Partial Least Squares Regression

Ordinary regression breaks down when you have dozens of highly correlated predictors and only a few hundred observations — think fifty overlapping macro and valuation signals trying to predict next month's return. The predictors are so tangled that ordinary least squares produces wild, unstable coefficients. One fix is principal component regression: compress the predictors into a handful of new variables first, then regress the outcome on those. Partial least squares (PLS) does something smarter — it builds those compressed variables while looking at the outcome, so the very first combined variable it constructs is chosen specifically because it correlates strongly with what you're trying to predict, not just because it captures a lot of variation among the predictors themselves.

Concretely, PLS finds a direction in predictor space that maximizes covariance with the target, regresses the target on that direction, removes what's explained, and repeats on the leftover variation to find a second direction, and so on — each new component squeezed for outcome-relevant signal rather than generic predictor variance.

In a return-prediction setting with fifty correlated signals and 200 monthly observations, PLS with just two or three components often out-predicts ordinary least squares on out-of-sample data, because it never lets the noisiest, least-relevant directions of predictor variation into the model in the first place. Choosing the right number of components is itself usually done by cross-validation: add components one at a time and stop once the held-out prediction error stops improving, since every extra component beyond that point is fitting noise in the training sample rather than adding real predictive signal.