Forecasting Exponential Smoothing Exponential SmoothingFor Stationary Models

the rest of the data is ignoredThe Last Period method uses only one period (the last) and the n-Period Moving Average and Weighted Moving methods use only the last n periods to make forecasts – the rest of the data is ignored. Exponential SmoothingallExponential Smoothing uses all the time series values to generate a forecast with lesser weights given to the observations further back in time. Exponential Smoothing

Basic Concept Exponential smoothing is actually a way of “smoothing” out the data by eliminating much of the “noise” (random effects). exponentially smoothed level, L t, estimate of the unknown constant level, β 0At each period t, an exponentially smoothed level, L t, is calculated which updates the previous level, L t-1, as the best current estimate of the unknown constant level, β 0, of the time series by the following formula: L t = αy t + (1-α)L t-1 Revised Estimate of the Level at time t Weight placed on current time series value Weight placed on last estimate for the Level Current time series value Last estimate for the Level

α in Exponential Smoothing The idea behind “smoothing” the data is to get a more realistic idea about what is “really going on”. smoothing constant, α, –The value of the smoothing constant, α, is selected by the modeler. Higher values of α allow the time series to be swayed quickly by the most recent observation. Lower values keep the smoothed time series “flatter” as not that much weight will be given to the most recent observation. –Usual values of α are between about.1 and.7 –See graphs for α =.1 and α =.7 later in this module. (1-α)damping factor. –The value (1-α) is called the damping factor.

Using Exponential Smoothing to Prepare Forecasts in Stationary Models The Level, L t, calculated at time period t is the best estimate at time t for the unknown constant, β 0. Since that is the best estimate of β 0, it will be the forecast for the next data value of the time series, F t+1. Since the model is stationary, it will be the forecast for all future time periods until more time series data is observed. F t+1 = L t

Once a value of α has been selected, the Level (or smoothed value) at time t depends on only two values --  –The current period’s actual value (y t ) with weight of . 1-  –The forecast value for the current period (which is the level at the previous period, L t-1 ) with weight of 1- . Calculations then, for L t (and hence for F t+1 ) are very simple. Initialization Step – –There is no L 0. So we cannot calculate L 1 by αy 1 + (1-α )L 0 –Since y 1 is the only value known after period 1, set: Exponential Smoothing Technique Initialization Step L 1 = y 1

Sample Calculations for First Four Periods of Yoho Data The first four values of the time series for the Yoho yoyo time series were: 415, 236, 348, 272 α =.1Suppose we have selected to use a smoothing constant of α =.1. Initialization – Period 1 L 1 = y 1 = the level for week 1 is 415 F 2 = L 1 = the forecast for week 2 is 415

Continued Week 2 L 2 =.1y 2 +.9L 1 =.1(236) +.9(415) = The smoothed (leveled) value for week 2 is F 3 = L 2 = The forecast for week 3 is Week 3 L 3 =.1y 3 +.9L 2 =.1(348) +.9(397.1) = The smoothed (leveled) value for week 3 is F 4 = L 3 = The forecast for week 4 is Week 4 L 4 =.1y 4 +.9L 3 =.1(272) +.9(392.19) = The smoothed (leveled) value for week 4 is F 5 = L 4 = The forecast for week 5 is

Excel – Exponential Smoothing Note: Rows 8-43 are hidden =B2 =.1*B3+.9*C2 =D54 Drag C3 down to C53 Drag D3 down to D54 Drag D55 down to D56 =C3

How Exponential Smoothing Uses All Previous Time Series Values Recall that the recursive formula used is: L t = αy t + (1-α)L t-1 This means: L t-1 = αy t-1 + (1-α)L t-2 L t-2 = αy t-2 + (1-α)L t-3 L t-3 = αy t-3 + (1-α)L t-4 Etc. Substituting, L t = αy t + (1-α)L t-1 = αy t + (1-α)(αy t-1 + (1-α)L t-2 ) = = αy t + α(1-α)y t-1 + (1-α) 2 L t-2 = = αy t + α(1-α)y t-1 + α(1-α) 2 y t-2 + (1-α) 3 L t-3 = αy t + α(1-α)y t-1 + α(1-α) 2 y t-2 + α(1-α) 3 y t-3 + (1-α) 4 L t-4 Etc. allThus all time series values, y t, y t-1, y t-2, y t-3, etc. will be included with successive weights reduced (dampened) by a factor of (1-α).

Exponential Smoothing (α =.1) How Much Smoothing Is There? We said the lower the value of α, the more “smooth” the time series will become. Actual Data Smoothed time series with α =.1 A “flat” smoothed series

What About Larger Values of α? Here is the “smoothed” series for α =.7: Exponential Smoothing (α =.7) Actual Data Smoothed time series with α =.7 Very sensitive to most recent time series value – not much smoothing

What Value of α Should Be Used? Up to the modeler If the modeler is considering several values of α, a forecast using each value could be prepared. –Only consider values of α that would give useful results (not α = 0, for instance) Then a performance measure (MSE, MAD, MAPE, LAD) could be used to determine which of the values of α that are being considered have the lowest value of the selected performance measure.

Review Exponential smoothing is a way to take some of the random effects out of the time series by using all time series values up to the current period. The smoothed value (Level) at time period t is: α(current value) + (1-α)(last smoothed value) Forecast for period t+1= Smoothed Value at t Initialization: First smoothed value = first actual time series value The smaller the value of α, the less movement in the time series. Excel approach to exponential smoothing