tailieunhanh - Lecture Advanced Econometrics (Part II) - Chapter 1: Review of least squares & likelihood methods

Lecture "Advanced Econometrics (Part II) - Chapter 1: Review of least squares & likelihood methods" presentation of content: Least quares methods, maximum likelihood estimation. | Advanced Econometrics - Part II Chapter 1 Review Of Least Squares Likelihood Methods Chapter 1 REVIEW OF LEAST SQUARES LIKELIHOOD METHODS I. LEAST QUARES METHODS 1. Model - We have N observations individuals forms . drawn randomly from a large population i 1 2 . N - On observation i Yi and K-dimensional column vector of explanatory variables Xi Xn Xi 2 . Xik and assume Xie 1for all i 1 2 . N. - We are interested in explaining the distribution of Y in terms of the explanatory variables Xi using linear model Y ft Xi e P P1 . Pk In matrix notation Y Xp e Y P P2 X12 P3 Xi3 . PkXik e -I- Assumption 1 Xi Yi 1 are independent and identically distributed -I- Assumption 2 ei I Xi N 0 Ơ2 ị Assumption 3 ei 1 Xi eiXy. 0 i 1 -I- Assumption 4 E ei I Xi 0 -I- Assumption 5 E et Xi 0 Nam T. Hoang UNE Business School 1 University of New England Advanced Econometrics - Part II Chapter 1 Review Of Least Squares Likelihood Methods The Ordinary Least Squares OLS estimator for p solves n min Y - PX 2 1-- I This leads to z 1 z - i 1 1 P l X X l X Y X X - XY i 1 J i 1 J The exact distribution of the OLS estimation under the normality assumption is P N P ở X1X 1 Without the normality of the it is difficult to derive the exact distribution of P. However we can establish asymptotic distribution đ PN P P N 0 XX 1 We do not know Ở2 we can consistently estimate it as in Ở2 Y P X 2 n k 11 In practice whether we have exact normality for the error terms or not we will use the following distribution for p ù _ P N P V where V C2. E XX 1 estimate V by V X 1 i 1 If we are interested in a specific coefficient Pk N Pk V V is the i j element of the matrix V Confidence intervals for Pk would be 95 Pk Pk kk Test a hypothesis whether Pk a Nam T. Hoang UNE Business School 2 University of New England Advanced Econometrics - Part II Chapter 1 Review Of Least Squares Likelihood Methods t Pk a N 0 1 2. Robust Variances If we don t have the homoscedasticity assumption then 4n P - P N 0 E XX -1