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Time Series and Forecasting Chapter 16 McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. 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 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 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 16-3 Secular Trend – Examples 16-4 Cyclical Variation – Sample Chart 1991 1996 2001 2006 2011 16-5 Seasonal Variation – Sample Chart 16-6 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 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 Moving Average Method - Constant duration of cycles 16-9 3-year and 5-Year Moving Averages Gas Sales 39 37 61 58 18 56 82 27 41 69 49 66 54 42 90 66 100 Data-> Data Analysis -> Moving Average 90 80 70 60 Gas Sales 50 3-period 5-period 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 16-10 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 Data-> Data Analysis -> Exponential smoothing, damping factor = 1-w Gas Sales 39 100 37 61 58 18 56 82 27 41 69 49 66 54 42 90 66 90 80 70 60 Gas Sales 50 damping=.8 damping=.3 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 16-12 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 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 threeyear moving average and a three-year weighted moving average with weights of 0.2, 0.3, and 0.5 for successive years. 16-14 Weighted Moving Average - Example 16-15 Weighed Moving Average – An Example 16-16 Linear Trend The long term trend of many business series often approximates a straight line Linear Trend Equation : Y a bt where : Y read "Y hat" , is the projected value of the Y variable for a selected value of t a the Y - intercept b the slope of the line t any value of time (coded) that is selected 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) 16-19 Linear Trend –An Example A hotel in Bermuda has recorded the occupancy rate for each quarter for the past 5 years. The data are shown here. Year 2006 2007 2008 2009 1010 Rate 0.561 0.702 0.800 0.568 0.575 0.738 0.868 0.605 0.594 0.738 0.729 0.600 0.622 0.708 0.806 0.632 0.665 0.835 0.873 0.670 Quarter 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 16-20 Linear Trend –An Example Using Excel Rate 1 0.9 y = 0.0052x + 0.6394 0.8 0.7 0.6 0.5 Rate 0.4 Linear (Rate) 0.3 0.2 0.1 0 0 5 10 15 20 25 Insert->Scatter->first option->Right-click on any marker->Add trendline->At the bottom: Display Equation on chart 16-21 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 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 Bermuda Hotel example Quarter Step (1) – Organize time series data in column form Step (2) Compute the 4quarter moving totals Step (3) Compute the 4quarter 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 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 16-24 Seasonal Index – An Example Year 1 2 2006 3 4 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 2010 0.895171 1.104497 1.155763 0.879457 Average 0.883308 1.077137 1.175223 0.873227 Index 0.883308 1.077137 1.175223 0.873227 16-25 Actual versus Deseasonalized Sales for Toys International Deseasonalized Series = Actual series / Seasonal Index Quarter 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Period t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Rate 0.561 0.702 0.800 0.568 0.575 0.738 0.868 0.605 0.594 0.738 0.729 0.600 0.622 0.708 0.806 0.632 0.665 0.835 0.873 0.670 4-quarter Centered moving moving average average 0.66 0.66 0.67 0.69 0.70 0.70 0.70 0.67 0.67 0.67 0.66 0.68 0.69 0.70 0.73 0.75 0.76 0.66 0.67 0.68 0.69 0.70 0.70 0.68 0.67 0.67 0.67 0.67 0.69 0.70 0.72 0.74 0.76 Ratio 1.21 0.85 0.85 1.07 1.24 0.86 0.87 1.11 1.09 0.90 0.92 1.03 1.16 0.88 0.90 1.10 Seasonal Seasonal adjusted Index rate 0.88 0.64 1.08 0.65 1.18 0.68 0.87 0.65 0.88 0.65 1.08 0.69 1.18 0.74 0.87 0.69 0.88 0.67 1.08 0.69 1.18 0.62 0.87 0.69 0.88 0.70 1.08 0.66 1.18 0.69 0.87 0.72 0.88 0.75 1.08 0.78 1.18 0.74 0.87 0.77 16-26 Actual versus Deseasonalized Series 1 0.9 0.8 0.7 0.6 Rate 0.5 Seasonal adjusted rate 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 16-27 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. Quarterly Forecast for 2011 Estimated Seasonal rat Index Quarterly Forecast Quarter Period 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 X SI = .76 X .87 Ŷ = .6371+ 0.0053(24) 16-28 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 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 The Linear Equation is : y 2.053805 0.153357t 16-31 Log Trend Equation – Gulf Shores Importers Example Estimate the Import f or the year 2014 using the linear trend y 2.053805 0.153357t Substitute into the linear equation above the code (19) f or 2014 y 2.053805 0.153357(19) y 4.967588 Then f ind the antilog of y 10 ^ Y 104.967588 92,809 16-32