Busting R² in Combined Regression

This quant interview question is about understanding how goodness-of-fit behaves when you move from simple to multiple linear regression. It focuses on how the explanatory power of individual predictors combines in a joint model and how this is reflected in the coefficient of determination. For quant prep, it reinforces the link between correlation structure and model fit, a staple of statistical modeling in quantitative finance interviews.

It trains your intuition for variance decomposition, conditional relationships between variables, and the geometry of projections in regression. You are pushed to think about overlapping information, dependence between predictors, and how shared signal versus noise influences model performance. This is core quant prep material, touching correlation, linear algebra, and basic statistical learning.

It matters for quant interviews because real-world alpha modeling, risk modeling, and factor construction all rely on combining signals. Interviewers want to see that you understand when adding a predictor truly helps, when it is redundant, and how to reason rigorously about limits on model improvement. Being able to reason about bounds on R² shows mature understanding of regression beyond plug-and-play formulas, which is exactly what high-level quant interviews aim to probe.

What it tests

In linear regression, the $R^2$ value measures the proportion of variance in the response variable explained by the predictors. When adding predictors to a model, the $R^2$ cannot decrease because the model can always ignore redundant predictors, so the fit is at least as good as with any subset. However, if two predictors are perfectly collinear, the combined model cannot explain more variance than either predictor alone, since they provide no independent information. The key is that the marginal gain in $R^2$ from adding a predictor depends on how much new, non-overlapping information it brings about the response. This is why the minimum $R^2$ for the combined model equals the maximum $R^2$ from the individual models when predictors are perfectly correlated.