MGT 3660
Chapter 3 Forecasting Trends: Exponential Smoothing
Notations:
t = Present time, end of time period t
Ft+1(h) = Forecast for period t+1 made “h” periods ahead
Example:
F13(1) = Forecast for period 13 made one period ahead, i.e. at the beginning of time period 13
F14(1) = Forecast for period 14 made one period ahead, i.e. at the beginning of time period 14
F14(2) = Forecast for period 14 made two periods ahead, i.e. at the beginning of time period 13
F15(1) = Forecast for period 15 made one period ahead, i.e. at the beginning of time period 15
F15(2) = Forecast for period 15 made two periods ahead, i.e. at the beginning of time period 14
F15(3) = Forecast for period 15 made three period ahead, i.e. at the beginning of time period 13
§3.2.1 Extrapolation of the mean value
Locally Constant Forecasts: Ft+1(h) = Constant (i.e. Mean of all available observations)
When a new observation becomes available, the constant is updated as follows:
New mean = Old mean +
(𝐃𝐢𝐟𝐟𝐞𝐫𝐞𝐧𝐜𝐞 𝐛𝐞𝐭𝐰𝐞𝐞𝐧 𝐧𝐞𝐰 𝐨𝐛𝐬𝐞𝐫𝐯𝐚𝐭𝐢𝐨𝐧 𝐚𝐧𝐝 𝐨𝐥𝐝 𝐦𝐞𝐚𝐧)
𝐍𝐞𝐰 𝐬𝐚𝐦𝐩𝐥𝐞 𝐬𝐢𝐳𝐞 (= 𝐨𝐥𝐝 𝐬𝐚𝐦𝐩𝐥𝐞 𝐬𝐢𝐳𝐞 + 𝟏)
§3.2.2 Use of Moving Averages
Ft+1 = MA(t/K) = Average of past K values calculated at the end of time period t, i.e. Forecast for period t+1
New Average = MA(t/K) +
𝒀𝒕+𝟏 −𝒀𝒕+𝟏−𝑲
𝑲
§3.3 Simple Exponential Smoothing (SES)
Lt = Local level computed at the end of time period t, used as forecast for period t+1, i.e. Ft+1.
Yt+1 = Actual value for period t+1
Lt+1 = Lt +  (Yt+1 - Lt )
In the above Lt = Lt-1 +  (Yt - Lt-1 ), and Lt-1 = Lt-2 +  (Yt-1 - Lt-2 ), and so on.
Thus, through series of substitutions, Lt+1 = (1 – )t+1 L0 + (1 – )t Y1 + (1 – )t-1 Y2 + ….  Yt+1
Value
Yt+1
Yt
Yt-1
…..
Y1
Weight



t
When Lt is used as forecast for period t+1, the SES formula can be written as follows:
Ft+1 =Ft +  (Yt - Ft) OR
Ft+1 =Ft + et
where 0 <=  <= 1
Starting Value:
1. For starting value for F1 use either (a) the first observation Y1, or (b) the average of the first 3 or 4
observations.
2. For a, start with any initial trial value, and then use Excel Solver to select the best value for  by
minimizing the MSE.
§3.4 Linear Exponential Smoothing (LES)
Define Lt = Level of series at time t, and
Tt = Trend of series at time t
Then, Ft+1 = Lt + Tt
Formulas:
Lt = (Lt-1 + Tt-1) +  et,
Tt = Tt-1 +  (Lt - Ft)
OR
Lt = Ft +  et, and
where 0 <=  <= 1 and 0 <=  <= 1
Starting Value:
1. Thumb rule for initial values of smoothing constants: 0.05 <=  <= 0.3, and 0.05 <=  <= 0.15. Use
Excel Solver to select the best value for  and  by minimizing the MSE.
2. For L and T use the following:
T3 =
(𝒀𝟑 −𝒀𝟏 )
𝟐
and
L3 =
𝒀𝟏 +𝒀𝟐 +𝒀𝟑
𝟑
+ T3
§3.5 Exponential Smoothing with Damped Trend (LES)
Define Lt = Level of series at time t, and
Tt = Trend of series at time t
Then, Ft+1 = Lt + Tt
Formulas:
where 0 <=  <= 1
Lt = (Lt-1 + Tt-1) +  et,
Tt = Tt-1 +  (Lt - Ft)
OR
Lt = Ft +  et, and
§3.6 Prediction Interval
Forecast ± Z/2 (RMSE)
§3.8 Use of Transformations
§3.8.1 The Log Transformation
1. Convert Yt into Ln(Yt), let us call it Zt.
2. Use Zt and fit a forecasting model.
3. Convert forecasted Zt values back to the original format using EXP(Zt)
4. Find RMSE, MAE, MAPE etc.
§3.8.2 Use of Growth Rates
1. Compute growth rate Gt = 𝟏𝟎𝟎 𝒙
𝒀𝒕 −𝒀𝒕−𝟏
𝒀𝒕−𝟏
2. Fit a suitable forecasting model to predict growth rate for the next period, gt+1
3. One-step ahead forecast Ft+1 = Yt (𝟏 +
4. Find RMSE, MAE, MAPE etc.
𝒈𝒕+𝟏
)
𝟏𝟎𝟎
§3.8.3 The Box-Cox Transformations
1. Convert Yt into square-root of Yt, i.e. =SQRT(Yt ), OR cube-root of Yt, i.e. =POWER(Yt,1/3)
2. Fit a suitable forecasting model using the transformed values and find the forecasts.
3. Convert forecasted values back to the original format, use Forcast^2 for square-root transformation
and Forecast^3 for the cube-root model.
4. Find RMSE, MAE, MAPE etc.