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Time Series
Forecasting Techniques
Back in the 1970s, we were working with a company in the major home
appliance industry. In an interview, the person in charge of quantitative
forecasting for refrigerators explained that their forecast was based on one time
series technique. (It turned out to be the exponential smoothing with trend
and seasonality technique that is discussed later in this chapter.) This tech-
nique requires the user to specify three “smoothing constants” called α , β , and
γ (we will explain what these are later in the chapter). The selection of these
values, which must be somewhere between 0 and 1 for each constant, can have
a profound effect upon the accuracy of the forecast.
As we talked with this forecast analyst, he explained that he had chosen
the values of 0.1 for α , 0.2 for β , and 0.3 for γ. Being fairly new to the world
of sales forecasting, we envisioned some sophisticated sensitivity analysis that
this analyst had gone through to find the right combination of the values for
the three smoothing constants to accurately forecast refrigerator demand.
However, he explained to us that in every article he read about this
technique, the three smoothing constants were always referred to as α , β ,
and γ , in that order. He finally realized that this was because they are the 1st,
2nd, and 3rd letters in the Greek alphabet. Once he realized that, he “simply
took 1, 2, and 3, put a decimal point in front of each, and there were my
smoothing constants.”
After thinking about it for a minute, he rather sheepishly said, “You know,
it doesn’t work worth a darn, though.”
INTRODUCTION
We hope that over the years we have come a long way from this type
of time series forecasting. First, it is not realistic to expect that each
product in a line like refrigerators would be accurately forecast by the
same time series technique—we probably need to select a different
time series technique for each product. Second, there are better ways
to select smoothing constants than our friend used in the previous
example. To understand how to better accomplish both of these, the
purpose of this chapter is to provide an overview of the many tech-
niques that are available in the general category of time series analy-
sis. This overview should provide the reader with an understanding
of how each technique works and where it should and should not
be used.
Time series techniques all have the common characteristic that
they are endogenous techniques. This means a time series technique
looks at only the patterns of the history of actual sales (or the series of
sales through time—thus, the term time series). If these patterns can
be identified and projected into the future, then we have our forecast.
Therefore, this rather esoteric term of endogenous means time series
techniques look inside (that is, endo) the actual series of demand
through time to find the underlying patterns of sales. This is in con-
trast to regression analysis, which is an exogenous technique that we
will discuss in Chapter 4. Exogenous means that regression analysis
examines factors external (or exo) to the actual sales pattern to look for
a relationship between these external factors (like price changes) and
sales patterns.
If time series techniques only look at the patterns that are part
of the actual history of sales (that is, are endogenous to the sales
history), then what are these patterns? The answer is that no matter
what time series technique we are talking about, they all examine one
or more of only four basic time series patterns: level, trend, seasonal-
ity, and noise. Figure 3.1 illustrates these four patterns broken out of a
monthly time series of sales for a particular refrigerator model. The
74 SALES FORECASTING MANAGEMENT
are high sales every summer for air conditioners, high sales of
agricultural chemicals in the spring, and high sales of toys in the fall.
The point is that the pattern of high sales in certain periods of the year
and low sales in other periods repeats itself every year. When broken
out of the time series in Figure 3.1, the seasonality line can be seen as a
regular pattern of sales increases and decreases around the zero line at
the bottom of the graph.
Noise is random fluctuation—that part of the sales history that
time series techniques cannot explain. This does not mean the fluctu-
ation could not be explained by regression analysis or some qualita-
tive technique; it means the pattern has not happened consistently in
the past, so the time series technique cannot pick it up and forecast
it. In fact, one test of how well we are doing at forecasting with time
series is whether the noise pattern looks random. If it does not have
a random pattern like the one in Figure 3.1, it means there are still
trend and/or seasonal patterns in the time series that we have not yet
identified.
We can group all time series techniques into two broad categories—
open-model time series techniques and fixed model time series techniques —
based on how the technique tries to identify and project these four
patterns. Open-model time series (OMTS) techniques analyze the
time series to determine which patterns exist and then build a unique
model of that time series to project the patterns into the future and,
thus, to forecast the time series. This is in contrast to fixed-model time
series (FMTS) techniques, which have fixed equations that are based
upon a priori assumptions that certain patterns do or do not exist in
the data.
In fact, when you consider both OMTS and FMTS techniques, there
are more than 60 different techniques that fall into the general category
of time series techniques. Fortunately, we do not have to explain each
of them in this chapter. This is because some of the techniques are very
sophisticated and take a considerable amount of data but do not pro-
duce any better results than simpler techniques, and they are seldom
used in practical sales forecasting situations. In other cases, several dif-
ferent time series techniques may use the same approach to forecasting
and have the same level of effectiveness. In these latter cases where
several techniques work equally well, we will discuss only the one that
is easiest to understand (following the philosophy, why make some-
thing complicated if it does not have to be). This greatly reduces the
number of techniques that need to be discussed.
76 SALES FORECASTING MANAGEMENT
Because they are generally easier to understand and use, we will
start with FMTS techniques and return to OMTS later in the chapter.
FIXED-MODEL TIME SERIES TECHNIQUES
FMTS techniques are often simple and inexpensive to use and require
little data storage. Many of the techniques (because they require little
data) also adjust very quickly to changes in sales conditions and, thus,
are appropriate for short-term forecasting. We can fully understand the
range of FMTS techniques by starting with the concept of an average as
a forecast (which is the basis on which all FMTS techniques are founded)
and move through the levels of moving average, exponential smoothing,
adaptive smoothing, and incorporating trend and seasonality.
The Average as a Forecast
All FMTS techniques are essentially a form of average. The sim-
plest form of an average as a forecast can be represented by the
following formula:
Forecast
t + 1
= Average Sales
1 to t
= ∑
N
t = 1
S
t
/N (1)
where: S = Sales
N = Number of Periods of Sales Data (t)
In other words, our forecast for next month (or any month in the
future, for that matter) is the average of all sales that have occurred in
the past.
The advantage to the average as a forecast is that the average is
designed to “dampen” out any fluctuations. Thus, the average takes
the noise (which time series techniques assume cannot be forecast
anyway) out of the forecast. However, the average also dampens out
of the forecast any fluctuations, including such important fluctuations
as trend and seasonality. This principle can be demonstrated with a
couple of examples.
Figure 3.2 provides a history of sales that has only the time
series components of level and noise. The forecast (an average) does
a fairly good job of ignoring the noise and forecasting only the level.
However, Figure 3.3 illustrates a history of sales that has the time series
components of level and noise, plus trend. As will always happen when
Time Series Forecasting Techniques 77
Time Series Forecasting Techniques 79
J
F01M01A01M
J01J01J
S01O01N01D
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F02M02A02M
J02J
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F03M03A03M
J03J
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Demand Forecast
Figure 3.3 Average as a Forecast: Level, Trend, and Noise
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Demand Forecast
Figure 3.4 Average as a Forecast: Level, Seasonality, and Noise
decreasing (seasonality). Therefore, sales should be the same (level) for
each period in the future. If nothing else, this demonstrates the rather
naïve assumption that accompanies the use of the average as a forecast.
The average as a forecasting technique has the added disadvantage
that it requires an ever-increasing amount of data storage. With each
successive month, an additional piece of data must be stored for the
calculation. With the data storage capabilities of today’s computers,
this may not be too onerous a disadvantage, but it does cause the aver-
age to be sluggish to changes in level of demand. One last example
should illustrate this point. Figure 3.5 shows a data series with little
noise, but the level changes. Notice that the average as a forecast never
really adjusts to this new level because we cannot get rid of the “old”
data (the data from the previous level).
Thus, the average as a forecast does not consider trend or season-
ality, and it is sluggish to react to changes in the level of sales. In fact,
it does little for us as a forecasting technique, other than give us
an excellent starting point. All FMTS techniques were developed
to overcome some disadvantage of the average as a forecast. We next
explore the first attempt at improvement, a moving average.
80 SALES FORECASTING MANAGEMENT
J01F
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Demand Forecast
Figure 3.5 Average as a Forecast: Level Change
82 SALES FORECASTING MANAGEMENT
J01F
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O01N01D
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Demand 3 Month MA 6 Month MA 12 Month MA
Figure 3.6 Moving Average as a Forecast: Level and Noise
However, for the time series with trend added (Figure 3.7), very
different results are obtained. The longer the moving average, the less
reactive the forecast, and the more the forecast lags behind the trend
(because it is more like the average). Again, this is because moving
averages were not really designed to deal with a trend, but the shorter
moving averages adjust better (are more reactive) than the longer in
this case.
An interesting phenomenon occurs when we look at the use of
moving averages to forecast time series with seasonality (Figure 3.8).
Notice that both the three-period and the six-period moving averages
lag behind the seasonal pattern (forecast low when sales are rising
and forecast high when sales are falling) and miss the turning points
in the time series. Notice also that the more reactive moving average
(three-period) does a better job of both of these. This is because in
the short run (defined here as between turning points), the seasonal
pattern simply looks like trend to a moving average.
However, the 12-period moving average simply ignores the
seasonal pattern. This is due to the fact that any average dampens out
Time Series Forecasting Techniques 83
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Demand 3 Month MA 6 Month MA 12 Month MA
Figure 3.7 Moving Average as a Forecast: Level, Trend, and Noise
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O01N01D
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DEMAND 3 Month MA 6 Month MA 12 Month MA
Figure 3.8 Moving Average as a Forecast: Level, Seasonality, and Noise
A final problem with the moving average is that the same weight
is put on all past periods of data in determining the forecast. It is more
reasonable to put greater weight on the more recent periods than
the older periods (especially when a longer moving average is used).
Therefore, the question when using a moving average becomes how
many periods of data to use and how much weight to put on each
of those periods. To answer this question about moving averages, a
technique called exponential smoothing was developed.
Exponential Smoothing
Exponential smoothing is the basis for almost all FMTS techniques
in use today. It is easier to understand this technique if we acknowl-
edge that it was originally called an “exponentially weighted moving
average.” Obviously, the original name was too much of a mouthful
for everyday use, but it helps us to explain how this deceptively
complex technique works. We are going to develop a moving average,
but we will weight the more recent periods of sales more heavily in
the forecast, and the weights for the older periods will decrease at
an exponential rate (which is where the “exponential smoothing” term
came from).
Regardless of that rather scary statement, we are going to accom-
plish this with a very simple calculation (Brown & Meyer, 1961).
F
t+ 1
= α S
t
t
(3)
where: F t
= Forecast for Period t
S
t
= Sales for Period t
0 < α < 1
In other words, our forecast for next period (or, again, any period in
the future) is a function of last period’s sales and last period’s forecast,
with this α thing thrown in to confuse us.
What we are actually doing with this exponential smoothing
formula is merely a weighted average. Because α is a positive fraction
(that is, between 0 and 1), 1 − α is also a positive fraction, and the two
of them add up to 1. Any time we take one number and multiply it
by a positive fraction, take a second number and multiply it by the
reciprocal of the positive fraction (another way of saying 1 − the first
Time Series Forecasting Techniques 85
fraction), and add the two results together, we have merely performed
a weighted average. Several examples should help:
- When we want to average two periods’ sales (Period 1 was 50
and Period 2 was 100, for example) and not put more weight on
one than the other, we are actually calculating it as ((0.5 × 50) +
(0.5 × 100)) = 75. We simply placed the same weight on each
period. Notice that this gives us the same result as if
we had done the simpler equal-weight average calculation of
(50 + 100)/2.
- When we want the same two periods of sales but want to
put three times as much weight on Period 2 (for reasons we
will explain later), the calculation would now be ((0.25 × 50) +
(0.75 × 100)) = 87.5. Notice that in this case α would be 0.25 and
1 − α would be 0.75.
- Finally, if we want nine times as much weight on Period 2, the
resultant calculation would be ((0.1 × 50) + (0.9 × 100)) = 95. Again,
notice that in this case α would be 0.1 and 1 − α would be 0.9.
Therefore, we can control how much emphasis in our forecast is
placed on what sales actually were last period. But what is the purpose
of using last period’s forecast as part of next period’s forecast? This is
where exponential smoothing is “deceptively complex” and requires
some illustration.
For the purpose of this illustration, let’s assume that on the
evening of the last day of each month, we make a forecast for the next
month. Let’s also assume that we have decided to use exponential
smoothing and to put 10% of the weight of our forecast on what hap-
pened last month. Further, let’s assume this is the evening of the last
day of June. Thus, our value for α would be 0.1 and our forecast for
July would be:
F
JULY
= .1 S
JUNE
JUNE
But where did we get the forecast for June? In fact, a month ago on
the evening of the last day of May, we made this forecast:
F
JUNE
= .1 S
MAY
MAY
86 SALES FORECASTING MANAGEMENT
exponential smoothing formula does it for us. We do need to remember,
however, that the higher the value of α, the more weight we are putting
on last period’s sales and the less weight we are putting on all the
previous periods combined. In fact, as α approaches one, exponential
smoothing puts so much weight on the past period’s sales and so little
on the previous periods combined, that it starts to look like our naïve
technique (F
t+ 1
= S
t
) from Chapter 2. Conversely, as α approaches zero,
exponential smoothing puts more equal weight on all periods and
starts to look much like the average as a forecast.
This leads us to some conclusions about what the value of α
should be:
- The more the level changes, the larger α should be, so that
exponential smoothing can quickly adjust.
- The more random the data, the smaller α should be, so that
exponential smoothing can dampen out the noise.
Several examples should help illustrate these conclusions. For
our first illustration, we can use the data pattern from Figure 3.9 for
the moving average, now Figure 3.10 for exponential smoothing.
88 SALES FORECASTING MANAGEMENT
J01F
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O01N01D
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O02N02D
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Demand Alpha = .1 Alpha = .5 Alpha =.
Figure 3.10 Exponential Smoothing as a Forecast: Level Change
Time Series Forecasting Techniques 89
J
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Demand Alpha = .1 Alpha = .5 Alpha =.
Figure 3.11 Exponential Smoothing as a Forecast: Low Noise
In Figure 3.10, we can see three exponential smoothing forecasts of the
time series. All three do a fairly good job when the level is stable, but
the higher the value of α in the forecast, the quicker it reacts to the
change in level. Because a low value of α is much like an average, the
forecast for the low α never quite reaches the new level.
However, a very different result is found when we observe the
forecasts of the time series in Figures 3.11 and 3.12. Figure 3.11 is a
reproduction of the data series used in Figures 3.2 and 3.6 and repre-
sents a time series with no trend and a low amount of noise. In this
series, the exponential smoothing forecasts with various levels of α all
perform fairly well. However, in the time series of Figure 3.12, which
has a stable level but a high amount of noise, the forecasts with the
higher values of α overreact to the noise and, as a result, jump around
quite a bit. The forecast with the lower level of α does a better job of
dampening out the noise.
Given these illustrations of our conclusions about the value of α
that should be used, we have in exponential smoothing a technique
that overcomes many of the problems with the average and the mov-
ing average as forecasting techniques. Exponential smoothing is less
the previous period’s forecast to adjust the value of α for the next
period’s forecast (Trigg & Leach, 1967). Thus, the original exponential
smoothing formula is still used:
F
t+ 1
= α S
t
t
(4)
but after each period’s sales are recorded, the value of α is adjusted for
the next period by the following formula:
α
t+ 2
= | (F
t+ 1
− S
t+ 1
)/S
t+ 1
| = |PE
t+ 1
| (5)
Because Equation (5) can produce values outside the range of α,
this calculation is adjusted by the following rules:
If |PE
t+ 1
| is equal to or greater than 1.0, then α
t+ 2
= 0.
If |PE
t+ 1
| is equal to 0.0, then α
t+ 2
= 0.
We can illustrate the adaptability of this technique by forecasting
the times series with level change in Figure 3.10, now Figure 3.13 for
Time Series Forecasting Techniques 91
J01F
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Demand Forecast
Figure 3.13 Adaptive Smoothing as a Forecast: Level Change
92 SALES FORECASTING MANAGEMENT
adaptive smoothing. To illustrate the changes in α that result in this
technique, the calculations are also reproduced in Table 3.1.
To get the process started, we used the usual convention of setting
the initial value of α at 0.1, although any value can be chosen without
changing the resultant forecasts. The reason for this is that we also
assume that the initial forecast was equal to the first period demand, so
the first forecast becomes:
F
= α S
So regardless of the initial value of α that is chosen, the forecast
for period two is always equal to sales from period one. The true
calculation of a forecast and the adapted values of α begin at that point.
Notice that the value of α stays low (well below 0.1) while the
time series is level (a low value of α dampens out the noise), but
as soon as the level changes, the value of α jumps dramatically to
adjust. Once the time series levels off, the value of α again returns
to a low level.
This adaptive smoothing technique overcomes one of the major
problems with exponential smoothing: what should be the value cho-
sen for α? However, all the techniques we have discussed so far have a
common problem: none of them considers trend or seasonality. Since
this technique assumes there is no trend or seasonality, our forecast of
January 2004 is 1950 and is also our forecast for every month in 2004—
we assume there will be no general increase or decrease in sales (trend),
nor will there be any pattern of fluctuation in sales (seasonality).
Because this is unrealistic for many business demand situations, we
need some way to incorporate trend and seasonality into our FMTS
forecasts. To do so, we temporarily set aside the concept of smoothing
constant adaptability and introduce first trend and then seasonality
into our exponential smoothing calculations.
Exponential Smoothing With Trend
Although we tend to think of trend as a straight or curving line
going up or down, for the purposes of exponential smoothing, it is
helpful to think of trend as a series of changes in the level. In other
words, with each successive period, the level either “steps up” or
“steps down.” This “step function,” or changing level pattern, of trend
is conceptually illustrated in Figure 3.14. Although demand is going up
