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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 . Review 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe September 18, 2007 Lecture 4 Spectrum Review Recall the spectrum is ∞ S ( ω ) = e − iωj γ j j = −∞ note that γ j = γ − j , so ∞ S ( ω ) = γ + 2 γ j cos( jω ) j =1 so S ( ω ) = S ( − ω ), (where the bar denotes the complex conjugate), and S ( ω ) = S ( ω + 2 π ). Recall that we can recover the covariances from the spectrum using the inverse Fourier transform 1 π iωj γ j = e S ( ω ) dω 2 π − π The cdf of the spectrum is F ( ω ) = ω S ( λ ) dλ − π 2 π π γ = dF ( ω ) − π Cramer’s Representation The spectrum is a unique way of representing a time series. We will sketch an argument to show this. Let λ j ∈ [ − π, π ] be evenly spaced fixed points with j = 1 , .., n . Let Z ( λ j ) = A ( λ j ) + iB ( λ j ), where A () and B () are two real-valued random processes. ( i.e. A ( λ j ) and B ( λ j ) are random variables for each j ) Suppose we have a process such that: 1. E Z ( λ j ) = 0 2. Var( Z ( λ j )) = E Z ( λ j ) Z ( λ j ) = E |Z ( λ j ) | 2 = σ 2 j 3. E Z ( λ j ) Z ( λ k ) = 0 for j = k 4. Z ( λ j ) = Z ( λ − j ) Cramer’s Representation 2 Remark 1 . Z is like a discrete Brownian motion (or more generally, a discrete orthogonal process). We will take the limit as n → ∞ and it becomes an orthogonal process. Suppose n x t = Z ( λ j ) e itλ j j =1 n/ 2 = cos( λ j t ) A ( λ j ) + sin( λ j t ) B ( λ j ) j =1 Condition 4 implies that x t is real-valued....
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lec4 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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