Let 0 , 1 , . . . , k be the OLS estimates from the regression of y i on x

Let β̂₀, β̂₁, . . . , β̂_k be the OLS estimates from the regression of y_i on x_i1, . . . , x_ik, i = 1, 2, . . . , n. For nonzero constants c₁, . . . , c_k, argue that the OLS intercept and slopes from the regression of c₀y_i on c₁x_i1, . . . , c_kx_ik, i = 1, 2, . . . , n, are given by β̃₀ = c₀ β̂₀, β̂₁ = (c₀/c₁) β̂₁, . . . , β̂_k = (c₀ /c_k) β̂_k.

Use the fact that the β̂_j solve the first order conditions in (3.13), and the β̂_j must solve the first order conditions involving the rescaled dependent and independent variables.