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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 . Notation and some Linear Algebra 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe September 27, 2007 Lecture 7 VARs Notation and some Linear Algebra Let p y t = X a j y t- j + e t (1) j =1 where y t and e t are k 1, and a j is k k . e t is white noise with Ee t e t = and Ee t e s = 0 Lemma 1. y t is stationary if det I k- p p j =1 a j z j = 0 for all | z | 1 , i.e. all roots of det I k- j =1 a j z j are outside the unit circle. Definition 2. Companion form of (1) is: a a y 1 2 t ... a p e I ... t y t- 1 Y I ... t = . , A = , E t = . . . . . . . . . y . . . . t- p +1 . I so Y t and E t are kp 1 and A is kp kp . We can rewrite (1) as Y t = AY t- 1 + E t Y = A Y A + E a 11 a 12 Definition 3. If A = a 21 a 22 , then a 31 a 32 a 11 a 12 vec( A ) = a 21 a 22 a 31 a 32 A useful property is Lemma 4. vec ( ABC ) = ( C A ) vec ( B ) 6 Estimation 2 so, vec( Y ) =vec ( A Y A + E ) =( A A )vec( Y ) + vec( E ) 1 vec( Y ) = ( I- ( A A ))- vec( E ) Estimation Lemma 5....
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lec7 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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