Question

Let β⁰​ and β¹ 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~​₀and 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 β¹​, and the relation between the formula of β~​0​ and that of β⁰.

Answer 1

In the OLS regression of the linear regression model, the estimated intercept is β⁰​ and the estimated slope is β¹. In the OLS regression of the transformed variables, the estimated intercept is β~₀ and the estimated slope is β~₁. The relation between the formula of β~​₁ and β¹​ is β~₁ = 10β¹​. The relation between the formula of β~​₀​ and β⁰ is β~₀ = 10β⁰​.

Step-by-step explanation:

In the OLS regression of the linear regression model Yi​=β₀​+β¹​Xi​+ui​, the estimated intercept is β⁰​ and the estimated slope is β¹. In the OLS regression of the transformed variables Y~i​=10⋅Yi​ and X~i​=10⋅Xi​, the estimated intercept is β~₀ and the estimated slope is β~₁.

To find the relation between the formula of β~​₁ and β¹​, we can consider the transformation. Since Y~i​=10⋅Yi​ and X~i​=10⋅Xi​, we can substitute β~₀ with 10β⁰​ and β~₁ with 10β¹​, which gives us β~₀ = 10β⁰​ and β~₁ = 10β¹​.

Similarly, to find the relation between the formula of β~​₀​ and β⁰, we substitute Y~i​=10⋅Yi​, X~i​=10⋅Xi​, and β~₀ with β~₀ = 10β⁰​, which gives us β~₀ = 10β⁰​.