To see if the variable X 2i belongs in the model Y i = 1 + 2 X i + u i ,

To see if the variable X_2i belongs in the model Y_i = β₁ + β₂X_i + u_i , Ramsey’s RESET test would estimate the linear model, obtaining the estimated Yi values from this model [i.e., Ŷ_i = β̂₁ + β̂₂X_i ] and then estimating the model Y_i = α₁ + α₂X_i+ α₃Ŷ_2i+ v_i and testing the significance of α₃. Prove that, if α̂₃ turns out to be statistically significant in the preceding (RESET) equation, it is the same thing as estimating the following model directly: Y_i = β₁ + β₂X_i + β₃X_2i + u_i.