Chapter 11
Forecasting
Outcome
By the end of this session you should be able to:
identify trends and patterns in time series graphs, using appropriate techniques
identify the components of a time series model
understand seasonal variations using both additive and multiplicative models
and explain when each is appropriate
calculate predicted values, given a time series model
identify the limitations of forecasting models
and answer questions relating to these areas.
The underpinning detail for this chapter in your Integrated Workbook can be
found in Chapter 11 of your Study Text
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Chapter 11
Overview
FORECASTING
Components of
time series
Forecasting
Establishing Using the
seasonal
the trend model
elements
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Forecasting
Introduction
1.1 Components and models of time series
the trend, T
the seasonal component, S
the cyclical component, C; and
the residual (or irregular, or random) component, R.
1.2 Additive model
Y=T+S+C+R
1.3 Multiplicative model
Y=T×S×C×R
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Chapter 11
Question 1
By calculating the predicted sales units, you are developing the technical
knowledge related to forecasting.
Additive model
If an underlying trend forecasts sales of 3,000 units but there are seasonal
variations of –600 applicable for this period along with a cyclical factor of –240
and a residual factor of +540, calculate the predicted sales units.
Y=T+S+C+R
Y = 3,000 – 600 – 240 + 540 = 2,700 units
Question 2
Multiplicative model
If an underlying trend forecasts sales of 3,000 units but there are seasonal
variations of 0.8 applicable for this period along with a cyclical factor of 0.9 and
a residual factor of 1.25, calculate the predicted sales units.
Y=T×S×C×R
Y = 3,000 × 0.8 × 0.9 × 1.25 = 2,700 units
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Forecasting
Question 3
By calculating the value of the residual element, you are demonstrating
the maths skill of solving mathematical problems.
Multiplicative backwards
In a time series analysis using the multiplicative model, at a certain time actual,
trend, seasonal and cyclical values are 196.35, 150, 1.1 and 0.85 respectively.
Calculate the value of the residual element.
Y=T×S×C×R
196.35 = 150 × 1.1 × 0.85 × R
R = 196.35/(150 × 1.1 × 0.85) = 1.4
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Chapter 11
Establishing the trend, T
2.1 linear regression
Only valid if we assume a linear trend
2.2 Moving averages
Useful when we cannot assume a linear trend
Example
The following table gives the takings ($000) of a shopkeeper in each quarter
of four successive years
Qtrs 1 2 3 4
20X1 13 22 58 23
20X2 16 28 61 25
20X3 17 29 61 26
20X4 18 30 65 29
The sales show the following pattern:
Sales
Time
There is a clear seasonal element and an upward underlying trend.
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Forecasting
Example continued
Takings 8 Qtr
Quarter ($’000) 4 Qtr total Average
Year =Y total (centred) =T
1 13 –
–
2 22 – –
20X1 116
3 58 235 29.4
119
4 23 244 30.5
125
1 16 253 31.6
128
2 28 258 32.2
20X2 130
3 61 261 32.6
131
4 25 263 32.9
132
1 17 264 33.0
132
2 29 265 33.1
20X3 133
3 61 267 33.4
134
4 26 269 33.6
135
1 18 274 34.2
139
2 30 281 35.1
20X4 142
3 65 – –
–
4 29 –
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Chapter 11
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Forecasting
Question 4
By calculating the predicted sales units, you are developing the technical
knowledge related to forecasting.
Moving averages
A time series analysis of past sales units shows the following actual figures:
Jan 107, Feb 176, Mar 197, Apr 182, May 251, Jun 272, Jul 257, Aug 326, Sep
347, Oct 332, Nov 401.
Calculate the 3 point moving averages for the data given
Y=T+S
Actual/prediction 3 point average
Month (Y) 3 point total (T)
Jan 107
Feb 176 480 160
Mar 197 555 185
Apr 182 630 210
May 251 705 235
Jun 272 780 260
Jul 257 855 285
Aug 326 930 310
Sep 347 1005 335
Oct 332 1080 360
Nov 401
Dec
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Chapter 11
Question 5
Moving averages
A time series analysis of past sales units shows the following actual and trend
figures. Calculate the seasonal variations for the data given and use the
information to predict a sales unit value for December.
Y=T+S
Actual/prediction 3 point average
Month (Y) 3 point total (T)
Jan 107
Feb 176 480 160
Mar 197 555 185
Apr 182 630 210
May 251 705 235
Jun 272 780 260
Jul 257 855 285
Aug 326 930 310
Sep 347 1005 335
Oct 332 1080 360
Nov 401
Dec
Y=T+S S=Y–T
Actual/prediction 3 point 3 point average Seasonal
Month (Y) total (T) variation (S)
Jan 107
Feb 176 480 160 16
Mar 197 555 185 12
Apr 182 630 210 –28
May 251 705 235 16
Jun 272 780 260 12
Jul 257 855 285 –28
Aug 326 930 310 16
Sep 347 1005 335 12
Oct 332 1080 360 –28
Nov 401 =360 + 25 = 385 16
Dec =410 + 12 = 422 =385 + 25 = 410 12
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Forecasting
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Chapter 11
Forecasting seasonal components
3.1 Two approaches to obtaining seasonal variations
The multiplicative model The additive model
S = Y/T S=Y–T
Example continued Example continued
Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4
20X1 – – 1.97 0.75 20X1 – – +29 –8
20X2 0.51 0.87 1.87 0.76 20X2 –16 –4 +28 –8
20X3 0.52 0.88 1.83 0.77 20X3 –16 –4 +28 –8
20X4 0.53 0.85 – – 20X4 –17 –5 – –
Average 0.52 0.87 1.89 0.76 Average –16 –4 28 –8
Adjusted 0.51 0.86 1.88 0.75 Adjusted –16 –4 28 –8
Note: Seasonal variations are adjusted Note: Seasonal variations are adjusted
so they add up to 4. so they add up to 0.
usually considered the better assumes that the seasonal
variations are a constant amount,
ensures that seasonal variations are and thus would constitute a
assumed to be a constant diminishing part of, say, an
proportion of the sales. increasing sales trend.
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Forecasting
Using the model
4.1 Forecasting
Typically ignore cyclical and random elements
Additive: Y=T+S
Multiplicative: Y=T×S
4.2 'De-seasonalising' data
Effectively the same as extracting the trend
Typically ignore cyclical and random elements
Additive: T=Y–S
Multiplicative: T=Y÷S
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Chapter 11
Question 6
By calculating the predicted sales units, you are developing the skill of
analysis.
Seasonal variation
A trend analysis reveals that the underlying trend of sales units can be
calculated using the equation:
Sales units = 1,350 + 6t where t is the month number.
The seasonal variations, using an additive model, for the sales units have been
calculated as follows:
Any month in quarter 1 +50 units
Any month in quarter 2 +15 units
Any month in quarter 3 –5 units
Any month in quarter 4 –60 units.
If t = 1 in March of 2016, calculate the predicted sales units for November 2017.
In November 17, t = 21
Trend sales units = 1,350 + 6 × 21 = 1,476
Applying seasonal variation Y = T + S
Y = 1,476 – 60 = 1,416 units predicted
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Forecasting
Question 7
Seasonal variation
A trend analysis reveals that the underlying trend of sales units can be
calculated using the equation:
Sales units = 990 + 19t where t is the quarter number.
The seasonal variations, using a multiplicative model, for the sales units have
been calculated as follows:
Quarter 1 95%
Quarter 2 85%
Quarter 3 120%
Quarter 4 100%.
If t = 1 in quarter 1 of 2016, calculate the predicted sales units for quarter 3 of
2018.
In Q3 2018, t = 11
Trend sales units = 990 + 19 × 11 = 1,199
Applying seasonal variation Y = T × S
Y = 1,199 × 120% = 1,439 units predicted
Illustrations and further practice
Now read illustrations 1 to 7 and try TYUs 1 to 13 from Chapter 11
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Chapter 11
You should now be able to answers all the questions from chapter 11 of the
Study Text and questions 204 – 212 from the Exam Practice Kit.
For further reading, visit Chapter 11 from the Study Text.
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Forecasting
Answers
Question 1
Y=T+S+C+R
Y = 3,000 – 600 – 240 + 540 = 2,700 units
Question 2
Y=T×S×C×R
Y = 3,000 × 0.8 × 0.9 × 1.25 = 2,700 units
Question 3
Y=T×S×C×R
196.35 = 150 × 1.1 × 0.85 × R
R = 196.35/(150 × 1.1 × 0.85) = 1.4
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Chapter 11
Question 4
Y=T+S
Actual/prediction 3 point average
Month (Y) 3 point total (T)
Jan 107
Feb 176 480 160
Mar 197 555 185
Apr 182 630 210
May 251 705 235
Jun 272 780 260
Jul 257 855 285
Aug 326 930 310
Sep 347 1005 335
Oct 332 1080 360
Nov 401
Dec
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Forecasting
Question 5
Y=T+S S=Y–T
Actual/prediction 3 point 3 point average Seasonal
Month (Y) total (T) variation (S)
Jan 107
Feb 176 480 160 16
Mar 197 555 185 12
Apr 182 630 210 –28
May 251 705 235 16
Jun 272 780 260 12
Jul 257 855 285 –28
Aug 326 930 310 16
Sep 347 1005 335 12
Oct 332 1080 360 –28
Nov 401 =360 + 25 = 385 16
Dec =410 + 12 = 422 =385 + 25 = 410 12
Question 6
In November 17, t = 21
Trend sales units = 1,350 + 6 × 21 = 1,476
Applying seasonal variation Y = T + S
Y = 1,476 – 60 = 1,416 units predicted
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Chapter 11
Question 7
In Q3 2018, t = 11
Trend sales units = 990 + 19 × 11 = 1,199
Applying seasonal variation Y = T × S
Y = 1,199 × 120% = 1,439 units predicted
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