Chapter 24 Forecasting ModelsDescription24.1 Moving-Average Forecasting MethodsChoice of NSlide 624.2 Simple Exponential SmoothingSlide 8Slide 924.3 Holt’s Method: Exponential Smoothing with TrendSlide 11Slide 12A Spreadsheet Interpretation of the Holt Method24.4 Winter’s Method: Exponential Smoothing with SeasonalitySlide 15Slide 16Slide 17Initialization of Winter’s MethodForecasting Accuracy24.5 Ad Hoc ForecastingSlide 21Slide 22Slide 23Slide 2424.6 Simple Linear RegressionSlide 26Slide 27How Good a Fit?Slide 29Slide 30Slide 31T-Tests in RegressionAssumptions Underlying the Simple Linear Regression ModelSlide 34Slide 35Slide 36Slide 37Running Regressions with ExcelSlide 39Obtaining a Scatterplot with Excel24.7 Fitting Nonlinear RelationshipsUsing a Spreadsheet to Fit a Nonlinear RelationshipUtilizing the Excel Trend CurveSlide 4424.8 Multiple RegressionEstimation of the βi’sGoodness of Fit RevisitedHypothesis TestingChoosing the Best Regression EquationMulticollinearityDummy VariablesInterpretation of Coefficients of Dummy VariablesMultiplicative ModelsHeteroscedasticity and AutoCorrelation in Multiple RegressionSlide 55Implementing Multiple Regression on a SpreadsheetSlide 57Chapter 24Forecasting Modelsto accompanyOperations Research: Applications and Algorithms 4th edition by Wayne L. WinstonCopyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.2DescriptionWe discuss two important types of forecasting methods: extrapolation methods and causal forecasting methods.Extrapolation methods (unlike the causal forecasting methods) don’t take into account what “caused” past data; they simply assume that past trends and patterns will continue in the future.Causal forecasting methods attempt to forecast future values of a variable (called the dependent variable) by using past data to estimate the relationship between the dependent variable and one or more independent variables.324.1 Moving-Average Forecasting MethodsLet x1, x2,…,xt,… be observed values of a time series, where xt is the value of the time series observed during period t.One of the most commonly used forecasting methods is the moving-average method.We define ft,1 to be the forecast period for period t+1 made after observing xt.For the moving-average method, ft,1 = average of the last N observations =average of xt, xt-1, xt-2,…,xt-N+1where N is a given parameter.4Choice of NWe will use the mean absolute deviation (MAD) as our measure of forecast accuracy.Before defining the MAD, we need to define the concept of a forecast error.Given a forecast for xt, we define et to be the error in our forecast for xt, to be given byet=xt-(forecast for xt)The MAD is simply the average of the absolute values of all the et’s.We begin with an explanation of the Excel OFFSET function.5 This function lets you pick out a cell range relative to a given location in the spreadsheet.The syntax of the OFFSET function is as follows:OFFSET (reference, rows, columns, height, width)Reference is the cell from which you base the row and column references.Rows helps locate the upper left-hand corner of the OFFSET range. Rows is measured by number of rows up or down from the cell reference.Columns helps locate the upper left-hand corner of the OFFSET range. Columns is measured by the number of columns left or right from the cell references.6 Height is the number of rows in the selected range.Width is the number of columns in the selected range.File Offsetexample.xls contains some examples of how the OFFSET function works.The nice thing about the OFFSET function is that it can be copied like any formula.Moving-average forecast perform well for a time series that fluctuates about a constant base level.We find that air conditioner sales is a good example. They exhibit seasonality: The peaks and valleys of the series repeat at regular 12 month intervals.724.2 Simple Exponential SmoothingIf a time series fluctuates about a base level, simple exponential smoothing may be used to obtain good forecasts for future values of the series.To describe simple exponential smoothing let At = smoothed average of a time series after observing xt.After observing xt, At is the forecast for the value of the time series during any future period.The key equation in simple exponential smoothing is At = αxt + (1- α)At-1.8 In the above equation, α is a smoothing constant that satisfies 0< α<1.As with moving-average forecasts, we let ft,k be the forecast for xt+k made at the end of period t. Then At=ft,kAssuming that we are trying to forecast one period ahead, our error for predicting xt is given by et=xt-ft-1,1 = xt-At-1.Thus, our new forecast At=ft, 1 is equal to our old forecast (At-1) plus a fraction of our period t error (et).9 This implies that if we “overpredict” xt, we lower our forecast, and if we “underpredict” xt, we raise our forecast.For larger values of the smoothing constant α, more weight is given to the most recent observation.1024.3 Holt’s Method: Exponential Smoothing with TrendIf we believe that a time series exhibits a linear trend, Holt’s method often yields good forecasts.At the end of the tth period, Holt’s method yields an estimate of the base level (Lt) and the per-period trend (Tt) of the series.To computer Lt, we take a weighted average of the following two quantities:Xt, which is an estimate of the period t base level from the current periodLt-1+Tt-1, which is an estimate of the period t base level based on previous data11 To computer Tt, we take a weighted average of the following two quantities:An estimate of trend from the current period given by the increase in the smoothed base from t-1 to period tTt-1, which is an estimate of the trendAs before, we define ft,k to be the forecast for xt+k made at the end of period t. Thenft,k=Lt,kTtTo initialize Holt’s method, we need an initial estimate of the base and an initial estimate (call it T0) of the trend.12 A multiplicative version of Holt’s method can be used to generate good forecasts for a series of the form xt=abtεtHere , the value of b represents the percentage growth in the base level of the series during each period.13A Spreadsheet Interpretation of the Holt MethodThe file Holt.xls contains an implementation of the Holt method.We can use an Excel one-way data table to determine values of α and β that