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

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

16-16

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

16-17

Linear Trend Plot

16-18

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

16-30

Log Trend Equation – Gulf Shores Importers Example

ty 15335700538052 ..

:is Equation Linear The

16-31

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

16-32