lec2 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 14.384 Time Series Analysis Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Limit Theorems 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe September 11, 2007 Lecture 2 Limit Theorems, OLS, and HAC Limit Theorems What are limit theorems? n 1. Law of large numbers (LLN) { x i } iid, Ex i = , Var( x 2 i ) = , then 1 i =1 x 2 i (in L , a.s., in n probability) n 2. Central limit theorems (CLT) n ( 1 ) N (0 , ) n i =1 x 2 i We stated these while assuming independence. In time series, we usually dont have independence. Proving LLN with independence n n 1 1 E x i =Var x i (1) n n i =1 i =1 n 1 = Var x i (2) n 2 i =1 n 1 = Var( x i ) (3) n 2 i =1 n 2 = 0 (4) n 2 We used independence to go from 2 to 3. Without independence, wed have n 1 1 Var x i = cov( x i , x j ) n n 2 i =1 1 = n 2 ( n + 2( n 1) 1 + 2( n 2) 2 + ... ) n 1 k = 2 k 1 + n n k =1 Assume absolute summability, i.e. j = | j | < , then n 1 k lim + = 0 n n k =1 n k 1 Thus, covariance stationarity and absolute summable covariances is enough for a law of large numbers. Remark 1 . Stationarity is not enough. Let z N (0 , 2 ). Suppose x t = z t . Then cov( x t , x s ) = 2 t, s , so we do not have absolute summability, and clearly we do not have a LLN for { x t } since it is equivalent to...
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lec2 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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