13.2 THE SIMPLE EXPONENTIAL SMOOTHING MODEL EXchange rate between Pound Sterling and german Mark Quarter 6 Ex Rate To forecast the exchange rate for time periods 12, 13&14
13.2 THE SIMPLE EXPONENTIAL SMOOTHING MODEL • Exchange rate between Pound Sterling and German Mark • To forecast the exchange rate for time periods 12, 13 & 14 Quarter 1 2 3 4 5 6 7 8 9 10 11 Ex.Rate 2.95 2.97 2.94 2.96 3.01 3.02 2.98 2.96 2.94 2.97 2.88
Exchange rate Number of Marks to￡1 3.5 3,4 3.3 2.8 No well-defined trend or seasonal variation Using simple exponential smoothing model
• No well-defined trend or seasonal variation • Using simple exponential smoothing model
This type of time-series data is described as a stationary time-series For a stationary time-series the forecast for the next time point is the average value of the time-series variable over the length of the series The estimate of the exchange rate at time point 12 Simple average Weighted average
– This type of time-series data is described as a stationary time-series. – For a stationary time-series the forecast for the next time point is the average value of the 'time-series variable' over the length of the series. – The estimate of the Exchange Rate at time point 12 • Simple average • Weighted average
13.2.1 A Notation the simple EWMA relationship a common abbreviated notation for this time-series Ⅹt,t=1,2,,n The estimate of the level made on the basis of the previous t observations is labelled as M,, then a weighted average can be calculated as follows M4=at+at×(1+a2t2+…+a×1 t+ at1+ at2+...+ a1=1
13.2.1 A Notation & the simple EWMA relationship • A common abbreviated notation for this time-series: – Xt , t=1,2,…,n. • The estimate of the level made on the basis of the previous t observations is labelled as Mt , then a weighted average can be calculated as follows: Mt = atXt + at-1Xt-1 + at-2Xt-2 +…+ a1X1 at+ at-1 + at-2 +…+ a1=1
Simple average a=at1=a+2=…=a1=1f weighted Average Heavier weighting is given to more current data points a+at1>at2>…>a1 (1-)j=1,2,3..0≤0≤1 M =ax+a(1-a)X, 7+a(1 t2+ Qx)3×3+
• Simple average – at= at-1 = at-2 =…= a1=1/t • weighted Average – Heavier weighting is given to more current data points at>at-1> at-2 >…>a1 at-j= (1-) j j=1,2,3… 0≤≤1 Mt = Xt + (1-) Xt-1 + (1-) 2Xt-2 + (1- ) 3Xt-3 +…