Multicollinearity and Interactions

In a multiple regression $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \varepsilon$, the coefficient $\beta_1$ is a partial coefficient: the change in $E[y]$ when $x_1$ rises by one unit and $x_2$ stays fixed. It is generally different from the slope you get by regressing $y$ on $x_1$ alone. The two agree when $x_1$ and $x_2$ are uncorrelated, or when the partial coefficient on $x_2$ happens to be zero, as in the example below.

The Frisch-Waugh-Lovell theorem makes "holding fixed" concrete. Regress $x_1$ on $x_2$ and keep the residual $\tilde x_1$, the part of $x_1$ that $x_2$ cannot explain. Then $\hat\beta_1$ equals the slope of $y$ on $\tilde x_1$. The regression learns about $x_1$ only from the variation it does not share with $x_2$.

With standardized variables there is a closed form. Let $r_{1y}$ and $r_{2y}$ be each predictor's correlation with $y$, and $\rho$ the correlation between the predictors:

Worked example: a 12-month and a 6-month momentum signal have $\rho = 0.9$. Their correlations with the target are $r_{1y} = 0.5$ and $r_{2y} = 0.45$. Then $\hat\beta_1 = (0.5 - 0.405)/0.19 = 0.5$ and $\hat\beta_2 = (0.45 - 0.45)/0.19 = 0$. The 6-month signal looks good on its own, yet once you know the 12-month signal it adds nothing: its entire correlation with the target is accounted for by $0.9 \times 0.5 = 0.45$. So the answer to "which one matters?" is the predictor whose unshared part still predicts $y$.