Katkat
Back

Bound R-Squared for Individual and Combined OLS Regressions

VoleonQuantitative Researcher · Technical Screen
MediumQuantitative ResearcherTechnical ScreenStatistics & Math

Last updated

Derive in-sample OLS R-squared bounds and the joint-model range from individual scores, including intercept assumptions, predictor correlation, and out-of-sample exceptions.

What range can R-squared take in ordinary least squares, and why? On the same response and observations, suppose the regressions y ~ x1 and y ~ x2 have R-squared values 0.1 and 0.2. What is the possible range for y ~ x1 + x2?

Constraints & Assumptions

For the main question, use in-sample OLS with an intercept, a nonconstant response, and the usual centered definition R² = 1 - SSE/SST. Compare models fitted on the same rows. Also explain what changes without an intercept or when scoring on different data.

Clarifying Questions

Is an intercept included? Are the data and response identical across fits? Is this ordinary R-squared or adjusted R-squared? Is evaluation in sample?

Follow-up questions

  • Can the joint model explain all variation even when each predictor has a small individual score? Can adding a predictor reduce ordinary training R-squared? Under what conditions can R-squared be negative or undefined?