Simple Average MS4102 Business Forecasting Methods Moving Average Methods n Suitable for no trend/horizontal series yt = β0 + εt, εt ~ N(0, σ2) where β0 may change slowly with time Lecturer : Room No : Tel No : Dr Iris Yeung P7509 2788 8566 n The first n data points are averaged and used to forecast the next period F t +1 = n ∑ y t n t =1 1 Simple Average n 2 Simple Moving Average § Suitable for no trend/ horizontal series Forecast sales of ovens for 2000 with simple average yt = β0 + εt, εt ~ N(0, σ2 ) 24 n where β0 may change slowly with time t 9800 = = 408.33 24 24 9800 + 850 = = 426 25 yˆ 25 = yˆ 26 ∑y t =1 Disadvantage uses the mean of all data to forecast and has not placed more weight to recent observations § The n-period moving average calculated at time period t-1 is the average of the n most recent observations M t −1 = y t −1 + yt −2 + L + yt −n +1 + yt −n n 3 Simple Moving Average Simple Moving Average § As each new observation becomes available, a new moving average can be computed by dropping the oldest value and including the newest one. yt + yt −1 + L + yt − n+ 1 n § Mt can also be calculated by Mt = M t = M t −1 + 4 n We use the moving average calculated at time t to forecast the y value at time t + 1 Ft +1 = M t = y t + y t −1 + L + y t −n +1 n y t − y t−n n 5 6 1 Example on Simple Moving Average (2) (3) (4) (5) Time Period Observed Values Forecast using Three-Month Moving Average Forecast using Five -Month Moving Average Jan Feb Mar Apr 1 2 3 4 200.0 135.0 195.0 197.5 ------176.7 --------- May June July Aug Sept Oct 5 6 7 8 9 10 310.0 175.0 155.0 130.0 220.0 277.0 175.8 234.2 227.5 213.3 153.3 168.3 --207.5 202.5 206.5 193.5 198.0 Nov Dec 11 12 235.0 --- 209.2 244.2 191.4 203.5 350 Shipments of electric openers (1) Month n 250 Y 200 n=3 ma 150 n=5 ma 100 50 0 0 1 2 3 4 5 6 7 8 9 1011 12 13 Time (months) Choice of n for Simple Moving Average n 300 If n = 1, the simple moving average is simply the naïve model. Double Moving Average n Suitable for linear trend series n 0 + β1t + ε(t ), yt be = βshown It can that If data have large randomness, use large n. Otherwise use small n. E (M t In general, we choose n such that MSE, MAPE, … is minimum 9 2 where M t′′ = t 2 )− n −1 β1 2 M t + M t −1 + M t − 2 + L + M t −n +1 n is the double moving average. Double Moving Average ) n − 1 1 β ) = E ( yt ) − E ( M t′′) = E ( M n ε(t ) ~ N (0,6 10 Double Moving Average 25 Time 1 2 3 4 5 6 7 8 9 10 11 Actual SMA(3) Act-SMA DMA(3) SMA-DMA 2 4 6 4 2 8 6 2 10 8 2 6 2 12 10 2 8 2 14 12 2 10 2 16 14 2 12 2 18 16 2 14 2 20 18 2 16 2 A B Forecast eForecast 20 10 12 14 16 18 20 2 2 2 2 2 2 12 14 16 18 20 22 0 0 0 0 0 15 Actual 10 SMA(3) DMA(3) 5 0 0 5 10 15 Time 11 2 Example on Double Moving Average Double Moving Average n The current level of the data can be estimated by n The slope β1 of the series can be estimated by a t = M t + (M t − M t′′) = 2 M t − M ′t ′ bt = n 2 (M t − M ′t′) n−1 The forecast for period t + τ is obtained by extrapolating the trend τ periods into the future Ft +m =at +bt m Period (1) Inventory Balance of Product E12 (2) Four-Month Moving Average of (1) (3) Four-Month Moving Average of (2) (4) One-Month Ahead Forecast 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 140.00 159.00 136.00 157.00 173.00 131.00 177.00 188.00 154.00 179.00 180.00 160.00 182.00 192.00 224.00 188.00 198.00 206.00 203.00 238.00 228.00 231.00 221.00 259.00 273.00 --- ----148.00 156.25 149.25 159.50 167.25 162.50 174.50 175.25 168.25 175.25 178.50 189.50 196.50 200.50 204.00 198.75 211.25 218.75 225.00 229.50 234.75 246.00 --- ------------153.25 158.06 159.62 165.93 169.87 170.12 173.31 174.31 177.87 184.93 191.25 197.62 199.93 203.62 208.18 213.43 221.12 227.00 233.81 --- --------------169.91 182.56 167.29 188.77 184.20 165.12 178.47 185.47 208.87 215.77 215.91 214.62 196.77 223.95 236.35 244.27 243.45 247.66 266.31 13 Summary Inventory 300 250 200 150 100 50 0 Y SMA DMA FORECAST 0 10 20 30 n Mean n Simple moving average n n n Time period Equal weight to all past observations Equal weight to the past n observations only Double moving average n Unequal weight 16 Determination of buy/sell signals n Use a moving average n n n Use two moving averages n n http://www.pcn.com.hk/ Buy if P > MA Sell if P < MA Buy if MA(short) > MA (long) Sell if MA(short) < MA (long) 18 3 19 4