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**Time Series and Forecasting**

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

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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 16-2

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**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 16-3

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**Secular Trend – Examples**

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**Cyclical Variation – Sample Chart**

1991 1996 2001 2006 2011 16-5

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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 16-7

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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 16-8

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**Moving Average Method - Constant duration of cycles**

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**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 16-10

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**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

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**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 16-12

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**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 16-13

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**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. 16-14

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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 16-17

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Linear Trend Plot 16-18

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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) 16-19

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**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. 16-20

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**Linear Trend –An Example Using Excel**

Insert->Scatter->first option->Right-click on any marker->Add trendline->At the bottom: Display Equation on chart 16-21

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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 16-22

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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 16-23

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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 16-24

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**Seasonal Index – An Example**

Year 1 2 3 4 2006 2007 2008 2009 2010 Average Index 16-25

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**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 16-26

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**Actual versus Deseasonalized Series**

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**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 16-28

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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 16-29

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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

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**Log Trend Equation – Gulf Shores Importers Example**

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**Log Trend Equation – Gulf Shores Importers Example**

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©The McGraw-Hill Companies, Inc. 2008McGraw-Hill/Irwin Time Series and Forecasting Chapter 16.

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