Best R² from Two Regressors

This question is about how much explanatory power you can squeeze out of a multiple linear regression when each feature looks weak in isolation. It focuses on how individual predictors relate both to the target and to each other, and how those relationships constrain the maximum goodness of fit. In quant prep terms, it is about understanding what information content really means in linear models.

It trains geometric intuition for least squares, mastery of variance decompositions, and a firm grasp of how correlations between predictors and the response interact. A candidate must reason precisely about R², projections, and dependence structures without resorting to brute-force computation. It is also a test of comfort with counterintuitive cases in regression.

This matters in quant interviews because portfolio construction, risk modeling, and signal combination all rely on combining individually weak predictors. Strong quants must judge when extra factors can meaningfully increase explanatory power, how collinearity limits incremental value, and how to think rigorously about model capacity. It is a direct test of the statistical modeling instincts needed for real quant research and systematic trading.

What it tests

The general structure at play here is that the explanatory power of a regression model (as measured by $R^2$) depends not only on the individual predictive strength of each regressor but also on how their combination spans the space of possible relationships with the response variable. When regressors are functionally related (for example, $x_2$ is a nonlinear transformation of $x_1$), a single regressor may capture only a limited aspect of the relationship, but their joint inclusion can model more complex patterns, such as polynomial relationships. The key is that the combined model's $R^2$ is not limited by the sum or maximum of the individual $R^2$ values; instead, it is determined by the total variance in $Y$ that the span of all regressors can explain. This is why, in cases where the regressors together can perfectly reconstruct $Y$, the combined $R^2$ can reach 1, even if each regressor alone explains very little.