Exponential smoothing uses all past time series values to generate forecasts, with less weight given to older values. It works by calculating a smoothed level (Lt) at each period t as a weighted average of the current value (yt) and the previous smoothed level (Lt-1). The forecast for the next period (Ft+1) is then set equal to the current smoothed level (Lt). Choosing the smoothing constant (α) determines how responsive the smoothed level is to recent changes in the data.

1. Forecasting
Exponential Smoothing
For
Stationary Models

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.
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:
Weight placed on last
Revised Estimate of
estimate for the Level
the Level at time t Lt = αyt + (1-α)Lt-1
Last estimate
Weight placed on current Current time for the Level
time series value series value

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.
Ft+1 = Lt
• Since the model is stationary, it will be the
forecast for all future time periods until
more time series data is observed.

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 α .
– The forecast value for the current period (which is the
level at the previous period, Lt-1) with weight of 1-α .
• 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 theInitialization Step after period 1, set:
only value known
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. .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
=.1*B3+.9*C2
=B2
=C3
Drag D3 down
to D54
=D54
Drag C3 down Drag D55 down
to C53 to D56
Note:
Rows 8-43
are hidden

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

11. How Much Smoothing Is There?
• We said the lower the value of α, the more
“smooth” the time series will become.
Exponential Sm oothing (.1)
Exponential Smoothing (α = .1)
700
600
Actual Data
500
400
300
200
Smoothed time
100
A “flat” smoothed series series with α = .1
0
0 10 20 30 40 50 60
Perio d

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

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