Chapter 17
Time series: Measurements taken over time. i.e. every
hour, day, week, month or year.
17.1 Time series components or patterns
TREND COMPONENT (T)
The gradual shifting of the time series over longer period of
time is referred to as the trend in the time series. It has a
smooth pattern and is longer than a year.
CYCLICAL COMPONENT (C)
Any recurring sequence of points above and below the trend
line lasting more than one year can be attributed to the
cyclical component of the time series. For example: Levels
of pessimism or optimism in the economy, trade unions,
world organisations etc.
Volume
Trend and cyclical components of a time series with
data points one year apart
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21
Time
Copyright Reserved
1
SEASONAL COMPONENT (S)
Identical variation patterns repeating daily, monthly,
quarterly. The time series show a regular pattern within
one-year periods. For example: Traffic flow/volume during
week days, special occurring events e.g. Rand Easter Show,
school holidays etc.
IRREGULAR COMPONENT (I)
The irregular component is caused by the unpredictable
occurrences like one-off events such as natural disasters
(floods, fires etc.) or man-made disasters (strikes, boycotts
or accidents).
17.3 Smoothing Methods
Moving Average Forecast
• Can only be used for stable time series – no significant
trend, cyclical or seasonal effects. In other words, using
the moving average forecast approach, the time series
must consist of only the irregular component.
Copyright Reserved
2
• 3-week moving average
Moving
Sales
Average
(in 1000) Forecast
Week
1
17
2
21
3
19
4
23
5
18
6
16
7
20
8
18
9
22
10
20
11
15
12
22
Total
Squared
Forecast Forecast
Error
Error
− − 0
92
= ∑ − =
• Average of the sum of squared errors (MSE)
=
• How many values should be included in the moving
average?
Copyright Reserved
3
• 5-week moving average
Moving
Sales
Average Forecast
(in 1000) Forecast Error
− Week
1
17
2
21
3
19
4
23
5
18
6
16
7
20
8
18
9
22
10
20
11
15
12
22
Total
-1.2
Squared
Forecast
Error
− 51.84
• Average of the sum of squared errors:
=
• The MSE of the 5-weekly moving average is the smallest
• Hence the 5-weekly moving average is the best for
forecasting
Copyright Reserved
4
• Use Excel’s AVERAGE function for section 17.3; selfstudy [also see Assignment Book]
17.4 Trend Projection
• Estimating the long term linear trend
= + = trend value of the time series in period t
= y-intercept of the trend line
= slope of the trend line
t = time
• Time scale
t = 1: time of the first observation on the time series data
t = 2: time of the second observation on the time series data
= value of the time series at period t.
Year
t
1994
1995
1996
1997
1998
1
2
3
4
5
Sales (in 1000)
13
15
20
21
23
Copyright Reserved
5
• Formulas for calculating the trend line
=
∑ − − ∑ − ∑ − ∑ ∑ /
=
∑ − ∑ /
and
= − Copyright Reserved
6
• Calculating the trend line
Year
Sales
− t
(in 1000)
1
13
2
15
3
20
4
21
5
23
15
92
=
=
− − − − =
∑ !!
∑!
"
=
= − =
• Estimated trend line: = + =
• Interpretation:
=
The estimated number of items sold has
increased by ________ per year.
=
The estimated number of items sold during
1993 was __________.
Copyright Reserved
7
• Forecasting:
The estimated number of items to be sold during the next
year:
Set: t = 6
Calculating the slope with the “calculation” formula
Year (t)
1
2
3
4
5
15
=
Sales in 1000
(Yt)
13
15
20
21
23
92
∑ !∑ ∑ /#
∑ " !∑ " /#
=
t Yt
t2
13
30
60
84
115
302
1
4
9
16
25
55
$!%&/%
%%!%" /%
=
• Note : The answer is the same
MULTIPLICATIVE MODEL
The multiplicative time series model: = × × (
We will illustrate the use of the multiplicative model with
the two examples below.
Copyright Reserved
8
EXAMPLE (ODD)
A movie theatre wants to determine the popularity of
specific weekdays of moviegoers in order to formulate an
advertising campaign during busier periods. The number of
people (in hundreds) visiting the movie theatre daily is
summarised in the table below:
Estimate of Week
t
Day
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Mon
Tues
Wed
Thurs
Fri
Mon
Tues
Wed
Thurs
Fri
Mon
Tues
Wed
Thurs
Fri
Mon
Tues
Wed
Thurs
2
3
4
Number of
moviegoers
(in hundreds)
Yt
5
15.2
7.4
6.7
21
4.1
14.9
6.7
6
19
2.1
14.2
6.5
6.1
18
(
5-day moving
average
Seasonal
Irregular
Value
10.68
10.54
10.14
9.74
9.6
9.56
9.58
9.38
0.66
0.62
1.98
0.22
***
0.69
)
Deseasonalised
moviegoers
(in hundreds)
9.68
9.69
7.00
9.79
9.70
9.84
9.18
Calculation for Question 7
Copyright Reserved
9
Seasonal irregular values (* + values):
Monday Tuesday Wednesday Thursday
Friday
Average seasonal irregular values:
Monday Tuesday Wednesday Thursday
Friday
Correction factor:
No correction factor is needed, since the sum of the average
seasonal irregular values =
Seasonal indices:
Monday Tuesday Wednesday Thursday
Friday
Note: The trend line fitted to the deseasonalised series is:
Copyright Reserved
10
Question 1: The 5-day moving average for Friday Week 1
is:
Question 2: The seasonal irregular value for a Tuesday
Week 2 is:
Question 3: The seasonal irregular value for a Tuesday
Week 3 is:
Question 4: The seasonal index for Tuesday is:
Question 5: The deseasonalised number of moviegoers (in
hundreds) for Wednesday Week 2 is:
OR The seasonally adjusted number of moviegoers (in
hundreds) for Wednesday Week 2 is:
Copyright Reserved
11
Question 6: The interpretation of the slope of the trend line
is:
Question 7: According to the seasonal index for
Wednesday the number of moviegoers is:
Question 8: The trend value of the number of moviegoers
for Wednesday Week 4 is:
Question 9: The estimated number of moviegoers (in
hundreds) for Thursday Week 4 is:
Hint: Use the multiplicative model
= × × (
Copyright Reserved
12
EXAMPLE (EVEN)
The quarterly sales data (2000-2002) for the PQR company
(in R1000) is given below:
Estimate of (
)
Year
Quarter
t
Sales
2000
1
1
54
69.43
2
2
58
***
3
3
94
1.3598
70.05
4
4
70
1.0054
68.88
1
5
55
***
70.71
2
6
61
0.9004
70.59
3
7
87
***
64.84
4
8
66
1.0154
64.94
1
9
49
0.751
63.00
2
10
55
0.8178
63.65
3
11
95
70.80
4
1
2
3
4
1
2
12
13
14
15
16
17
18
74
72.81
2001
2002
2003
2004
4-quartly
Moving
Average
Centered
Moving
Average
Seasonal
Irregular
Value
Deseasonalised
Sales
Calculation of Questions 8 and 9
Copyright Reserved
13
Seasonal irregular values:
Quarter 1
Quarter 2
Quarter 3
Quarter 4
Average seasonal irregular values (Also called
unadjusted seasonal indexes):
Quarter 1
Quarter 2
Quarter 3
Quarter 4
Correction factor:
The sum of the unadjusted seasonal indexes is
Therefore, the correction factor =
Seasonal Index:
Quarter 1
Quarter 2
Quarter 3
Quarter 4
Given: The trend line fitted for the deseasonalised sales is:
Copyright Reserved
14
Question 1: The centered moving average for the third
quarter of 2000 is:
Question 2: The seasonal irregular value for the first
quarter of 2001 is:
Question 3: The seasonal irregular value for the third
quarter of 2001 is:
Question 4: The unadjusted seasonal index for the fourth
quarter is:
Question 5: If the sum of the unadjusted seasonal indices is
3.977, then the correction factor is:
Copyright Reserved
15
Question 6: The adjusted seasonal index for the second
quarter is 0.8641. This indicates that:
Question 7: The deseasonalised sales (in R1000) for the
second quarter of 2000 is:
Question 8: The trend value (in 1000) for the third quarter
of 2003 is:
Question 9: The estimated sales (in 1000) for the second
quarter of 2004 is:
Copyright Reserved
16