Orthogonal explanatory variables. Suppose in the model Y i = 1 + 2 X 2i

Orthogonal explanatory variables. Suppose in the model

Y_i = β₁ + β₂X_2i + β₃X_3i + · · ·+β_k X_ki + u_i

X₂ to X_k are all uncorrelated. Such variables are called orthogonal variables. If this is the case:

a. What will be the structure of the (X'X) matrix?

b. How would you obtain β̂ = (X'X)⁻¹X'y?

c. What will be the nature of the var–cov matrix of β̂?

d. Suppose you have run the regression and afterward you want to introduce another orthogonal variable, say, X_k+1 into the model. Do you have to recomputed all the previous coefficients β̂₁ to β̂_k ? Why or why not?