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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. Filtering 14.384 Time Series Analysis, Fall 2007 Recitation by Paul Schrimpf Supplementary to lectures given by Anna Mikusheva September 21, 2007 1 Recitation 3 Filtering In lecture 4, we introduced filtering. Here we'll spend a bit more time deriving some common filters and showing how to use them. Recall that an ideal band-pass filter has 1 [l , h ] B (ei ) = 0 otherwise and can be written as B (e ) = where j i - j e-ij 1 = eij d 2 ||[l ,h ] 1 ij = e d + eij d 2 l ,h -h ,-l i j 1 = e h - eil j + e-il j - e-ih j 2ij sin(jh )-sin(jl ) j = 0 = h -l j j=0 Baxter-King Baxter and King (1999) proposed approximating the ideal filter with one of order J by solving 1 min |B(ei - B (ei )|2 d B() 2 - s.t. B(1) = Baxter-King 2 where the constraint may or may not be present. We might want to impose B(1) = 0 so that the filtered series is stationary, or if we're constructing a low-pass filter, we might want B(1) = B(ei0 ) = 1 to preserve the lowest frequency movements. The Lagrangian is 1 B (ei - L= bj e-ij B (ei - bj e-ij d + ( bj - ) 2 - |j|J |j|J |j|J The first order conditions are 1 1 B (ei - [bk ] : 0 = bj e-ij eik d + e-ik B (ei - bj e-ij d + 2 - 2 - |j|J |j|J : 0 = j - |j|J Using the fact that conditions for bj are 1 2 ei(j-k) = - 1 j=k and 0 j=k 2 1 2 - B ()eij = j , the first order bj = j + Using the constraint to solve for gives: = j + |j|J 2J + 1 2 j = To summarize: the Baxter King filter of order J on [l , h ] constrained to have B(0) = is given by bj =j + 2 - 2J + 1 |j|J where j = sin(jh )-sin(jl ) j h -l = 1 - j 2J + 1 |j|J j = 0 j=0 Christiano-Fitzgerald 3 Christiano-Fitzgerald Christiano and Fitzgerald (1999) propose a generalization of the Baxter-King filter. The advocate choosing a finite approximation to the ideal filter by solving min E[(Bt (L)xt - B (L)xt )2 ] Bt () where xt is some chosen process. Bt (L) is allowed to use all data available in your sample of length T . Note that Bt (L) will generally not be symmetric and will change with t. As shown in problem set 1, this is equivalent to solving, 1 min |Bt (ei ) - B (ei )|2 Sx ()d Bt () 2 - where Sx () is the spectrum of xt . The Baxter-King filter can be considered a special case of this approach where xt is white noise, and we restrict Bt () to be time-invariant and only have bj = 0 for |j| J. Christiano and Fitzgerald argue that having xt a random walk works well for macro time-series. Hodrick-Prescott Recall that the Hodrick-Prescott filter solves: min{ (t - xt )2 + (t+1 - 2t + t-1 )2 } t For 1 < t < T - 1, the first order conditions are: 0 = 2(t - xt ) + 2( - 2(t+1 - 2t + t-1 )+ + t - 2t-1 + t-2 + + t+2 - 2t+1 + t ) 0 = - xt + t-2 - 4t-1 + (6 + 1)t - 4t+1 + t+2 The first order conditions for t = 0, 1, T - 1, T are we have -2 + 1 -2 5 + 1 -4 = -4 6 + 1 . . . similar. Writing them all in matrix form, -1 0 0 ... 0 0 ... 0 -4 ... 0 X .. . We can then form our cycle as ct = xt - t . ...
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