KVANLI PAVUR KEELING. Chapter 4 Time Series Analysis and Forecasting. Chapter Objectives. At the completion of this chapter, you should be able to: ∙ Deseasonalize a time series by first calculating the seasonal indexes
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∙ Deseasonalize a time series by first calculating the
seasonal indexes
∙ Discuss the nature of additive and multiplicative
seasonality
∙ Estimate the trend, cyclical and noise components
∙ Forecast a time series containing trend
∙ Calculate price indexes, including a Laspeyres and
Paasche index
1 1985 1.7
2 1986 2.4
3 1987 2.8
4 1988 3.4
.
.
22 2006 9.6
23 2007 10.7
This is y1
This is an example of annual data
This is y23
∙ annual (one value for each year)
∙ quarterly (4 values for each year)
∙ monthly (12 values for each year)
∙ Trend (TR) ∙ Seasonality (S)
∙ Cyclic (C) ∙ Irregular or noise (I)
Monthly or quarterly data only
11.0 –
10.0 –
9.0 –
8.0 –
7.0 –
6.0 –
5.0 –
4.0 –
3.0 –
2.0 –
1.0 –
Trend
Number of employees (thousands)

2000

2001

2002

2003

2004

2005

2006

2007
t
Employees ExampleWe’ll take a closer look at this example in the slides to follow
300 –
200 –
100 –
Power consumption (million kwh)

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007
t (time)
An Illustration of Curvilinear TrendThe increase in power consumption slows down over time and the trend is not linear
Figure 4.1
∙ Additive seasonality
∙ Multiplicative seasonality
∙ Usually the case
∙ Assumed in the
Excel macros
4.0 –
3.5 –
3.0 –
2.5 –
2.0 –
1.5 –
1.0 –
Corporate taxes
(millions of dollars)
1
2
3

1980

1990

2000

2005
Textile Exampleyt = St+TRt + Ct+ It
yt = St ∙ TRt ∙ Ct ∙ It
The seasonal component (St) is omitted for annual data
Multiplicative seasonality is the usual situation and assumed in the Excel macros
This is the trend line
yt= b0+ b1t
1 1.1 1.1
2 2.4 4.8
3 4.6 13.8
.
.
8 11.289.6
48.3 276.3
Let A = the sum of the time series values
So, A = 48.3
Let B = the sum of the righthand column
So, B = 276.3
Let T = the number of time periods. So, T = 8
Carry a lot of decimal places
The trend line is = .279 + 1.404t
OK to round now
The forecast for 2008
The forecast period
sample data
yt = TRt ∙ Ct ∙ It
= .279 + 1.404(1) = 1.125
= .279 + 1.404(2) = 2.529
= .279 + 1.404(3) = 3.933
= .279 + 1.404(8) = 10.953
tytytyt/yt
^
^
1 1.1 1.125 .978
2 2.4 2.529 .949
3 4.6 3.933 1.169
4 5.4 5.337 1.012
5 5.9 6.741 .875
6 8.0 8.145 .982
7 9.7 9.549 1.016
8 11.2 10.953 1.022
Trend and Cyclical ActivityTrend activity
Cyclical activity
Ct
1.15 –
1.10 –
1.05 –
1.00 –
.95 –
.90 –
Start
End

1

2

3

4

5

6

7

8
t
2000
2002
2004
2007
Plot of Cyclical ComponentsYt
11.0 –
10.0 –
9.0 –
8.0 –
7.0 –
6.0 –
5.0 –
4.0 –
3.0 –
2.0 –
1.0 –
Actual yt
^
yt = −.279 + 1.404t
(trend line)
Number of employees (thousands)

2000

2001

2002

2003

2004

2005

2006

2007
t
Cyclical ActivityYt
Trend
2000 –
1500 –
1000 –
500 –
100 units
100 units
Actual time series
Units sold
100 units

Winter
2005

Winter
2006

Winter
2007
t
Additive Seasonal VariationFigure 4.16
Yt
700 –
600 –
500 –
400 –
300 –
200 –
100 –
TRt = 100 + 20t
Sales (tens of thousands of dollars)
Estimated sales using trend and seasonality

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20
t
JetskiSales – Additive SeasonalityFigure 4.17
Yt
2000 –
1500 –
1000 –
500 –
250 units
Trend
180 units
Units sold
Actual time series
100 units

Winter
2005

Winter
2006

Winter
2007
t
Heat Pump Sales – Multiplicative SeasonalityFigure 4.18
Yt
700 –
600 –
500 –
400 –
300 –
200 –
100 –
TRt = 100 + 20t
Sales (tens of thousands of dollars)
Estimated sales using trend and seasonality

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20
t
Jetski Sales Multiplicative Season VariationFigure 4.19