Class Note for ECON 520 with Professor Hirano at UA
Class Note for ECON 520 with Professor Hirano at UA
Popular in Course
Popular in Department
This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 17 views.
Reviews for Class Note for ECON 520 with Professor Hirano at UA
Report this Material
What is Karma?
Karma is the currency of StudySoup.
You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!
Date Created: 02/06/15
Economics 522A Spring 2007 Lecture Note 12 Large Sample Properties of OLS 1 Motivation and Model Assumptions Relaxing the assumption of normality of yi is useful because the normality assumption might be too strong in many practical situations However it becomes dif cult or impossible to get exact distribution theory and carry out exact inference So we often turn to large sample asymptotic approximations In order to use large sample results we will make some additional assumptions to the neoclassical regression model We will assume 0 The vector yi is HD from some joint distribution Eiyilmil 90257 Vlyilmil 027 The fourth moments of yi exist The matrix is invertible 2 Recap Basic Asymptotic Results We will focus on convergence in probability and convergence in distribution These concepts were de ned in Econ 520 LNQ and are also explained in Ruud 134 We state some key results about convergence of sample averages1 For the most part these are the same results in Econ 520LN9 and Ruud 134 but in some cases we give slightly more powerful versions of the results which will simplify the later arguments In general we will give results for sequences of random vectors not just scalar random variables The notation denotes the Euclidean norm of X Here X and Y denote arbitrary random vectors not necessarily the X and y in the neoclassical regression model Weak Law of Large Numbers WLLN Let X1X2 be a sequence of iid random variables such that lt 00 Then 7 1 X E ZX i EX1 l Multivariate Central Limit Theorem CLT Let X1 X2 be iid random vectors in Rk with nite mean u EX1 and nite covariance matrix 2 EX1 7 uX1 7 u 1These results are drawn from A van der Vaart Asymptotic Statistics Cambridge University Press Then n 1 7 d W 2109 i M ltXn M H N07E Continuous Mapping Theorem CMT Let be a function from Rk to R and suppose g is continuous at every point in a set C st PX E C 1 Then m gX m y ii If X 3 X then gX 3 9X Note that matrix addition and multiplication are continuous functions There are various useful facts about convergence in probability and convergence in distri bution Result i Convergence in probability implies convergence in distribution amp3X3amp3X ii Convergence in distribution to a constant implies convergence in probability X g a a constant 3 X g a iii If X 3 X and lan 3 KM 3 0 then Yn 3 X iv lf Xn g X and Yn g a where a is a constant then the vector Xn Y 3 X c v If Xn g X and Y i Y then XmYn g X Y This is not true for convergence in distribution A useful corollary of the previous result Slutsky7s Lemma Let X X and Yn be random vectors or matrices lf Xn g X and d i i Y a c where c 1s a constant vector or matrix then i XnYnXa ii 1an 3 0X iii Yan g c lX provided 0 is invertible Delta Method Let Xn be a sequence of didimensional random vectors such that ltXn e p 3 No2 where E is positive de nite and nite Let 9 denote a continuously differentiable function from Rd into Rk and let Cm agam denote the k x 1 matrix of partial derivatives Then d W79Xn 901 N07 MOSCOW 3 Asymptotic Theory for B and 52 31 Consistency of B Consider B X X 1X y Note that 951 mg XX x1z2xn 39 11 954 Similarly 71 X y Z 11 So 71 71 n 6 Z 11 11 We can also write this as 1 A 1 n 1 n 5 11 11 By the law of large numbers and the assumption that the relevant moments exist 1 TL P E 901901 HEl90139901l7 n 39 1 l 1 p Z 901141 Elmiyil n 11 Note that E myil Using Slutsky s Lemma and the assumption that is invertible B 1 Elm l lElm l B So we have shown that B is consistent 32 Asymptotic Normality 0f Useful trick de ne 6139 24139 15 Then we can write yi 23 617 and E 1l1l Elyi Elyilml 9025 0 Also7 Vl51l1l Vlyilmil 02 Now7 write the OLS estimator as A 1 11 11 11 s g i 2 11 11 1 11 1 11 1 11 7 Z g 2 161 11 11 11 1 n 1 1 5 gzzm 29019 11 11 So And then we can write 1 1 1 M 7 m lt7 m n 11 11 Let us focus on the term n 1 7 2 Mel 11 We can think of mm as a k x 1 random vector7 call it 10139 where wi is HD and Elwil El151l ElElMEilmill ElmiEleilmill 0 Vlwil Elwiw l Elvm elzl Elv ElEllell UzElmz l So7 by the multivariate CLT7 The term izj converges in probability as we have already shown so by Slutsky s lemma g 7 g 3 Ezz 1NO a2Em N N0 azEp irl 1 So this is saying that the approximate unconditional distribution of B is multivariate normal with mean 3 and variance equal to 1 702Em 71 02Enxix71 n In other words using i to mean approximately distributed as77 3 1 NW 02En Note the similarity to the exact result under normality of yi There we had 31X NB702X X 17 where X X 1 ELI 33 Consistency of 52 Recall A 2 55 219 9525 3 7 n 7 k n 7 k Now if we knew B we could form 6 y 7 mg and estimate 02 by 2 52 221 5139 71 By the law of large numbers we would have 52 g Vb 0392 But in practice we have to use 5 7 6 so this argument doesn t quite work However since we know that BA 3 3 we expect that B should be close to the true 3 in large samples and this in turn suggests that using 5139 in place of q might still lead to the same result To show this formally we need to try to get 52 in the form of a sample averages over llD variables We can write 82 mikee 1 A A nikoixm wixm 1 n 7 k My 7 Xmy y X B X X Note that So 1 72 y 1 EM 711 l 1 p i1 And we have already shown that 3 g 3 Now consider We can write yi yi 7 23 23 to get EM E 2 7 9526 2026 E w 7 mm 26 2w 7 z mzm EH EMBV whim 02 E 39010523 E 610023 But Ekiz EWMMMB 0 S0 EM 02 B Ez 2 S0 7 122137 B lt3 Z 3 3 a B Engm B Emz m 02 i1
Are you sure you want to buy this material for
You're already Subscribed!
Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'