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# rec01 - 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 . Stationarity 1 14.384 Time Series Analysis, Fall 2008 Recitation by Paul Schrimpf Supplementary to lectures given by Anna Mikusheva September 5, 2008 Recitation 1 Stationarity Definition 1. White noise { e 2 2 t } s.t. Ee t = 1, Ee t e s = 0, Ee t = σ Remark 2 . { e t } can be white noise without being independent. Definition 3. strict stationarity A process, { y t } , is strictly stationarity if for each k , the distribution of { y t , ..., y t + k } is the same for all t Definition 4. 2nd order stationarity { y t } , is 2nd order stationary if Ey t , Ey 2 t , and cov( y t , y t + k ) do not depend on t Remark 5 . 2nd order stationarity is also called covariance stationarity or weak stationarity Example 6 . ARCH : Let y t = σ t e t σ 2 t = α + θy 2 t − 1 with e t ∼ iid (0 , σ 2 ). This is an ARCH(1) process. It is covariance stationary. To show this, we first need to note that Eσ 2 t is finite Eσ 2 α t = 1 − θσ 2 assuming that θσ 2 ∈ [0 , 1). Now, we know that E [ y t ] = E [ σ t e t ] = 0 and cov( y t , y t + k ) = E [ σ t e t σ t + k e t + k ] = 0 if k = 0 ασ 2 if k = 0 1 − θσ 2 So this process is white noise. σ 2 = α + θ ( σ t 2 − 1 e 2 t − 1 ) t = α + θ ( α + θσ t 2 − 2 e t 2 − 2 ) e t 2 − 1 ∞ j = α θ j ( e t 2 − k ) j =0 k =1 ⇒ ARMA 2 ARMA ARMA ( p, q ) : a ( L ) y t = b ( L ) e t , where a(L) is order p and b ( L ) is order q , and a ( L ) and b ( L ) are relatively prime....
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rec01 - MIT OpenCourseWare http/ocw.mit.edu 14.384 Time...

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