Question

Let β^​0​ and β^​1​ be the estimated intercept and slope from the OLS regression of the linear regression model: Yi​=β₀+β₁​Xi​+ui​,i=1,…,n. Let β~​₀​ and β~​₁ be the estimated intercept and slope from the OLS regression of Y~i​ on X~i​, where Y~i​=10⋅Yi​ and X~i​=10⋅Xi​,i=1,…,n. Show the relation between the formula of β~​₁​ and that of β^​₁​, and the relation between the formula of β~​0​ and that of β^​₁

Answer 1

In linear regression, when both the dependent and independent variables are scaled by the same factor, the estimated slope remains unchanged, but the intercept will differ due to the scaling's impact on the value of the dependent variable when the independent variable is zero.

Step-by-step explanation:

The question is related to linear regression, specifically regarding the estimation and interpretation of the intercept and slope in a regression model.

According to the given linear regression model Yi​=β₀+β₁​Xi​+ui​, the estimated slope (βˆ​¹) represents the average change in the dependent variable Y for a one-unit change in the independent variable X, and the estimated intercept (βˆ​°) is the value of Y when X equals zero.

When we multiply both Y and X by 10, resulting in Y~i​=10⋅Yi​ and X~i​=10⋅Xi​, the new slope (β∼​¹) remains the same as the original slope (βˆ​¹) because scaling both variables by the same factor does not change their relationship. However, the new intercept (β∼​°) will be different because the scaling affects the value of Y when X is zero.