rec04

# rec04 - 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 . 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 iω t } with covariances γ j j and spectrum S ( ω ) = ∞ j = γ j e − . We want to −∞ estimate S ( ω ). Using Covariances As in lecture 5, we can estimate the spectrum in the same way that we estimate the long-run variance. Na¨ ıve approach We cannot estimate all the covariances from a finite sample. Let’s just estimate all the covariances that we can 1 T γ ˆ j = z j z j − k T j = k +1 and use them to form S ˆ ( ω ) = T − 1 γ ˆ j e − iωj j = − ( T − 1) This estimator is not consistent. It convergers to a distribution instead of a point. To see this, let y ω = √ 1 T t T =1 e − iωt z t , so that S ˆ ( ω ) = y ω y ¯ ω If ω = 0 2 S ˆ ( ω ) S ( ω ) χ 2 (2) ⇒ Kernel Estimator S T S ˆ ( ω ) = 1 − | j | γ ˆ j e − iωj 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 ω ∈ [ − π,π ] | S ˆ ( ω ) − S ( ω ) | > → Using Covariances 2 Proof. This is an informal “proof” that sketches the ideas, but isn’t 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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