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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 . HAC 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe September 13, 2007 Lecture 3 More HAC and Intro to Spectrum HAC Continuing with our setup from last time: • { z t } stationary • γ k = cov( z t , z t + k ) • J = ∑ ∞ k =∞ γ k we want to estimate J . Last time, we first considered just summing all the sample covariances. This was inconsistent because high order covariances will always be noisily estimated. Then we considered a truncated sum of covariances with bandwidths slowly increasing to infinity. This was consistent, but could lead to a ˆ not positive definite J . To fix this, we considered the kernel estimator: S T J ˆ = X k T ( j ) γ ˆ j S T ˆ ˆ We want to choose S T and k T () such that (1) J is consistent and (2) J is always positive definite. Consistency Theorem 1. J ˆ is consistent if we assume: • ∑ ∞∞  γ j  < ∞ • k T ( j ) → 1 as T → ∞ and  k T ( j )  < 1 ∀ j • ξ t,j (defined below) are sationary for all j and sup j ∑ k  cov ( ξ t,j , ξ t + k,j  < C for some constant C (limited dependence) • S T → ∞ S 3 and T T → Proof. This is an informal “proof” that sketches the ideas, but isn’t completely rigorous. S T S T J ˆ J = X γ j + X ( k T ( j ) 1) γ j + X k T ( j )( γ ˆ j γ j  j  >S T j = S T j = S T We can interprete these three terms as follows; 1....
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