©The McGraw-Hill Companies, Inc. 2008McGraw-Hill/IrwinTime Series and ForecastingChapter 16 GoalslDefine the components of a time serieslCompute moving averagelDetermine a linear trend equationlCompute a trend equation for a nonlinear trendlUse a trend equation to forecast future time periods and to develop seasonally adjusted forecastslDetermine and interpret a set of seasonal indexeslDeseasonalize data using a seasonal indexlTest for autocorrelation2 Time SeriesWhat is a time series?–a collection of data recorded over a period of time (weekly, monthly, quarterly)–an analysis of history, it can be used by management to make current decisions and plans based on long-term forecasting–Usually assumes past pattern to continue into the future3 Components of a Time SerieslSecular Trend – the smooth long term direction of a time serieslCyclical Variation – the rise and fall of a time series over periods longer than one yearlSeasonal Variation – Patterns of change in a time series within a year which tends to repeat each yearlIrregular Variation – classified into:Episodic – unpredictable but identifiableResidual – also called chance fluctuation and unidentifiable4 Cyclical Variation – Sample Chart5 Seasonal Variation – Sample Chart6 Secular Trend – Home Depot Example7 Secular Trend – EMS Calls Example8 Secular Trend – Manufactured Home Shipments in the U.S.9 The Moving Average MethodlUseful in smoothing time series to see its trendlBasic method used in measuring seasonal fluctuationlApplicable when time series follows fairly linear trend that have definite rhythmic pattern10 Moving Average Method - Example11 Three-year and Five-Year Moving Averages12 Weighted Moving AveragelA simple moving average assigns the same weight to each observation in averaginglWeighted moving average assigns different weights to each observationlMost recent observation receives the most weight, and the weight decreases for older data valueslIn either case, the sum of the weights = 113 Cedar Fair operates seven amusement parks and five separately gated water parks. Its combined attendance (in thousands) for the last 12 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.Weighted Moving Average - Example14 Weighted Moving Average - Example15 Weighed Moving Average – An Example16 Linear TrendlThe long term trend of many business series often approximates a straight line17 Linear Trend Plot18 Linear Trend – Using the Least Squares MethodlUse the least squares method in Simple Linear Regression (Chapter 13) to find the best linear relationship between 2 variableslCode time (t) and use it as the independent variablelE.g. let t be 1 for the first year, 2 for the second, and so on (if data are annual)19 YearSales($ mil.)2002720031020049200511200613The sales of Jensen Foods, a small grocery chain located in southwest Texas, since 2002 are:Linear Trend – Using the Least Squares Method: An ExampleYeartSales($ mil.)20021720032102004392005411200651320 Linear Trend – Using the Least Squares Method: An Example Using Excel21 Nonlinear TrendslA linear trend equation is used when the data are increasing (or decreasing) by equal amountslA nonlinear trend equation is used when the data are increasing (or decreasing) by increasing amounts over timelWhen data increase (or decrease) by equal percents or proportions plot will show curvilinear pattern22 Log Trend Equation – Gulf Shores Importers ExamplelTop graph is plot of the original datalBottom graph is the log base 10 of the original data which now is linear(Excel function: =log(x) or log(x,10)lUsing Data Analysis in Excel, generate the linear equation lRegression output shown in next slide23 Log Trend Equation – Gulf Shores Importers Example24 Log Trend Equation – Gulf Shores Importers Example25 Seasonal VariationlOne of the components of a time serieslSeasonal variations are fluctuations that coincide with certain seasons and are repeated year after yearlUnderstanding seasonal fluctuations help plan for sufficient goods and materials on hand to meet varying seasonal demandlAnalysis of seasonal fluctuations over a period of years help in evaluating current sales26 Seasonal IndexlA number, usually expressed in percent, that expresses the relative value of a season with respect to the average for the year (100%)lRatio-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 series27 The table below shows the quarterly sales for Toys International for the years 2001 through 2006. The sales are reported in millions of dollars. Determine a quarterly seasonal index using the ratio-to-moving-average method.Seasonal Index – An Example28 Step (1) – Organize time series data in column form Step (2) Compute the 4-quarter moving totalsStep (3) Compute the 4-quarter moving averagesStep (4) Compute the centered moving averages by getting the average of two 4-quarter moving averagesStep (5) Compute ratio by dividing actual sales by the centered moving averages29 Seasonal Index – An Example30 Actual versus Deseasonalized Sales for Toys InternationalDeseasonalized Sales = Sales / Seasonal Index31 Actual versus Deseasonalized Sales for Toys International – Time Series Plot using Minitab32 Seasonal Index – An Example Using Excel33 Seasonal Index – An Example Using Excel34 Seasonal Index – An Excel Example using Toys International Sales35 Seasonal Index – An Example Using ExcelGiven the deseasonalized linear equation for Toys International sales as Ŷ=8.109 + 0.0899t, generate the seasonally adjusted forecast for the each of the quarters of 2007QuartertŶ(unadjusted forecast)Seasonal IndexQuarterly Forecast(seasonally adjusted forecast)Winter2510.356750.7657.923Spring2610.446660.5756.007Summer2710.536571.14112.022Fall2810.626481.51916.142Ŷ = 8.109 + 0.0899(28)Ŷ X SI = 10.62648 X 1.51936 Durbin-Watson StatisticlTests the autocorrelation among the residualslThe Durbin-Watson statistic, d, is computed by first determining the residuals for each observation: et = (Yt – Ŷt)lThen compute d using the following equation:37 Durbin-Watson Test for Autocorrelation – Interpretation of the StatisticlRange of d is 0 to 4d = 2 No autocorrelationd close to 0Positive autocorrelationd beyond 2Negative autocorrelationlHypothesis Test:H0: No residual correlation (ρ = 0)H1: Positive residual correlation (ρ > 0)lCritical values for d are found in Appendix B.10 usinglα - significance levelln – sample sizelK – the number of predictor variables38 Durbin-Watson Critical Values ( =.05)39 Durbin-Watson Test for Autocorrelation: An ExampleThe Banner Rock Company manufactures and markets its own rocking chair. The company developed special rocker for senior citizens which it advertises extensively on TV. Banner’s market for the special chair is the Carolinas, Florida and Arizona, areas where there are many senior citizens and retired people The president of Banner Rocker is studying the association between his advertising expense (X) and the number of rockers sold over the last 20 months (Y). He collected the following data. He would like to use the model to forecast sales, based on the amount spent on advertising, but is concerned that because he gathered these data over consecutive months that there might be problems of autocorrelation.MonthSales (000)Ad ($millions)11535.521565.531535.341475.551595.461605.371475.581475.791525.9101606.2111696.3121765.9131766.1141796.2151846.2161816.5171926.7182056.9192156.5202096.440 Durbin-Watson Test for Autocorrelation: An ExamplelStep 1: Generate the regression equation41 Durbin-Watson Test for Autocorrelation: An ExamplelThe resulting equation is: Ŷ = - 43.802 + 35.95XlThe coefficient (r) is 0.828lThe coefficient of determination (r2) is 68.5%(note: Excel reports r2 as a ratio. Multiply by 100 to convert into percent)lThere is a strong, positive association between sales and advertisinglIs there potential problem with autocorrelation?42 ∑(ei -ei-1)2∑(ei)2=E4^2=(E4-F4)^2=-43.802+35.95*C3=B3-D3=E3Durbin-Watson Test for Autocorrelation: An Example43 lHypothesis Test:H0: No residual correlation (ρ = 0)H1: Positive residual correlation (ρ > 0)lCritical values for d given α=0.5, n=20, k=1 found in Appendix B.10dl=1.20 du=1.41dl=1.20du=1.41Reject H0Positive AutocorrelationInconclusiveFail to reject H0No AutocorrelationDurbin-Watson Test for Autocorrelation: An Example44 END OF CHAPTER 1645 。