Swapping the regression variables

MediumStatistics~5m

Two datasets $X$ and $Y$ have $\operatorname{Var}(X) = 10$ and $\operatorname{Var}(Y) = 20$ . Regressing $Y$ on $X$ (fitting $y = \alpha_x + \beta_x\,x$ ) gives slope $\beta_x = 1$ . If you instead regress $X$ on $Y$ (fitting $x = \alpha_y + \beta_y\,y$ ), what slope $\beta_y$ do you get?

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The OLS slope equals covariance divided by the variance of the predictor. Use $\beta_x$ to back out $\operatorname{Cov}(X,Y)$ .
Then divide that covariance by $\operatorname{Var}(Y)$ - not by $\beta_x$ . (Sanity check: $\beta_x\beta_y = \rho^2$ .)

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Swapping the regression variables

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