1. Time Series and Forecasting
Chapter 16
McGraw-Hill/Irwin
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

2. Learning Objectives
LO1 Define the components of a time series LO2 Compute Moving average, weighted moving average and exponential smoothing LO3 Determine a linear trend equation LO4 Use a trend equation to compute forecasts LO5 Determine and interpret a set of seasonal indexes LO6 Deseasonalize data using a seasonal index LO7 Calculate seasonally adjusted forecasts LO8 Use a trend equation for a nonlinear trend
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3. Time Series and its Components
TIME SERIES is a collection of data recorded over a period of time (weekly, monthly, quarterly), an analysis of history, that can be used by management to make current decisions and plans based on long-term forecasting. It usually assumes past pattern to continue into the future
Components of a Time Series
Secular Trend – the smooth long term direction of a time series
Cyclical Variation – the rise and fall of a time series over periods longer than one year
Seasonal Variation – Patterns of change in a time series within a year which tends to repeat each year
Irregular Variation – classified into:
Episodic – unpredictable but identifiable
Residual – also called chance fluctuation and unidentifiable
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4. Secular Trend – Examples
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5. Cyclical Variation – Sample Chart
1991
1996
2001
2006
2011
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6. Seasonal Variation – Sample Chart
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7. Irregular variation
Caused by irregular and unpredictable changes in a times series that are not caused by other components
Exists in almost all time series
Needs to reduce irregular variation to make accurate predictions
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8. The Moving Average Method
Useful in smoothing time series to see its trend
Basic method used in measuring seasonal fluctuation
Applicable when time series follows fairly linear trend that have definite rhythmic pattern
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9. Moving Average Method - Constant duration of cycles
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10. 3-year and 5-Year Moving Averages
Data-> Data Analysis -> Moving Average
Gas Sales 39 37 61 58 18 56 82 27 41 69 49 66 54 42 90
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11. Exponential Smoothing
Overcome some drawbacks of moving average:
No moving averages for the first and last time periods.
“Forgets” most of the previous values.
St = wyt + (1 – w)St-1 (for t ≥ 2)
where:
St = Exponentially smoothed time series at time t
yt = Time series at time period t
St-1 = Exponentially smoothed time series at time t–1
w = Smoothing constant, 0 ≤ w ≤ 1

12. Exponential smoothing
Data-> Data Analysis -> Exponential smoothing, damping factor = 1-w
Gas Sales 39 37 61 58 18 56 82 27 41 69 49 66 54 42 90
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13. Weighted Moving Average
A simple moving average assigns the same weight to each observation in averaging
Weighted moving average assigns different weights to each observation
Most recent observation receives the most weight, and the weight decreases for older data values
In either case, the sum of the weights = 1
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14. Weighted Moving Average - Example
Cedar Fair operates seven amusement parks and five separately gated water parks. Its combined attendance (in thousands) for the last 17 years is given in the following table. A partner asks you to study the trend in attendance. Compute a three-year moving average and a three-year weighted moving average with weights of 0.2, 0.3, and 0.5 for successive years.
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15. Weighted Moving Average - Example
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16. Weighed Moving Average – An Example
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17. Linear Trend
The long term trend of many business series often approximates a straight line
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18. Linear Trend Plot
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19. Linear Trend – Using the Least Squares Method
Use the least squares method in Simple Linear Regression (Chapter 13) to find the best linear relationship between 2 variables
Code time (t) and use it as the independent variable
E.g. let t be 1 for the first year, 2 for the second, and so on (if data are annual)
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20. Linear Trend –An Example
Year
Rate
Quarter
2006
0.561 1
0.702 2
0.800 3
0.568 4
2007
0.575
0.738
0.868
0.605
2008
0.594
0.729
0.600
2009
0.622
0.708
0.806
0.632
1010
0.665
0.835
0.873
0.670
A hotel in Bermuda has recorded the occupancy rate for each quarter for the past 5 years. The data are shown here.
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21. Linear Trend –An Example Using Excel
Insert->Scatter->first option->Right-click on any marker->Add trendline->At the bottom: Display Equation on chart
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22. Seasonal Variation
Fluctuations that coincide with certain seasons; repeated year after year
Understanding seasonal fluctuations help plan for sufficient goods and materials on hand to meet varying seasonal demand
Analysis of seasonal fluctuations over a period of years help in evaluating current sales
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23. Seasonal Index
A number, usually expressed in percent, that expresses the relative value of a season with respect to the average for the year (100%)
Ratio-to-moving-average method
The method most commonly used to compute the typical seasonal pattern
It eliminates the trend (T), cyclical (C), and irregular (I) components from the time series
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24. Bermuda Hotel example
Quarter
Period t
Rate
4-quarter moving averages
Centered moving averaged
Ratio of sales to centered moving averages 1
0.561 2
0.702 3
0.800
0.6595 4
0.568 5
0.575 6
0.738
0.6965 7
0.868 8
0.605 9
0.594
0.6665 10 11
0.729 12
0.600
0.6685 13
0.622
0.684 14
0.708
0.692
0.688 15
0.806 16
0.632
0.7345 17
0.665 18
0.835
0.756 19
0.873 20
0.670
Step (1) – Organize time series data in column form Step (2) Compute the 4-quarter moving totals Step (3) Compute the 4-quarter moving averages Step (4) Compute the centered moving averages by getting the average of two 4-quarter moving averages Step (5) Compute ratio by dividing actual rate by the centered moving averages
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25. Seasonal Index – An Example
Year 1 2 3 4
2006
2007
2008
2009
2010
Average
Index
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26. Actual versus Deseasonalized Sales for Toys International
Deseasonalized Series = Actual series / Seasonal Index
Quarter
Period t
Rate
4-quarter moving average
Centered moving average
Ratio
Seasonal Index
Seasonal adjusted rate 1
0.561
0.88
0.64 2
0.702
0.66
1.08
0.65 3
0.800
1.21
1.18
0.68 4
0.568
0.67
0.85
0.87 5
0.575
0.69 6
0.738
0.70
1.07 7
0.868
1.24
0.74 8
0.605
0.86 9
0.594 10
1.11 11
0.729
1.09
0.62 12
0.600
0.90 13
0.622
0.92 14
0.708
1.03 15
0.806
1.16 16
0.632
0.73
0.72 17
0.665
0.75 18
0.835
0.76
1.10
0.78 19
0.873 20
0.670
0.77
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27. Actual versus Deseasonalized Series
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28. Seasonally Adjusted Forecast
(1) Obtain the linear equation using the deseasonalized data:
Ŷ= t
(2) Use the linear equation to predict the dependent variable, rate.
(3) Use the predicted rate times the corresponding seasonal index to obtain the seasonally adjusted forecast.
Quarterly Forecast for 2011
Quarter
Period
Estimated rat
Seasonal Index
Quarterly Forecast 1 21
0.75
0.88
0.66 2 22
1.08
0.81 3 23
0.76
1.18
0.89 4 24
0.87
0.67
Ŷ = (24)
Ŷ X SI = .76 X .87
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29. Nonlinear Trends
A linear trend equation is used when the data are increasing (or decreasing) by equal amounts
A nonlinear trend equation is used when the data are increasing (or decreasing) by increasing amounts over time
When data increase (or decrease) by equal percents or proportions plot will show curvilinear pattern
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30. Log Trend Equation – Gulf Shores Importers Example
Graph on right is the log base 10 of the original data which now is linear
(Excel function:
=log(x) or log(x,10)
Using Data Analysis in Excel, generate the linear equation
Regression output shown in next slide
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31. Log Trend Equation – Gulf Shores Importers Example.
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32. Log Trend Equation – Gulf Shores Importers Example
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