Time Series and Forecasting
Chapter 16
McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
Learning ObjectivesLO1 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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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
Time Series and its Components
Components of a Time Series
1. Secular Trend – the smooth long term direction of a time series
2. Cyclical Variation – the rise and fall of a time series over periods longer than one year
3. Seasonal Variation – Patterns of change in a time series within a year which tends to repeat each year
4. Irregular Variation – classified into:
Episodic – unpredictable but identifiable
Residual – also called chance fluctuation and unidentifiable
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Secular Trend – Examples
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Cyclical Variation – Sample Chart
1991 1996 2001 2006 2011
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Seasonal Variation – Sample Chart
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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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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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Moving Average Method - Constant duration of cycles
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3-year and 5-Year Moving Averages
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Gas Sales39376158185682274169496654429066
Data-> Data Analysis -> Moving Average
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160
10
20
30
40
50
60
70
80
90
100
Gas Sales
3-period
5-period
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
Exponential smoothing
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Gas Sales39
376158185682274169496654429066
Data-> Data Analysis -> Exponential smoothing, damping factor = 1-w
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160
10
20
30
40
50
60
70
80
90
100
Gas Sales
damping=.8
damping=.3
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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Weighted Moving Average - ExampleCedar 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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Weighted Moving Average - Example
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Weighed Moving Average – An Example
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Linear Trend The long term trend of many business series often
approximates a straight line
selected is that (coded) timeof any value
line theof slope the
intercept - the
of valueselected afor ariable v
theof valueprojected theis ,hat" " read
:where
:Equation TrendLinear
t
b
Ya
t
YYY
btaY
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Linear Trend Plot
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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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A hotel in Bermuda has recorded the occupancy rate for each quarter for the past 5 years. The data are shown here.
Linear Trend –An Example
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Year Rate Quarter2006 0.561 1
0.702 20.800 30.568 4
2007 0.575 10.738 20.868 30.605 4
2008 0.594 10.738 20.729 30.600 4
2009 0.622 10.708 20.806 30.632 4
1010 0.665 10.835 20.873 30.670 4
Linear Trend –An Example Using Excel
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0 5 10 15 20 250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f(x) = 0.00524586466165414 x + 0.639368421052632
Rate
Rate
Linear (Rate)
Insert->Scatter->first option->Right-click on any marker->Add trendline->At the bottom: Display Equation on chart
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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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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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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Quarter Period t Rate
4-quarter moving averages
Centered moving averaged
Ratio of sales to centered moving averages
1 1 0.561
2 2 0.702 0.65775
3 3 0.800 0.66125 0.6595 1.21304
4 4 0.568 0.67025 0.66575 0.853173
1 5 0.575 0.68725 0.67875 0.847145
2 6 0.738 0.6965 0.691875 1.066667
3 7 0.868 0.70125 0.698875 1.241996
4 8 0.605 0.70125 0.70125 0.862745
1 9 0.594 0.6665 0.683875 0.86858
2 10 0.738 0.66525 0.665875 1.108316
3 11 0.729 0.67225 0.66875 1.090093
4 12 0.600 0.66475 0.6685 0.897532
1 13 0.622 0.684 0.674375 0.922335
2 14 0.708 0.692 0.688 1.02907
3 15 0.806 0.70275 0.697375 1.155763
4 16 0.632 0.7345 0.718625 0.879457
1 17 0.665 0.75125 0.742875 0.895171
2 18 0.835 0.76075 0.756 1.104497
3 19 0.873
4 20 0.670
Bermuda Hotel example
Seasonal Index – An Example
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Year 1 2 3 4
2006 1.21304 0.853173
2007 0.847145 1.066667 1.241996 0.862745
2008 0.86858 1.108316 1.090093 0.897532
2009 0.922335 1.02907 1.155763 0.879457
2010 0.895171 1.104497
Average 0.883308 1.077137 1.175223 0.873227
Index 0.883308 1.077137 1.175223 0.873227
Actual versus Deseasonalized Sales for Toys International
Deseasonalized Series = Actual series / Seasonal Index
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Quarter Period t Rate
4-quarter moving average
Centered moving average Ratio
Seasonal Index
Seasonal adjusted
rate1 1 0.561 0.88 0.642 2 0.702 0.66 1.08 0.653 3 0.800 0.66 0.66 1.21 1.18 0.684 4 0.568 0.67 0.67 0.85 0.87 0.651 5 0.575 0.69 0.68 0.85 0.88 0.652 6 0.738 0.70 0.69 1.07 1.08 0.693 7 0.868 0.70 0.70 1.24 1.18 0.744 8 0.605 0.70 0.70 0.86 0.87 0.691 9 0.594 0.67 0.68 0.87 0.88 0.672 10 0.738 0.67 0.67 1.11 1.08 0.693 11 0.729 0.67 0.67 1.09 1.18 0.624 12 0.600 0.66 0.67 0.90 0.87 0.691 13 0.622 0.68 0.67 0.92 0.88 0.702 14 0.708 0.69 0.69 1.03 1.08 0.663 15 0.806 0.70 0.70 1.16 1.18 0.694 16 0.632 0.73 0.72 0.88 0.87 0.721 17 0.665 0.75 0.74 0.90 0.88 0.752 18 0.835 0.76 0.76 1.10 1.08 0.783 19 0.873 1.18 0.744 20 0.670 0.87 0.77
Actual versus Deseasonalized Series
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Rate
Seasonal adjusted rate
Seasonally Adjusted Forecast(1) Obtain the linear equation using the deseasonalized data:
Ŷ= .6371+.0053t
(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.
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Quarterly Forecast for 2011
Quarter PeriodEstimated
ratSeasonal
IndexQuarterly Forecast
1 21 0.75 0.88 0.66
2 22 0.75 1.08 0.81
3 23 0.76 1.18 0.89
4 24 0.76 0.87 0.67
Ŷ = .6371+ 0.0053(24)
Ŷ X SI = .76 X .87
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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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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Log Trend Equation – Gulf Shores Importers Example
ty 15335700538052 ..
:is Equation Linear The
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Log Trend Equation – Gulf Shores Importers Example
80992
10
10
9675884
1915335700538052
15335700538052
9675884
,
of antilog the find Then
.
)(..
2014 for (19) code the above equation linear the into Substitute
..
trend linear the using 2014 yearthe for Import the Estimate
.
^
Yy
y
y
ty
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