rec01 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 = 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 = t e t 2 t = + y 2 t 1 with e t iid (0 , 2 ). This is an ARCH(1) process. It is covariance stationary. To show this, we first need to note that E 2 t is finite E 2 t = 1 2 assuming that 2 [0 , 1). Now, we know that E [ y t ] = E [ t e t ] = 0 and cov( y t , y t + k ) = E [ t e t t + k e t + k ] = 0 if k = 0 2 if k = 0 1 2 So this process is white noise. 2 = + ( t 2 1 e 2 t 1 ) t = + ( + t 2 2 e t 2 2 ) e t 2 1 j = 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....
View Full Document

This note was uploaded on 03/20/2012 for the course 14 14.02 at MIT.

Page1 / 5

rec01 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online