Exam 16: Time-Series Forecasting

TABLE 16-13 Given below is the monthly time-series data for U.S. retail sales of building materials over a specific year. Month Retail Sales 1 6,594 2 6,610 3 8,174 4 9,513 5 10,595 6 10,415 7 9,949 8 9,810 9 9,637 10 9,732 11 9,214 12 9,201 The results of the linear trend, quadratic trend, exponential trend, first-order autoregressive, second-order autoregressive and third-order autoregressive model are presented below in which the coded month for the first month is 0: $\text { Linear trend model: }$ Coefficients Standard Error t Stat P-value Intercept 7950.7564 617.6342 12.8729 0.0000 Coded Month 212.6503 95.1145 2.2357 0.0494 Quadratic trend model: Coefficients Standard Error t Stat P-value Intercept 6358.2473 417.2692 15.2378 0.0000 Coded Month 1168.1558 176.3526 6.6240 0.0001 Coded Month 2 -86.8641 15.4474 -5.6232 0.0003 Exponential trend model: Coefficients Standard Error t Stat P-value Intercept 3.8912 0.0315 123.3674 0.0000 Coded Month 0.0116 0.0049 2.3957 0.0376 $\text { First-order autoregressive: }$ Coefficients Standard Error t Stat P-value Intercept 3132.0951 1287.2899 2.4331 0.0378 YLag1 0.6823 0.1398 4.8812 0.0009 Second-order autoregressive:: Coefficients Standard Error t Stat P -value Intercept 4968.5789 766.9416 6.4784 0.0003 YLag1 0.9333 0.1547 6.0316 0.0005 YLag2 -0.4487 0.1238 -3.6235 0.0085 Third-order autoregressive:: Coefficients Standard Error t Stat P-value Intercept 6782.7567 2105.7115 3.2211 0.0234 YLag1 0.5481 0.3918 1.3990 0.2207 YLag2 0.0198 0.4034 0.0490 0.9628 YLag3 -0.2749 0.2234 -1.2308 0.2731 Below is the residual plot of the various models: -Referring to Table 16-13, what is your forecast for the 13ᵗʰ month using the exponential-trend model?

After estimating a trend model for annual time-series data, you obtain the following residual plot against time. The problem with your model is that

TABLE 16-5 The number of passengers arriving at San Francisco on the Amtrak cross-country express on 6 successive Mondays were: 60, 72, 96, 84, 36, and 48. -Referring to Table 16-5, the number of arrivals will be exponentially smoothed with a smoothing constant of 0.1. Then the forecast for the seventh Monday will be ________.

TABLE 16-5 The number of passengers arriving at San Francisco on the Amtrak cross-country express on 6 successive Mondays were: 60, 72, 96, 84, 36, and 48. -Referring to Table 16-5, the number of arrivals will be exponentially smoothed with a smoothing constant of 0.1. The smoothed value for the second Monday will be ________.

TABLE 16-11 The manager of a health club has recorded mean attendance in newly introduced step classes over the last 15 months: 32.1, 39.5, 40.3, 46.0, 65.2, 73.1, 83.7, 106.8, 118.0, 133.1, 163.3, 182.8, 205.6, 249.1, and 263.5. She then used Microsoft Excel to obtain the following partial output for both a first- and second-order autoregressive model. SUMMARY OUTPUT - 2nd Order Model Regression Statistics Multiple R 0.993 R Square 0.987 Adjusted R Square 0.985 Standard Error 9.276 Observations 15 Coefficients Intercept 5.86 X Variable 1 0.37 X Variable 2 0.85 SUMMARY OUTPUT - 1 st Order Model Regression Statistics Multiple R 0.993 R Square 0.987 Adjusted R Square 0.985 Standard Error 9.150 Observations 15 Coefficients Intercept 5.66 X Variable 1 1.10 -Referring to Table 16-11, using the first-order model, the forecast of mean attendance for month 16 is ________.

TABLE 16-4 The number of cases of merlot wine sold by a Paso Robles winery in an 8-year period follows. Year Cases of Wine 2003 270 2004 356 2005 398 2006 456 2007 358 2008 500 2009 410 2010 376 -Referring to Table 16-4, a centered 3-year moving average is to be constructed for the wine sales. The moving average for 2004 is ________.

TABLE 16-4 The number of cases of merlot wine sold by a Paso Robles winery in an 8-year period follows. Year Cases of Wine 2003 270 2004 356 2005 398 2006 456 2007 358 2008 500 2009 410 2010 376 -Referring to Table 16-4, exponentially smooth the wine sales with a weight or smoothing constant of 0.2.

The principle of parsimony indicates that the simplest model that gets the job done adequately should be used.

TABLE 16-9 Given below are Excel outputs for various estimated autoregressive models for a company's real operating revenues (in billions of dollars) from 1985 to 2008. From the data, you also know that the real operating revenues for 2006, 2007, and 2008 are 11.7909, 11.7757 and 11.5537, respectively. First-Order Autoregressive Model: Coefficients Standard Error t Stat p -value Intercept 0.1802 0.3980 0.4528 0.6553 XLag1 1.0112 0.0497 20.3526 0.0000 Second-Order Autoregressive Model: Coefficients Standard Error t Stat p -value Intercept 0.3005 0.4408 0.6817 0.5036 X Lag 1 1.1732 0.2347 4.9980 0.0001 Lag 2 -0.1830 0.2507 -0.7300 0.4743 Third-Order Autoregressive Model: Coefficients Standard Error t Stat p -value Intercept 0.3130 0.5144 0.6085 0.5509 1 1.1737 0.2465 4.7617 0.0002 2 -0.0694 0.3731 -0.1860 0.8547 3 -0.1221 0.2820 -0.4330 0.6704 -Referring to Table 16-9, if one decides to use the Third-Order Autoregressive model, what will the predicted real operating revenue for the company be in 2011?

TABLE 16-12 A local store developed a multiplicative time-series model to forecast its revenues in future quarters, using quarterly data on its revenues during the 4-year period from 2005 to 2009. The following is the resulting regression equation: log₁₀ $\hat Y$ = 6.102 + 0.012 X - 0.129 Q₁ - 0.054 Q₂ + 0.098 Q₃ where $\hat Y$ is the estimated number of contracts in a quarter. X is the coded quarterly value with X = 0 in the first quarter of 2005. Q₁ is a dummy variable equal to 1 in the first quarter of a year and 0 otherwise. Q₂ is a dummy variable equal to 1 in the second quarter of a year and 0 otherwise. Q₃ is a dummy variable equal to 1 in the third quarter of a year and 0 otherwise. -Referring to Table 16-12, to obtain a fitted value for the fourth quarter of 2006 using the model, which of the following sets of values should be used in the regression equation?

TABLE 16-10 Business closures in Laramie, Wyoming from 2005 to 2010 were: 2005 10 2006 11 2007 13 2008 19 2009 24 2010 35 Microsoft Excel was used to fit both first-order and second-order autoregressive models, resulting in the following partial outputs: SUMMARY OUTPUT - 2nd Order Model Coefficient Intercept -5.77 X Variable 1 0.80 X Variable 2 1.14 SUMMARY OUTPUT - 1 st Order Model Coefficients Intercept -4.16 X Variable 1 1.59 -Referring to Table 16-10, the fitted values for the first-order autoregressive model are ________, ________, ________, ________, and ________.

TABLE 16-10 Business closures in Laramie, Wyoming from 2005 to 2010 were: 2005 10 2006 11 2007 13 2008 19 2009 24 2010 35 Microsoft Excel was used to fit both first-order and second-order autoregressive models, resulting in the following partial outputs: SUMMARY OUTPUT - 2nd Order Model Coefficient Intercept -5.77 X Variable 1 0.80 X Variable 2 1.14 SUMMARY OUTPUT - 1 st Order Model Coefficients Intercept -4.16 X Variable 1 1.59 -Referring to Table 16-10, the fitted values for the second-order autoregressive model are ________, ________, ________, and ________.

The MAD is a measure of the mean of the absolute discrepancies between the actual and the fitted values in a given time series.

Which of the following is not an advantage of exponential smoothing?

TABLE 16-5 The number of passengers arriving at San Francisco on the Amtrak cross-country express on 6 successive Mondays were: 60, 72, 96, 84, 36, and 48. -Referring to Table 16-5, the number of arrivals will be exponentially smoothed with a smoothing constant of 0.1. The smoothed value for the sixth Monday will be ________.

The overall upward or downward pattern of the data in an annual time series will be contained in the ________ component.

TABLE 16-3 The following table contains the number of complaints received in a department store for the first 6 months of last year. Month Complaints January 36 February 45 March 81 April 90 May 108 June 144 -Referring to Table 16-3, if a three-month moving average is used to smooth this series, what would be the last calculated value?

TABLE 16-4 The number of cases of merlot wine sold by a Paso Robles winery in an 8-year period follows. Year Cases of Wine 2003 270 2004 356 2005 398 2006 456 2007 358 2008 500 2009 410 2010 376 -Referring to Table 16-4, exponential smoothing with a weight or smoothing constant of 0.2 will be used to smooth the wine sales. The value of E₂, the smoothed value for 2004 is ________.

TABLE 16-13 Given below is the monthly time-series data for U.S. retail sales of building materials over a specific year. Month Retail Sales 1 6,594 2 6,610 3 8,174 4 9,513 5 10,595 6 10,415 7 9,949 8 9,810 9 9,637 10 9,732 11 9,214 12 9,201 The results of the linear trend, quadratic trend, exponential trend, first-order autoregressive, second-order autoregressive and third-order autoregressive model are presented below in which the coded month for the first month is 0: $\text { Linear trend model: }$ Coefficients Standard Error t Stat P-value Intercept 7950.7564 617.6342 12.8729 0.0000 Coded Month 212.6503 95.1145 2.2357 0.0494 Quadratic trend model: Coefficients Standard Error t Stat P-value Intercept 6358.2473 417.2692 15.2378 0.0000 Coded Month 1168.1558 176.3526 6.6240 0.0001 Coded Month 2 -86.8641 15.4474 -5.6232 0.0003 Exponential trend model: Coefficients Standard Error t Stat P-value Intercept 3.8912 0.0315 123.3674 0.0000 Coded Month 0.0116 0.0049 2.3957 0.0376 $\text { First-order autoregressive: }$ Coefficients Standard Error t Stat P-value Intercept 3132.0951 1287.2899 2.4331 0.0378 YLag1 0.6823 0.1398 4.8812 0.0009 Second-order autoregressive:: Coefficients Standard Error t Stat P -value Intercept 4968.5789 766.9416 6.4784 0.0003 YLag1 0.9333 0.1547 6.0316 0.0005 YLag2 -0.4487 0.1238 -3.6235 0.0085 Third-order autoregressive:: Coefficients Standard Error t Stat P-value Intercept 6782.7567 2105.7115 3.2211 0.0234 YLag1 0.5481 0.3918 1.3990 0.2207 YLag2 0.0198 0.4034 0.0490 0.9628 YLag3 -0.2749 0.2234 -1.2308 0.2731 Below is the residual plot of the various models: -Referring to Table 16-13, if a five-month moving average is used to smooth this series, what would be the first calculated value?

TABLE 16-13 Given below is the monthly time-series data for U.S. retail sales of building materials over a specific year. Month Retail Sales 1 6,594 2 6,610 3 8,174 4 9,513 5 10,595 6 10,415 7 9,949 8 9,810 9 9,637 10 9,732 11 9,214 12 9,201 The results of the linear trend, quadratic trend, exponential trend, first-order autoregressive, second-order autoregressive and third-order autoregressive model are presented below in which the coded month for the first month is 0: $\text { Linear trend model: }$ Coefficients Standard Error t Stat P-value Intercept 7950.7564 617.6342 12.8729 0.0000 Coded Month 212.6503 95.1145 2.2357 0.0494 Quadratic trend model: Coefficients Standard Error t Stat P-value Intercept 6358.2473 417.2692 15.2378 0.0000 Coded Month 1168.1558 176.3526 6.6240 0.0001 Coded Month 2 -86.8641 15.4474 -5.6232 0.0003 Exponential trend model: Coefficients Standard Error t Stat P-value Intercept 3.8912 0.0315 123.3674 0.0000 Coded Month 0.0116 0.0049 2.3957 0.0376 $\text { First-order autoregressive: }$ Coefficients Standard Error t Stat P-value Intercept 3132.0951 1287.2899 2.4331 0.0378 YLag1 0.6823 0.1398 4.8812 0.0009 Second-order autoregressive:: Coefficients Standard Error t Stat P -value Intercept 4968.5789 766.9416 6.4784 0.0003 YLag1 0.9333 0.1547 6.0316 0.0005 YLag2 -0.4487 0.1238 -3.6235 0.0085 Third-order autoregressive:: Coefficients Standard Error t Stat P-value Intercept 6782.7567 2105.7115 3.2211 0.0234 YLag1 0.5481 0.3918 1.3990 0.2207 YLag2 0.0198 0.4034 0.0490 0.9628 YLag3 -0.2749 0.2234 -1.2308 0.2731 Below is the residual plot of the various models: -Referring to Table 16-13, what is your forecast for the 13ᵗʰ month using the third-order autoregressive model?