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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 . Spectrum Estimation 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe September 20, 2007 Lecture 5 Spectrum Estimation and Information Criteria Spectrum Estimation Same setup as last time. We have a stationary series, { z t } with covariances j and spectrum S ( ) = i j e j j = . We want to estimate S ( ). Na ve approach We cannot estimate all the covariances from a finite sample. Lets just estimate all the covariances that we can T 1 j = T z j z j k j = k +1 and use them to form T 1 S ( ) = j e ij j = ( T 1) This estimator is not consistent. It convergers to a distribution instead of a point. To see this, let y = 1 T t T =1 e it z t , so that S ( ) = y y If = 2 S ( ) S ( ) 2 (2) Kernel Estimator S T j e ij S ( ) = 1 | j | S T j = S T Under appropriate conditions on S T ( S T , but more slowly than T ), this estimator is consistent 1 This can be shown in a way similar to the way we showed the Newey-West estimator is consistent. Information Criteria Suppose you want to estimate an AR ( p ), but you dont know the right p . As a first case, lets say you know an upper bound, p p . Then, well consider the case where p is not known or p = . Well begin by assuming p is known. 1 In a uniform sense, i.e. P sup [ , ] | S ( ) S ( ) | &gt; Using information criteria 2 (1) (1) A bad way: testing down We test H 0 : AR ( p 1) vs H : AR ( p ) If accept, then test p 2 vs p A 1....
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lec5 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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