Question

Due: Jan 22 ECN 702 Econometrics II 1. Given data on (x) for i,n, consider the following least square problem for a simple linear regression. bobi We assume the four linear regression model assumptions dicussed in class hold. 6) Compute the partial derivatives of the objective function. (ii) Put the derived partial derivatives in (i) equal to zeros. Explain why the resulting equa- tions are called ‘normal equation. (Hint: two n-dimesional vectors (voi-i and (wi)-1 are normal(-orthogonal) if Σ(-1 uiv.-0.) (iii) The solution to the above equation system is OLS estimator (Bo. B1). Derive them. (iv) Suppose the estimated equation is 1.4+ 1.7x Test the hypothesis H0 : ß-0 against Hi : A yb 0 at 95% significance level.

Answer 1

Consider the given problem have the error sum of square is given by e = Sigma^n_i = 1 (v_i - b_0 - b_1 x_i)^2 rightarrow partial differential e/partial differential b_0 = Sigma (y_i - b_0 - b_1 x_i) 2 (- 1) Ret partial differential e /partial differential b_0 = Sigma (- 2)(v_i -b_0 -b_1 x_i) rightarrow (1) similarly partial differential e/ partial differential b_1 = Sigma (v_i - b_0 - b_1 x_i) (2) (- x_I) = Sigma (- 2) (v_i - b_0 -b_1 x_i) rightarrow (2) 2) from (1) we have �� partial differential e/partial differential b_0 = 0 �� rightarrow Sigma (- 2) (v_i - b_0 - b_1 x_i) = 0 rightarrow sigma (v_i - b_0 - b_1 x_i) = 0 rightarrow Sigma y_i - Sigma b_0 - b_1 Sigma x_i = 0 rightarrow (1) form (2) we have �� partial differential v /partial differential u 1 = 0 �� rightarrow Sigma (-2) (v_i - b_0 - b_1 x_i) x_i = 0 rightarrow Sigma (y_i x_i - b_0 x_i - b_i x_i)^2 = 0 rightarrow Sigma y_i x_i = b_0 Sigma x_i - b_1 Sigma x^2_1 = 0 rightarrow (2)