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Bài giảng Chapter 3: Stochastic regression model

Bài giảng Chapter 3: Stochastic regression model hướng đến trình bày các vấn đề cơ bản như: Consistency; classical stochastic regression model; limiting distributions and asymptotic distributions; asymptotic distribution of;. Mời các bạn cùng tìm hiểu và tham khảo nội dung thông tin tài liệu. | Advanced Econometrics Chapter 3 Stochastic Regression Model Chapter 3 STOCHASTIC REGRESSION MODEL I. CONSISTENCY 1. Definition Let 6n be a random variable. If for any Ve 0 we have lim 6n -ớ 0 Othen 6 is probability limit of 6n. If 6n is an estimator for 6 then 6n is said a consistent estimator of 6 . notation p lim 6n 6 n w Note A sufficient condition for this to hold is if Bias 6n 6 and Var 6n 6 when n w 2. Cramer Theorem i lim EưL 6 If. n w ii lim Var 6n 0 n w . 1 1 zi Zh then p lim6n 6 n w Nam T. Hoang University of New England - Australia 1 University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 3 Stochastic Regression Model Example xi N p ơ2 i 1 2 3 n sample size. r - Get X 1 - a n i 1 e X p flim Var X 0 _ _ So 5 n then X is a consistent estimator of Li plim X p I lim E X p n x In Note If an estimator is inconsistent then it is a useless estimator unreliable . There are many situations where OLS estimator is inconsistent. Need to be clear with this. 3. Slutsky Theorem Let F be a continuous function then p liH F 4 n 4 n -.- kn. F p lim 4 n p JM n . p l im Ể k n n n n n EX if p lim 4 C p lim F dn F C p lim 1 Ể n 1 c p lim 0 3 c3 p lim exp n ec p lim 4n A n p lim 4n p lim An A and B are stochastic matrices p lim AS p lim . p lim B also p lim .J p lim A -1 if A is non-singular. Nam T. Hoang University of New England - Australia 2 University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 3 Stochastic Regression Model II. CLASSICAL STOCHASTIC REGRESSION MODEL Now consider the LS model first under our standard assumption. However we will relax some of them. Don t need normality. X can be random just assume that Xi si is a random independent sequence. Model 1 Y X p s nxk 2 X and s are generated independently of each other and Rank X k . 3 E sX 0 4 E ss X Ớ2Ỉ 5 X consists of stationary random variables with E I XIX Y XX i XX Xnxk 1xkJ 1 1 A ____ _ and plim nXX plimnsXX E X X X Because X now is random . 1 Stationary random variable Xi x 2 Xi3