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1 Chapter 7 My interest is in the future because I am going to spend the rest of my life there.— Charles F. Kettering Forecasting

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2 Time-Series Analysis A time series is numerical sequence of values generated over regular time intervals. The classical time-series model involves four components: Secular trend (T t ). Cyclical movement (C t ). Seasonal fluctuation (S t ). Irregular variation (I t ). The multiplicative model determine the level of the forecast variable Y t : Y t = T t × C t × S t × I t

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3 Classical Time-Series Model

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4 Exponential Smoothing Finding the components is difficult. A direct approach averages past Y t values by exponential smoothing. The forecast value is computed from F t+1 = Y t + (1 )F t The above involves a single parameter, the smoothing constant alpha. All previous time periods are reflected in the Fs, and greater weight is given to the more recent.

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5 Forecasts with Single-Parameter Exponential Smoothing

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6 Single-Parameter Forecasts The preceding slide shows single-parameter forecasts of Blitz Beer sales. These were generated by computer. The level for was.20. A greater will assign more weight to the present. Quality of forecasts may be measured. Most common is the mean squared error: which averages errors over all forecasts made. Other measures are the mean absolute deviation (MAD) and mean absolute percent error (MAPE). MSE = (Y t – F t ) 2 n

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7 Two-Parameter Exponential Smoothing The smoothing constant can be tuned to the past, possibly providing better forecasts. But single-parameter forecasts may still lead or lag actuals, as seen for Blitz Beer, because the impact of trends is delayed. Trend T t can be incorporated with a second trend smoothing constant (gamma): T t = Y t + (1 – )(T t –1 + b t –1 ) b t = (T t – T t –1 ) + (1 – )b t –1 F t+1 = T t + b t That greatly reduces Blitz Beer’s MSE.

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8 Forecasts with Two-Parameters

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9 Seasonal Exponential Smoothing with Three Parameters Many time series have regular seasonal patterns to be incorporated into forecasts. The three-parameter model incorporates a seasonal smoothing constant (beta): T t = Y t /S t –p ) + (1 – )(T t –1 + b t –1 ) b t = (T t – T t –1 ) + (1 – )b t –1 S t = Y t /T t ) + (1 – )S t –p F t+1 = (T t + b t ) S t –p+1

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10 Forecasting with Three Parameters

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11 Forecasting with Three Parameters The above works for p = 4 quarters or p = 12 months. The preceding slide needs 6 quarters to generate the first (very bad) forecast. The process settles quickly, providing good forecasts p periods into the future.

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12 Forecasting Trend Using Regression To forecast years in advance, regression analysis provides a trend line. Ŷ(X) = a + bX The independent variable X is years beyond the base period, the forecast or dependent variable is Y. The regression coefficients a (intercept) and b (slope) are found (with the computer) by applying the least squares method. The following applies to toothpaste sales Y.

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13 Forecasting Trend Using Regression

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14 Regression in Causal Models Regression analysis can make forecasts with with a non-time independent variable. A simple regression employs a straight line. Ŷ(X) = a + bX The dependent variable is not time periods, such as: store size order amount weight For 10 rail shipments, the transportation time Y was forecast for specific distance X.

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15 Forecasting Transportation Time from Distance

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16 Multiple Regression in Forecasting Regression fits data employing a multiple regression equation with several predictors: Ŷ = a + b 1 X 1 + b 2 X 2 Floorspace X 1 and advertising expense X 2 make forecasts of hardware outlet sales Y: Ŷ = 22,979 + 11.42X 1 + 23.41X 2 The above was obtained in a computer run using 10 data points. Forecast with X 1 =2,500 sq.ft. and X 2 =$750: Ŷ = 22,979+11.42(2,500)+23.41(750)=$23,129

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17 Forecasting Using Seasonal Indexes The classical time-series model provides a rationale for isolating seasonal components: The procedure is the ratio-to-moving- average method: (1) Compute moving averages from Ys. (2) Center above. (3) Express Ys as % of moving average. (4) Determine medians by season and adjust. Above (done on computer) leaves only S t. YtYt = T t × C t × S t × I t = S t × I t Moving averageT t × C t

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18 Excel Forecasting Templates and Tools Single-parameter exponential smoothing Two-parameter exponential smoothing Three-parameter exponential smoothing Regression Classical time series analysis

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19 Exponential Smoothing Tool Single-parameter exponential smoothing is easy with Excel’s ToolPak. Click on Tools on the menu bar, select the Data Analysis option, and then in the Data Analysis dialog box, click on Exponential Smoothing.

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20 Single-Parameter Exponential Smoothing (Figure 7-4 ) 1. Enter the smoothing constant in D2. 2. Enter problem information in B6:D25. Notice D26 does not have a value because it is to be forecast. 3. If more than 20 periods of data are available, expand columns B, C, and D to include all the data by inserting rows. Then copy the formula in E26 down to all the cells in the expanded table. 4. Click on Tool, Data Analysis, and the Exponential Smoothing to get the Exponential Smoothing dialog box shown next.

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21 Exponential Smoothing Dialog Box (Figure 7-5) 1. In the Input Range line enter the range of the data. The result shown is $D$6:$D$25 2. Enter the Damping factor. It is 1 - 3. In the Output Range enter the location of the results. 4. Click the OK button to get the results shown previously in Figure 7-4.

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22 Two-Parameter Exponential Smoothing (Figure 7-6) 3. If more than 20 periods of data are available, expand columns A, B, and C to include all the data by inserting rows. Copy the formulas in D25:E25 down to the next to last row in the expanded table. Finally, copy F26 down to all the rows in the expanded table. 1. Enter the smoothing constants in C2 and E2. 2. Enter problem information in A6:C25. Notice C26:E25 do not have values because they correspond to the period for which the forecast is being made.

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23 Finding the Best Smoothing Parameters with Solver (Figure 7-7) 2. To minimize the MSE, Min is selected in the Equal To line. 4. In the Subject to the Constraints box the two smoothing parameters are restricted to being equal to or less than one. 3. The Changing Cells line has the location of the two smoothing parameters. 5. Click the Options button to verify that nonnegative smoothing parameters are specified. 6. Click the Solve button to get Figure 7-6, shown previously. NOTE: Normally all these entries appear in the Solver Parameter dialog box so you only need to click on the Solve button. However, you should always check to make sure the entries are correct for the problem you are solving. 1. The location of the MSE is in the Set Target Cell line.

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24 The Add Constraint Dialog Box (Figure 7-8) The Add Constraint dialog box is used to specify the constraints on the variable in the Changing Cells line. Enter a smoothing parameter such as C2 in the Cell Reference line. Select a sign by clicking on the down arrow in the middle, and then enter the right-hand side in the Constraint line (1 here). Click OK when finished. Normally, all these entries already appear. You will need to use this dialog box only if you need to add a constraint. If you need to change a constraint, the Change Constraint dialog box functions just like this one.

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25 Three-Parameter Exponential Smoothing (Figure 7-11) 4. Finally, copy the four forecasting formulas in G34:G37 to the last four rows of the expanded table (the last year). Make sure that these formulas refer back to last row of data. In this spreadsheet the forecasting formulas in cells G34:G37 refer back to row 33 which contains the data for the fall of 2000, the last period. Thus, if the four forecasting formulas in the expanded table are in G44:G47 then they should refer back to row 43, the last row of data. 1. Enter the smoothing constants in C2, E2, and G2. 2. Enter problem information in A6:C33. 3. If less or more than 7 years of data are available, shorten or expand columns A, B, and C to include all the data by deleting or inserting rows. Copy the formulas in D33:G33 down through the last quarter of the next to last year in the expanded table.

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26 Regression Regression is easy with Excel’s Regression Tool. Click on Tools on the menu bar, select the Data Analysis option, and then in the Data Analysis dialog box select Regression. This yields the Regression dialog box shown next.

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27 Regression Dialog Box (Figure 7-18) 3. Click on the OK button to get the Regression Summary Output shown next. 2. In the Input X Range line enter the range of the X data. The result here shown is $B$7:$B$16 1. In the Input Y Range line enter the range of the Y data. The result shown here is $C$7:$C$16

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28 Excel’s Regression Tool (Figure 7-16) The slope and intercept are read from E15:E16 and yield the regression equation below. The multiple R, R squared, adjusted R, standard error, and F and t statistics are shown also.

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29 Multiple Regression (Figure 7-21) Excel’s regression tool can be used to do multiple regression. Just list ALL the X variables when designating the Input X Range; C7:D16 in this example.

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30 Classical Time Series Analysis (Figure 7-22) 1. Enter problem information in A19:C46. 3. (cont’d) Copy the formula in H44 down through the end of the data in the expanded table. Make sure that the forecasting formulas in the expanded table (the ones currently in H47:H50) refer back to the last period of data (like the formulas in H47:H50 refer back to H46). 2. If more than 7 years of data are available, expand columns A, B, and C to include all the data by inserting rows. Copy the formulas in D44:F44 down through the first two quarters of the last year of the data in the expanded table. The Seasonal Indices in column G are the same every year so enter their values in the expanded table (for the rows corresponding to data and also the rows for the year for which forecasts are being calculated).

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31 Other Forecasting Templates on the CD-ROM The three parameter-exponential smoothing and classical time series analysis templates described previously are for quarterly data. The CD-ROM also contains three- parameter exponential smoothing and classical time series analysis templates for monthly data.

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Analysis of Time Series and Forecasting

Analysis of Time Series and Forecasting

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