CHAPTER 3
TIME SERIES FORECASTING OF A PATTERNED PROCESS
INTRODUCTION
A time series measures an attribute, like sales, at equal intervals of time. When a
forecaster plots the sales history of a product, the plot often has a recurring pattern.
Usually time is on the horizontal axis, and sales in dollars or units is on the vertical axis.
Particular months may always be lower than other months year after year. Slow sales
may seem to grow each year.
Figure 3-1 Here
Figure 3-1 is an example of a sales time series that is a patterned process.
January and July are always higher than other periods. (This pattern is called seasonal).
Most consumer products are seasonal time series. The growth year after year is referred
to as trend.
SEASONAL PROCESSES
Seasonal processes prevail in almost every sales time series. A process which
detects a regular pattern of activity each year distinguishes them. The pattern is likely to
be predictable from year to year, thus allowing for the extrapolation of the time series over
time. Usually we assume a seasonal time series consists of monthly data, but the data
3-1
may also be grouped in quarterly information or weekly reports. In many cases the usual
assumption is that the seasonal period (pattern) repeats itself in one complete year.
Some causes of seasonal effects on sales appear below.
Weather:
Sunshine
increased demand for ice cream, sunglasses, beer,
soft drinks, camera film, and lotion
Cool weather heating fuels, clothing, anti-freeze, ski equipment, snow tires,
and tire chain
Rainfall
raincoats, umbrellas, and taxi service
Holidays:
Christmas
food, toys, trees, cards, travel, hotel accommodation,
sporting equipment, and clothing of all types
Easter
baskets, eggs, and fashion clothing
Mother's Day confectionery, certain clothing, cards,
and appliances
In each of the examples, the forecaster expects sales to increase during the
period, as compared to the rest of the year. Marketing people refer to "consumption
holidays" and run special sales on Labor Day or on President's Day. Some people who
have a day off from work because of the holiday spend the time shopping. This shopping
raises average sales for months with holidays.
Knowing this, forecasters must
accommodate the seasonal pattern in their forecasts. A first step in forecasting is to plot
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the data and observe if a seasonal pattern exists. Next the forecaster must find any
natural, repetitive factors that influence the sales of the product and cause it to sell in a
seasonal pattern. Holidays, weather and promotions like January white sales can create
the seasonal pattern. Many personal care products have a fall seasonal pattern because
some individuals "stock up" for the winter.
Again, since the seasonal effect is recurrent and periodic, it is predictable.
Seasonality is a very common forecasting problem, especially as a component of a firm's
sales pattern. In most retailing enterprises, sales are higher in December; breweries and
soft drink companies have sales patterns that follow the temperature--high sales in warm
weather and low sales in cool weather; ski manufacturers have a sales pattern that peaks
every winter. One could create a virtually endless list of time series that exhibit seasonal
patterns. In fact, it is difficult to find products that are not seasonal.
The seasonal pattern in figure 3-1 fits many clothing and household products.
Shoe retailers typically run January and July shoe sales every year, causing sales to be
greater in these two months than in other months. Many retail stores have annual white
sales every January and July, which also causes sales to surge during these two months.
Other traditional sales promotions that create seasonal patterns are post-Christmas,
President's Day, Easter, Lent, Fourth of July, back-to-school, and store anniversary sales.
In addition to traditional sales promotions, consumer habits alone often produce
seasonal patterns. Products such as pumpkin and mince pies, ham, turkey and corned
beef, eggnog, chocolates, ice creams, and wedding dresses all have seasonal sales
patterns because of consumer buying and consumption habits.
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Another cause of seasonality in sales is availability. Some products, like fresh
salmon, fresh strawberries, etc., are only available during certain seasons.
Whatever the cause, most businesses know their sales are patterned on a
seasonal basis.
Seasonality for sales in manufac-turing firms causes management
problems because it requires the firm to hire and then lay off in order to build large
inventories before the peak season. This increases the cost of storage and frequently
generates a cash pinch because of inventory and payroll costs.
A related problem is the sudden cash accumulation that occurs after the season.
Many seasonal industries note that they borrow when funds are dear and loan when
funds are plentiful.
Most production managers believe the most efficient use of production facilities
occurs when production operates at a constant level. Accordingly, managers often offer
price concessions during low sales periods to achieve more uniform operations by having
some sales occur before or after the season.
Another technique to smooth out the seasonal pattern is strong advertising during
nonseasonal periods. In recent years, the turkey growers' organization have tried this
approach in an effort to influence consumers to eat turkey during the spring, winter, and
summer seasons. These efforts may complicate the effort to forecast sales because the
seasonal pattern may shift over time.
A third approach to lessening the impact of seasonality on the company's
operations is to manufacture and market products that have complementary seasonal
patterns.
For example, a firm manufacturing passenger lawn mowers may add
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snowmobiles to its product mix to smooth operations by smoothing the demand for its
products.
Since seasonality can have such a dramatic impact on sales, the forecaster must
consider it in developing his forecasting system. Several approaches to seasonality have
been developed and are discussed later in this chapter.
TREND
In addition to seasonality, a second prominent pattern exhibited by most time
series is trend. Trend refers to a pattern where sales seem to increase or decrease over
a period of time.
Figure 3-2 Here
Figure 3-2 shows a time series with a trend pattern. Sales grow at a very consistent rate.
The trend shown above can be described as positive and linear. Some time series have
a nonlinear pattern.
Both linear and nonlinear trends can be expressed in a mathematical formula. For
example: Sales = constant + (slope x months) represents the mathematical pattern of
linear trend in figure 3-2. This is the simple regression model explained in the prior
chapter. The following values apply:
3-5
constant = 2400
1
slope = 100
If the forecaster wants to know sales at any period, he just substitutes it into the
formula. For example, for the 10th month, October:
Sales = 2400 + 100 x 10 = 3400 2
And for the 22nd month, October, one year later:
Sales = 2400 + 100 x 22 = 4600 3
Many times series have linear trend or near linear trend.
Those which are
nonlinear can often be segmented, so that over various ranges or segments of the time
series, the trend appears to be linear.
Figure 3-3 Here
The time series in figure 3-3 can be defined using three different linear equations. Many
nonlinear time series can also be represented with a single equation.
Figure 3-4 Here
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Figure 3-4 is an example of a nonlinear trend which can be represented by
Sales = 0 + 1000 x time2 4
(1)
Causes of Trend
Several factors cause trend.
One basic factor is population growth.
As the
number of potential buyers increases, the sales can increase. Trend can also be the
result of more people becoming aware of the product. More or fewer competitors can
cause a time series to develop a trend pattern. When the time series is expressed in
dollars, the trend is often caused by inflation.
Trends are significant factors in most time series, and the forecaster must
therefore determine what the trend is and how to account for it in the forecasting system.
Another pattern that may appear in the time series is cyclicality.
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Figure 3-5 Here
Figure 3-5 shows a time series with cyclicality. Cycles are really a special trend,
and some forecasting models can handle cycles as part of trend. When the forecaster
chooses to treat cyclic variations separately from trend, he must take cyclicality into
account with an equation that includes sine and cosine terms to match the pattern.
Complex Time Series
Most times series are not just a trend or a seasonal pattern, but a combination of
patterns. This complicates the forecasting efforts, since the forecaster must determine
the effects of seasons and trends to do a good job. The effects of trend and seasonality
may be interactive and difficult to separate. In such a case, forecasting can be very
difficult.
Decomposing the Time Series
One approach to creating a forecasting model is to decompose the time series into
its components or various patterns of trend and seasonality. Decomposing a time series
is the process of removing each component or pattern step by step using mathematical
procedures.
If the forecaster correctly identifies the mathematical equation for trend and then
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detrends the time series, he removes a source of variance and simplifies the time series.
If he then removes seasonality by determining the seasonal mathematical relationships
and adjusting the data for seasonality, he removes a second source of complexity in the
time series. What does the forecaster gain by this process? He mathematically models
various components of the time series. He can then put these mathematical relationships
into his forecasting system.
In theory and practice, the easiest time series to forecast is flat, linear, and stable.
Figure 3-6 Here
Figure 3-6 is such a time series. It is easy to forecast the next month. It is the
same as last month and the month before, etc. It is referred to as a stationary time
series.
Figure 3-7, on the other hand, is a complex time series with both seasonal and
trend patterns. It is quite difficult to forecast. The forecaster must decompose the series
by identifying each component and writing a mathematical function to express its
relationship to the others. He then combines the mathematical relationships to form a
forecasting equation or model.
The time series in figure 3-7 is tourists' buying airplane tickets to Hawaii. Figure 38 shows this time series with trend removed. Notice that it is still not a flat straight line,
because two additional components of the time series are to be removed. These are (1)
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atypical events, and (2) what forecasters call noise.
Atypical events are onetime sudden increases in orders and surges in sales
caused by atypical events, like promotions, temporary price changes, strikes, tornadoes,
legal actions, price freezes, etc.
The forecaster usually uses company records or
management discussions to identify atypical events. Once identified, the atypical event is
replaced--and the time series thus modified--with what managers or the forecaster feels
sales would have been without the atypical event.
Building forecasting models for
atypical periods is explained in chapter 5. Figure 3-10 shows the Hawaii-tourist time
series adjusted for seasonality, trend, and atypical events. Although the time series is not
a flat, repetitive line, it is nearly so. Forecasters attribute minor variations from a flat line
to noise.
Figure 3-7 Here
Figure 3-8 Here
Figure 3-9 Here
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Figure 3-10 Here
Noise is a random process of the dynamic environment. The underlying process
of seasonality, trend, cycles, and atypical events generates a time series. Noise is minor
variances that slightly shift the underlying process from month to month. These minor
shifts average to zero over the long run, so the expected value of noise is zero. The
forecaster does not write a mathematical expression to model noise. He generally checks
to see that noise (random error) is zero.
BUILDING THE FORECASTING SYSTEM OR MODEL
In developing a model, the first step is generally to identify what components exist.
There are three common approaches used to identify the components of a time series.
One approach is to plot the data and estimate which components are present from
the appearance of the plot.
A second approach is to use spectral analysis to determine the presence of
seasonality, trend, and cycles. Spectral analysis decomposes the total variations from
the stationary time series into contributions at a continuous range of frequencies. This
procedure assumes the time series is made up of sine and cosine waves with different
frequencies. Spectral analysis converts the time series from the time domain into the
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frequency domain. It then measures the intensity of various frequencies to determine
how significant and important they are to the time series.
A frequency of .085 is
equivalent to 1/12, or a yearly frequency when the time series is monthly observations.
If the intensity of the .085 frequency is high, the time series has a seasonal
component. Most spectral programs remove linear trend before calculating the intensities
at each frequency. A regression approach is used to remove the trend. The program
identifies the trend and the mathematical expression that models the trend. If the trend is
nonlinear, the regression procedure does not remove all the trend, and the intensity of the
low frequencies is very high.
A third method of determining the components of a time series is auto-correlation
and partial autocorrelation values.
This is the basis of ARIMA forecasting models.
ARIMA methods and autocorrelations are discussed in chapter 4.
Model Building
After forecasts have determined the components in the time series, they then have
to determine the model that includes these variables and the values for these variables.
The forecaster evaluates past sales to determine the model to use.
He calculates
necessary parameter values from past sales data. One of the advantages of time series
forecasting is the elimination of complex and expensive efforts at gathering external data
such as housing starts or interest rates. Also, time series forecasting generally produces
the most accurate sales forecasts. For many products, the past is an excellent predictor
of the future. Several studies have shown that for some products time series forecasting
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outperforms other forecasting techniques. Further, when the firm has many products to
forecast, it may be possible to use one type of time series model successfully with all the
products which results in a lower cost of forecasting sales. Using econometric forecasting
procedures, which were discussed in Chapter 2, frequently requires a different model for
each product, which greatly increases the data gathering and forecasting efforts and
costs. Finally, time series forecasts can be an automated process of an inventory control
system and the forecasts are automatically produced each period with very little cost.
TIME SERIES FORECASTING MODELS
The Running Average Forecasting Technique
This forecasting technique works well if trend is fairly constant and if seasonality is
not present in the data. The procedure takes past periods and averages them. The
average is the sales forecast. Each period, the forecast drops the oldest period and adds
the most recent period. Figure 3-11 is a three-month running average for times series
showing sales of airline tickets to Hawaii.
Figure 3-11 Here
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Table 3-1 Here
The major cause for error in this example is the seasonality of the time series, the
running average model doesn't model seasonality. Figure 2-12 shows the three-month
running average forecast with a seasonal factor incorporated into the model.
Incorporating seasonal factors greatly improves accuracy. The model has a bias toward
underforecasting because only a lagged trend is incorporated into the model. A special
case of the running average is the naive model. The forecast is last period's value (yt+1 =
yt). As discussed in Chapter 2, the naive model is used as a bench mark, and if other
models do not out perform the naive model, they are discarded.
Figure 3-12 Here
Table 3-2 Here
Seasonal Factors
Seasonal factors are most often multiplicative variables in the forecasting model.
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Sometimes an additive seasonal factor is calculated. Each month, forecasts use one of
twelve different seasonal factors--one for each month of the year. Seasonal factors
modify forecasts to match the peaks and valleys that seasonality causes in a time series.
The seasonal factor relates a particular month to the average month. One way to
calculate seasonal factors is to take each month's sales for a year and then average the
data. This average is a base by which each of the months' actual sales is divided. A
seasonal factor of .80 indicates that a month's sales are 20% less than the average
month's. A seasonal factor of 1.15 indicates that that month's sales are 15% greater than
the average month's sales. Figure 3-13 shows a plot of the calculated seasonal factors
used in the forecast shown in figure 3-12.
Figure 3-13 Here
Dividing each month by the average of the twelve months gives the values in figure
3-13.
Table 3-3 Here
The above technique has two problems. First, since only one year is used, one
atypical month causes erroneous seasonal factors to be calculated. An atypical event,
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such as a strike or price promotion, affects sales; and the seasonal factor reflects the
atypical event rather than the seasonal nature of the time series. In the above procedure,
atypical events are interpreted as seasonal influences.
To eliminate this problem,
forecasters generally use at least two years' data in calculating seasonal factors, or they
modify or change the atypical data to reduce the impact of the atypical event. Adjusting
for atypical periods is discussed in Chapter 5.
The second problem with the procedure is that the date has trend; that is, sales
are increasing over time. This trend effect is interpreted as seasonal effect. Removing
the trend in the data before seasonal factors are calculated eliminates this problem.
Figure 3-14 is a plot of the calculated seasonal factor using two years of detrended data.
Figure 3-14 Here
Table 3-4 is the calculation of trend, or monthly rate of change, in the time series
showing sales of airline tickets to Hawaii.
Column 2 indicates the month and year;
column 3 (labeled X) is the consecutive numbering of each month through twenty-four.
The average of column X is 12.5. Column 4 is actual sales. Column 5 is the difference
between each value of column 3 and X. Column 6 is column 4 multiplied by column 5.
The total of column 6 is 580,000. Column 7 is the values in column 5 squared. Column
7's total is 1150. Dividing the total in column 6 by the total in column 7 gives trend. In this
case trend is 504 per month. This means that because of trend, sales have grown by 504
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after two months and by 11,592 after 23 months.
The increase in sales due to trend is shown in column 1. Subtracting column 1
values from actual sales (column 4) one obtains the detrend-sales needed to calculate
seasonal factors. These values appear in column 8. It is this column's figures that are
used to calculate seasonal factors. The average of this column is 138,000.
Table 3-5 is the procedure for calculating the seasonal factors.
The two
deseasonalized values for each month are averaged, and this average is divided by the
overall average for all the detrended months.
Table 3-4 Here
Table 3-5 Here
An Additional Method of Calculating Seasonality
A second method of calculating seasonal factors, which doesn't require
detrending, is to divide the actual sales by the sum of that month and the 5½ months
before and after it. Using the actual data in table 3-4, one can calculate the seasonal
factor for July:
3-17
188,000
110,000+123,000+148,000+125,000+131,000+180,000+188,000
+219,000+135,000+149,000+123,000+162,000+1/2(115,000)
Taking equal amounts of data before and after the month whose trend is in
question eliminates the problem of interpreting trend for seasonality. Unfortunately, this
procedure doesn't lessen the problem of having atypical data periods interpreted as
seasonality.
This procedure is similar to the one used by the Census Bureau in
calculating seasonal factors in their X-11 program, which is the process most forecasters
use to calculate seasonal factors.
Additional Limitations of Running Averages
An additional limitation of the running average forecast is that the forecast always
lags behind the actual sales when trend is present.
To eliminate this problem, the
forecaster adds the trend variable to the forecast of sales. For sales of airline tickets to
Hawaii, the trend value is 504 per month. Thus if forecasters forecasted one month
ahead, they would calculate the running average and add 504 to it. If they forecasted two
months ahead, they would add 1008 to their forecast.
A Trend Forecasting Model
If seasonality is not present in the time series, it is possible to dispense with the
running average and simply calculate a trend variable. Adding the trend variable to last
month's actual sales allows next month's sales to be forecasted. The procedure for
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calculating trend is found in table 2-4. This procedure is applicable to both positive and
negative trends. If the trend is positive, the trend variable is a positive value. If the time
series slopes downward, the trend variable is a negative value.
The Exponential Smoothing Model
This model has received widespread acceptance among American business firms
that employ sales forecasts for managerial planning and control. Most computer software
packages for inventory control use exponential smoothing for forecasting. Exponential
smoothing is generally the most accurate of the time series forecasting models.
Exponential smoothing models use special weighted moving averages and a
seasonal factor that is multiplied by the weighted moving average to calculate the
forecast. These weighted moving averages are referred to as smoothing statistics. The
exponential smoothing models are an extension of the running average model.
Generally, exponential smoothing uses three smoothed statistics that are
weighted, so that the more recent the data, the more weight given the data in producing a
forecast. These three averages are referred to as single, double, and triple smoothing
statistics and are running averages that are weighted in an exponential declining method.
Most forecasting systems use three separate forecasting equations: one model
called a constant model, a second called a linear model, and a third called a quadratic
model. The constant forecast uses only the single smoothed statistic and is best when
the time series has little trend. The linear forecast model uses the single and double
smoothed statistics and is best when there is a linear trend in the time series. The
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quadratic forecast model uses all three statistics--single, double, and triple smoothed.
A forecasting system using the exponential smoothing model continually evaluates
forecasting accuracy and selects the forecasting equation with the most accurate
forecasting history. If the equation no longer forecasts as accurately as one of the other
equations because the environment has changed, the system automatically switches to
the equation that will forecast best.
adaptive forecasting.
This automatic switching of equations is called
Other adapting techniques sometimes built into the model are
adapting the weighing scheme (changing the smoothing constant used) and smoothing
past errors into future forecasts to reduce bias.
If a forecast is consistently
underforecasted, adding the average past error into the forecast, will eliminate the bias.
An example of exponential smoothing is given in both table 3-6 and Table 3-7.
Table 3-6 is the retail department store sales for Phoenix from January 1974 to
December 1980. These values are used to establish an exponential smoothing model to
forecast future retail department store sales. Table 3-7 shows the forecast's results.
When the actual sales for 1981 are compared with the forecasted sales, it is clear that the
model can accurately forecast future department store sales.
Table 3-6 Here
Table 3-7 Here
3-20
The Smoothing Process
Each month all three smoothing statistics are updated by the most recent month's
sales. The process of updating is called smoothing because a fixed percent, alpha (), of
the most recent sales is added to (1-) times the old single smoothing statistic. For
example, if  were .3 and sales were X, a new single smoothing statistic St[1](X) would be
calculated by taking .3X + (1 - .3) times St-1[1](X) = St[1](X). Note that St-1[1](X) refers to last
month's single smoothing statistic. If t-2 were used in the statistic, it would refer to the
statistic two months ago. Next month's statistic would be t + 1. The t in St[1]X is the
current statistic. This is a convenient notational technique to designate what statistic is
being referred to. The [1] in St[1] does not refer to an exponential power, but rather
identifies that this is the single smooth statistic.
St[2](X) refers to the double smoothed statistic. The (X) in both statistics signifies
that the statistic is calculated from the time series being examined (sales).
It was explained in Chapter 2 that to signify our forecast, we useX
t+1.
The ^ is
referred to as hat. This signifies that the value under the hat is an estimate rather than an
actual value.X
t+1
is an estimate of X. The t+1 signifies that the equation is the estimate
for t, the current period plus 1, which is next month. The error of estimation is calculated
for next month by (Xt - Xt). In one month, Xt+1 will have become Xt.
The double smooth statistic is calculated like the single smooth statistic with one
exception: actual sales are replaced by the new single smoothing statistic. Last month's
double smoothing statistic St-1[2](X) is multiplied by (1-) and added to St-1[1](X), giving
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St[2](X). The box 2, [2], indicates this is the double smoothing statistic.
The triple smoothing statistic continues the process. The new triple smoothed
statistic is calculated by adding St[2](X) to (1-) St-1[3](X). The smoothing process takes
into account trend and cycles but ignores seasonality. Therefore, a seasonality variable
is included in the forecasting equation. To keep from confounding or combining seasonal
effects and trends or cycles, we divide the actual sales data (Xt) by the seasonal factor for
month t before it is used in the smoothing process. If the seasonal factor is signified by
t, the single smoothed statistic for seasonal data is calculated
Xt
[1]
[1]
S t (X) = d   + (1 -  ) S t -1 (X) 5

t
 
(2)
The procedure for calculating seasonal factors is given in an earlier section of this
chapter.
The Forecasting Equations
The three forecasting equations are designed to generate forecasts that follow
three different patterns. The constant equation models a flat, constant pattern. This
forecasting equation isX
t+1
= t+1(St[1](X)). Figure 3-15 shows the patterns of time series
forecasted best by the constant equation.
Figure 3-15 Here
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The linear equation isX
t+1
= t+1[d1St[1](X)-d2St[2]]. The patterns that this equation
models best are those showing a linear trend. In the equation, d1 and d2 are lag factors.
The smoothing statistics lag the actual values. The value of d1 is 2 +
for d2 is 1 +
 (1 -  )
6. The lag

 (1 -  )
7. Figure 2-15 also illustrates the patterns of time series forecasted

best by the linear equation. The number of future periods for which the forecast is being
made is indicated by . If we are forecastingX t+1,  equals one; if we are forecasting Xt+2,
 equals 2; etc.
The quadratic forecasting equation is designed to forecast time series with
nonlinear trend. The equation is as follows:
[1]
[2]
[3]
Xˆ t+1 =  t+1 T 1 S t (X) - T 2 S t (X) + T 3 S t (X) . 8
(3)
T1, T2, and T3 are lag correction variables. The calculation of lag coefficients is
somewhat mathematically complicated and is as follows:
T1=
6(1 -  )2 + (6 - 5 )   +  2  2
9
2(1 -  )2
(4)
T2=
6(1 -  )2 + 2(5 - 4 )   +  2  2
10
2(1 -  )2
(5)
T3=
2(1 -  )2 + (4 - 3 )   +  2  2
11
2(1 -  )2
(6)
An example of exponential smoothing forecasting can be made using the seasonal
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factors in figure 2-13. If the initial smoothing statistics, after smoothing in the December
1971, were
[3]
[2]
[1]
S t (X) = 184,000 , S 1 (X) = 185,000 , and S t (X) = 190,000 12
and the smoothing constant =.1 and January seasonal factor equals .80, the January
1972 forecasts would be:
Constant:
[1]
Xˆ t+1 =  t+1 ( S t (x))= .80(190,000) = 152,000 13
(7)
Linear:
[1]
[2]
Xˆ t+1 =  t+1 ( d 1 S t (X) - d 2 ( S t (X))
= .80(2.1111(190,000) - 1.1111(85,000)) 14
= .80(401,109 - 205,553) = 156,444
(8)
Quadratic:
(9)
[1]
[2]
[3]
Xˆ t+1 =  t+1 ( T 1 S t X - T 2 S t X + T 3 S t X)
= .80(3.34568 x 190,000 - 3,58025 x 185,000 + 1.23457 x 184,000) 15
= .80(633,566 - 662,344 + 227,160) = .80(198,392) = 158,705
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If actual sales for January 1972 were 159,097, to make a forecast for February
1972, the forecasting program would smooth in the January sales:
Xt
[1]
[1]
S t =    + (1 -  )( S t (X))
 t 
159,097
=(
) + .9(190,000) 16
.80
= 19,887 + 171,000 = 190,887
(10)
[2]
[1]
[2]
S t (X) =  S t (X) + (1 -  ) S t -1 (X)
17
= .1(190,887) + (185,000)
= 19,089 + 166,500 = 185,589
(11)
[3]
[2]
[3]
S t (X) =  S t (X) + (1 -  ) S t -1 (X)
= .1(185,589) + .9(184,000) 18
= 18,559 + 165,000 = 184,159
(12)
February forecasts would be,
Constant:
.82(190,887) = 156,527 19
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Linear:
.82[(2.11111)(190,88 7) - (1.11111)(185,589)] = .82(1996,774)
20
= 161,354
Quadratic:
.82[(3.34568)(190,887) - 3.580251(185,589) + 1.23457(184,159)] = 165,271 21
Actual sales in February 1972 were 191,424.
To forecast March 1972 sales, the program would smooth in the February sales
and then make a new forecast. For April, (Xt+2), the lag coefficients d1, d2, T1, T2, and T3
would be calculated with =2.
Gardner (1985) discusses various methods used to estimate initial smoothing
statistic values.
Makridakis and Hibon (1991) have looked at the effect that initial
smoothing statistic values have on forecasting accuracy.
Selecting Alpha
Alpha is selected by a simulation method. Various alphas between .01 and .99 are
used to forecast an initial period. Usually, the same years used to calculate seasonal
factors are also used to determine which alpha predicts best.
The criterion for
determining the alpha is accuracy. Usually the constant linear and quadratic equations
are tested with various alphas. The forecast with the lowest error determines which alpha
3-26
and model to use for future forecasts. The larger the alpha, the greater the weight of
recent data and the less the weight of earlier data. The formula for determining the
weight of data (past months) is (1-)k, where K is the data's age. The weight of this
month is equal to  in the smoothing statistic since k=1; the weight of this month in the
smoothing statistics is (1-). Theoretically, this way of calculating smoothing statistics is
appropriate because the most recent data have the greatest impact on the forecast.
Figure 3-16 Here
Figure 3-16 shows the weights for various values of alpha. Note how quickly the
effects of past data diminish as alpha increases. When alpha equals .1, the data affect
the smoothing statistic for eighteen months. This is in contrast to an alpha of .3, which
only uses nine months of data to calculate the statistic.
The initial values of the smoothing statistic also are often determined with the initial
data from which seasonal factors were calculated. The procedure is to start each of the
three smoothing statistics at the earliest month's sales value and then to smooth in actual
sales up to the period for which forecasts will be made. The resulting smoothing statistics
are used to start forecasting. Another way of establishing smoothing statistics is to use
the average monthly sales for all three smoothing statistics as the initial smoothing
statistic values. A third procedure is to forecast backwards the past values from a point in
time--this is called backcasting. The firm then forecasts from the last backcasted value to
3-27
the present value. The forecasting system updates smoothing values ('s) each time new
forecasts are made.
Broze and Me'lard (1990) have suggested a maximum likelihood approach to
estimating alpha. The alpha used is an important part of building an accurate forecasting
model. Research has been done to determine the best alpha level (for example, see
Newbold and Bos [1989] and Gardner [1985]).
CORRECTING BIAS
If a forecast does not have an average error value of zero over time, the forecast is
biased. The forecast is consistently either an over-forecast or an under-forecast. Adding
or subtracting a constant increases forecasting accuracy. Exponential smoothing models
often include the addition of an exponentially smoothed error term to the forecast to
correct bias. The process smooths the errors, just as the actual data is smoothed. An
error smoothing statistic is calculated. Each time a forecast is made, an error is calculated
by subtracting the actual forecast, and the error is smoothed into the old error smoothing
statistic. This error smoothing statistic is then added to the next forecast. If the model
underforecasts, the errors are positive and the error smoothing statistic is also positive; so
the next forecast will be increased. If the forecasts are consistently higher than the
actual, the errors are negative, and the error smoothing statistic is negative. Thus, adding
the error smooth statistic will reduce the next forecast. Usually only the constant (single
smoothed) error smoothing statistic is calculated, and usually a small alpha, such as .1, is
used. When a forecast is biased over time, it is most often or usually because the wrong
3-28
model or wrong parameter values are used. Thus the forecaster should use a different
model or different parameter values when a bias exists. However, when the bias is small,
the forecaster may have the best model and best parameter values. The error correction
smoothing statistic makes minor adjustments that modestly improve forecasts. Cipra
(1992) has suggested a method of making exponential smoothing robust to outliers which
reduces some bias errors.
Advantages of Exponential Smoothing Forecasting
The advantages of the exponential model are many. Although the process is
somewhat tedious when done by hand, the process is straightforward and easy when
programmed for a computer.
Many computer programs have been written that use
exponential smoothing, and they are readily available.
Chapter 11 discusses many
computer programs that use exponential smoothing forecasting models. Exponential
smoothing models adapt to environmental changes and are self-correcting. The firm with
many products needs only one program with three equations to forecast all its products'
sales. The procedure has proved effective for a wide spectrum of product forecasts. Only
past data are used; thus the problem and expense of collecting external data are
eliminated.
SUMMARY
Time series data contain lots of information about the sales processes of the
company. Some factors hidden in the data are seasonality, trend, and cyclicality of the
sales. Seasonality is dependant on the characteristics of the product and the people
3-29
buying it. Managers attempt to decrease the seasonality of sales. A time series forecast
can mathematically explain products' seasonality and show how the seasonality changes
over time. Trends and cycles also occur with sales. Trends can be modeled as linear,
segmented to linear, or nonlinear. (Cyclical sales can be modeled by an equation using
sine or cosine functions).
These sales patterns make up the underlying process of sales for the company.
Each component of the sales pattern can be detected with the time series data detection
which more fully describes the underlying process of sales.
Three approaches to
breaking the time series data into components are (1) to plot the data and estimate the
components, (2) to do a spectral analysis on the data, and (3) to use ARIMA forecasting
models (see Chapter Four).
Two models explained in the chapter are the running average forecasting
technique and the exponential smoothing model.
The running average forecast
calculates seasonal factors, which are multiplied by the average sales of the previous
three months to determine the monthly sales forecast. A trend factor is used to increase
the accuracy of the running average method.
The exponential smoothing technique is a self-adjusting weighted running
averages model that uses a special weighted moving average factor and the seasonal
factor to calculate the forecast. The three types of exponential smoothing forecasts are
the constant forecast, the linear forecast, and the quadratic forecast.
Each uses a
different equation with a different weighted moving average (smoothing statistic). Alpha
values and bias correction factors adjust exponential smoothing forecasts.
3-30
3-31
Chapter Three Questions
1.
What are the three types of variability within time series data?
2.
Give three examples of how managers can decrease seasonality effects.
3.
Give four factors which can cause trends in sales.
4.
After one decomposes a time series and subtracts out the components, why does
the remaining data often still show variability?
5.
What are the three approaches discussed in the chapter for building a forecasting
model?
6.
Explain the running average forecasting technique. What kind of data does it work
well with?
7.
What is a seasonality factor? How is it calculated?
8.
Give two problems associated with the running average forecasting technique.
9.
What are the three forecasting equations? What kind of data does each equation
forecast?
10.
What are the smoothing statistics? How are they used in the three forecasting
equations?
11.
Which variable--trend or seasonality--is included in the smoothing process and
which isn't?
12.
How does the alpha variable affect the forecast?
13.
How is bias subtracted from an exponential smoothing forecast?
3-32
REFERENCES
Broze, L. and G. Me'lard (1990), "Exponential Smoothing: Estimation by Maximum
Likelihood," Journal of Forecasting Vol. 9, 445-455.
Cipra, T. (1992), "Robust Exponential Smoothing," Journal of Forecasting 11, 57-69.
Gardner, E. S., Jr. (1983), "Exponential Smoothing: The State of the Art," Journal of
Forecasting 4, 1-28.
Granger, C. W. J. and P. Newbold (1972), "Economic Forecasting--The Atheist's
Viewpoint," Nottingham University Forecasting Project, Note 11.
Jenkins, G. M. and G. C. Watts (1968), Spectral Analysis and Its Applications (San
Francisco: Holden-Day).
Makridakis, S. and M. Hibon (1991), "Exponential Smoothing: The Effect of Initial Values
and Loss Functions on Post-Sample Forecasting Accuracy," International Journal
of Forecasting 7, 317-330.
Naylor, T. H., T. H. Seaks, and D. W. Wichern (1972), "Box-Jenkins Methods: An
Alternative to Econometric Models," International Statistics Review Vol. 40, No. 2.
Newbold, P. and T. Bos (1989), "On Exponential Smoothing and the Assumption of
Deterministic Trend Plus White Horse Data-Generating Models," International
Journal of Forecasting 5, 523-527.
Parsons, L. J. and W. A. Henry (1972), "Testing Equivalence of Observed and
Generated Time Series Data by Spectral Methods," Journal of Marketing
Research 9, 391-395.
3-33
Table 3-1
AIRLINE TICKETS SOLD TO HAWAII
Three-Month Running Average to Forecast 1970
Previous
3 Months
Month
Error
Jan., 1970
D,N,O,/3 -
Feb.
J,D,N,/3 -
127
122
5
Mar.
F,J,D,/3 -
129
148
19
Apr.
M,F,J,/3 -
129
124
5
May
A,M,F,/3 -
131
130
1
June
M,A,M,/3 -
134
180
46
July
J,M,A,/3 -
145
188
43
Aug.
J,J,M,/3 -
166
217
51
Sept.
A,J,J,/3 -
195
134
61
Oct.
S,A,J,/3 -
180
149
31
Nov.
O,S,A,/3 -
167
123
44
Dec.
N,O,S,/3 -
135
163
28
*All figures in thousands.
Forecast
125
Actual
118
7
Table 3-2
AIRLINE TICKETS TO HAWAII TIME SERIES
Three-Month Running Average with Seasonal Factors
1970 Forecast
S.F.
Jan.
.85
.86
.89
.84
.97
1.21
1.19
1.20
.94
1.02
.90
Dec. 1.14
x
x
x
x
x
x
x
x
x
S.F. x 3 Mo. Ave.
3 Mo. Ave.
Forecast
131
127
129
129
131
134
145
166
195
180
167
135
*All figures in thousands.
-
111
109
114
108
127
162
173
199
183
184
144
159
Actual
118
122
148
124
130
180
188
217
134
149
123
162
Error
- 7
-13
-34
-16
- 3
-18
- 7
-18
+49
+35
+21
- 3
Table 3-3
SEASONAL FACTORS FOR VISITORS TO HAWAII
.12 Seasonal factors are developed for a year.
.Forecast is equal to last 3 month average x S.F.
Develop seasonal factors by
1.
2.
3.
Sum of 12 month sales
Divide sum by 12 giving average
Divide each month's sales by average seasonal factors
1969 J
110
110/129 =
.85
F
111
111/129 =
.86
M
115
115/129 =
.89
A
108
108/129 =
.84
M
125
125/129 =
.97
J
156
156/129 =
1.21
J
152
152/129 =
1.19
A
155
155/129 =
1.20
S
121
121/129 =
.94
O
131
131/129 =
1.02
N
116
116/129 =
.90
D
147
147/129 =
1.14
Total =
1547
1547
= 129
12
The symbol used for seasonal factors is .
Table 3-4
TREND CALCULATIONS**
(REGRESSION)
Column 1
2
Detrended
3
X1
4
Y1
5
X1-X
6
Y1(X1-X)
3024
115000
3528
119000
4032
144000
4536
120000
5040
126000
5544
174000
6048
*182000
6552
210000
7056
128000
7560
141000
8064
119000
8
(X1-X)
Sales
Dat
0
132000
504
154000
1008
120000
1512
129000
2016
114000
2520
144000
7
1969
Jul.
1
152000
-11.5
-1748000
132.25
Aug.
2
155000
-10.5
-1627500
110.25
Sep.
3
121000
-9.5
-1149500
90.25
Oct.
4
131000
-8.5
-1113500
72.25
Nov.
5
116000
-7.5
-870000
56.25
Dec.
6
147000
-6.5
-955500
42.25
1970
Jan.
7
116000
-5.5
-638000
30.25
Feb.
8
123000
-4.5
-553500
20.25
Mar.
9
148000
-3.5
-518000
12.25
Apr.
10
125000
-2.5
-312500
6.25
May
11
131000
-1.5
-196500
2.25
Jun.
12
180000
-0.5
-90000
0.25
Jul.
13
188000
0.5
94000
0.25
Aug.
14
217000
1.5
325500
2.25
Sep.
15
135000
2.5
337500
6.25
Oct.
16
149000
3.5
521500
12.25
Nov.
17
123000
4.5
553500
20.25
8568
153000
9072
106000
9576
117000
10080
126000
10584
133000
11088
133000
11592
153000
Dec.
18
162000
5.5
891000
30.25
1971
Jan.
19
115000
6.5
747500
42.25
Feb.
20
127000
7.5
952500
56.25
Mar.
21
136000
8.5
1156000
72.25
Apr.
22
144000
9.5
1368000
90.25
May
23
144000
10.5
1512000
110.25
Jun.
24
165000
11.5
1897500
132.25
 = 300
X
138000
* Mid-point
**
584000
1150
3312000
= 12.5
Rate of growth of time series
b = 584000 / 1150 = 507.83
Table 3-5
CALCULATION OF SEASONAL FACTORS*
Total
Jul.
Ave.X
SF
152 + 182 = 334
167
138
1.21
154 + 210 = 364
182
138
1.32
Sep.
120 + 128 = 248
124
138
0.90
Oct.
129 + 141 = 270
135
138
0.98
Nov.
114 + 115 = 229
115
138
0.83
Dec.
144 + 153 = 297
149
138
1.08
Jan.
115 + 106 = 221
111
138
0.80
Feb.
119 + 117 = 236
113
138
0.82
Mar.
144 + 126 = 270
135
138
0.98
Apr.
120 + 133 = 253
127
138
0.92
May
126 + 133 = 259
130
138
0.94
Jun.
174 + 153 = 327
164
138
1.19
Aug.
*All figures in thousands.
Table 3-6
DATA USED TO BUILD THE MODELS
(Phoenix Department Store Sales)
Date
Actual Sales
1974
January
February
March
April
May
June
July
August
September
October
November
December
27356
26162
32315
34335
36267
32430
30522
33921
29202
34380
39132
62252
1975
January
February
March
April
May
June
July
August
September
October
November
December
27356
26678
33820
33095
37616
33733
31402
33895
30903
37299
43412
74401
1976
January
February
March
April
May
June
July
August
September
October
November
December
31884
31282
38242
39452
39883
38204
37335
37563
35274
40087
48239
84307
1977
January
February
March
April
May
34738
36300
44711
43917
42765
Date
Actual Sales
July
August
September
October
November
December
40222
40880
39368
44684
54551
94653
1978
January
February
March
April
May
June
July
August
September
October
November
December
40454
38627
49760
47368
49245
49616
44119
46646
45027
49026
63309
106068
1979
January
February
March
April
May
June
July
August
September
October
November
December
47214
45922
56728
58598
56725
55832
52031
57071
54709
58701
74400
119907
1980
January
February
March
April
May
June
July
August
September
October
November
52168
53264
59909
60377
62341
56022
53273
58744
55393
62947
74187
June
40288
December
124151
Table 3-7
Department Store Forecast and Actual Sales
1981
Actual
sales
Forecasted
sales Xt+1
Error
%Error
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
54169.0
54739.0
69447.0
69628.0
68359.0
64448.0
59743.0
63727.0
58453.0
67251.0
78180.0
54309.4
52566.0
65526.8
67644.1
68500.5
63625.9
59907.5
63641.4
59163.4
65978.7
78506.6
140.4
-2173.0
-3920.2
-1983.9
141.5
-822.1
164.5
-85.6
710.4
-1272.3
326.6
0.26
-3.97
-5.64
-2.85
0.21
-1.28
0.28
-0.13
1.22
-1.89
0.42
Chapter 3
Time Series Forecasting of a Patterned Process
May 12, 1992