Suppose in the model Y i = 1 + 2 X 2i + 3

Suppose in the model

Y_i = β₁ + β₂X_2i + β₃X_3i + u_i

that r₂₃, the coefficient of correlation between X₂ and X₃, is zero. Therefore, someone suggests that you run the following regressions:

Y_i = α₁ + α₂X_2i + u_1i

Y_i = γ₁ + γ₃X_3i + u_2i

a. Will α̂₂ = β̂₂ and γ̂₃ = β̂₃? Why?

b. Will β̂₁ equal α̂₁ or γ̂₁ or some combination thereof?

c. Will var (β̂₂) = var (α̂₂) and var (β̂₃) = var (γ̂₃)?