Max Min R-Squared in Combined Regression

This linear regression question is about how goodness-of-fit behaves when you move from separate models to a combined model in ordinary least squares. It focuses on how the proportion of variance explained changes when you introduce multiple sets of predictors together, a core idea in statistical modeling used in systematic trading and risk modeling.

It trains your understanding of explained variance, model comparison, and how overlapping or complementary information in predictors affects the overall fit. It reinforces geometric intuition about projections in feature space, and deepens your feel for what an R-squared value really measures in practical quant prep for interviews.

It matters for quant interviews because you must reason precisely about models, not just run them. Strong candidates can discuss bounds, dependencies, and interaction of features, which is critical in high-dimensional alpha research, portfolio construction, and robust backtesting.

What it tests

The structure underlying this problem class is the monotonicity and additivity of explained variance in linear regression models. When you add predictors to an OLS regression, the $R^2$ (proportion of variance explained) cannot decrease, because the model can always assign a zero coefficient to any new variable, thus retaining the fit of the smaller model. The minimum $R^2$ for the combined regression is therefore the maximum of the individual $R^2$ values, since the new variable might add nothing if it is perfectly collinear with the first. The maximum $R^2$ is achieved when the predictors explain non-overlapping (orthogonal) components of $y$, so their contributions to explained variance are additive up to a theoretical maximum of 1. This principle holds because $R^2$ measures the fraction of variance captured by the best linear combination of the predictors, and the geometry of projections in vector spaces ensures that adding more directions (predictors) cannot reduce the length of the projection (explained variance).