Chapter 8 Appendix: Forecasting
Think of what you could accomplish if you could see the future. Of course, no one can
truly predict future events. However, in many situations a forecast can provide a good idea
of the possibilities. Of course, forecasting patterns for the near future is much easier than
predicting what might happen several years out.
Forecasting is used in many areas of business. Marketing forecasts future sales, the
effect of various sales strategies, and changes in buyer preferences. Finance forecasts future
cash flows, interest rate changes, and market conditions. The HRM department builds
forecasts of various job markets, the amount of absenteeism, and labor turnover. Strategic
managers forecast technological changes, actions by rivals, and various market conditions.
Sometimes these forecasts are built on intuition and rules of thumb. But, it is better to use
statistical techniques whenever possible.
The science of forecasting is dominated by
P
S
two major approaches: time-series forecasting that
identifies trends over time, and structural modeling
that identifies relationships among the underlying
D’
D
variables. Many forecasts require the use of both
Q
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Structural model forecast. The underlying
model helps explain the causal relationships,
making it easier to forecast the effect of
changes. Here, an increase in income shifts
the demand curve, which causes increased
sales (Q) even at higher prices.
techniques.
As shown in Figure 8.1A, by focusing on the
underlying model, structural modeling seeks to
identify the cause of changes. For example, if
consumer income increases, the demand for our
product shifts out, which results in more sales. If we know the shapes of the supply and
demand curves, it is straightforward to predict how sales will increase.
Figure 8.2A shows a time-series-approach to
sales
trend
the same issue of sales forecasting. In some ways it
is simpler. We know nothing about the underlying
model and have simply collected sales data for the
past few months. The data is plotted over time. By
time
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Time series forecast. Sales change over time.
The trend line provides a forecast of future
values. But it assumes that underlying factors
behave as they did in earlier times.
fitting a trend line to the data, it is clear that sales
are increasing. Assuming that this trend continues,
it is easy to forecast sales for the next period.
It is often tempting to say that structural
models are “better” than time series forecasts. After all, the model provides an
understanding of the causal relationships. So we know that the increase in sales is
occurring because of increasing consumer income. While this knowledge is valuable, the
structural model does not tell us how fast income will increase in the next period. Hence,
we end up using time-series techniques to estimate the trends in consumer income.
Consequently, we need both techniques. As much as possible, find a structural model to
explain the problem. Then use time-series methods to estimate the underlying trends. Plug
these forecasts into the structural model to determine the outcome of the desired variables.
Structural Modeling
Modeling an underlying structure provides the most information and knowledge
about a problem. Consider a simple physics problem: If you throw a ball at a certain angle,
with a certain force, how far will it travel? You could try several experiments, timing each
event and measuring the outcome. You could then use this data to make a forecast of future
attempts. However, if you know the underlying model of gravity (e.g., Newton’s equations),
then it is easy to determine the outcome of any attempt.
Many economic models have been developed to determine relationships that apply
to business decisions. For instance, cost models are used to determine supply relationships,
and consumer preferences generate demand curves. Demand for a product can be expressed
as a function of several variables: price, income, and prices of related products. These
relationships can be estimated with common statistical techniques (e.g., multiple
regression).
Figure 8.3A presents the
Model
QD = b0 + b1 Price + b2 Income + b3 Substitute
basic steps in using a structural
Quantity Price
Income
Substitute
1
24926
134
20000
155
Data
2
26112
150
21000
155
model to forecast sales demand.
3
27313
142
22000
135
4
26143
141
21000
150
5
26741
144
21500
150
First you need a model—in this
Estimate
QD = 11146 - 0.126149
Price + 1.2 Income
-21000
1.0 Substitute
137
155
7
27893
140
22500
143
case, a basic economic model.
Forecast
33318 = 1114
- 26397
0.1 (155) + 1.2
- 1.0 (160)
8
142 (20000)
21200
153
9
24895
147
20000
155
Figure Error! No text
specified148
style 23000
in
10 of 28501
160
Then you need to collect data for
11
29747
150
24000
165
document..3A
12
29175
134
23500
153
Forecasting process using 13
a structural
Collect23700
data and use158
each of the variables in the
29413model.142
regression to estimate the 14
underlying
plug in 162
30348parameters.
147 Then
24500
estimates of the future values
the independent
15 of 32148
150 variables
26000 to forecast
163
model. It is best if the underlying
33327
152
27000
164
the value of the dependent16variable
(quantity).
17
34532
158
28000
165
34775
28200 observations
160
variables change over18time,
and you162
will need
from several points in time. It is
19
33372
157
27000
155
20
32681
155
26500
160
Time
best to have at least 40 observations, but you get better results with more data. Next you
use regression analysis to estimate the values of the model parameters. Finally, you plug in
estimates of the independent variables to obtain a forecast of the future sales.
Time-Series Forecasts
When you do not have a structural model, or when you need to forecast the value of
an underlying variable, you can use time-series techniques to examine how variables
change over time. The basic process is to collect data over time, identify any patterns that
exist, and then extrapolate this pattern for the future. The approach assumes that the
underlying pattern will remain the same. For example, if income has been gradually
increasing over time, it assumes that this increase will continue.
Figure 8.4A shows a time
sales
series consists of a number of
observations made over a period of
Seasonal
Trend
time. Four types of patterns often
arise in time-series data: (1) trends,
time
Dec
Dec
Dec
(2) cycles, (3) seasonal variations,
Dec
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Time series components. A trend is a gradual change over time.
Seasonal patterns show up as peaks and troughs at annual
intervals. Other cycles are similar but cover longer time periods
and are usually less regular. Random change is shown because
the cycle and trend are not perfect.
and (4) random changes. A trend is
a gradual increase or decrease over
time. A cycle consists of up and
down movements relative to the
trend. Seasonal variations arise in many disciplines. For instance, agricultural production
increases in the summer and fall seasons, and many industries experience an increase in
sales in November and December due to holiday sales. Random components are variations
that we cannot explain through other means. In some cases, the random component
dominates the others, and forecasting is virtually impossible. For example, many people
believe this situation exists for stock market prices.
Exponential Smoothing
Random variations make it difficult to see the underlying trend, seasonal, and
cyclical components. One solution is to remove these variations with exponential
smoothing. Exponential smoothing computes a new data point based on the previously
computed value and the newly observed data value. The weight given to each component is
called the smoothing factor. The higher the smoothing factor, the more weight that is given
to the new data point. Typical values range from 0.2 to 0.30, although it is possible to use
factors up to 1.0 (which would consider only the new value and ignore the old ones). Lower
values (down to 0.01) put more weight on previous computations and result in a smoother
estimate.
Current spreadsheets (e.g.,
Exponential Smoothing
Microsoft Excel) have procedures
1600
1500
that will quickly estimate the
1400
1300
Raw Data
1200
1100
1000
Smooth:0.20
moving average for a range of data.
You simply mark the range of data,
900
800
1
3
5
7
9
11 13
15
17
19
21
highlight the output range, and
supply the smoothing factor. As
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Exponential smoothing. The smoothed data is computed by Excel
by applying a smoothing factor to each new data point.
shown in Figure 8.5A, it is then
easy to graph the original and the
smoothed data.
How do you choose the smoothing factor? The best method is to apply several
smoothing factors (start with 0.10, 0.20, and 0.30), and then examine the accuracy of the
result. The accuracy is typically measured as the sum-of-squared errors. For each smoothed
column of data, compute (actual – smoothed)*(actual – smoothed) to get the squared-error
on each row. Add these values to get the
Double Exponential Smoothing
total. Now compare these totals for each
34000
of the smoothing factors. The smoothing
32000
30000
28000
Raw Data
26000
Smooth:0.20
factor with the smallest error is the one
24000
22000
to use.
20000
1
3
5
7
9
11
13
15
17
19
In practice, most data will have a
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Double exponential smoothing. When there is a trend, you
need to smooth the data a second time.
trend component. In these cases, you
need to use double exponential
smoothing. Perform the first smoothing as usual and find the best smoothing factor. Then
perform a second smoothing on the new (smoothed) column of data using the same
smoothing factor. Figure 8.6A shows that the result follows the basic trend line but still
incorporates the cyclical variations in the data.
The smoothed data can be used
Forecast for time T+
 
  [ 2]



yT    2 
 S T  1 
 ST
 1 
 1 
T = 20
=1
 = 0.2
S20 = 32,064
last of the raw data
forecast one period ahead
smoothing factor
(value at time 20, after one smoothing)
S[2] = 33,141
(value at time 20, after second smoothing)
Y21 = (2.25)32,064 - (1.25)33,141
= 30,718
to forecast the dependent variable for
future periods. It is wise to stick with
forecasts only one or two periods ahead.
Longer range forecasts are less likely to
be accurate. The basic formula is given
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in Figure 8.7A. You need the smoothing
Forecasting with double exponential smoothing. Plug your
values into this formula to estimate future values of the data.
factor () and the number of time
periods ahead to forecast (). Then you take the smoothed values at the last data point
(single and double) and plug them into the formula. The result is a forecast that
incorporates a linear trend
Time Quantity Trend Difference
1
24917 24484
432
2
26152 24983
1169
3
27297 25482
1816
4
26157 25980
177
5
26710 26479
231
6
26103 26977
-874
7
27981 27476
505
8
26327 27975
-1647
9
24913 28473
-3560
10
28524 28972
-448
11
29774 29470
303
12
29136 29969
-833
13
29332 30468
-1136
14
30306 30966
-660
15
32133 31465
669
16
33329 31963
1366
17
34522 32462
2060
18
34769 32961
1808
19
33355 33459
-104
20
32684 33958
-1274
21
34456
22
34955
23
35454
24
35952
along with the basic cyclical
Yt = b0 + b1(t)
Use regression to estimate b0 and b1.
Intercept
Time
Coefficients Std Error t Stat P-value
23985.81
652.48
36.76 2.2E-18
498.60
54.47
9.15 3.4E-08
Plug t into equation to estimate new value (on trend):
variations in the data.
If you want a
forecast that utilizes only
the trend (and none of the
Y21 = 23,986 + 498.6 * (21)
= 34,456
cyclical variations), you can
Result is the prediction on the trend, with no
random factors and no cycles.
use a simple regression
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Forecasting with linear trends. Use regression to estimate the two parameters.
Then plug in the desired time value to obtain the predicted trend value for
any new time period.
technique. Use standard
regression techniques with
the observed data as the
dependent (Y) variable and time (t) as the independent variable. Then use the computed
parameters to estimate the predicted value at any future point in time. Again remember
that linear trends may not continue for extended time periods, so keep your predictions
down to a few periods. Figure 8.8A illustrates the process, using Excel to obtain the
regression coefficients. The result is the prediction along the trend line, which ignores
cyclical variations.
Once you compute the trend factor, you can subtract it from the original series to
identify the cyclical, seasonal, and random components. Use the spreadsheet to plug in
values for each time period and compute the trend line. Then subtract these values from
the original series. If you plot the new series, you will see the data without the trend. It
should be easier to see seasonal and cyclical patterns on this new chart.
Seasonal and Cyclical Components
Two powerful methods exist to decompose time series data into its trend, seasonal,
and cyclical components. They are Box-Jenkins and Fourier analysis. You can purchase
tools that will perform the complex calculations for these methods. Unfortunately, it would
take many pages to describe either one of these techniques, so they are beyond the scope of
this appendix. Just remember that it is possible to perform much more detailed analyses
(and forecasts) of time series data. If you need them, hire an expert, or take a separate class
in time series forecasting.
Exercises
1. Obtain a three-year set of monthly data from the Bureau of Labor Statistics Web site
(http://stats.bls.gov) that is not seasonally adjusted (e.g., Producer Price Index).
Transfer the data to a spreadsheet. Plot the data and include a trend line.
Sales data for the remaining exercises
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
1998 414 382 396 530 551 396 365 415 424 485 684 802
1999 457 432 465 598 632 424 392 476 489 555 768 883
2000 505 477 534 636 696 466 442 506 531 610 825 973
2. Plot the sales data from the table. Draw one graph with a trend line and a second chart
with three-period exponential smoothing.
3. Using the regression functions in the spreadsheet, estimate the trend line and produce a
forecast for four periods ahead.
4. Use double exponential smoothing (damping of 0.3) and plot the new data.
5. Use the formula in Figure 8.7A to forecast sales for the next four periods using double
exponential smoothing.