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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 2008 Recitation by Paul Schrimpf Supplementary to lectures given by Anna Mikusheva September 26, 2008 Recitation 4 Spectrum Estimation We have a stationary series, { z i t } with covariances j j and spectrum S ( ) = j = j e . We want to estimate S ( ). Using Covariances As in lecture 5, we can estimate the spectrum in the same way that we estimate the long-run variance. Na ve approach We cannot estimate all the covariances from a finite sample. Lets just estimate all the covariances that we can 1 T j = z j z j k T j = k +1 and use them to form S ( ) = T 1 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 = 0 2 S ( ) S ( ) 2 (2) Kernel Estimator S T S ( ) = 1 | j | j e ij 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. 1 In a uniform sense, i.e. P sup [ , ] | S ( ) S ( ) | > Using Covariances 2 Proof. This is an informal proof that sketches the ideas, but isnt completely rigorous. It is nearly identical to the proof of HAC consistency in lecture 3....
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rec04 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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