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1、ForecastingChapter 18ForecastingLearning Objectives1.Understand that the long-run success of an organization is often closely related to how well management is able to predict future aspects of the operation. 2.Know the various components of a time series.3.Be able to use smoothing techniques such a
2、s moving averages and exponential smoothing.4.Be able to use the least squares method to identify the trend component of a time series.5.Understand how the classical time series model can be used to explain the pattern or behavior of the data in a time series and to develop a forecast for the time s
3、eries.6.Be able to determine and use seasonal indexes for a time series.7.Know how regression models can be used in forecasting.8.Know the definition of the following terms:time seriesmean squared errorforecastmoving averagestrend componentweighted moving averagescyclical componentsmoothing constant
4、seasonal componentseasonal constantirregular componentSolutions:1.a.WeekTime-SeriesValue Forecast Forecast Error (Error)21 82133154171252551615116 916-749 75Forecast for week 7 is (17 + 16 + 9 ) / 3 = 14b.MSE = 75 / 3 = 25c.Smoothing constant = .3.Week tTime-Series ValueYtForecast FtForecast Error Y
5、t - Ft Squared Error (Yt - Ft)218213 8.005.0025.00315 9.006.0036.0041710.206.8046.2451611.564.4419.716912.45-3.4511.90 138.85138.85Forecast for week 7 is .2(9) + .8(12.45) = 11.76d.For the a = .2 exponential smoothing forecast MSE = 138.85 / 5 = 27.77. Since the three-week moving average has a small
6、er MSE, it appears to provide the better forecasts.e.Smoothing constant = .4.Week tTime-Series ValueYtForecast FtForecast Error Yt - Ft Squared Error (Yt - Ft)218213 8.0 5.0 25.0031510.0 5.0 25.0041712.0 5.0 25.0051614.0 2.0 4.006914.8 -5.833.64112.64MSE = 112.64 / 5 = 22.53. A smoothing constant of
7、 .4 appears to provide better forecasts.Forecast for week 7 is .4(9) + .6(14.8) = 12.482.a.WeekTime-Series Value4-Week Moving Average Forecast(Error)25-Week Moving Average Forecast(Error)211722131942351820.004.0061620.2518.0619.6012.9672019.001.0019.400.3681819.251.5619.201.4492218.0016.0019.009.001
8、02019.001.0018.801.44111520.0025.0019.2017.64122218.7510.5619.00 9.0077.1851.84b.MSE(4-Week) = 77.18 / 8 = 9.65MSE(5-Week) = 51.84 / 7 = 7.41c.For the limited data provided, the 5-week moving average provides the smallest MSE.3.a.WeekTime-SeriesValueWeighted Moving Average ForecastForecastError(Erro
9、r)211722131942319.333.6713.4751821.33-3.3311.0961619.83-3.8314.6772017.832.174.7181818.33-0.330.1192218.333.6713.47102020.33-0.330.11111520.33-5.3328.41122217.834.17 17.39103.43b.MSE = 103.43 / 9 = 11.49Prefer the unweighted moving average here.c.You could always find a weighted moving average at le
10、ast as good as the unweighted one. Actually the unweighted moving average is a special case of the weighted ones where the weights are equal. 4.WeekTime-Series ValueForecastError(Error)211722117.004.0016.0031917.401.602.5642317.565.4429.5951818.10-0.100.0161618.09-2.094.3772017.882.124.4981818.10-0.
11、100.0192218.093.9115.29102018.481.522.31111518.63-3.6313.18122218.273.73 13.91 101.72101.72MSE = 101.72 / 11 = 9.25a = .2 provided a lower MSE; therefore a = .2 is better than a = .15.a.F13 = .2Y12 + .16Y11 + .64(.2Y10 + .8F10) = .2Y12 + .16Y11 + .128Y10 + .512F10F13 = .2Y12 + .16Y11 + .128Y10 + .51
12、2(.2Y9 + .8F9) = .2Y12 + .16Y11 + .128Y10 + .1024Y9 + .4096F9F13 = .2Y12 + .16Y11 + .128Y10 + .1024Y9 + .4096(.2Y8 + .8F8) = .2Y12 + .16Y11 + .128Y10 + .1024Y9 + .08192Y8 + .32768F8b.The more recent data receives the greater weight or importance in determining the forecast. The moving averages metho
13、d weights the last n data values equally in determining the forecast.6.a.MonthYt3-Month Moving Averages Forecast(Error)2a = 2Forecast(Error)218028280.004.0038480.4012.9648382.00 1.0081.123.5358383.00 0.0081.502.2568483.33 0.4581.804.8478583.33 2.7982.247.6288484.00 0.0082.791.4698284.33 5.4383.031.0
14、6108383.67 0.4582.830.03118483.00 1.0082.861.30128383.00 0.0083.09 0.0111.1239.06MSE(3-Month) = 11.12 / 9 = 1.24MSE(a = .2) = 39.06 / 11 = 3.55Use 3-month moving averages.b.(83 + 84 + 83) / 3 = 83.37.a.MonthTime-Series Value3-Month Moving Average Forecast(Error)24-Month Moving Average Forecast(Error
15、)219.529.339.449.69.400.0459.89.430.149.450.1269.79.600.019.530.0379.89.700.019.630.03810.59.770.539.730.5999.910.000.019.950.00109.710.070.149.980.08119.610.030.189.970.14129.69.730.029.920.101.081.09MSE(3-Month) = 1.08 / 9 = .12MSE(4-Month) = 1.09 / 8 = .14Use 3-Month moving averages.b. Forecast =
16、 (9.7 + 9.6 + 9.6) / 3 = 9.63c. For the limited data provided, the 5-week moving average provides the smallest MSE.8.a.MonthTime-Series Value3-Month Moving Average Forecast(Error)2a = .2Forecast(Error)212402350240.0012100.003230262.001024.004260273.33177.69255.6019.365280280.000.00256.48553.19632025
17、6.674010.69261.183459.797220286.674444.89272.952803.708310273.331344.69262.362269.579240283.331877.49271.891016.9710310256.672844.09265.511979.3611240286.672178.09274.411184.0512230263.331110.89267.53 1408.5017,988.5227,818.49MSE(3-Month) = 17,988.52 / 9 = 1998.72MSE(a = .2) = 27,818.49 / 11 = 2528.
18、95Based on the above MSE values, the 3-month moving averages appears better. However, exponential smoothing was penalized by including month 2 which was difficult for any method to forecast. Using only the errors for months 4 to 12, the MSE for exponential smoothing is revised toMSE(a = .2) = 14,694
19、.49 / 9 = 1632.72Thus, exponential smoothing was better considering months 4 to 12.b. Using exponential smoothing,F13 = aY12 + (1 - a)F12 = .20(230) + .80(267.53) = 260 9.a.Smoothing constant = .3.Month tTime-Series ValueYtForecast FtForecast Error Yt - Ft Squared Error (Yt - Ft)21105.02135.0105.003
20、0.00900.003120.0114.006.0036.004105.0115.80-10.80116.64590.0112.56-22.56508.956120.0105.7914.21201.927145.0110.0534.951221.508140.0120.5419.46378.699100.0126.38-26.38695.901080.0118.46-38.461479.1711100.0106.92-6.9247.8912110.0104.855.15 26.52 Total5613.18MSE = 5613.18 / 11 = 510.29Forecast for mont
21、h 13: F13 = .3(110) + .7(104.85) = 106.4b.Smoothing constant = .5Month tTime-Series ValueYtForecast FtForecast Error Yt - FtSquared Error (Yt - Ft)211052135105 30.00 900.003120.5(135) + .5(105) = 120 0.00 0.004105.5(120) + .5(120) = 120-15.00 225.005 90.5(105) + .5(120) = 112.50-22.50 506.256120.5(9
22、0) + .5(112.5) = 101.25 18.75 351.567145.5(120) + .5(101.25) =110.63 34.371181.308140.5(145) + .5(110.63) = 127.81 12.19 148.609100.5(140) + .5(127.81) = 133.91-33.911149.8910 80.5(100) + .5(133.91) = 116.95-36.951365.3011100.5(80) + .5(116.95) = 98.48 1.52 2.3112110.5(100) + .5(98.48) = 99.24 10.76
23、 115.78 5945.99MSE = 5945.99 / 11 = 540.55Forecast for month 13: F13 = .5(110) + .5(99.24) = 104.62Conclusion: a smoothing constant of .3 is better than a smoothing constant of .5 since the MSE is less for 0.3.10.a/b.WeekTime-Series Valuea = .2Forecast(Error)2a = .3Forecast(Error)217.3527.407.35.002
24、57.35.002537.557.36.03617.36.036147.567.40.02567.42.019657.607.43.02897.46.019667.527.46.00367.50.000477.527.48.00167.51.000187.707.48.04847.51.036197.627.53.00817.57.0025107.557.55.00007.58.0009.1548.1178c. MSE(a = .2) = .1548 / 9 = .0172MSE(a = .3) = .1178 / 9 = .0131Use a = .3.F11 = .3Y10 + .7F10
25、 = .3(7.55) + .7(7.58) = 7.5711.a.MethodForecastMSE3-Quarter80.732.534-Quarter80.552.81The 3-quarter moving average forecast is better because it has the smallest MSE.b.MethodForecastMSEa = .480.402.40a = .580.572.01The a = .5 smoothing constant is better because it has the smallest MSE.c.The a = .5
26、 is better because it has the smallest MSE.12.The following values are needed to compute the slope and intercept:Tt = 4.7 + 2.1tForecast: T6 = 4.7 + 2.1(6) = 17.313.The following values are needed to compute the slope and intercept:Computation of slope:Computation of intercept:Equation for linear tr
27、end: Tt = 207.467 - 3.514tForecast: T6 = 207.467 - 3.514(7) = 182.8714.The following values are needed to compute the slope and intercept:Computation of slope:Computation of intercept:Equation for linear trend: Tt = 20.7466 - 0.3514tConclusion: enrollment appears to be decreasing by an average of ap
28、proximately 351 students per year.15.The following values are needed to compute the slope and intercept:Computation of slope:Computation of intercept:Equation for linear trend: Tt = 28,800 + 421.429 t16.A linear trend model is not appropriate. A nonlinear model would provide a better approximation.1
29、7.a. A linear trend appears to be reasonable.b. The following values are needed to compute the slope and intercept:Computation of slope:Computation of intercept:Equation for linear trend: Tt = 19.993 + 1.774 tConclusion: The firm has been realizing an average cost increase of $1.77 per unit per year
30、.18.a. The following values are needed to compute the slope and intercept:Computation of slope:Computation of intercept:Equation for linear trend: Tt = .365 + .193 tForecast: Tt = .365 + .193(11) = $2.49b. Over the past ten years the earnings per share have been increasing at the average rate of $.1
31、93 per year. Although this is a positive indicator of Walgreens performance. More information would be necessary to conclude “good investment.”19.a. The following values are needed to compute the slope and intercept:Computation of slope:Computation of intercept:7.5833 - 0.0514(3.5) = 7.4033Equation
32、for linear trend: Tt = 7.4033 + 0.0514 tThe number of applications is increasing by approximately 1630 per year.b. 1996: Tt = 7.4033 + 0.0514(7) = 7.7633 or about 7.76%1997: Tt = 7.4033 + 0.0514(8) = 7.8148 or about 7.81%20.a. The following values are needed to compute the slope and intercept:Comput
33、ation of slope:Computation of intercept:4184.1 - 397.545(5.5) = 1997.6Equation for linear trend: Tt = 1997.6 + 397.545 tb. T11 = 1997.6 + 397.545(11) = 6371 T12 = 1997.6 + 397.545(12) = 676821.a.The following values are needed to compute the slope and intercept:Computation of slope:Computation of in
34、tercept:(118.2/6) - 7.7714(21/6) = -7.5Equation for linear trend: Tt = -7.5 + 7.7714tb.7.7714 ($M) per yearc.1998 forecast: T8 = -7.5 + 7.7714 (7) = 46.922. a.YearQuarterYtFour-QuarterMoving AverageCenteredMoving Average114223.50333.7504.00454.1254.252164.5004.75235.0005.25355.3755.50475.8756.253176
35、.3756.50266.6256.753648 b.YearQuarterYtCenteredMoving AverageSeasonal-IrregularComponent11422333.7500.8000 454.1251.2121 2164.5001.3333 235.0000.6000 355.3750.9302 475.8751.1915 3176.3751.0980 266.6250.9057 3648QuarterSeasonal-IrregularComponent ValuesSeasonal IndexAdjusted Seasonal Index11.3333,1.0
36、9801.21571.20502.60000,.90570.75290.74633.80000,.90320.86510.867541.2121,1.19151.20181.19124.0355Note: Adjustment for seasonal index = 4.000 / 4.0355 = 0.991223.a.Four quarter moving averages beginning with(1690 + 940 + 2625 + 2500) / 4 = 1938.75Other moving averages are 1966.252002.501956.252052.502025.002060.001990.002123.75b.QuarterSeasonal-IrregularComponent ValuesSeasonal IndexAdjusted Seasonal Index10.9040.9000.90200.90020.4480.5260.49700.48631.3441.4531.39851.39641.2751.1641.21951.217 4.0070Note: Adjustment for seasonal index = 4.000