lec6 - 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 . Wold Decomposition Theorem 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe September 25, 2007 Lecture 6 Introduction to VARs Wold Decomposition Theorem Definition 1. The span { y t , y t- 1 , ... } is the set of all linear combinations of y t , i.e. { z : z = j =0 j y t- j } Remark 2 . The span { y t , y t- 1 , ... } is a subset of I t , which is the sigma-algebra generated by { y t , y t- 1 , ... } Theorem 3 (Wold decomposition). Let y t be a 2nd order stationary process with Ey t = 0 . Then y t could be written as y t = v t + c ( L ) e t , where 1. e t : Ee t = 0 , Ee t e t = , and e t span { y t , y t- 1 , ... } = I t ( i.e. e t are fundamental) 2. Ee t e s = 0 for all t = s 3. 2 j < =0 k c j k 4. v t is deterministic, that is v t I- = j =0 I t- j Remark 4 . The span { y t , y t- 1 , ... } = I t is the space of all linear combinations of { y t , y t- 1 , ... } . Proof. The theorem is stated for vector processes, but well just prove it in the scalar case. We will prove the theorem by constructing e t and showing that it satisfies the conditions stated. Let e t = y t- E ( y t | I t- 1 ) = y t- a ( L ) y t- 1 , where E ( y t | I t- 1 ) is the best linear forecast of y t from past y . Note that the form of this forecast ( a ( L )) does not depend on t . This is because y t is second order stationary,...
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lec6 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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