TIME SERIES AND FORECASTING

CHAPTER 16
Bob Levey/Getty Images
TIME SERIES AND FORECASTING
LEARNING OBJECTIVES
When you have completed this chapter, you
will be able to:
Team Sports Inc. sells sporting goods to high schools and colleges via
a nationally distributed catalogue. Management at Team Sports
estimates it will sell 2000 Wilson Model A2000 catcher’s mitts next
year. The deseasonalized sales are projected to be the same for each
of the four quarters next year. The seasonal factor for the second
quarter is 145. Determine the seasonally adjusted sales for the
second quarter of next year. (Exercise 12, LO7)
LO1
Define and describe the components of a
time series.
LO2
Smooth a time series by computing a
moving average.
LO3
Use regression analysis to fit a linear
trend line to a time series.
LO4
Use a trend equation to compute
forecasts.
LO5
Use regression analysis to fit a nonlinear
trend equation.
LO6
Compute and apply seasonal indexes to
make seasonally adjusted forecasts.
LO7
Deseasonalize a time series using
seasonal indexes.
16.1 INTRODUCTION
The emphasis in this chapter is on time series analysis and forecasting. A time series is a
collection of data recorded over a period of time—weekly, monthly, quarterly, or yearly.
Two examples of time series are the sales by quarter of the Microsoft Corporation since
1985 and the annual production of sulphuric acid since 1970.
An analysis of history—a time series—can be used by management to make current
decisions and plans based on long-term forecasting. We usually assume past patterns
will continue into the future. Long-term forecasts extend more than one year into the
future; 2-, 5-, and 10-year projections are common. Long-range predictions are essential
to allow sufficient time for the procurement, manufacturing, sales, finance, and other departments of a company to develop plans for possible new plants, financing, development of new products, and new methods of assembling.
Forecasting the level of sales, both short-term and long-term, is practically dictated by the very nature of business organizations. Competition for the consumer’s
dollar, stress on earning a profit for the stockholders, a desire to procure a larger share
of the market, and the ambitions of executives are some of the prime motivating
forces in business. Thus, a forecast (a statement of the goals of management) is necessary to have the raw materials, production facilities, and staff available to meet the
projected demand.
This chapter deals with the use of data to forecast future events. First, we look at the
components of a time series. Then, we examine some of the techniques used in analyzing data. Finally, we project forecast events.
515
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Chapter 16
LO1
16.2 COMPONENTS OF A TIME SERIES
There are four components to a time series: the trend, the cyclical variation, the seasonal variation, and the irregular variation.
Secular Trend
Secular trend The
smoothed long-term direction of a time series.
The long-term trends of sales, employment, stock prices, and other business and economic
series follow various patterns. Some move steadily upward, others decline, and still others stay
the same over time.
The following are several examples of a secular trend.
• The following chart shows the amount of Canadian exports (in millions) from 1997 to
2013. The value has increased from $302 541.4 million in 1997 to $479 363.8 million in
2013. This is an increase of $176 822.4 million, or 58.45%. The long-run trend is
increasing.
Exports (in millions) 1997–2013
Exports (in millions)
600 000
500 000
400 000
300 000
200 000
100 000
0
1995
Statistics in Action
2000
2005
Year
2010
2015
Source: statcan.gc.ca/tables-tableaux/sum-som/l01/cst01/gblec04-eng.htm.
• The next chart is an example of a declining long-run trend. In 1981, the total number of
farms in Canada was 318 361. By 2011, the number decreased to 205 730.
Number of Farms 1981–2011
Number of Farms
Statisticians, economists,
and business executives are
constantly looking for variables that will forecast the
country’s economy. The
production of crude oil, price
of gold on world markets,
the S&P/TSX Composite
Index average, as well as
many published government
indexes are variables that
have been used with some
success. Variables, such as
the length of hemlines and
the winner of the Super
Bowl, have also been tried.
The variable that seems
overall to be the most successful is the price of scrap
metal. Why? Scrap metal is
the beginning of the manufacturing chain. When its
demand increases, this is
an indication that manufacturing is also increasing.
350 000
300 000
250 000
200 000
150 000
100 000
50 000
0
1970
1980
1990
2000
2010
2020
Year
Source: statcan.gc.ca/daily-quotidien/120510/dq120510a-eng.htm.
Cyclical Variation
Cyclical variation The rise
and fall of a time series over
periods longer than one year.
The second component of a time series is cyclical variation. A typical business cycle consists
of a period of prosperity followed by periods of recession, depression, and then recovery. There
are sizable fluctuations unfolding over more than one year in time above and below the
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Time Series and Forecasting
secular trend. In a recession, for example, employment, production, the S&P/TSX Composite
Index, and many other business and economic series are below the long-term trend lines.
Conversely, in periods of prosperity they are above their long-term trend lines.
Chart 16–1 shows the annual unit sales of batteries sold by National Battery Retailers Inc.
over a period of 20 years. The cyclical nature of business is highlighted. There are periods of
recovery, followed by prosperity, and then recession, and finally, the cycle bottoms out with
depression.
Battery
Sales (000)
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
24.0
30.0
31.0
26.5
27.0
27.5
34.0
35.0
31.0
32.0
35.5
Year
Battery
Sales (000)
60
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
41.0
42.0
38.0
39.0
46.0
52.0
53.5
47.0
51.0
48.0
50
Battery Sales (000)
Year
Contraction
Prosperity
40
Long-term
secular trend
30
20
Recovery
Depression
10
0
1990
1995
2005
2000
2010
2015
Year
CHART 16–1 Battery Sales by National Battery Retailers Inc.1993–2013.
Seasonal Variation
The third component of a time series is the seasonal variation. Many sales, production,
and other series fluctuate with the seasons. The unit of time reported is either quarterly or
monthly.
Almost all businesses tend to have recurring seasonal patterns. Men’s and boys’ clothing,
for example, have extremely high sales just prior to Christmas and relatively low sales just after
Christmas and during the summer. Toy sales are another example with an extreme seasonal
pattern. More than half of the business for the year is usually done in the months of November
and December. Many businesses try to even out the seasonal effects by engaging in an offsetting seasonal business. At ski resorts throughout the country, you will often find golf courses
nearby. The owners of the lodges try to rent to skiers in the winter and golfers in the summer.
This is an effective method of spreading their fixed costs over the entire year rather than a few
months.
Chart 16–2 shows the quarterly sales, in millions of dollars, of Hercher Sporting Goods Inc.
The sporting goods company specializes in selling baseball and softball equipment to high
Year
Quarter
Code
Sales (mil)
2011
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
1
2
3
4
5
6
7
8
9
10
11
12
9.0
10.2
3.5
5.0
12.0
14.1
4.4
7.3
16.0
18.3
4.6
8.6
2012
2013
Sales ($ million)
Seasonal variation Patterns
of change in a time series
within a year. These patterns
tend to repeat themselves
each year.
20
18
16
14
12
10
8
6
4
2
0
Q1
Trend Line
Q3
Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4
2011
2012
2013
CHART 16–2 Sales of Baseball and Softball Equipment, Hercher Sporting Goods, 2011–2013 by Quarter
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Chapter 16
schools, colleges, and youth leagues. It also has several retail outlets in some of the larger
shopping malls. There is a distinct seasonal pattern to its business. Most of its sales are in the
first and second quarters of the year, when schools and organizations are purchasing equipment for the upcoming season. During the early summer, it keeps busy by selling replacement
equipment. It does some business during the holidays (fourth quarter). The late summer (third
quarter) is its slow season.
Irregular Variation
Many analysts prefer to subdivide the irregular variation into episodic and residual variations.
Episodic variations are unpredictable, but they can be identified. The initial impact on the economy of a major strike or a war can be identified, but a strike or war cannot be predicted. After
the episodic fluctuations have been removed, the remaining variation is called the residual
variation. The residual fluctuations, often called chance fluctuations, are unpredictable, and they
cannot be identified. Of course, neither episodic nor residual variation can be projected into
the future.
LO2
Statistics in Action
Investors frequently use
regression analysis to study
the relationship between a
particular stock and the
general condition of the
market. The dependent variable is the monthly percentage change in the value of
the stock, and the independent variable is the monthly
percentage change in a
market index, such as the
S&P/TSX 300 Composite
Index. The value of b in the
regression equation is the
particular stock’s beta coefficient or just the beta. If b
is greater than 1, the implication is that the stock is
sensitive to market
changes. If b is between 0
and 1, the implication is
that the stock is not sensitive to market changes. This
is the same concept that
economists refer to as
elasticity.
16.3 A MOVING AVERAGE
A moving average is useful in smoothing a time series to see its trend. It is also the basic
method used in measuring the seasonal fluctuation, described later in the chapter. In
contrast to the least squares method, which expresses the trend in terms of a mathematical equation (Y ' = a + bt), the moving-average method merely smooths the fluctuations
in the data. This is accomplished by “moving” the arithmetic mean values through the
time series.
To apply the moving-average method to a time series, the data should follow a fairly
linear trend and have a definite rhythmic pattern of fluctuations (repeating, say, every three
years). The data in the following example have three components—trend, cycle, and irregular
variation, abbreviated T, C, and I. There is no seasonal variation because the data are recorded
annually. What the moving-average method does, in effect, is average out C and I. What is left
is the trend.
If the duration of the cycles is constant, and if the amplitudes of the cycles are equal, the
cyclical and irregular fluctuations can be removed entirely using the moving average. The result is a line. For example, in the following time series the cycle repeats itself every seven
years, and the amplitude of each cycle is 4; that is, there are exactly four units from the trough
(lowest period) to the peak. The seven-year moving average, therefore, averages out the cyclical
and irregular fluctuations perfectly, and the residual is a linear trend.
The first step in computing the seven-year moving average is to determine the seven-year
moving totals. The total sales for the first seven years (1988–1994 inclusive) are $22 million,
found by 1 + 2 + 3 + 4 + 5 + 4 + 3. (See Table 16–1.) The total of $22 million is divided by
7 to determine the arithmetic mean sales per year. The seven-year total (22) and the sevenyear mean (3.143) are positioned opposite the middle year for that group of seven, namely,
1991, as shown in Table 16–1. Then, the total sales for the next seven years (1989–1995
inclusive) are determined. (A convenient way of doing this is to subtract the sales for 1988
[$1 million] from the first seven-year total [$22 million] and add the sales for 1995 [$2 million],
to give the new total of $23 million.) The mean of this total, $3.286 million, is positioned
opposite the middle year, 1992. The sales data and seven-year moving average are shown
graphically in Chart 16–3.
The number of data values to include in a moving average depends on the character of the
data collected. If the data are quarterly, then four terms might be typical, since there are four
quarters in a year. If the data are daily, then seven terms might be appropriate, since there are
seven days in a week. You might also use trial and error to determine a number that best levels
out the chance fluctuations.
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Time Series and Forecasting
T A B L E 1 6 – 1 The Computations for the Seven-Year Moving Average
Year
Sales
($ millions)
Seven-Year
Moving Total
Seven-Year
Moving Average
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
$1
2
3
4
5
4
3
2
3
4
5
6
5
4
3
4
5
6
7
6
5
4
5
6
7
8
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
3.143
3.286
3.429
3.571
3.714
3.857
4.000
4.143
4.286
4.429
4.571
4.714
4.857
5.000
5.143
5.286
5.429
5.571
5.714
5.857
9
8
Sales
7
Sales
6
5
4
3
2
1
0
1985
Seven-Year
Moving Average
1990
1995 2000 2005 2010
Year
2015
CHART 16–3 Sales and Seven-Year Moving Average
A three-year and a five-year moving average for some production data are shown in Table 16–2
and depicted in Chart 16–4.
excel
Computer commands to create a moving average can be found at the end of the chapter.
Chapter 16
T A B L E 1 6 – 2 A Three-Year Moving Average and a Five-Year Moving Average
Year
Production
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
5
6
8
10
5
3
7
10
12
11
9
13
15
18
15
11
14
17
22
Three-Year
Moving Total
Three-Year
Moving Average
Five-Year
Moving Total
Five-Year
Moving Average
19
24
23
18
15
20
29
33
32
33
37
46
48
44
40
42
53
6.3
8.0
7.7
6.0
5.0
6.7
9.7
11.0
10.7
11.0
12.3
15.3
16.0
14.7
13.3
14.0
17.7
34
32
33
35
37
43
49
55
60
66
70
72
73
75
79
6.8
6.4
6.6
7.0
7.4
8.6
9.8
11.0
12.0
13.2
14.0
14.4
14.6
15.0
15.8
25
20
Production
520
15
10
Production
3-Year
5-Year
5
0
1994
1996
1998 2000 2002 2004 2006 2008 2010
Year
2012
2014
CHART 16–4 A Three-Year Moving Average and a Five-Year Moving Average
Sales, production, and other economic and business series usually do not have (1) periods
of oscillation that are of equal length or (2) oscillations that have identical amplitudes. Thus, in
actual practice, the application of the moving-average method to data does not result precisely
in a line. For example, the production series in Table 16–2 repeats about every five years, but
the amplitude of the data varies from one oscillation to another. The trend appears to be upward and somewhat linear. Both moving averages—the three-year and the five-year—seem to
adequately describe the trend in production since 1995.
Four-year, six-year, and other even-numbered-year moving averages present one minor
problem with regard to the centring of the moving totals and moving averages. Note in
Table 16–3 that there is no centre period, so the moving totals are positioned between two periods.
The total for the first four years ($42) is positioned between 2006 and 2007. The total for the
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Time Series and Forecasting
T A B L E 1 6 – 3 Calculations of a Four-Year Moving Average
Year
Sales,
Y ($)
2005
2006
8
11
2007
9
2008
2009
2010
2011
2012
2013
Four-Year
Moving
Total ($)
Four-Year
Moving
Average ($)
42 (8 + 11 + 9 + 14)
10.50 (42 ÷ 4)
43 (11 + 9 + 14 + 9)
10.75 (43 ÷ 4)
42
10.50
43
10.75
37
9.25
40
10.00
Centred
Moving
Average ($)
10.625
14
10.625
9
10.625
10
10.000
10
9.625
8
12
next four years is $43. The averages of the first four years and the second four years ($10.50 and
$10.75, respectively) are averaged, and the resulting figure is centred on 2007. This procedure
is repeated until all possible four-year averages are computed.
self-review 16–1
LO3
Determine a three-year moving average for the following production series. Plot both the original
data and the moving average.
Year
Number Produced
(thousands)
Year
Number Produced
(thousands)
2009
2
2012
5
2010
6
2013
3
2011
4
2014
10
16.4 LINEAR TREND
The long-term trend of many business series, such as sales, exports, and production, often approximates a straight line. If so, the equation to describe this growth is:
LINEAR TREND EQUATION
Y' = a + bt
[16–1]
where:
Y' (read Y prime) is the projected value of the Y variable for a selected value of t.
a is the Y-intercept. It is the estimated value of Y when t = 0. Another way to put it is: a
is the estimated value of Y, where the line crosses the Y-axis when t is zero.
b is the slope of the line, or the average change in Y' for each change of one unit in t.
t is any value of time that is selected.
To illustrate the meaning of Y', a, b, and t in a time-series problem, a line has been drawn
in Chart 16–5 to represent the typical trend of sales. Assume that this company started in
business in 2005. This beginning year (2005) has been arbitrarily designated as year 1. Note
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Chapter 16
that sales increased by $2 million, on average, every year; that is, based on the straight line
drawn through the sales data, sales increased from $3 million in 2005 to $5 million in
2006, to $7 million in 2007, to $9 million in 2008, and so on. The slope, or b, is therefore 2.
Note, too, that the line intercepts the Y-axis (when t = 0) at $1 million. This point is a.
Another way of determining b is to locate the starting place of the straight line in year 1. It
is 3 for 2005 in this problem. Then, locate the value on the straight line for the last year. It
is 19 for 2013.
Year
Sales
Trend Line
25
1
2
3
4
5
6
7
8
9
2005
2006
2007
2008
2009
2010
2011
2012
2013
6
2
9
8
16
10
12
21
18
3
5
7
9
11
13
15
17
19
20
Sales ($ millions)
Time Period
Sales
15
10
Trend Line
5
0
2004
2006
2008
2010
Year
2012
2014
CHART 16–5 A Straight Line Fitted to Sales Data
The equation for the line in Chart 16–5 is:
Y' = 1 + 2t
where:
Sales are in millions of dollars. The origin, or year 0, is 2004.
t increases by one unit for each year.
In Chapter 12, we drew a line through points on a scatter diagram to approximate the regression line. We stressed, however, that this method for determining the regression equation
has a serious drawback—namely, the position of the line depends on the judgment of the individual who drew the line. Three people would probably draw three different lines through the
scatter plots. Likewise, the line we drew through the sales data in Chart 16–5 might not be the
“best-fitting” line. Because of the subjective judgment involved, this method should be used
only when a quick approximation of the straight-line equation is needed or to check the reasonableness of the least squares line, which is discussed next.
16.5 LEAST SQUARES METHOD
In the discussion of simple linear regression in Chapter 12, we showed how the least-squares
method is used to find the best linear relationship between two variables. In forecasting methods, time is the independent variable, and the value of the time series is the dependent variable. Furthermore, we often code the independent variable, time, to make the equation easier
to interpret. In other words, we let t be 1 for the first year, 2 for the second, and so on. When
time is coded, we use the following equations to find the slope, b, and the intercept, a, to substitute into the linear trend equation (16–1).
THE SLOPE
THE INTERCEPT
b=
n©tY − ( ©Y) ( ©t)
n©t2 − ( ©t) 2
[16–2]
©Y
©t
− b( )
n
n
[16–3]
a=
523
Time Series and Forecasting
Many calculators can compute a and b directly from data. If the number of years is large—say,
15 or more—and the magnitude of the numbers is also large, use of a computer software program is recommended.
Example
The sales of Jensen Foods, a small grocery chain, since 2009 are as follows:
Year
Sales
($ millions)
2009
2010
2011
2012
2013
$7
10
9
11
13
Determine the least squares trend line equation. How much are sales increasing each year?
What is the sales forecast for 2016?
Solution
To determine the trend equation,, the years are replaced by coded values. That is, we let
2009 be 1, 2010 be 2, and so forth. This reduces the size of the values of πt, πt2, and πtY. (See
Table 16–4.) This is often referred to as the coded method.
T A B L E 1 6 – 4 Calculations for Determining the Trend Equation
Year
2009
2010
2011
2012
2013
Sales ($
millions),
Y
t
tY
t2
$7
10
9
11
13
$50
1
2
3
4
5
15
7
20
27
44
65
163
1
4
9
16
25
55
Determining a and b using formulas (16–2) and (16–3):
n©tY − ( ©Y) ( ©t)
5(163) − 50(15)
=
= 1.3
2
2
n©t − ( ©t)
5(55) − (15) 2
©Y
©t
50
15
a=
− b( ) =
− 1.3( ) = 6.1
n
n
5
5
b=
The trend equation is, therefore, Y' = 6.1 + 1.3t, where:
Sales are in millions of dollars.
The origin, or year 0, is 2008, and t increases by one unit for each year.
How do we interpret the equation? The value of 1.3 indicates sales increased at a rate of
$1.3 million per year. The value 6.1 is the estimated value of sales in the year 0. That is, the
estimated sales amount for 2008 (the base year) is $6.1 million. So, to determine the point on
the line for 2016, insert the value of 4 for t in the equation. Then, Y' = 6.1 + 1.3(4) = 11.3.
excel
Computer commands to find the linear trend equation for this example can be found at the end of
the chapter.
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Chapter 16
Plotting the Line
The least squares equation can be used to find points on the line through the data. The sales
data from Table 16–4 are repeated in Table 16–5 to show the procedure. The equation determined earlier is Y' = 6.1 + 1.3t. To get the coordinates of the point on the line for 2012, insert
the t-value of 4 in the equation. Then, Y' = 6.1 + 1.3(4) = 11.3.
T A B L E 1 6 – 5 Calculations Needed for Determining the Points on the Straight Line Using the
Coded Method
Year
Sales($
millions),
Y
T
Y’
2009
2010
2011
2012
2013
$7
10
9
11
13
1
2
3
4
5
7.4
8.7
10.0
11.3
12.6
Found by
←
←
←
←
←
6.1 + 1.3(1)
6.1 + 1.3(2)
6.1 + 1.3(3)
6.1 + 1.3(4)
6.1 + 1.3(5)
Sales ($ million)
The sales and trend in sales as represented by the line are shown in the following output:
14
13
12
11
10
9
8
7
6
Trend Equation
y' = 6.1 + 1.3t
1
2
3
Time
4
5
CHART 16–6 Trend Line for Jensen Food, 2009–2013
LO4
Estimation
If sales, production, or other data approximate a linear trend, the equation developed by the
least squares technique can be used to estimate future values. It is reasonable that the sales for
Jensen Foods follow a linear trend. So, we can use the trend equation to forecast future sales.
As the year 2009 is coded 1, the year 2010 is coded 2, and so on, and then the year 2016 will
be coded as 8. So, we substitute 8 into the trend equation and solve for Y'.
Y¿ = 6.1 + 1.3(8) = 16.5
Thus, on the basis of past sales, the estimate for 2016 is $16.5 million.
In this time series example, there were five years of sales data. On the basis of those
five sales figures, we estimated sales for 2016. Many researchers suggest that we do not
project sales, production, and other business and economic series more than n/2 periods
into the future, where n is the number of data points. If, for example, there are 10 years of
data, we would make estimates only up to five years into the future (n/2 = 10/2 = 5). Others
suggest the forecast may be for no longer than two years, especially in rapidly changing
economic times.
excel
Computer commands to estimate the sales for period 8 can be found at the end of the chapter.
525
Time Series and Forecasting
self-review 16–2
Annual production of king-size rockers by Wood Rockers Inc. since 2007 is as follows:
Year
Production
(thousands)
Year
Production
(thousands)
2007
2008
2009
2010
4
8
5
8
2011
2012
2013
2014
11
9
11
14
(a) Plot the production data.
(b) Determine the least squares equation.
(c) Determine the points on the line for 2007 and 2013. Connect the two points to arrive at the line.
(d) On the basis of the linear trend equation, what is the estimated production for 2017?
EXERCISES
1. The numbers of bank failures for the years 2010 through 2014 are given below. Determine the least
squares equation and estimate the number of failures in 2016.
Year
Code
Number of Failures
2010
2011
2012
2013
2014
1
2
3
4
5
79
120
138
184
200
2. The CPI (Consumer Price Index) for Canada for the six years from 2005 to 2013 is listed below. On
the basis of these data, determine the least squares equation, and estimate the value of the CPI for
2017. See Connect for the complete file.
2005
2006
2007
2006
2009
2010
2011
2012
2013
107.0
109.1
111.5
114.1
114.4
116.5
119.9
121.7
122.8
3. The following table lists the average price of home listings in Canada from January 2007 to January
2014. (See Connect for the CREA data set).
a.
b.
c.
d.
e.
Year
Amount
Year
Amount
Jan-14
Jan-13
Jan-12
Jan-11
$388 553
354 951
348 178
344 118
Jan-10
Jan-09
Jan-08
Jan-07
328 728
274 711
309 448
282 420
Provide a time-series graph.
Determine the least-squares trend equation.
Interpret the coefficients in the trend equation.
Estimate the amount of the average house listing for January 2018.
Use the least squares trend equation to calculate the trend value for 2012? Why is this value not
the same as the given value 348 178? Explain.
526
Chapter 16
4. Listed below are the net sales in millions of dollars for Home Depot Inc. and its subsidiaries from
1993 to 2012:
Year
Net Sales
Year
Net Sales
Year
Net Sales
1993
1994
1995
1996
1997
1998
1999
$ 9 239
12 477
15 470
19 535
24 156
30 219
38 434
2000
2001
2002
2003
2004
2005
2006
$45 738
53 553
58 247
64 816
73 094
81 511
90 837
2007
2008
2009
2010
2011
2012
$77 349
71 288
66 176
67 997
70 395
74 754
Determine the least-squares equation. According to this information, what are the estimated sales
for 2014 and 2015?
5. The following table lists the annual amounts of glass cullet produced by Kimble Glass Works Inc.:
Year
Code
Scrap
(tonnes)
2008
2009
2010
2011
2012
2013
2014
1
2
3
4
5
6
7
2.0
4.0
3.0
5.0
6.0
7.0
6.0
Determine the least-squares trend equation. Estimate the amount of scrap for the year 2016.
6. The sales by Walker’s Milk and Dairy products, in millions of dollars, for the period from 2008
to 2014 are reported below. Determine the least-squares trend equation, and estimate sales
for 2015.
LO5
Year
Code
Sales
($ millions)
2008
2009
2010
2011
2012
2013
2014
1
2
3
4
5
6
7
$17.5
19.0
21.0
22.7
24.5
26.7
27.3
16.6 NONLINEAR TRENDS
The emphasis in the previous discussion was on a time series whose growth or decline approximated a straight line. A linear trend equation is used to represent the time series when it
is believed that the data are increasing (or decreasing) by equal amounts, on the average, from
one period to another.
Data that increase (or decrease) by increasing amounts over a period of time appear curvilinear when plotted on paper having an arithmetic scale. To put it another way, data that increase
(or decrease) by equal percentages or proportions over a period of time appear curvilinear. (See
Chart 16–7.)
527
Year
Sales ($000)
Year
Sales ($000)
1999
2000
2001
2002
2003
2004
2005
2006
124.2
175.6
306.9
524.2
714.0
1 052.0
1 638.3
2 463.2
2007
2008
2009
2010
2011
2012
2013
3 358.2
4 181.3
5 388.5
8 027.4
10 587.2
13 537.4
17 515.6
Sales (000)
Time Series and Forecasting
20 000
18 000
16 000
14 000
12 000
10 000
8 000
6 000
4 000
2 000
0
1995
2000
2005
Year
2010
2015
CHART 16–7 Sales for Gulf Shores Importers, 1999–2013
The trend equation for a time series that does approximate a curvilinear trend, such as the
one portrayed in Chart 16–7, is computed by using the logarithms of the data and the leastsquares method. The general equation for the logarithmic trend equation is:
LOG TREND EQUATION
log Y¿ = log a + log b(t)
[16–4]
The logarithmic trend equation can be determined for the Gulf Shores Importers data in
Chart 16–7 using Excel. The first step is to enter the data, and the next step is to find the log
base 10 of each year’s imports. Finally, use the regression procedure to find the least-squares
equation. To put it another way, we take the log of each year’s data, then use the logs as the
dependent variable and the coded year as the independent variable.
The regression equation is: Y ' = 2.0538 + 0.1534t. This equation is the log form. We now
have a trend equation in terms of percentage change. That is, the value 0.1534 is the percentage change in Y ' for each unit increase in t. This value is similar to the geometric mean described in Chapter 3.
The log of b is 0.1534 and its antilog or inverse is 1.424. If we subtract 1 from this value, as
we did in Chapter 3, the value 0.424 indicates the geometric mean rate of increase from 1997
to 2011. We conclude that imports increased at a rate of 42.4% annually during the period.
We can also use the logarithmic trend equation to make estimates of future values. Suppose that we want to estimate the imports in the year 2017. The first step is to determine the
528
Chapter 16
code for the year 2017, which is 19. To explain, the year 2013 has a code of 15, and the year
2017 is four years later, so 15 + 4 = 19. The log of imports for the year 2017 is:
Y¿ = 2.0538 + 0.1534(t) = 2.0538 + 0.1534(19) = 4.9684
To find the estimated imports for the year 2017, we need the antilog of 4.9684. It is 92 982.239.
This is our estimate of the sales of imports for 2017. Recall that the data were in thousands of
dollars, so the estimate is $92 982 239.
excel
self-review 16–3
Computer commands to produce the output for this example can be found at the end of the chapter.
Sales at Tomlin Manufacturing since 2010 are as follows:
(a)
(b)
(c)
Year
Sales
($ millions)
2010
2011
2012
2013
2014
$2.13
18.10
39.80
81.40
112.00
Determine the logarithmic trend equation for the sales data.
Sales increased by what percentage annually?
What is the projected sales amount for 2015?
EXERCISES
7. Sally’s Software Inc. is a rapidly growing supplier of computer software. Sales for the last five years
are given below:
Year
2010
2011
2012
2013
2014
Sales
($ millions)
$1.1
1.5
2.0
2.4
3.1
a. Determine the logarithmic trend equation.
b. By what percentage did sales increase, on average, during the period?
c. Estimates sales for the year 2017.
8. It appears that the imports of carbon black have been increasing by about 10% annually.
Year
Imports
of Carbon Black
(thousands of tonnes)
Year
Imports
of Carbon Black
(thousands of tonnes)
2007
2008
2009
2010
92.0
101.0
112.0
124.0
2011
2012
2013
2014
135.0
149.0
163.0
180.0
a. Determine the logarithmic trend equation.
b. By what percentage did imports increase, on average, during the period?
c. Estimate imports for the year 2017.
529
Time Series and Forecasting
16.7 SEASONAL VARIATION
We mentioned that seasonal variation is another of the components of a time series. Business
series, such as automobile sales, shipments of soft drinks, and residential construction, have
periods of above-average and below-average activity each year.
In the area of production, one of the reasons for analyzing seasonal fluctuations is to have
a sufficient supply of raw materials on hand to meet the varying seasonal demand. The glass
container division of a large glass company, for example, manufactures nonreturnable beer
bottles, iodine bottles, aspirin bottles, bottles for rubber cement, and so on. The production
scheduling department must know how many bottles to produce and when to produce each
kind. A run of too many bottles of one kind may cause a serious storage problem. Production
cannot be based entirely on orders on hand because many orders are telephoned in for immediate shipment. Since the demand for many of the bottles varies according to the season, a
forecast a year or two in advance, by month, is essential to good scheduling.
An analysis of seasonal fluctuations over a period of years can also help in evaluating
current sales. The typical sales of department stores, excluding mail-order sales, are expressed as indexes in Table 16–5. Each index represents the average sales for a period of
several years. The actual sales for some months were above average (which is represented by
an index over 100.0), and the sales for other months were below average. The index of 126.8
for December indicates that typically, sales for December are 26.8% above an average month;
the index of 86.0 for July indicates that department store sales for July are typically 14% below an average month.
T A B L E 1 6 – 5 Typical Seasonal Indexes for Department Store Sales, Excluding Mail-Order Sales
January
February
March
April
May
June
87.0
83.2
100.5
106.5
101.6
89.6
July
August
September
October
November
December
86.0
99.7
101.4
105.8
111.9
126.8
Suppose that an enterprising store manager, in an effort to stimulate sales during December,
introduced a number of unique promotions, including bands of carollers strolling through
the store singing holiday songs, large mechanical exhibits, and clerks dressed in Santa Claus
costumes. When the index of sales was computed for that December, it was 150.0. Compared with the typical sales of 126.8, it was concluded that the promotional program was a
huge success.
LO6
Determining a Seasonal Index
A typical set of monthly indexes consists of 12 indexes that are representative of the data for a
12-month period. Logically, there are four typical seasonal indexes for data reported quarterly.
Each index is a percentage, with the average for the year equal to 100.0; that is, each monthly
index indicates the level of sales, production, or another variable in relation to the annual average of 100.0. A typical index of 96.0 for January indicates that sales (or whatever the variable is)
are usually 4% below the average for the year. An index of 107.2 for October means that the
variable is typically 7.2% above the annual average.
Several methods have been developed to measure the typical seasonal fluctuation in a time
series. The method most commonly used to compute the typical seasonal pattern is called the
ratio-to-moving-average method. It eliminates the trend, cyclical, and irregular components from
the original data (Y). In the following discussion, T refers to trend, C to cyclical, S to seasonal,
and I to irregular variation. The numbers that result are called the typical seasonal index.
We will discuss in detail the steps followed in arriving at typical seasonal indexes using
the ratio-to-moving-average method. The data of interest might be monthly or quarterly. To illustrate, we have chosen the quarterly sales of Toys International. First, we will show the steps
needed to arrive at a set of typical quarterly indexes. Then, we use MegaStat to calculate the
seasonal indexes.
530
Chapter 16
Example
Table 16–6 shows the quarterly sales for Toys International for the years 2008 through 2013.
The sales are reported in millions of dollars. Determine a quarterly seasonal index using the
ratio-to-moving-average method.
T A B L E 1 6 – 6 Quarterly Sales of Toys International ($ millions)
Solution
Spring
Summer
Fall
6.7
6.5
6.9
7.0
7.1
8.0
4.6
4.6
5.0
5.5
5.7
6.2
10.0
9.8
10.4
10.8
11.1
11.4
12.7
13.6
14.1
15.0
14.5
14.9
Sales
($ Million)
2008
1
2
3
4
1
2
3
4
1
2
3
4
6.7
4.6
10.0
12.7
6.5
4.6
9.8
13.6
6.9
5.0
10.4
14.1
Year
Quarter
Sales
($ Million)
2011
1
2
3
4
1
2
3
4
1
2
3
4
7.0
5.5
10.8
15.0
7.1
5.7
11.1
14.5
8.0
6.2
11.4
14.9
2012
2013
Sales ($ millions)
Quarter
2010
Winter
2008
2009
2010
2011
2012
2013
Chart 16–8 depicts the quarterly sales for Toys International over the six-year period. Note
the seasonal nature of the sales. For each year, the fourth-quarter sales are the largest, and
the second-quarter sales are the smallest. Also, there is a moderate increase in the sales
from one year to the next. To observe this feature, look only at the six fourth-quarter sales
values. Over the six-year period, the sales in the fourth quarter increased. If you connect
these points in your mind, you can visualize fourth-quarter sales increasing for 2014.
Year
2009
Year
16
14
12
10
8
6
4
2
0
0
4
8
12
Quarter
16
20
24
CHART 16–8 Quarterly Sales of Toys International, 2008–2013
There are six steps to determining the quarterly seasonal indexes.
Step 1: For the following discussion, refer to Table 16–7. The first step is to determine the
four-quarter moving total for 2008. Starting with the winter quarter of 2008, we
add $6.7, $4.6, $10.0, and $12.7. The total is $34.0 (million). The four-quarter total
is “moved along” by adding the spring, summer, and fall sales of 2008 to the winter sales of 2009. The total is $33.8 (million), found by: $4.6 + $10.0 + $12.7 +
$6.5. This procedure is continued for the quarterly sales for each of the six years.
Column 2 of Table 16–7 shows all of the moving totals. Note that the moving total $34.0 is positioned between the spring and summer sales of 2008. The next
moving total, $33.8, is positioned between sales for summer and fall of 2008, and
so on. Check the totals frequently to avoid arithmetic errors.
Step 2: Each quarterly moving total in column 2 is divided by 4 to give the four-quarter
moving average. (See column 3.) All the moving averages are still positioned between the quarters. For example, the first moving average (8.500) is positioned
between spring and summer of 2008.
Step 3: The moving averages are then centred. The first centred moving average is found by:
($8.500 + $8.450)/2 = $8.475 and centred opposite summer 2008. The second moving average is found by: ($8.450 + $8.450)/2 = $8.45. The others are found similarly.
Note in column 4 that a centred moving average is positioned on a particular quarter.
531
Time Series and Forecasting
T A B L E 1 6 – 7 Computations Needed for the Specific Seasonal Indexes
(1)
(2)
(3)
(4)
(5)
Sales
($ millions)
Four-Quarter
Total
Four-Quarter
Moving
Average
Centred
Moving
Average
Specific
Seasonal
34.0
8.500
8.475
1.180
33.8
8.450
8.450
1.503
33.8
8.450
8.425
0.772
33.6
8.400
8.513
0.540
34.5
8.625
8.675
1.130
34.9
8.725
8.775
1.550
35.3
8.825
8.900
0.775
35.9
8.975
9.038
0.553
36.4
9.100
9.113
1.141
36.5
9.125
9.188
1.535
37.0
9.250
9.300
0.753
37.4
9.350
9.463
0.581
38.3
9.575
9.588
1.126
38.4
9.600
9.625
1.558
38.6
9.650
9.688
0.733
38.9
9.725
9.663
0.590
38.4
9.600
9.713
1.143
39.3
9.825
9.888
1.466
39.8
9.950
9.888
0.801
40.1
10.025
10.075
0.615
40.5
10.125
Year
Quarter
2008
Winter
6.7
Spring
4.6
Summer
10.0
Fall
2009
Winter
Spring
Summer
Fall
2010
Winter
Spring
Summer
Fall
2011
Winter
Spring
Summer
Fall
2012
Winter
Spring
Summer
Fall
2013
Winter
Spring
12.7
6.5
4.6
9.8
13.6
6.9
5.0
10.4
14.1
7.0
5.5
10.8
15.0
7.1
5.7
11.1
14.5
8.0
6.2
Summer
11.4
Fall
14.9
Step 4: The specific seasonal for each quarter is then computed by dividing the sales in
column 1 by the centred moving average in column 4. The specific seasonal reports the ratio of the original time-series value to the moving average. To explain
further, if the time series is represented by TSCI and the moving average by TCI,
then, algebraically, if we compute TSCI/TCI, the result is the seasonal component.
The specific seasonal for the summer quarter of 2008 is 1.180, found by: 10.0/8.475.
532
Chapter 16
Step 5: The specific seasonals are organized in Table 16–8. This table will help us locate
the specific seasonals for the corresponding quarters. The values 1.180, 1.130,
1.141, 1.126, and 1.143 all represent estimates of the typical seasonal index for
the summer quarter. A reasonable method to find a typical seasonal index is to
average these values in order to eliminate the irregular component. So, we find
the typical index for the summer quarter by: (1.180 + 1.130 + 1.141 + 1.126 +
1.143)/5 = 1.144. We used the arithmetic mean, but the median or a modified
mean can also be used.
T A B L E 1 6 – 8 Calculations Needed for Typical Quarterly Indexes
Year
Winter
Spring
2008
2009
2010
2011
2012
2013
0.772
0.775
0.753
0.733
0.801
0.540
0.553
0.581
0.590
0.615
3.834
0.767
0.765
76.5
2.879
0.576
0.575
57.5
Total
Mean
Adjusted
Index
Summer
Fall
1.180
1.130
1.141
1.126
1.143
1.503
1.550
1.535
1.558
1.466
5.720
1.144
1.141
114.1
7.612
1.522
1.519
151.9
4.009
4.000
Step 6: The four quarterly means (0.767, 0.576, 1.144, and 1.522) should theoretically total 4.00 because the average is set at 1.0. The total of the four quarterly means
may not exactly equal 4.00 due to rounding. In this problem the total of the
means is 4.009. A correction factor is therefore applied to each of the four means
to force them to total 4.00.
CORRECTION FACTOR FOR
ADJUSTING QUARTERLY MEANS
Correction factor =
4.00
Total of four means
[16–5]
In this example,
Correction factor =
4.00
= 0.9978
4.009
The adjusted winter quarterly index, therefore, is: 0.767(0.9978) = 0.765. Each of the means is
adjusted downward so that the total of the four quarterly means is 4.00. Usually, indexes are
reported as percentages, so each value in the last row of Table 16–8 has been multiplied by
100. So, the index for the winter quarter is 76.5, and for the fall, it is 151.9. How are these values
interpreted? Sales for the fall quarter are 51.9% above the typical quarter, and for winter they are
23.5% below the typical quarter (100.0 − 76.5). These findings should not surprise you. The
period prior to Christmas (the fall quarter) is when toy sales are brisk. After Christmas (the winter quarter), sales of the toys decline drastically.
The computer output follows. As we noted earlier, the use of software will greatly reduce
the computational time and the chance of an error in arithmetic, but you should understand
the steps in the process, as outlined earlier. There can be slight differences in the results due to
the number of digits carried in the calculations.
533
Time Series and Forecasting
Centred Moving Average and Deseasonalization
Centred
Moving
Average
Ratio to
CMA
Seasonal
Indexes
Deseasonalized
Sales
8.475
8.450
1.180
1.503
0.765
0.575
1.141
1.519
8.759
8.004
8.761
8.361
6.50
4.60
9.80
13.60
8.425
8.513
8.675
8.775
0.772
0.540
1.130
1.550
0.765
0.575
1.141
1.519
8.498
8.004
8.586
8.953
1
2
3
4
6.90
5.00
10.40
14.10
8.900
9.038
9.113
9.188
0.775
0.553
1.141
1.535
0.765
0.575
1.141
1.519
9.021
8.700
9.112
9.283
2011
2011
2011
2011
1
2
3
4
7.00
5.50
10.80
15.00
9.300
9.463
9.588
9.625
0.753
0.581
1.126
1.558
0.765
0.575
1.141
1.519
9.151
9.570
9.462
9.875
17
18
19
20
2012
2012
2012
2012
1
2
3
4
7.10
5.70
11.10
14.50
9.688
9.663
9.713
9.888
0.733
0.590
1.143
1.466
0.765
0.575
1.141
1.519
9.282
9.918
9.725
9.546
21
22
23
24
2013
2013
2013
2013
1
2
3
4
8.00
6.20
11.40
14.90
9.988
10.075
0.801
0.615
0.765
0.575
1.141
1.519
10.459
10.788
9.988
9.809
t
Year
Quarter
Sales
1
2
3
4
2008
2008
2008
2008
1
2
3
4
6.70
4.60
10.00
12.70
5
6
7
8
2009
2009
2009
2009
1
2
3
4
9
10
11
12
2010
2010
2010
2010
13
14
15
16
Calculation of Seasonal Indexes
1
2008
2009
2010
2011
2012
2013
Mean:
Adjusted:
0.772
0.775
0.753
0.733
0.801
0.767
0.765
2
0.540
0.553
0.581
0.590
0.615
0.576
0.575
3
4
1.180
1.130
1.141
1.126
1.143
1.503
1.550
1.535
1.558
1.466
1.144
1.141
1.522
1.519
4.009
4.000
MegaStat commands to produce the above output can be found at the end of the chapter.
Now we briefly summarize the reasoning underlying the preceding calculations. The
original data in column 1 of Table 16–8 contain trend (T), cyclic (C), seasonal (S), and irregular (I) components. The ultimate objective is to remove seasonal (S) from the original
sales valuation.
Columns 2 and 3 in Table 16–8 are concerned with deriving the centred moving average
given in column 4. Basically, we “average out” the seasonal and irregular fluctuations from the
original data in column 1. Thus, in column 4 we have only trend and cyclic (TC).
Next, we divide the sales data in column 1 (TCSI) by the centred fourth-quarter moving
average in column 4 (TC) to arrive at the specific seasonals in column 5 (SI). In terms of letters,
TCSI/TC = SI. We multiply SI by 100.0 to express the typical seasonal in index form.
Finally, we take the mean of all the winter typical indexes, all the spring indexes, and so
on. This averaging eliminates most of the irregular fluctuations from the seasonals, and the resulting four indexes indicate the typical seasonal sales pattern.
534
Chapter 16
self-review 16–4
The Lake Louise and Banff National Park in Alberta contains accommodations, shops, and restaurants and has many activities, including festivals. There are two peak seasons—winter, for skiing, and
summer, for tourists visiting the area. The specific seasonals with respect to the total sales volume
for recent years are as follows:
Quarter
(a)
(b)
Year
Winter
Spring
Summer
Fall
2010
2011
2012
2013
2014
117.0
118.6
114.0
120.7
125.2
80.7
82.5
84.3
79.6
80.2
129.6
121.4
119.9
130.7
127.6
76.1
77.0
75.0
69.6
72.0
Develop the typical seasonal pattern for the Lake Louise and Banff National Park using the ratioto-moving-average method.
Explain the typical index for the winter season.
EXERCISES
9. Victor Anderson, the owner of Anderson Belts Inc., is studying absenteeism among his employees.
His workforce is small, consisting of only five employees. For the last three years, he recorded the
following number of employee absences, in days, for each quarter:
Quarter
Year
I
II
III
IV
2012
2013
2014
4
5
6
10
12
16
7
9
12
3
4
4
Determine a typical seasonal index for each of the four quarters.
10. The Appliance Centre sells a variety of electronic equipment and home appliances. For the last four
years the following quarterly sales (in $ millions) were reported:
Quarter
Year
I
II
III
IV
2011
2012
2013
2014
$5.3
4.8
4.3
5.6
$4.1
3.8
3.8
4.6
$6.8
5.6
5.7
6.4
$6.7
6.8
6.0
5.9
Determine a typical seasonal index for each of the four quarters.
LO7
Deseasonalizing Data
A set of typical indexes is very useful in adjusting a sales series, for example, for seasonal fluctuations. The resulting sales series is called deseasonalized sales or seasonally adjusted sales.
The reason for deseasonalizing the sales series is to remove the seasonal fluctuations so that
the trend and cycle can be studied. To illustrate the procedure, the quarterly sales totals of Toys
International from Table 16–7 are repeated in column 1 of Table 16–9.
To remove the effect of seasonal variation, the sales amount for each quarter (which contains
trend, cyclical, irregular, and seasonal effects) is divided by the seasonal index for that quarter, that
is, TSCI/S. For example, the actual sales for the first quarter of 2008 were $6.7 million. The seasonal
index for the winter quarter is 76.5, using the results shown previously. The index of 76.5 indicates
that sales for the first quarter are typically 23.5% below the average for a typical quarter. By dividing
the actual sales of $6.7 million by 76.5 and multiplying the result by 100, we find the deseasonalized
sales value for the first quarter of 2008. It is $8 758 170, found by: ($6 700 000/76.5)(100). We continue this process for the other quarters in column 3 of Table 16–9, with the results reported in
millions of dollars. Because the seasonal component has been removed (divided out) from the
quarterly sales, the deseasonalized sales figure contains only the trend (T), cyclical (C), and
535
Time Series and Forecasting
T A B L E 1 6 – 9 Actual and Deseasonalized Sales for Toys International
(1)
(2)
(3)
Year
Quarter
Sales
Seasonal
Index
Deseasonalized
Sales
2008
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
6.7
4.6
10.0
12.7
6.5
4.6
9.8
13.6
6.9
5.0
10.4
14.1
7.0
5.5
10.8
15.0
7.1
5.7
11.1
14.5
8.0
6.2
11.4
14.9
0.765
0.575
1.141
1.519
0.765
0.575
1.141
1.519
0.765
0.575
1.141
1.519
0.765
0.575
1.141
1.519
0.765
0.575
1.141
1.519
0.765
0.575
1.141
1.519
8.759
8.004
8.761
8.361
8.498
8.004
8.586
8.953
9.021
8.700
9.112
9.283
9.151
9.570
9.462
9.875
9.282
9.918
9.725
9.546
10.459
10.788
9.988
9.809
2009
2010
2011
2012
2013
irregular (I) components. Scanning the deseasonalized sales in column 3 of Table 16–9, we see
that the sales of toys showed a moderate increase over the six-year period. The chart shows both
the actual sales and the deseasonalized sales. It is clear that removing the seasonal factor allows
us to focus on the overall long-term trend of sales. We will also be able to determine the regression equation of the trend data and use it to forecast future sales. The regression equation and
coefficient of determination are automatically added to the chart.
2008
2009
2010
1
2
3
4
5
6
7
8
9
10
11
12
6.7
4.6
10.0
12.7
6.5
4.6
9.8
13.6
6.9
5.0
10.4
14.1
8.759
8.004
8.761
8.361
8.498
8.004
8.586
8.953
9.021
8.700
9.112
9.283
2011
2012
2013
13
14
15
16
17
18
19
20
21
22
23
24
7.0
5.5
10.8
15.0
7.1
5.7
11.1
14.5
8.0
6.2
11.4
14.9
9.151
9.570
9.462
9.875
9.282
9.918
9.725
9.546
10.459
10.788
9.988
9.809
Sales ($ million)
Sales Deseasonalized
Sales Deseasonalized
($
Sales
($
Sales
Year Time Million)
($ Million)
Year Time Million)
($ Million)
16
14
12
10
8
6
4
2
0
Sales
($ million)
0
4
8
12
Time
Deseasonalized
Sales ($ million)
16
20
24
CHART 16–9 Actual and Deseasonalized Sales for Toys International for 2008–2013
LO7
Using Deseasonalized Data to Forecast
The procedure for identifying trend and the seasonal adjustments can be combined to yield seasonally adjusted forecasts. To identify the trend, we determine the least-squares trend equation on the
deseasonalized historical data. Then, we project this trend into future periods, and finally, we adjust
these trend values to account for the seasonal factors. The following example will help to clarify this:
536
Chapter 16
Example
Toys International would like to forecast its sales for each quarter of 2014. Use the information
in Table 16–9 to determine the forecast.
Solution
The deseasonalized data shown in Chart 16–9 seems to follow a straight line. Hence, it is reasonable to develop a linear trend equation based on these data. The deseasonalized trend
equation is:
Y¿ = a + bt
where:
Statistics in Action
Forecasts are not always
correct. The reality is that a
forecast may just be a best
guess as to what will happen. What are the reasons
forecasts are not correct?
One expert lists eight common errors: (1) failure to
carefully examine the assumptions,(2) limited expertise, (3) lack of imagination,
(4) neglect of constraints,
(5) excessive optimism,
(6) reliance on mechanical
extrapolation, (7) premature
closure, and (8) over
specification.
Y¿
a
b
t
is the estimated trend for Toys International sales for period t.
is the intercept of the trend line at time 0.
is the slope of the trend line.
is the coded period
The winter quarter of 2008 is the first quarter, so it is coded 1, the spring quarter of 2008 is
coded 2, and so on. The last quarter of 2013 is coded 24. (See column 1 in Table 16–10.) The
sums needed to compute a and b are also shown in Table 16–10.
n©tY − ( ©Y) ( ©t)
24(2873.4) − (221.60) (300)
2481.6
=
=
= 0.0899
2
2
2
n©t − ( ©t)
24(4900) − (300)
27600.0
©Y
©t
221.60
300
a=
− b( ) =
− 0.0899(
) = 8.1096
n
n
24
24
b=
The trend equation is:
Y¿ = 8.1096 + 0.0899t
T A B L E 1 6 – 1 0 Deseasonalized Sales for Toys International: Data Needed for Determining Trend Line
Year
2008
2009
2010
2011
2012
2013
Total
Quarter
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
(1)
(2)
(3)
(4)
t
Y
tY
t2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
300
8.76
8.00
8.76
8.36
8.50
8.00
8.59
8.95
9.02
8.70
9.11
9.28
9.15
9.57
9.47
9.87
9.28
9.91
9.73
9.55
10.46
10.78
9.99
9.81
221.60
8.76
16.00
26.28
33.44
42.50
48.00
60.13
71.60
81.18
87.00
100.21
111.36
118.95
133.98
142.05
157.92
157.76
178.38
184.87
191.00
219.66
237.16
229.77
235.44
2873.40
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
441
484
529
576
4900
537
Time Series and Forecasting
The slope of the trend line is 0.0899. This shows that over the 24 quarters, the deseasonalized sales increased at a rate of 0.0899 ($ millions) per quarter, or $89 900 per quarter. The value
of 8.1096 is the intercept of the trend line on the Y-axis (i.e., for t = 0). The results are depicted on
the following chart. Note that the values for a and b can be slightly different due to rounding.
Sales ($ million)
Fitted and Deseasonalized Sales Data for
Toys International 2008 to 2013
11
10.5
10
9.5
9
8.5
8
7.5
Deseasonalized
Sales Data
Y' = 8.1096 + 0.0899t
R 2 = 0.785
0
4
8
12
Time
16
20
24
The above chart also shows the coefficient of determination. This value, R2, is 0.785, or
78.5%. We can use this value as an indication of the fit of the data. Because this is not sample
information, technically, we should not use R2 for judging a regression equation. However, it
will serve to quickly evaluate the fit of the deseasonalized sales data. In this instance, because
R2 is rather large, we conclude the deseasonalized sales of Toys International are effectively
explained by a linear trend equation.
If we assume that the past 24 periods are a good indicator of future sales, we can use the
trend equation to estimate future sales. For example, for the winter quarter of 2014, the value of
t is 25. Therefore, the estimated sale of that period is 10.3571, found by:
Y¿ = 8.1096 + 0.0899t = 8.1096 + 0.0899(25) = 10.3571
The estimated deseasonalized sales for the winter quarter of 2014 are $10 357 100. This is the
sales forecast, before we consider the effects of seasonality.
We use the same procedure and an Excel spreadsheet to determine a forecast for each of
the four quarters of 2014. A partial Excel output is as follows:
Now that we have the forecasts for the four quarters of 2014, we can seasonally adjust them.
The index for a winter quarter is 0.765. So, we can seasonally adjust the forecast for the winter
quarter of 2014 by: 10.3565(0.765) = 7.923. The estimates for each of the four quarters of 2014
are in the right-hand column of the Excel output. Note how the seasonal adjustments drastically increase the sales estimates for the last two quarters of the year. Also note that the values
for the estimated sales may be slightly different due to rounding.
538
Chapter 16
self-review 16–5
The Westberg Electric Company sells electric motors. The monthly trend equation, based on five
years of monthly data, is:
Y¿ = 4.4 + 0.5t
The seasonal factor for the month of January is 120, and it is 95 for February. Determine the seasonally adjusted forecast for January and February of the sixth year.
EXERCISES
11. The planning department of Padget and Kure Shoes, the manufacturer of an exclusive brand of women’s shoes, developed the following trend equation, in millions of pairs, based on five years of quarterly data:
Y¿ = 3.30 + 1.75t
The following table gives the seasonal factors for each quarter:
Quarter
I
II
Index 110.0 120.0
III
IV
80.0
90.0
Determine the seasonally adjusted forecast for each of the four quarters of the sixth year.
12. Team Sports Inc. sells sporting goods to high schools and colleges via a nationally distributed catalogue. Management at Team Sports estimates that it will sell 2000 Wilson Model A2000 catcher’s
mitts next year. The deseasonalized sales are projected to be the same for each of the four quarters
next year. The seasonal factor for the second quarter is 145. Determine the seasonally adjusted sales
for the second quarter of next year.
13. Refer to Exercise 9, regarding the absences at Anderson Belts Inc. Use the seasonal indexes you
computed to determine the deseasonalized absences. Determine the linear trend equation based on
the quarterly data for the three years. Forecast the seasonally adjusted absences for 2015.
14. Refer to Exercise 10, regarding sales at the Appliance Centre. Use the seasonal indexes you computed to determine the deseasonalized sales. Determine the linear trend equation based on the
quarterly data for the four years. Forecast the seasonally adjusted sales for 2015.
Chapter Summary
I. A time series is a collection of data over a period of time.
A. The trend is the long-run direction of the time series.
B. The cyclical component is the fluctuation above and below the long-term trend line over a longer period
of time.
C. The seasonal variation is the pattern in a time series within a year. These patterns tend to repeat themselves from year to year for most businesses.
D. The irregular variation is divided into two components.
1. The episodic variations are unpredictable, but they can usually be identified. A flood is an example.
2. The residual variations are random in nature.
II. A moving average is used to smooth the trend in a time series.
539
Time Series and Forecasting
III. The linear trend equation is Y = a + bt, where a is the Y-intercept, b is the slope of the line, and t is the coded time.
A. We use least squares to determine the trend equation.
B. If the trend is not linear, but rather the increases tend to be a constant percentage, the Y values are converted to logarithms, and a least squares equation is determined using the logarithms.
IV. A seasonal factor can be estimated using the ratio-to-moving-average method.
A. The six-step procedure yields a seasonal index for each period.
1. Seasonal factors are usually computed on a monthly or a quarterly basis.
2. The seasonal factor is used to adjust forecasts, taking into account the effects of the season.
Chapter Exercises
15. Refer to the following diagram:
a. Estimate the linear trend equation for the production series by drawing a line through the data.
b. What is the average annual decrease in production?
c. On the basis of the trend equation, what is the forecast for the year 2019?
20 000
Production
18 000
16 000
14 000
12 000
10 000
0
1993
1998
2003
2008
2013
Personal income (in dollars)
16. Refer to the following diagram:
a. Estimate the linear trend equation for the personal income series.
b. What is the average annual increase in personal income?
18 000
16 000
14 000
12 000
10 000
8 000
6 000
4 000
2 000
0
1998
2003
2008
2013
17. The asset turnovers, excluding cash and short-term investments, for the RNC Company from 2004 to 2014 are
as follows:
a.
b.
c.
d.
e.
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
1.11
1.28
1.17
1.10
1.06
1.14
1.24
1.33
1.38
1.50
1.65
Plot the data.
Determine the least squares trend equation.
Calculate the points on the trend line for 2007 and 2012, and plot the line on the graph.
Estimate the asset turnover for 2019.
How much did the asset turnover increase per year, on average, from 2004 to 2014?
540
Chapter 16
18. The sales, in billions of dollars, of Keller Overhead Door Inc. for 2009 to 2014 are as follows:
Year
Sales
($ billions)
Year
Sales
($ billions)
2009
2010
2011
$7.45
7.83
8.07
2012
2013
2014
$7.94
7.76
7.90
a. Plot the data.
b. Determine the least squares trend equation.
c. Use the trend equation to calculate the points for 2011 and 2014. Plot them on the graph, and
draw the regression line.
d. Estimate the net sales for 2017.
e. By how much have sales increased (or decreased) per year, on average, during the period?
19. The numbers of employees, in thousands, of Keller Overhead Door Inc. for the years 2009 to 2014
are as follows:
Year
Employees
(thousands)
Year
Employees
(thousands)
2009
2010
2011
45.6
42.2
41.1
2012
2013
2014
39.3
34.0
30.0
a. Plot the data.
b. Determine the least squares trend equation.
c. Use the trend equation to calculate the points for 2011 and 2014. Plot them on the graph, and
draw the regression line.
d. Estimate the number of employees in 2017.
e. By how much has the number of employees increased (or decreased) per year, on average, during
the period?
20. Listed below is the selling price for a share of PepsiCo Inc. at the close of the calendar year:
Year
Price
Year
Price
Year
Price
Year
Price
1990
1994
1998
2002
2006
2010
$13.00
18.13
40.88
42.22
62.55
65.33
1991
1995
1999
2003
2007
2011
16.94
27.94
35.25
46.62
75.90
66.35
1992
1996
2000
2004
2008
2012
20.75
29.25
49.56
52.20
54.77
68.43
1993
1997
2001
2005
2009
2013
20.44
36.25
48.69
59.08
60.80
82.94
Source: finance.yahoo.com/q/hp?s=PEP&a=11&b=31&c=2010&d=02&e=31&f=2011&g=m
a.
b.
c.
d.
Plot the data.
Determine the least-squares trend equation.
Calculate the points for the years 1998 and 2003.
Estimate the selling price in 2015. Does this seem like a reasonable estimate based on historical
data?
e. By how much has the stock price increased or decreased (per year), on average, during the period?
21. If plotted, the following sales series would appear curvilinear. This indicates that sales are increasing
at a somewhat constant annual rate (percentage). To fit the sales, therefore, a logarithmic straight-line
equation should be used.
Year
Sales
($ millions)
2004
2005
2006
2007
2008
2009
$8.0
10.4
13.5
17.6
22.8
29.3
Year
Sales
($ millions)
2010
2011
2012
2013
2014
$39.4
50.5
65.0
84.1
109.0
541
Time Series and Forecasting
a.
b.
c.
d.
Determine the logarithmic equation.
Determine the coordinates of the points on the logarithmic straight line for 2007 and 2012.
By what percentage did sales increase per year, on average, during the period from 2004 to 2014?
On the basis of the equation, what are the estimated sales for 2015?
22. Reported below are the amounts spent on advertising ($ millions) by a large firm from 2004 to 2014:
Year
Amount
Year
Amount
2004
2005
2006
2007
2008
2009
$88.1
94.7
102.1
109.8
118.1
125.6
2010
2011
2012
2013
2014
$132.6
141.9
150.9
157.9
162.6
a. Determine the logarithmic trend equation.
b. Estimate the advertising expenses for 2017.
c. By what percentage per year did advertising expense increase during the period?
23. Listed below is the selling price for a share of Oracle Inc. stock at the close of the calendar year:
Year
Price($)
Year
Price($)
Year
Price($)
Year
Price($)
1990
1994
1998
2002
2006
2010
0.1944
2.1790
7.1875
10.80
19.11
31.30
1991
1995
1999
2003
2007
2011
0.3580
3.1390
28.0156
13.23
20.23
25.69
1992
1996
2000
2004
2008
2012
0.7006
4.6380
29.0625
13.72
17.73
33.32
1993
1997
2001
2005
2009
2013
1.4197
3.7180
13.81
12.21
24.53
38.26
Source: nasdaq.com/aspx/historical_quotes.aspx?symbol=ORCL&selected=ORCL.
a. Plot the data.
b. Determine the least squares trend equation. Use both the actual stock price and the logarithm of
the price. Which seems to yield a more accurate forecast?
c. Calculate the points for the years 1996 and 2001.
d. Estimate the selling price in 2016. Does this seem like a reasonable estimate based on historical data?
e. By how much has the stock price increased or decreased (per year), on average, during the
period? Use your best answer from part (b).
24. The production of the Reliable Manufacturing Company for 2013 and part of 2014 is as follows:
Month
2013
Production
(thousands)
2014
Production
(thousands)
January
February
March
April
May
June
6
7
12
8
4
3
7
9
14
9
5
4
Month
2013
Production
(thousands)
2014
Production
(thousands)
3
5
14
6
7
6
4
July
August
September
October
November
December
a. Using the ratio-to-moving-average method, determine the specific seasonals for July, August, and
September 2013.
b. Assume that the specific seasonal indexes in the following table are correct. Insert in the table
the specific seasonals you computed in part (a) for July, August, and September 2013, and determine the 12 typical seasonal indexes.
Year
2013
2014
2015
2016
2017
Jan.
88.9
87.6
79.8
89.0
Feb.
102.9
103.7
105.6
112.1
Mar.
178.9
170.2
165.8
182.9
Apr.
118.2
125.9
124.7
115.1
May
60.1
59.4
62.1
57.6
c. Interpret the typical seasonal index.
June
43.1
48.6
41.7
56.9
July
Aug.
Sept.
Oct.
Nov.
Dec.
?
44.0
44.2
48.2
?
74.0
77.2
72.1
?
200.9
196.5
203.6
92.1
90.0
89.6
80.2
106.5
101.9
113.2
103.0
92.9
90.9
80.6
94.2
542
Chapter 16
25. The sales of Andre’s Boutique for 2013 and part of 2014 are as follows:
Month
2013 Sales
($thousands)
January
February
March
April
May
June
2014 Sales
($thousands)
$78
72
80
110
92
86
2013 Sales
($thousands)
Month
$65
60
72
97
86
72
July
August
September
October
November
December
2014 Sales
($thousands)
$81
85
90
98
115
130
$65
61
75
a. Using the ratio-to-moving-average method, determine the specific seasonals for July, August, September, and October 2013.
b. Assume that the specific seasonals in the following table are correct. Insert in the table the specific seasonals you computed in part (a) for June, July, August, and September 2013, and determine the 12 typical seasonal indexes.
Year
2013
2014
2015
2016
2017
Jan.
83.9
86.7
85.6
77.3
Feb.
77.6
72.9
65.8
81.2
Mar.
86.1
86.2
89.2
85.8
Apr.
May
118.7
121.3
125.6
115.7
99.7
96.6
99.6
100.3
June
July
?
87.0
85.5
88.9
92.0
92.0
94.4
89.7
Aug.
Sept.
Oct.
Nov.
Dec.
?
91.4
93.6
90.2
?
97.3
98.2
100.2
?
105.4
103.2
102.7
123.6
124.9
126.1
121.6
150.9
140.1
141.7
139.6
c. Interpret the typical seasonal index.
26. The quarterly production of pine lumber, in millions of board feet, by Northwest Lumber since 2010
is as follows:
Quarter
Year
Winter
2010
2011
2012
2013
2014
7.8
6.9
8.9
10.7
9.2
Spring
Summer
Fall
10.2
11.6
9.7
12.4
13.6
14.7
17.5
15.3
16.8
17.1
9.3
9.3
10.1
10.7
10.3
a. Determine the typical seasonal pattern for the production data using the ratio-to-moving-average
method.
b. Interpret the pattern.
c. Deseasonalize the data, and determine the linear trend equation.
d. Project the seasonally adjusted production for the four quarters of 2015.
27. Work Gloves Corp. is reviewing its quarterly sales of Toughie, the most durable glove it produces.
The numbers of pairs produced (in thousands) by quarter are as follows:
Quarter
Year
I
Jan.–Mar.
II
Apr.–June
III
July–Sept.
IV
Oct.–Dec.
2009
2010
2011
2012
2013
2014
142
146
160
158
162
162
312
318
330
338
380
362
488
512
602
572
563
587
208
212
187
176
200
205
543
Time Series and Forecasting
a. Using the ratio-to-moving-average method, determine the four typical quarterly indexes.
b. Interpret the typical seasonal pattern.
28. Sales of roof material, by quarter, since 2008 for Carolina Home Construction Inc. are shown below
(in $ thousands):
Quarter
Year
I
II
III
IV
2008
2009
2010
2011
2012
2013
2014
$210
214
246
258
279
302
321
$180
216
228
250
267
290
291
$60
82
91
113
116
114
120
$246
230
280
298
304
310
320
a. Determine the typical seasonal patterns for sales using the ratio-to-moving-average method.
b. Deseasonalize the data, and determine the trend equation.
c. Project the sales for 2015, and then seasonally adjust each quarter.
29. The inventory turnover rates for Bassett Wholesale Enterprises, by quarter, are as follows:
Quarter
Year
I
II
III
IV
2010
2011
2012
2013
2014
4.4
4.1
3.9
5.0
4.3
6.1
6.6
6.8
7.1
5.2
11.7
11.1
12.0
12.7
10.8
7.2
8.6
9.7
9.0
7.6
a. Arrive at the four typical quarterly turnover rates for the Bassett Company using the ratio-tomoving-average method.
b. Deseasonalize the data, and determine the trend equation.
c. Project the turnover rates for 2015, and seasonally adjust each quarter of 2015.
30. The following are the net sales of Mike’s Surf Shoppe from 2005 to 2014:
a.
b.
c.
d.
e.
Year
Sales
Year
Sales
Year
Sales
2005
2006
2007
2008
$58 436
59 994
61 515
63 182
2009
2010
2011
2012
$67 989
70 448
72 601
75 482
2013
2014
$78 341
81 111
Plot the data.
Determine the least squares trend equation. Use a linear equation.
Calculate the points for the years 2007 and 2012.
Estimate the sales for 2017. Does this seem like a reasonable estimate based on historical data?
By how much have sales increased or decreased (per year), on average, during the period?
31. Ray Anderson, owner of the Anderson Ski Lodge, is interested in forecasting the number of visitors
for the upcoming year. The following data are available, by quarter, since 2008. Develop a seasonal
index for each quarter. How many visitors would you expect for each quarter of 2015, if Ray projects
that there will be a 10% increase from the total number of visitors in 2014? Determine the trend
equation, project the number of visitors for 2015, and seasonally adjust the forecast. Which forecast
would you choose?
544
Chapter 16
Year
Quarter
Visitors
Year
Quarter
Visitors
2008
I
II
III
IV
I
II
III
IV
I
II
III
IV
I
II
III
IV
86
62
28
94
106
82
48
114
140
120
82
154
162
140
100
174
2012
I
II
III
IV
I
II
III
IV
I
II
III
IV
188
172
128
198
208
202
154
220
246
240
190
252
2009
2010
2011
2013
2014
32. The enrollment in the School of Business at the local college by quarter since 2010 is as follows:
Quarter
Year
Winter
Spring
Summer
Fall
2010
2011
2012
2013
2014
2033
2174
2370
2625
2803
1871
2069
2254
2478
2668
714
840
927
1136
—
2318
2413
2704
3001
—
Using the ratio-to-moving-average method:
a. Determine the four quarterly indexes.
b. Interpret the quarterly pattern of enrollment. Does the seasonal variation surprise you?
c. Compute the trend equation, and forecast the 2015 enrollment by quarter.
33. The Jamie Farr Kroger Classic is an LPGA (women’s professional golf) tournament played in Toledo,
Ohio, each year. Listed below are the total purse and the prize for winner for the 22 years from 1988
through 2009. Develop a trend equation for both variables. Which variable is increasing at a faster
rate? Project both the amount of the purse and the prize for the winner in 2011. Find the ratio of the
winner’s prize to the total purse. What do you find? Which variable can we estimate more accurately, the size of the purse or the winner’s prize?
Year
Purse ($)
Prize ($)
Year
Purse ($)
Prize ($)
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
$275 000
275 000
325 000
350 000
400 000
450 000
500 000
500 000
575 000
700 000
800 000
$41 250
41 250
48 750
52 500
60 000
67 500
75 000
75 000
86 250
105 000
120 000
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
$800 000
1 000 000
1 000 000
1 000 000
1 000 000
1 200 000
1 200 000
1 200 000
1 300 000
1 300 000
1 400 000
$120 000
150 000
150 000
150 000
150 000
180 000
180 000
180 000
195 000
195 000
210 000
545
Time Series and Forecasting
34. Sales for the La Parisienne Cosmetics Company (in $millions) are as follows:
a.
b.
c.
d.
e.
Year
Sales
($millions)
Year
Sales
($millions)
1996
1997
1998
1999
2000
2001
2002
2.3
2.9
3.4
4.5
3.4
4.0
4.2
2003
2004
2005
2006
2007
2008
2009
4.5
4.7
5.0
5.2
5.5
4.3
4.4
Year
Sales
($millions)
2010
2011
2012
2013
2014
4.5
4.9
5.2
5.5
5.8
Develop a three-year moving average.
Develop a four-year moving average.
Develop a five-year moving average.
Develop a seven-year moving average.
Interpret your findings.
35. Listed below is the number of movie tickets sold at the Library Cinema Complex, in thousands, for the period
from 2000 to 2014. Compute a three-and five-year moving average. Describe the trend.
Year
Tickets
(thousands)
Year
Tickets
(thousands)
Year
Tickets
(thousands)
2000
2001
2002
2003
2004
5.6
6.7
6.8
7.3
7.5
2005
2006
2007
2008
2009
6.5
6.2
6.3
7.2
7.5
2010
2011
2012
2013
2014
7.4
8.1
8.3
8.3
8.5
36. Listed below is the number of rooms rented at Spruce Falls Resort in Bobcaygeon from 2004 to 2014. Determine the least squares equation. According to this information, what is the estimated number of rentals for the
next five years?
Year
Rentals
Year
Rentals
Year
Rentals
2004
2005
2006
2007
6 714
7 991
9 075
9 775
2008
2009
2010
2011
9 762
10 180
8 334
8 272
2012
2013
2014
6 162
6 897
8 285
37. The average salary of accountants working for the XYZ Corp. is listed below:
a.
b.
c.
d.
Year
Salary
Year
Salary
Year
Salary
Year
Salary
1993
1994
1995
1996
1997
1998
58 436
59 994
61 515
63 182
67 989
70 448
1999
2000
2001
2002
2003
2004
72 601
75 482
78 341
81 111
83 918
86 666
2005
2006
2007
2008
2009
2010
89 257
92 574
95 843
99 248
102 771
106 099
2011
2012
2013
2014
109 031
112 483
117 138
112 136
Plot the data.
Determine the least squares trend equation. Use a linear equation.
Estimate the average salary for 2016.
By how much has the average salary increased (per year), on average, during the time period?
546
Chapter 16
38. Banner Rocker Company manufacturers rocking chairs. The company developed a special rocker for children
and would like to create a model to forecast sales for the next five years on the basis of the number sold in the
past years. The data are as follows:
a.
b.
c.
d.
Year
Sales
($000)
2004
2005
2006
2007
2008
2009
153
156
153
147
159
160
Year
Sales
($000)
2010
2011
2012
2013
2014
167
168
170
169
173
Plot the data.
Determine the least squares trend equation. Use a linear equation.
Forecast sales for the next five years.
By how much have sales increased (per year), on average, during the period?
39. Sales per quarter for Sunshine Shores Boat and Ski Rentals are as follows:
Year
Winter
Spring
Summer
Fall
2012
2013
2014
13.5
14.6
15.0
10.8
10.9
11.1
8.7
8.8
8.9
5.2
4.9
5.4
a. Determine a typical seasonal index for each of the four quarters.
b. Comment on the indexes.
40. Sales per quarter for Sunshine Shores Boat and Ski Rentals along with the seasonal indexes are as follows:
a.
b.
c.
d.
t
Year
Quarter
Sales
($000’s)
Seasonal
Indexes
1
2
3
4
5
6
7
8
9
10
11
12
2012
2012
2012
2012
2013
2013
2013
2013
2014
2014
2014
2014
1
2
3
4
1
2
3
4
1
2
3
4
13.5
10.8
8.7
5.2
14.6
10.9
8.8
4.9
15.0
11.1
8.9
5.4
149.046
110.505
89.421
51.028
149.046
110.505
89.421
51.028
149.046
110.505
89.421
51.028
Determine the deseasonalized sales for each of the four quarters.
Use the deseasonalized sales to develop a linear trend equation.
How much did sales increase (on average) over the time period?
Estimate sales for the four quarters of 2015.
41. Nine Thirty Jewellery has forecasted sales of $825 000 next year. The sales index for February is 126.9, and the
sales index for March is 86.3. Estimate the dollar value of sales for each month.
42. The number of desks produced per quarter for Office Elegance is as follows:
Year
Winter
Spring
Summer
Fall
2011
2012
2013
2014
1234
1334
1511
1434
1300
1234
1496
1576
1225
1134
1300
1356
1323
1450
1545
1598
547
Time Series and Forecasting
a. Determine a typical seasonal index for each of the four quarters.
b. Comment on the indexes.
43. Units produced per quarter for Office Elegance along with the seasonal indexes are as follows:
a.
b.
c.
d.
t
Year
Quarter
Units
Seasonal
Indexes
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
2011
2011
2011
2011
2012
2012
2012
2012
2013
2013
2013
2013
2014
2014
2014
2014
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1234
1300
1225
1323
1334
1234
1134
1450
1511
1496
1300
1545
1434
1576
1356
1598
102.89
101.89
90.32
104.89
102.89
101.89
90.32
104.89
102.89
101.89
90.32
104.89
102.89
101.89
90.32
104.89
Determine the deseasonalized number of units produced for each of the four quarters.
Use the deseasonalized number of units produced to develop a linear trend equation.
How much have the number of units produced increased (on average) over the time period?
Estimate number of units produced for the four quarters of 2015.
Data Set Exercises
Practise and learn online with Connect. Questions and tables with online data sets are marked with
.
44. Use the file Stock Indexes on Connect to determine the following:
a. Develop a trend line for the S&P/TSX Index from 2001 to 2013. Does the trend line follow a linear or log
equation? Use the best fit to estimate the value of the index in 2015.
b. Develop a trend line for the DJIA Index from 2001 to 2013. Does the trend line follow a linear or log equation? Use the best fit to estimate the value of the index in 2015.
c. Develop a trend line for the S&P/TSX Venture Index from 2001 to 2013. Does the trend line follow a linear
or log equation? Use the best fit to estimate the value of the index in 2015.
d. Develop a trend line for the DASDAQ index from 2001 to 2013. Does the trend line follow a linear or log
equation? Use the best fit to estimate the value of the index in 2015.
e. Develop a trend line for the S&P 500 index from 2001 to 2013. Does the trend line follow a linear or log
equation? Use the best fit to estimate the value of the index in 2015.
Practice Test
Part I Objective
1. The long-term direction of a time series is called the
2. One method used to smooth the trend in a time series is a
3. Irregular variation in a time series that is random in nature is called
4. In a three-year moving average, the weights given to each period are
(the same, oldest year has most weight, oldest year has the least weight)
.
.
.
.
548
Chapter 16
Part II Problems
1. The monthly linear trend equation for the Hoopes ABC Beverage Store is:
Y¿ = 5.50 + 1.25t
The equation is based on four years of monthly data and is reported in thousands of dollars. The index for
January is 105.0, and for February, it is 98.3. Determine the seasonally adjusted forecast for January and February of the fifth year.
Computer Commands
1. MegaStat steps to create moving averages are as follows:
a. Open the file Table 16-2 Moving Average from Connect.
b. Select MegaStat, Time Series/Forecasting, and Moving
Average.
c. In the Input range for data, enter B1:B20 and 3 (three-year
moving average) in the Number of Periods box. Click OK.
Note: for a five-year moving average, enter 5 in the
Number of Periods box.
2. MegaStat steps to produce the linear trend equation and estimate sales for period 8 for the Jensen Food example are as
follows:
a. Select MegaStat, Time Series/Forecasting, and Trendline Curve Fit.
b. In the Input range of time series, enter the range of data,
select Linear for the Type of trendline, 1 for the Beginning period, and enter 1 in the Forecast for period box
and 8 in the starting with period box. Click OK.
3. MegaStat steps for the Gulf Shores example are as follows:
b. Select MegaStat, Time Series/Forecasting, and Trendline Curve Fit.
c. In the Input range of time series, enter B1:B16,
select Exponential (Log) for the Type of trendline,
1 for the Beginning period, and enter 1 in the
Forecast for period box and 19 in the starting
with period box. Click OK.
Note: answers may be slightly different due to rounding.
4. MegaStat steps for the Toys International example are as
follows:
5. a. Open the file Table 16-6 Toys International Quarterly
Sales from Connect, Data Files and Tables.
b. Select MegaStat, Time Series/Forecasting, and
Deseasonalization.
c. In the Input range of seasonal data, enter B1:B25, select
Quarterly, for the First data periods: enter 1st Quarters,
and enter 1 for Year. Click OK.
Note: answers may be slightly different due to rounding.
a. Open the file Ch 16 Gulf Shores from Connect, Data
Files and Exercises.
Answers to Self-Reviews
16–1
Year
Production
(thousands)
2009
2010
2011
2012
2013
2014
2
6
4
5
3
10
Three-Year
Moving
Total
Three-Year
Moving
Average
12
15
12
18
Number produced (000)
12
9
Original data
Moving average
6
3
0
2008
2009 2010 2011
2012 2013
4
5
4
6
16–2 (a)
X =7
Y = 11.7263
16
12
8
X=1
Y = 4.5833
4
0
2007 2008 2009 2010 2011 2012 2013 2014 Years
1
2
3
4
5
6
7
8 Codes
(b) Y ' = a + bt = 3.3928 + 1.1905t (in thousands).
b=
50
8(365) − 36(70)
=
= 1.1905.
8(204) − (36) 2
42
a=
36
70
− 1.1905( ) = 3.3928.
8
8
549
Time Series and Forecasting
(c) For 2007:
16–4 (a) The following values are from a software program. Due
to rounding, your figures might be slightly different.
Y¿ = 3.3928 + 1.1905(1) = 4.5833
for 2013:
Y¿ = 3.3928 + 1.1905(7) = 11.7263
(d) For 2017, t = 11, so
Y¿ = 3.3928 + 1.1905(11) = 16.4883
or 16 488 king-size rockers.
16–3 (a)
Year
Y
log Y
t
t log Y
t2
2010
2011
2012
2013
2014
2.13
18.10
39.80
81.40
112.00
0.3284
1.2577
1.5999
1.9106
2.0492
7.1458
1
2
3
4
5
15
0.3284
2.5154
4.7997
7.6424
10.2460
25.5319
1
4
9
16
25
55
1
2
3
4
5
mean:
adjusted:
1
2
3
4
1.178
1.152
1.205
1.239
1.194
1.192
1.283
0.827
0.855
0.789
0.794
0.817
0.815
0.750
1.223
1.209
1.298
0.778
0.755
0.687
1.253
1.251
0.742
0.741
4.006
4.000
No correction is needed.
(b) Total sales for the winter season are typically 19.4%
above the annual average.
16–5 The forecast value for January of the sixth year is 34.9,
found by:
Y¿ = 4.40 + 0.5(61) = 34.9
20.4725
5(25.5319) − (7.1458)(15)
=
= 0.40945.
b=
5(55) − (15) 2
50
15
7.1458
− 0.40945( ) = 0.20081.
a=
5
5
(b) About 156.7%. The antilog of 0.40945 is 2.567. Subtracting 1 yields 1.567.
(c) About 454.5, found by: Y ' = 0.20081 + 0.40945(6) =
2.65751. The antilog of 2.65751 is 454.5.
Seasonally adjusting the forecast, 34.9(120)/100 = 41.88.
For February, Y ' = 4.40 + 0.50(62) = 35.4. Then,
(35.4)95/100 = 33.63.