Optimal filtering
نویسنده
چکیده
This paper presents short summary about commonly used filters in macroeconomics. We consider Hodrick-Prescott filter, Butterworth filter, Ideal Bandpass filter, Baxter-King filter and Christiano-Fitzgerald filter. 1 The filtering problem Given raw data, {xt}t=1, we are interested in isolating component of xt, denoted by {yt}t=1, with period of oscillation between pl and pu, where 2 ≤ pl < pu < ∞. In the rest of the paper we closely follow Schleicher (2003), Christiano and Fitzgerald (2003), and Iacobucci and Noullez (2004). 2 The Ideal Bandpass Filter Consider the following decomposition of the stochastic process, xt: xt = yt + x̄t The process, yt, has power only in the frequencies belonging to the interval {(a, b)∪(−b,−a)} ∈ (−π, π). The process, x̄t, has power only in the complement of this interval in (−π, π). Here, 0 < a < b < π, and a = 2π pu , b = 2π pl In case of infinite amount of data we can use the ideal bandpass filter yt = B(L)xt where the filter, B(L), has the following structure:
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