Suppose that we have two datasets X and Y with Var(X) = 10 and Var(Y) = 20. We perform the linear regression
\( y \sim \alpha_x + \beta_x x \)
and obtain beta_x = 1
\(beta_x = 1 \)
Suppose now that we perform the regression
\( x \sim \alpha_y + \beta_y y \)
Find beta_y.
Try to solve this problem yourself before moving on to the solution below
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Let r be the Pearson Correlation Coefficient of X and Y. Then
\( \beta_x = \frac{\sigma_y}{\sigma_x} \)
\(\beta_y = r \frac{\sigma_x}{\sigma_y} \)
Therefore
\( \beta_y = \frac{\sigma_x^2}{\beta_x \sigma_y^2} \Longrightarrow \beta_y = \beta_x \frac{\sigma_x^2}{\sigma_y^2} \)
Putting in the values we get
\(\beta_y = 1 \cdot \frac{10}{20} = \frac{1}{2}\)
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