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Ec 178 – ECON & BUS FORECASTING LECTURE NOTES Foster, UCSD February 25, 2010 TOPIC 2 - SMOOTHING & TRENDS A. Filters and Smoothing 1. Filters: a) A filter is a function which transforms a raw data series into a new series. 1) For raw data y t , let x t = f(y t ) be the transformed series. 2) x t is a linear filter if f(y t ) is a linear function. Example: x t = α 1 y t+1 + α 2 y t + α 3 y t-1 . 3) If Σα i = 1, then x t is an averaging transformation. b) Some common filters. Series mean Simple and Exponentially-weighted moving averages Hodrick-Prescott and other non-linear filters Consecutive differencing c) Uses of filters. [Fig. 1] 1) Smooth data to reduce randomness and reveal underlying patterns. 2) Remove seasonal component to pro- duce seasonally-adjusted series. 3) Generate simple, cheap forecasts. 4) Create stationary ARMA series. 2. Simple Moving Average of Length m: a) Let (m) t denote a simple moving average, the average of the m most recent values of y available at time t. (Note that t is the smoothed level of series y t .) b) Numerical example and notes. 1) A simple MA is appropriate for smoothing out a data series y t that has no up or down trend and no recurring seasonal pattern. 2) A larger m gives a smoother result, but we always lose m-1 values at the beginning of the smoothed series (m) t . 3) Simple MA has given way to EWMA and nonlinear filters. Simple MA t y t (2) t (4) t 1 2 3 4 5 6 3 5 2 5 2 4 /// 4.0 3.5 3.5 3.5 3.0 /// /// /// 3.7 5 3.5 0 3.2 5 y t x t Fig. 1 ) 2 {int 1 ) ( 1 1 1 0 + + + = = + - - - = - m m y y y y m m m t t t m s s t t
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Ec 178 SMOOTHING/TRENDS p. 2 3. Centered Moving Averages of Length m: a) Let CMA(m) t be a centered moving average of length m, an average of m consecutive values of series y centered on time period t. b) Numerical examples and notes. 1) A CMA is appropriate for smoothing series with trend or recurring seasonal variation (repeating every m periods). 2) Note that some observations are always lost at the beginning and end of the smoothed series CMA(m) t . 3) In the example at right: CMA(3) t = [y t-1 + y t + y t+1 ]/3 CMA(6) t = [y t-3 + 2(y t-2 + y t-1 + y t + y t+1 + y t+2 ) + y t+3 ]/ 12 4. Exponentially-Weighted Moving Averages: a) For smoothing parameter , let ( ) α ℓ α t or just t denote an exponentially-weighted moving average of series y at time t. b) An EWMA is a weighted average of all past observations on y t going back forever, with more weight given to recent observations than to those from the remote past. 1) By backward substitution we obtain the result below: 2) Since 0 < (1 ) < 1, as s −α , (1 ) s 0, so the weights are declining to zero as we go back in time. c) For recursive equations like EWMA, we must provide initial values. 1) One way to initialize is to set ( ) ℓ α 1 = y 1 . This implies a “run-in period” as t gradually forgets the arbitrary starting value and settles in to the average level of y t .
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