tailieunhanh - Lecture Advanced Econometrics (Part II) - Chapter 2: Hypothesis testing

Lecture "Advanced Econometrics (Part II) - Chapter 2: Hypothesis testing" presentation of content: Maximum likelihood estimators, wald test, likelihood ratio test, lagrange multiplier test, application of tests procedures to linear models, hausman specification test, power and size of tests. | Advanced Econometrics - Part II Chapter 2 Hypothesis Testing Chapter 2 HYPOTHESIS TESTING I. MAXIMUM LIKELIHOOD ESTIMATORS n e nf Zi e eMLE argmax e 1 e n L e In 0 In f Z e i 1 Asymptotic normality Solve e jfor 1 0 MLE de d MLE N e -E I -1-1 d2 L J I 0 E dL dẽ dL dẽi dL cẽ2 dL M k d2 L dede 0i e. e vector k xl e e J d2L d2 L . dL 1 e defie2 dexdek d2L d2 L d2 L d2 L de2de1 de22 de2dek dede d2 L d2 L . d2L dek ổ01 dek de2 dek J For the linear model Y nx1 XP . nxk kx1 nx1 Nam T. Hoang UNE Business School 1 University of New England Advanced Econometrics - Part II Chapter 2 Hypothesis Testing Y Xp e s N 0 ơ21 L P Ơ2 - n ln2n-n lnơ2 1r Y - XP Y - XP 2ơ dL --1 -X Y X X P dp Ơ2 0 XYĨ s 1 Y -XP Y -Xp ơơ 2a 2a 0 e p X X -1X Y 1 . 1 _ . . e e J - Y - Xp Y - Xp e e2 -E dede n n Ơ2 X X -1 0 0 en d2 L n We consider maximum likelihood estimator e the hypothesis c 0 q II. WALD TEST Let e be the vector of parameter estimator obtained without restrictions. We test the hypothesis H0 c 0 q 0 is restriction MLE of 0 If the restriction is valid then c 0 -q should be close to zero. We reject the hypothesis of this value significantly different from zero. The Wald statistic is W c 6 - q V ar c ể - q l cl 0 - q Under H0 c 0 q W has chi-squared distribution with degree of freedom equal to the number of restrictions number of equations in c 0 - q 0 W X 2j Nam T. Hoang UNE Business School 2 University of New England Advanced Econometrics - Part II Chapter 2 Hypothesis Testing III. LIKELIHOOD RATIO TEST H0 c Q q J Let Gv be the maximum likelihood estimator of Q obtained without restriction. J Let Q be the MLE of Q with restrictions. R J If L Lr are the likelihood functions evaluated at these two estimate. J The likelihood ratio Ĩ A L . .X L 0 2 1 J If the restriction c G q is valid then LR should be close to Ĩ- . Under H0 c Q q -2ln2 X2j is chi-squared with degree of freedom equal to the number of restrictions imposed. LR -2ln2 X 2 IV. LAGRANGE MULTIPLIER TEST OR SCORE TEST H0 c Q q .