Exponential smoothing uses all time series values to generate forecasts, with lesser weights given to older observations. It calculates a smoothed level (Lt) at each period (t) as a weighted average of the current value (yt) and the previous smoothed level (Lt-1). This smoothed level then becomes the forecast for the next period. The smoothing constant (α) determines the weights, with lower α producing a "flatter" smoothed series that changes less over time.

1. EXPONENTIAL SMOOTHING

2. Exponential Smoothing
• The 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 Smoothing uses all the time series
values to generate a forecast with lesser weights
given to the observations further back in time.

3. Basic Concept
• Exponential smoothing is actually a way of
“smoothing” out the data by eliminating much of
the “noise” (random effects).
• At each period t, an exponentially smoothed level,
Lt, is calculated which updates the previous level, Lt-
1, as the best current estimate of the unknown
constant level, β0, of the time series by the
following formula:
Lt = αyt + (1-α)Lt-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

4. α in Exponential Smoothing
• The idea behind “smoothing” the data is to get a
more realistic idea about what is “really going on”.
– 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.
– The value (1-α) is called the damping factor.

5. Using Exponential Smoothing to Prepare Forecasts in
Stationary Models
• The Level, Lt, calculated at time period t is the best
estimate at time t for the unknown constant, β0.
• Since that is the best estimate of β0, it will be the
forecast for the next data value of the time series,
Ft+1.
• Since the model is stationary, it will be the forecast
for all future time periods until more time series
data is observed.
Ft+1 = Lt

6. Exponential Smoothing Technique
• 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 (yt) with weight of a.
– The forecast value for the current period (which is the level at the
previous period, Lt-1) with weight of 1-a.
• Calculations then, for Lt (and hence for Ft+1) are very simple.
• Initialization Step –
– There is no L0. So we cannot calculate L1 by αy1+ (1-α )L0
– Since y1 is the only value known after period 1, set:
Initialization Step
L1 = y1

7. 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
• Suppose we have selected to use a
smoothing constant of α = .1.
Initialization – Period 1
L1 = y1 = 415 -- the level for week 1 is 415
F2 = L1 = 415 -- the forecast for week 2 is 415

8. Continued
Week 2
L2 = .1y2 + .9L1 = .1(236) + .9(415) = 397.1
The smoothed (leveled) value for week 2 is 397.1
F3 = L2 = 397.1 The forecast for week 3 is 397.1
Week 3
L3 = .1y3 + .9L2 = .1(348) + .9(397.1) = 392.19
The smoothed (leveled) value for week 3 is 392.19
F4 = L3 = 392.19 The forecast for week 4 is 392.19
Week 4
L4 = .1y4 + .9L3 = .1(272) + .9(392.19) = 380.171
The smoothed (leveled) value for week 4 is 380.171
F5 = L4 = 380.171 The forecast for week 5 is 380.171

9. 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

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

11. Exponential Smoothing (.1)
0
100
200
300
400
500
600
700
0 10 20 30 40 50 60
Period
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 = .1A “flat” smoothed series

12. What About Larger Values of α?
• Here is the “smoothed” series for α = .7:
Exponential Smothing ( .7)
0
100
200
300
400
500
600
700
0 10 20 30 40 50 60
Period
Exponential Smoothing (α = .7)
Actual Data
Smoothed time
series with α = .7Very sensitive to most recent time
series value – not much smoothing

13. 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.

14. 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