what circumstances is each of the methods more appropriate? Please leave any references should you have any.
It depends how large the overlapping interval is. Conceptually an infinite rolling window is equivalent to the level, and no one would suggest to 'regress on levels and apply Newey West'.
I think NW is 'robust' in the presence of relatively mild autocorrelation, not a panacea that will give the correct standard errors.
If you use, say dailly returns aggregated to montly rolling ones, then there is 90% overlap in your consecutive observations (and expected autocorrelation).
My ordered preferences would be: 4. Newey West or similar 3. Bootstrap standard errors 2. Cast as state space and apply Kalman Filter 1. Drop overlapping observations and use only the non-overlapping ones
What is your goal? I assume it is to find if/how y is caused by x. You really ought to make sure that y and x are stationary. There are only a few cases in which linear regression makes sense when they are not stationary (e.g. x is a deterministic trend term) and even in those cases, the statistical properties of the coefficient estimators are different than usual.
If the regressor is not stationary, the results can just be wrong, as in the case of spurious regression. The Newey-West method is not used to solve problems of stationarity but rather of serial correlation which is not the main issue in your case. It just changes the estimated errors, but if x is not stationary, the coefficient estimates may well be biased.
My hunch is that it'd be enough to inspect if changes in x are related to changes in y. Do this by differencing the two series. However, if you suspect a long term common trend, do a test of cointegration too (if x is just one variable and not a set of many variables, look for the Engle-Granger test, it's very simple).