Relate Y-on-X and X-on-Y coefficients
Company: Citadel
Role: Data Scientist
Category: Statistics & Math
Difficulty: medium
Interview Round: Technical Screen
In simple linear regression with intercepts, suppose you regress Y on X and obtain slope β_Y|X and coefficient of determination R^2 (for that regression). Without refitting, what is the slope β_X|Y from regressing X on Y? Express it in terms of β_Y|X and R^2, state any assumptions, and prove the relationship. Then extend your answer to standardized variables (mean 0, variance 1) and discuss the sign when β_Y|X < 0.
Quick Answer: This question evaluates understanding of simple linear regression theory, specifically the algebraic relationship between the slope of Y on X and the slope of X on Y, interpretation of R^2, sign implications, and behavior under standardization, testing statistical reasoning competency in the Statistics & Math domain.