rec02

# rec02 - MIT OpenCourseWare http/ocw.mit.edu 14.384 Time...

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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 .

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HAC 1 14.384 Time Series Analysis, Fall 2007 Recitation by Paul Schrimpf Supplementary to lectures given by Anna Mikusheva September 11, 2008 Recitation 2 HAC Goal : estimate J = β k (or, more generally, ˆ do inference on , which has asymptotic variance J ) Methods ± : 1. Parametric : estimate ARMA(p,q) for z t : A ( L ) z t = B ( L ) e t Recall the relationship between the spectrum and J . The spectral density is 1 S ( ± ) = e i j β j 2 ω j = so, 2 ωS (0) = J Also, remember that the spectrum of an ARMA 1 is: S ( ± ) = 1 α 2 B ( e i ) 2 | | A ( e i ) | | 2 2 ω so, for an ARMA, B (1) 2 J = α 2 A (1) 2 Thus, we can estimate J by estimating B ˆ ( L ) and A ˆ ( L ) using standard methods (OLS if the ARMA has a ﬁnite order AR representation, the Kalman ﬁlter otherwise), and then estimate J as ˆ = α ˆ 2 B ˆ (1) 2 (1) ˆ J A (1) 2 As Anna said, in practice this is often non-parametric since people tend to increase p and q with sample size 1 Anna showed this for the covariance function at the end of lecture 1.
HAC 2 For multivariate series, this is called

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rec02 - MIT OpenCourseWare http/ocw.mit.edu 14.384 Time...

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