Relate Y-on-X and X-on-Y coefficients

Q: Relate Y-on-X and X-on-Y coefficients

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.

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Relating Slopes When Reversing Simple Linear Regression

Context

You fit an ordinary least squares (OLS) simple linear regression with an intercept of Y on X, obtaining the slope β_Y|X and the coefficient of determination R^2 (for that regression). Without refitting a new model, determine the slope β_X|Y from regressing X on Y.

Task

Express β_X|Y in terms of β_Y|X and R^2.
State any assumptions needed.
Prove the relationship.
Extend the result to standardized variables (mean 0, variance 1).
Discuss the sign of β_X|Y when β_Y|X < 0.

Relate Y-on-X and X-on-Y coefficients

Relating Slopes When Reversing Simple Linear Regression

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Relate Y-on-X and X-on-Y coefficients

Overview

Relating Slopes When Reversing Simple Linear Regression

Context

Task

Solution

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