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# problems - 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 . 14.384: Time Series Analysis. Bank of sample problems for 14.384 Time series Disclaimer. The problems below do not constitute the full set of problems given as homework assignments for the course. Some of the problems are well-known folklore, some were inspired by the problem sets given at different times at Harvard, Upenn and Duke. I am thankful to Jim Stock, Frank Schorfheide and Barbara Rossi for giving me access to their course materials. 1. Transforming AR(p) to MA. If a p-order autoregressive process φ ( L ) y t = ε t is stationary, with moving average representation y t = ψ ( L ) ε t , show that p 0 = φ j ψ k − j = φ ( L ) ψ k , k = p,p + 1 ,... j =0 i.e., show that the moving average coeﬃcients satisfy the autoregressive differ- ence equation. 2. Sims’ formula for spectrum. Assume that we have a sample { y t ,x t } t T =1 from in- finite distributed lag model y t = B ( L ) x t + e t , B ( L ) = ∞ j =1 b j L j with absolutely summable coeﬃcients | b j | < ∞ (here e t is a white-noise, x t is stationary and weakly exogenous). Assume that one estimates (misspecified) model with q lags, that is, he regresses y t on to x t − 1 ,...,x t − q − 1 and obtains Þ a 1 ,..., Þ a q . As the sam- ple size increases to infinity (but q is kept constant), the estimated coeﬃcients converge to some non-random limits: Þ a j p a j . Let A ( L ) = a 1 L + ... + a p L p . → Show that A ( ) is a solution to the following problem: · 1 π A ( e − iω ) − B ( e − iω ) S X ( ω ) A ( e iω ) − B ( e iω ) , min a 1 ,...,a q 2 π − π where S X ( ) is the spectrum of the process x t . That is, one minimizes the · quadratic form in the differences between true and estimated polynomial, as- signing the greatest weights to the frequencies for which spectral density is the greatest. 1 3. Spectrum and filters. This is your first empirical problem. Choose a software package you feel comfortable using (I would recommend MatLab).You may use any users-written codes you find on Internet. Always make sure that the code is doing what you think it is doing. Please, do not forget to cite whatever you are using....
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problems - MIT OpenCourseWare http/ocw.mit.edu 14.384 Time...

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