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CHAPTER 16: TIME-SERIES FORECASTING AND INDEX NUMBERS 1. The effect of an unpredictable, rare event will be contained in the ___________ component. a) trend b) cyclical c) irregular d) seasonal ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: component factors, properties 2. The overall upward or downward pattern of the data in an annual time series will be contained in the ____________ component. a) trend b) cyclical c) irregular d) seasonal ANSWER: a TYPE: MC DIFFICULTY: Easy KEYWORDS: component factors, properties 3. The fairly regular fluctuations that occur within each year would be contained in the ____________ component. a) trend b) cyclical c) irregular d) seasonal ANSWER: d TYPE: MC DIFFICULTY: Easy KEYWORDS: component factors, properties 4. The annual multiplicative time-series model does not possess _______ component. a) a trend b) a cyclical c) an irregular d) a seasonal ANSWER: d TYPE: MC DIFFICULTY: Easy KEYWORDS: component factors, properties

Time-Series Forecasting and Index Numbers 5. Based on the following scatter plot, which of the time-series components is not present in this quarterly time series? 350 300 Stock Returns

216

250 200 150 100 50 0 0

10

20

30

40

50

60

Quarters

a. b. c. d.

trend seasonal cyclical irregular

ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: component factors, properties 6. The method of moving averages is used a) to plot a series. b) to exponentiate a series. c) to smooth a series. d) in regression analysis. ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: moving averages 7. When using the exponentially weighted moving average for purposes of forecasting rather than smoothing, a) the previous smoothed value becomes the forecast. b) the current smoothed value becomes the forecast. c) the next smoothed value becomes the forecast. d) none of the above ANSWER: b TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential smoothing, properties

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8. In selecting an appropriate forecasting model, the following approaches are suggested: a) Perform a residual analysis. b) Measure the size of the forecasting error. c) Use the principle of parsimony. d) all of the above ANSWER: d TYPE: MC DIFFICULTY: Moderate KEYWORDS: model selection 9. To assess the adequacy of a forecasting model, one measure that is often used is a) quadratic trend analysis. b) the MAD. c) exponential smoothing. d) moving averages. ANSWER: b TYPE: MC DIFFICULTY: Easy KEYWORDS: model selection 10. Which of the following methods should not be used for short-term forecasts into the future? a) exponential smoothing b) moving averages c) linear trend model d) autoregressive modeling ANSWER: b TYPE: MC DIFFICULTY: Moderate KEYWORDS: moving averages, properties 11. A model that can be used to make predictions about long-term future values of a time series is a) linear trend. b) quadratic trend. c) exponential trend. d) all of the above ANSWER: d TYPE: MC DIFFICULTY: Easy KEYWORDS: least-squares trend fitting, properties

218

Time-Series Forecasting and Index Numbers 12. You need to decide whether you should invest in a particular stock. You would like to invest if the price is likely to rise in the long run. You have data on the daily average price of this stock over the past 12 months. Your best action is to a) compute moving averages. b) perform exponential smoothing. c) estimate a least square trend model. d) compute the MAD statistic. ANSWER: c TYPE: MC DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, properties 13. When a time series appears to be increasing at an increasing rate, such that the percentage difference from observation to observation is constant, the appropriate model to fit is the a. linear trend. b. quadratic trend. c. exponential trend. d. none of the above ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: least-squares trend fitting, model selection 14. The method of least squares is used on time-series data for a) eliminating irregular movements. b) deseasonalizing the data. c) obtaining the trend equation. d) exponentially smoothing a series. ANSWER: c TYPE: MC DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, properties

15. Which of the following statements about the method of exponential smoothing is NOT true? a) b) c) d)

It gives greater weight to more recent data. It can be used for forecasting. It uses all earlier observations in each smoothing calculation. It gives greater weight to the earlier observations in the series.

ANSWER: d TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential smoothing, properties

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16. Which of the following is NOT an advantage of exponential smoothing? a) b) c) d)

It enables us to perform one-period ahead forecasting. It enables us to perform more than one-period ahead forecasting. It enables us to smooth out seasonal components. It enables us to smooth out cyclical components.

ANSWER: b TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential smoothing, properties

17. Which of the following statements about moving averages is NOT true? a) b) c) d)

They can be used to smooth a series. They give equal weight to all values in the computation. They are simpler than the method of exponential smoothing. They give greater weight to more recent data.

ANSWER: d TYPE: MC DIFFICULTY: Moderate KEYWORDS: moving averages, properties 18. 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: a) the cyclical component has not been accounted for. b) the seasonal component has not been accounted for. c) the trend component has not been accounted for. d) the irregular component has not been accounted for. 1 0.8 0.6 Residuals

0.4 0.2 0 -0.2 0

2

4

6

-0.4 -0.6 -0.8 Tim e (Year)

ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: component factors

8

10

12

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Time-Series Forecasting and Index Numbers TABLE 16-1 The number of cases of chardonnay wine sold by a Paso Robles winery in an 8-year period follows: Year Cases of Wine 1991 270 1992 356 1993 398 1994 456 1995 438 1996 478 1997 460 1998 480

19. Referring to Table 16-1, set up a scatter diagram (i.e., a time-series plot) with year on the horizontal X-axis. ANSWER: 500

Number of Cases of Wine

450 400 350 300 250 200 150 100 50 0 1991

1992

1993

1994

TYPE: PR DIFFICULTY: Easy KEYWORDS: scatter plot

Year

1995

1996

1997

1998

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20. Referring to Table 16-1, does there appear to be a relationship between the year and the number of cases of wine sold? a) No, there appears to be no relationship between the year and the number of cases of wine sold by the vintner. b) Yes, there appears to be a slight negative linear relationship between the year and the number of cases of wine sold by the vintner. c) Yes, there appears to be a slight positive relationship between the year and the number of cases of wine sold by the vintner. d) Yes, there appears to be a negative nonlinear relationship between the year and the number of cases of wine sold by the vintner. ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: scatter plot, component factor 21. 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 a) the cyclical component has not been accounted for. b) the seasonal component has not been accounted for. c) the trend component has not been accounted for. d) the irregular component has not been accounted for. 1

Residuals

0.5 0 0

2

4

6

8

10

12

-0.5 -1 Time (Year)

ANSWER: c TYPE: MC DIFFICULTY: Moderate KEYWORDS: component factor 22. The cyclical component of a time series a) represents periodic fluctuations which reoccur within 1 year. b) represents periodic fluctuations which usually occur in 2 or more years. c) is obtained by adding up the seasonal indexes. d) is obtained by adjusting for calendar variation. ANSWER: b TYPE: MC DIFFICULTY: Easy KEYWORDS: component factor

222

Time-Series Forecasting and Index Numbers 23. Which of the following terms describes the overall long-term tendency of a time series? a) trend b) cyclical component c) irregular component d) seasonal component ANSWER: a TYPE: MC DIFFICULTY: Easy KEYWORDS: component factor 24. Which of the following terms describes the up and down movements of a time series that vary both in length and intensity? a) trend b) cyclical component c) irregular component d) seasonal component ANSWER: b TYPE: MC DIFFICULTY: Easy KEYWORDS: component factor 25. The following is the list of MAD statistics for each of the models you have estimated from time-series data: Model MAD Linear Trend 1.38 Quadratic Trend 1.22 Exponential Trend 1.39 AR(2) 0.71 Based on the MAD criterion, the most appropriate model is a) linear trend. b) quadratic trend. c) exponential trend. d) second order autoregressive model. ANSWER: d TYPE: MC DIFFICULTY: Easy KEYWORDS: model selection

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TABLE 16-2 The monthly advertising expenditures of a department store chain (in $1,000,000s) were collected over the last decade. The last 14 months of this time series follows: Month Expenditures ($) 1 1.4 2 1.8 3 1.6 4 1.5 5 1.8 6 1.7 7 1.9 8 2.2 9 1.9 10 1.9 11 2.1 12 2.4 13 2.8 14 3.1

26. Referring to Table 16-2, set up a scatter diagram (i.e., time-series plot) with months on the horizontal X-axis. ANSWER: 3.5

Advertising Expenditures

3 2.5 2 1.5 1 0.5 0 0

2

4

6 8 Number of Months

10

12

14

TYPE: PR DIFFICULTY: Easy KEYWORDS: scatter plot 27. True or False: Referring to Table 16-2, advertising expenditures appear to be increasing in a linear rather than curvilinear manner over time. ANSWER: False TYPE: TF DIFFICULTY: Easy KEYWORDS: least-squares trend fitting

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Time-Series Forecasting and Index Numbers 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 28. Referring to Table 16-3, if a three-term moving average is used to smooth this series, what would be the second calculated term? a) 36 b) 40.5 c) 54 d) 72 ANSWER: d TYPE: MC DIFFICULTY: Easy KEYWORDS: moving averages 29. Referring to Table 16-3, if a three-term moving average is used to smooth this series, what would be the last calculated term? a) 72 b) 93 c) 114 d) 126 ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: moving averages 30. Referring to Table 16-3, if a three-term moving average is used to smooth this series, how many terms would it have? a) 2 b) 3 c) 4 d) 5 ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: moving averages, properties

Time-Series Forcasting and Index Numbers

31. Referring to Table 16-3, if this series is smoothed using exponential smoothing with a smoothing constant of 1/3, how many terms would it have? a) 3 b) 4 c) 5 d) 6 ANSWER: d TYPE: MC DIFFICULTY: Easy KEYWORDS: exponential smoothing, properties 32. Referring to Table 16-3, if this series is smoothed using exponential smoothing with a smoothing constant of 1/3, what would be the first term? a) 36 b) 39 c) 42 d) 45 ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential smoothing 33. Referring to Table 16-3, if this series is smoothed using exponential smoothing with a smoothing constant of 1/3, what would be the second term? a) 39 b) 42 c) 45 d) 53 ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential smoothing 34. Referring to Table 16-3, if this series is smoothed using exponential smoothing with a smoothing constant of 1/3, what would be the third term? a) 53 b) 65.33 c) 68 d) 81 ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential smoothing

225

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Time-Series Forecasting and Index Numbers 35. Referring to Table 16-3, suppose the last two smoothed values are 81 and 96. (Note: they are not.) What would you forecast as the value of the time series for July? a) 81 b) 86 c) 91 d) 96 ANSWER: d TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential smoothing, forecasting 36. Referring to Table 16-3, suppose the last two smoothed values are 81 and 96. (Note: they are not.) What would you forecast as the value of the time series for September? a) 81 b) 86 c) 91 d) 96 ANSWER: d TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential smoothing, forecasting 37. If you want to recover the trend using exponential smoothing, you will choose a weight (W) that falls in the range a) [ 0, 0.2] b) c)

d)

[ 0.2, 0.4] [ 0.6, 0.8] [ 0.8,1.0]

ANSWER: a TYPE: MC DIFFICULTY: Easy KEYWORDS: exponential smoothing, forecasting, properties

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TABLE 16-4 Given below are Excel outputs for various estimated autoregressive models for Coca-Cola's real operating revenues (in billions of dollars) from 1975 to 1998. From the data, we also know that the real operating revenues for 1996, 1997, and 1998 are 11.7909, 11.7757 and 11.5537, respectively. First Order Autoregressive Model: Coefficients Standard Error t Stat p-value Intercept 0.1802077 0.39797154 0.452815546 0.655325119 XLag1 1.011222533 0.049685158 20.35260757 2.64373E-15 Second Order Autoregressive Model: Coefficients Standard Error t Stat p-value Intercept 0.30047473 0.4407641 0.681713257 0.503646149 X Lag 1 1.17322186 0.234737881 4.998008229 7.98541E-05 X Lag 2 -0.183028189 0.250716669 -0.730020026 0.474283347 Third Order Autoregressive Model: Coefficients Standard Error Intercept 0.313043288 0.514437257 XLag1 1.173719587 0.246490594 XLag2 -0.069378567 0.373086508 XLag3 -0.122123515 0.282031297

t Stat 0.608515972 4.761721601 -0.185958391 -0.433014053

p-value 0.550890271 0.000180926 0.854678245 0.670448392

38. Referring to Table 16-4 and using a 5% level of significance, what is the appropriate AR model for Coca-Cola's real operating revenue? a) AR(1) b) AR(2) c) AR(3) d) any of the above ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: autoregressive model, t test on slope 39. Referring to Table 16-4, if one decides to use AR(3), what will the predicted real operating revenue for Coca-Cola be in 2001? a) $11.59 billion b) $11.68 billion c) $11.84 billion d) $12.47 billion ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: autoregressive model, forecasting

228

Time-Series Forecasting and Index Numbers TABLE 16-5 A contractor developed a multiplicative time-series model to forecast the number of contracts in future quarters, using quarterly data on number of contracts during the 3-year period from 1996 to 1998. The following is the resulting regression equation: ln Yˆ = 3.37 + 0.117 X – 0.083 Q1 + 1.28 Q2 + 0.617 Q3 where Yˆ is the estimated number of contracts in a quarter X is the coded quarterly value with X = 0 in the first quarter of 1996. 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. 40. Referring to Table 16-5 , the best interpretation of the constant 3.37 in the regression equation is: a) the fitted value for the first quarter of 1996, prior to seasonal adjustment, is ln 3.37. b) the fitted value for the first quarter of 1996, after seasonal adjustment, is ln 3.37. c) the fitted value for the first quarter of 1996, prior to seasonal adjustment, is e3.37. d) the fitted value for the first quarter of 1996, after seasonal adjustment, is e3.37. ANSWER: c TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, intercept, interpretation

41. Referring to Table 16-5, the best interpretation of the coefficient of X (0.117) in the regression equation is: a) the quarterly growth rate in contracts is around 11.7%. b) the annual growth rate in contracts is around 11.7%. c) the quarterly growth rate in contracts is around 17%. d) the annual growth rate in contracts is around 17%. ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, slope, interpretation

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42. Referring to Table 16-5, the best interpretation of the coefficient of Q3 (0.617) in the regression equation is: a) the number of contracts in the third quarter of a year is approximately 62% higher than the average over all 4 quarters. b) the number of contracts in the third quarter of a year is approximately 62% higher than it would be during the fourth quarter. c) the number of contracts in the third quarter of a year is approximately 85% higher than the average over all 4 quarters. d) the number of contracts in the third quarter of a year is approximately 85% higher than it would be during the fourth quarter. ANSWER: d TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, slope, interpretation 43. Referring to Table 16-5, to obtain a forecast for the first quarter of 1999 using the model, which of the following sets of values should be used in the regression equation? a) X = 12, Q1 = 0, Q2 = 0, Q3 = 0 b) X = 12, Q1 = 1, Q2 = 0, Q3 = 0 c) X = 13, Q1 = 0, Q2 = 0, Q3 = 0 d) X = 13, Q1 = 1, Q2 = 0, Q3 = 0 ANSWER: b TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, forecasting 44. Referring to Table 16-5, to obtain a forecast for the fourth quarter of 1999 using the model, which of the following sets of values should be used in the regression equation? a) X = 15, Q1 = 0, Q2 = 0, Q3 = 0 b) X = 15, Q1 = 1, Q2 = 0, Q3 = 0 c) X = 16, Q1 = 0, Q2 = 0, Q3 = 0 d) X = 16, Q1 = 1, Q2 = 0, Q3 = 0 ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, forecasting

230

Time-Series Forecasting and Index Numbers

45. Referring to Table 16-5, using the regression equation, which of the following values is the best forecast for the number of contracts in the third quarter of 1999? a) 228 b) 252 c) 277 d) 311 ANSWER: c TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, forecasting 46. Referring to Table 16-5, using the regression equation, which of the following values is the best forecast for the number of contracts in the second quarter of 2000? a) 212 b) 272 c) 592 d) 764 ANSWER: d TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, forecasting

47. Referring to Table 16-5, in testing the coefficient of X in the regression equation (0.117), the results were a t-statistic of 9.08 and an associated p-value of 0.0000. Which of the following is the best interpretation of this result? a) The quarterly growth rate in the number of contracts is significantly different than 0% ( = 0.05). b) The quarterly growth rate in the number of contracts is not significantly different than 0% ( = 0.05). c) The quarterly growth rate in the number of contracts is significantly different than 100% ( = 0.05). d) The quarterly growth rate in the number of contracts is not significantly different than 100% ( = 0.05).

α

α

α

α

ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, t test on slope, decision, conclusion, interpretation

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48. Referring to Table 16-5, in testing the coefficient for Q1 in the regression equation (–0.083), the results were a t-statistic of –0.66 and an associated p-value of 0.530. Which of the following is the best interpretation of this result? a) The number of contracts in the first quarter of the year is significantly different than the number of contracts in an average quarter ( = 0.05). b) The number of contracts in the first quarter of the year is not significantly different than the number of contracts in an average quarter ( = 0.05). c) The number of contracts in the first quarter of the year is significantly different than the number of contracts in the fourth quarter for a given coded quarterly value of X ( = 0.05). d) The number of contracts in the first quarter of the year is not significantly different than the number of contracts in the fourth quarter for a given coded quarterly value of X ( = 0.05).

α

α

α

α

ANSWER: d TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, t test on slope, decision, conclusion, interpretation 49. True or False: A trend is a persistent pattern in annual time-series data that has to be followed for several years. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: component factor 50. True or False: Given a data set with 15 yearly observations, a 3-year moving average will have fewer observations than a 5-year moving average. ANSWER: False TYPE: TF DIFFICULTY: Easy KEYWORDS: moving averages, properties 51. True or False: Given a data set with 15 yearly observations, there are only thirteen 3-year moving averages. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: moving averages, properties 52. True or False: Given a data set with 15 yearly observations, there are only seven 9-year moving averages. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: moving averages, properties

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Time-Series Forecasting and Index Numbers

53. True or False: MAD is the summation of the residuals divided by the sample size. ANSWER: False TYPE: TF DIFFICULTY: Moderate KEYWORDS: model selection 54. True or False: A least squares linear trend line is just a simple regression line with the years recoded. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: least-squares trend fitting 55. True or False: If a time series does not exhibit a long-term trend, the method of exponential smoothing may be used to obtain short-term predictions about the future. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: exponential smoothing 56. True or False: The method of least squares may be used to estimate both linear and curvilinear trends. ANSWER: True TYPE: TF DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting 57. True or False: The principle of parsimony indicates that the simplest model that gets the job done adequately should be used. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: model selection 58. True or False: In selecting a forecasting model, we should perform a residual analysis. ANSWER: True TYPE: TF DIFFICULTY: Moderate KEYWORDS: model selection

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59. True or False: Each forecast using the method of exponential smoothing depends on all the previous observations in the time series. ANSWER: True TYPE: TF DIFFICULTY: Moderate KEYWORDS: exponential smoothing, properties

60. True or False: The MAD is a measure of the average of the absolute discrepancies between the actual and fitted values in a given time series. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: model selection TABLE 16-6 The number of cases of merlot wine sold by a Paso Robles winery in an 8-year period follows: Year Cases of Wine 1991 270 1992 356 1993 398 1994 456 1995 358 1996 500 1997 410 1998 376 61. Referring to Table 16-6, a centered 3-year moving average is to be constructed for the wine sales. The result of this process will lead to a total of __________ moving averages. ANSWER: 6 TYPE: FI DIFFICULTY: Easy KEYWORDS: moving averages, properties 62. Referring to Table 16-6, a centered 3-year moving average is to be constructed for the wine sales. The moving average for 1992 is __________. ANSWER: 341.33 TYPE: FI DIFFICULTY: Easy KEYWORDS: moving averages, properties

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Time-Series Forecasting and Index Numbers

63. Referring to Table 16-6, a centered 3-year moving average is to be constructed for the wine sales. The moving average for 1995 is __________. ANSWER: 438 TYPE: FI DIFFICULTY: Easy KEYWORDS: moving averages, properties 64. Referring to Table 16-6, construct a centered 3-year moving average for the wine sales. ANSWER: Period Cases MA 1 270 * 2 356 341.333 3 398 403.333 4 456 404.000 5 358 438.000 6 500 422.667 7 410 428.667 8 376 * TYPE: PR DIFFICULTY: Moderate KEYWORDS: moving averages 65. Referring to Table 16-6, a centered 5-year moving average is to be constructed for the wine sales. The number of moving averages that will be calculated is __________. ANSWER: 4 TYPE: FI DIFFICULTY: Easy KEYWORDS: moving averages 66. Referring to Table 16-6, a centered 5-year moving average is to be constructed for the wine sales. The moving average for 1993 is __________. ANSWER: 367.6 TYPE: FI DIFFICULTY: Moderate KEYWORDS: moving averages 67. Referring to Table 16-6, a centered 5-year moving average is to be constructed for the wine sales. The moving average for 1996 is __________. ANSWER: 420.0 TYPE: FI DIFFICULTY: Moderate KEYWORDS: moving averages

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68. Referring to Table 16-6, construct a centered 5-year moving average for the wine sales. ANSWER: Period Cases MA 1 270 * 2 356 * 3 398 367.6 4 456 413.6 5 358 424.4 6 500 420.0 7 410 * 8 376 * TYPE: PR DIFFICULTY: Moderate KEYWORDS: moving averages

69. Referring to Table 16-6, 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 1992, is __________. ANSWER: 287.2 TYPE: FI DIFFICULTY: Moderate KEYWORDS: exponential smoothing

70. Referring to Table 16-6, exponential smoothing with a weight or smoothing constant of 0.2 will be used to smooth the wine sales. The value of E4, the smoothed value for 1994, is __________. ANSWER: 338.7 TYPE: FI DIFFICULTY: Moderate KEYWORDS: exponential smoothing 71. Referring to Table 16-6, exponential smoothing with a weight or smoothing constant of 0.2 will be used to forecast wine sales. The forecast for 1999 is __________. ANSWER: 380.2 TYPE: FI DIFFICULTY: Moderate KEYWORDS: exponential smoothing, forecasting

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Time-Series Forecasting and Index Numbers

72. Referring to Table 16-6, exponentially smooth the wine sales with a weight or smoothing constant of 0.2. ANSWER: Time CaseWine Smooth 1 270 270.000 2 356 287.200 3 398 309.360 4 456 338.688 5 358 342.550 6 500 374.040 7 410 381.232 8 376 380.186 TYPE: PR DIFFICULTY: Difficult KEYWORDS: exponential smoothing

73. Referring to Table 16-6, exponential smoothing with a weight or smoothing constant of 0.4 will be used to smooth the wine sales. The value of E2, the smoothed value for 1992, is __________. ANSWER: 304.4 TYPE: FI DIFFICULTY: Moderate KEYWORDS: exponential smoothing

74. Referring to Table 16-6, exponential smoothing with a weight or smoothing constant of 0.4 will be used to smooth the wine sales. The value of E5, the smoothed value for 1995, is __________. ANSWER: 375.7 TYPE: FI DIFFICULTY: Moderate KEYWORDS: exponential smoothing 75. Referring to Table 16-6, exponential smoothing with a weight or smoothing constant of 0.4 will be used to forecast wine sales. The forecast for 1999 is __________. ANSWER: 401.95 TYPE: FI DIFFICULTY: Moderate KEYWORDS: exponential smoothing, forecasting

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76. Referring to Table 16-6, exponentially smooth the wine sales with a weight or smoothing constant of 0.4. ANSWER: Time CaseWine Smooth 1 270 270.000 2 356 304.400 3 398 341.840 4 456 387.504 5 358 375.702 6 500 425.421 7 410 419.253 8 376 401.952 TYPE: PR DIFFICULTY: Difficult KEYWORDS: exponential smoothing 77. Referring to Table 16-6, the Holt-Winters method for forecasting with a smoothing constant of 0.2 for both level and trend will be used to smooth the wine sales. The smoothed values of the level and trend for 1992 are ______ and ______, respectively. ANSWER: 356; 86 TYPE: FI DIFFICULTY: Easy KEYWORDS: Holt-Winters method 78. Referring to Table 16-6, the Holt-Winters method for forecasting with a smoothing constant of 0.2 for both level and trend will be used to smooth the wine sales. The smoothed values of the level and trend for 1993 are ______ and ______, respectively. ANSWER: 406.8; 57.84 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Holt-Winters method 79. Referring to Table 16-6, the Holt-Winters method for forecasting with a smoothing constant of 0.2 for both level and trend will be used to smooth the wine sales. The smoothed values of the level and trend for 1998 are ______ and ______, respectively. ANSWER: 383.82; -42.70 TYPE: FI DIFFICULTY: Difficult KEYWORDS: Holt-Winters method

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Time-Series Forecasting and Index Numbers

80. Referring to Table 16-6, the Holt-Winters method for forecasting with a smoothing constant of 0.2 for both level and trend will be used to forecast the wine sales. The forecast for 1999 is _____. ANSWER: 341.12 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Holt-Winters method, forecasting 81. Referring to Table 16-6, the Holt-Winters method for forecasting with a smoothing constant of 0.2 for both level and trend will be used to forecast the wine sales. The forecast for 2002 is _____. ANSWER: 213.01 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Holt-Winters method, forecasting 82. Referring to Table 16-6, use the Holt-Winters method of fitting wine sales to compute the smoothed level and trend with a smoothing constant of 0.2 for both level and trend. ANSWER: Year Series (Y) Level (E) Trend (T) 1991 270 1992 356 356 86 1993 398 406.8 57.84 1994 456 457.728 52.3104 1995 358 388.4077 -44.9942 1996 500 468.6827 55.22118 1997 410 432.7808 -17.6773 1998 376 383.8207 -42.7035

TYPE: PR DIFFICULTY: Difficult KEYWORDS: Holt-Winters method

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83. Referring to Table 16-6, plot both the wine sales series and the series of Holt-Winters forecasts for 1999 to 2002 using a smoothing constant of 0.2 for both level and trend. ANSWER: 600 500 400 Prediction

300

Sales (Y)

200 100 0 1

2

3

4

5

6

7

8

9 10 11 12

TYPE: PR DIFFICULTY: Difficult KEYWORDS: Holt-Winters method, forecasting, scatter plot 84. Referring to Table 16-6, the Holt-Winters method for forecasting with a smoothing constant of 0.8 for both level and trend will be used to smooth the wine sales. The smoothed values of the level and trend for 1992 are ______ and ______, respectively. ANSWER: 356; 86 TYPE: FI DIFFICULTY: Easy KEYWORDS: Holt-Winters method 85. Referring to Table 16-6, the Holt-Winters method for forecasting with a smoothing constant of 0.8 for both level and trend will be used to smooth the wine sales. The smoothed values of the level and trend for 1993 are ______ and ______, respectively. ANSWER: 433.2; 84.24 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Holt-Winters method 86. Referring to Table 16-6, the Holt-Winters method for forecasting with a smoothing constant of 0.8 for both level and trend will be used to smooth the wine sales. The smoothed values of the level and trend for 1998 are ______ and ______, respectively. ANSWER: 609.11; 46.46 TYPE: FI DIFFICULTY: Difficult KEYWORDS: Holt-Winters method

240

Time-Series Forecasting and Index Numbers 87. Referring to Table 16-6, the Holt-Winters method for forecasting with a smoothing constant of 0.8 for both level and trend will be used to forecast the wine sales. The forecast for 1999 is _____. ANSWER: 655.56 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Holt-Winters method, forecasting 88. Referring to Table 16-6, the Holt-Winters method for forecasting with a smoothing constant of 0.8 for both level and trend will be used to forecast the wine sales. The forecast for 2002 is _____. ANSWER: 794.93 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Holt-Winters method, forecasting 89. Referring to Table 16-6, use the Holt-Winters method of fitting wine sales to compute the smoothed level and trend with a smoothing constant of 0.8 for both level and trend. ANSWER: Year Series (Y) Level (E) Trend (T) 1991 270 1992 356 356 86 1993 398 433.2 84.24 1994 456 505.152 81.7824 1995 358 541.1475 72.62502 1996 500 591.018 68.07412 1997 410 609.2737 58.11044 1998 376 609.1073 46.45507

TYPE: PR DIFFICULTY: Difficult KEYWORDS: Holt-Winters method

Time-Series Forcasting and Index Numbers

241

90. Referring to Table 16-6, plot both the wine sales series and the series of Holt-Winters forecasts for 1999 to 2002 using a smoothing constant of 0.8 for both level and trend. ANSWER: 900 800 700 600 500

Prediction

400

Sales (Y)

300 200 100 0 1

2

3

4

5

6

7

8

9 10 11 12

TYPE: PR DIFFICULTY: Difficult KEYWORDS: Holt-Winters method, forecasting, scatter plot TABLE 16-7 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. 91. Referring to Table 16-7, the number of arrivals will be smoothed with a 3-term moving average. There will be a total of __________ smoothed values. ANSWER: 4 TYPE: FI DIFFICULTY: Easy KEYWORDS: moving averages, properties 92. Referring to Table 16-7, the number of arrivals will be smoothed with a 3-term moving average. The first smoothed value will be __________. ANSWER: 76 TYPE: FI DIFFICULTY: Easy KEYWORDS: moving averages

242

Time-Series Forecasting and Index Numbers

93. Referring to Table 16-7, the number of arrivals will be smoothed with a 3-term moving average. The last smoothed value will be __________. ANSWER: 56 TYPE: FI DIFFICULTY: Easy KEYWORDS: moving averages 94. Referring to Table 16-7, the number of arrivals will be smoothed with a 5-term moving average. The first smoothed value will be __________. ANSWER: 69.6 TYPE: FI DIFFICULTY: Moderate KEYWORDS: moving averages 95. Referring to Table 16-7, the number of arrivals will be smoothed with a 5-term moving average. The last smoothed value will be __________. ANSWER: 67.2 TYPE: FI DIFFICULTY: Moderate KEYWORDS: moving averages 96. Referring to Table 16-7, the number of arrivals will be exponentially smoothed with a smoothing constant of 0.1. The smoothed value for the second Monday will be __________. ANSWER: 61.2 TYPE: FI DIFFICULTY: Moderate KEYWORDS: exponential smoothing 97. Referring to Table 16-7, the number of arrivals will be exponentially smoothed with a smoothing constant of 0.1. The smoothed value for the sixth Monday will be __________. ANSWER: 62.0 TYPE: FI DIFFICULTY: Moderate KEYWORDS: exponential smoothing 98. Referring to Table 16-7, the number of arrivals will be exponentially smoothed with a smoothing constant of 0.1. The forecast for the seventh Monday will be __________. ANSWER: 62.0 TYPE: FI DIFFICULTY: Moderate KEYWORDS: exponential smoothing, forecasting

99. Referring to Table 11.7, exponentially smooth the number of arrivals using a smoothing constant of 0.1.

Time-Series Forcasting and Index Numbers

243

ANSWER: Time Arrivals Smooth 1 60 60.0000 2 72 61.2000 3 96 64.6800 4 84 66.6120 5 36 63.5508 6 48 61.9957 TYPE: PR DIFFICULTY: Difficult KEYWORDS: exponential smoothing 100. Referring to Table 16-7, the number of arrivals will be exponentially smoothed with a smoothing constant of 0.25. The smoothed value for the second Monday will be __________. ANSWER: 63.0 TYPE: FI DIFFICULTY: Moderate KEYWORDS: exponential smoothing 101. Referring to Table 16-7, the number of arrivals will be exponentially smoothed with a smoothing constant of 0.25. The smoothed value for the third Monday will be __________. ANSWER: 71.25 TYPE: FI DIFFICULTY: Moderate KEYWORDS: exponential smoothing 102. Referring to Table 16-7, the number of arrivals will be exponentially smoothed with a smoothing constant of 0.25. The forecast of the number of arrivals on the seventh Monday will be __________. ANSWER: 60.6 TYPE: FI DIFFICULTY: Moderate KEYWORDS: exponential smoothing, forecasting

244

Time-Series Forecasting and Index Numbers 103. Referring to Table 16-7, exponentially smooth the number of arrivals using a smoothing constant of 0.25. ANSWER: Time Arrivals Smooth 1 60 60.0000 2 72 63.0000 3 96 71.2500 4 84 74.4375 5 36 64.8281 6 48 60.6211 TYPE: PR DIFFICULTY: Difficult KEYWORDS: exponential smoothing 104. Referring to Table 16-7, the Holt-Winters method for forecasting with a smoothing constant of 0.3 for both level and trend will be used to smooth the number of arrivals. The smoothed values of the level and trend for the second Monday are ______ and ______, respectively. ANSWER: 72; 12 TYPE: FI DIFFICULTY: Easy KEYWORDS: Holt-Winters 105. Referring to Table 16-7, the Holt-Winters method for forecasting with a smoothing constant of 0.3 for both level and trend will be used to smooth the number of arrivals. The smoothed values of the level and trend for the sixth Monday are ______ and ______, respectively. ANSWER: 42.43; -15.74 TYPE: FI DIFFICULTY: Easy KEYWORDS: Holt-Winters 106. Referring to Table 16-7, the Holt-Winters method for forecasting with a smoothing constant of 0.3 for both level and trend will be used to forecast the number of arrivals. The forecast for the seventh Monday is _____. ANSWER: 26.70 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Holt-Winters, forecasting 107. Referring to Table 16-7, the Holt-Winters method for forecasting with a smoothing constant of 0.3 for both level and trend will be used to forecast the number of arrivals. The forecast for the twelfth Monday is _____. ANSWER: -51.98 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Holt-Winters, forecasting 108. Referring to Table 16-7, use the Holt-Winters method of fitting number of arrivals to compute the smoothed level and trend with a smoothing constant of 0.3 for both level and trend.

Time-Series Forcasting and Index Numbers

245

ANSWER: Mondays Arrivals Level (E) Trend (T) 1 60 2 72 72 12 3 96 92.4 17.88 4 84 91.884 5.0028 5 36 54.26604 -24.8317 6 48 42.43029 -15.7345

TYPE: PR DIFFICULTY: Difficult KEYWORDS: Holt-Winters 109. Referring to Table 16-7, plot both the number of arrivals series and the series of HoltWinters forecasts for the seventh through twelfth Mondays using a smoothing constant of 0.3 for both level and trend. ANSWER: 120 100 80 60 40

Prediction

20

Arrivals

0 -20

1

2

3

4

5

6

7

8

9 10 11 12

-40 -60

TYPE: PR DIFFICULTY: Difficult KEYWORDS: Holt-Winters, forecasting, scatter plot 110. Referring to Table 16-7, the Holt-Winters method for forecasting with a smoothing constant of 0.9 for both level and trend will be used to smooth the number of arrivals. The smoothed values of the level and trend for the second Monday are ______ and ______, respectively. ANSWER: 72; 12 TYPE: FI DIFFICULTY: Easy KEYWORDS: Holt-Winters

246

Time-Series Forecasting and Index Numbers 111. Referring to Table 16-7, the Holt-Winters method for forecasting with a smoothing constant of 0.9 for both level and trend will be used to smooth the number of arrivals. The smoothed values of the level and trend for the sixth Monday are ______ and ______, respectively. ANSWER: 105.64; 10.63 TYPE: FI DIFFICULTY: Easy KEYWORDS: Holt-Winters 112. Referring to Table 16-7, the Holt-Winters method for forecasting with a smoothing constant of 0.9 for both level and trend will be used to forecast the number of arrivals. The forecast for the seventh Monday is _____. ANSWER: 116.27 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Holt-Winters, forecasting 113. Referring to Table 16-7, the Holt-Winters method for forecasting with a smoothing constant of 0.9 for both level and trend will be used to forecast the number of arrivals. The forecast for the twelfth Monday is _____. ANSWER: 169.40 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Holt-Winters, forecasting 114. Referring to Table 16-7, use the Holt-Winters method of fitting the number of arrivals to compute the smoothed level and trend with a smoothing constant of 0.9 for both level and trend. ANSWER: Mondays Arrivals Level (E) Trend (T) 1 60 2 72 72 12 3 96 85.2 12.12 4 84 95.988 11.9868 5 36 100.7773 11.26705 6 48 105.6399 10.62661

TYPE: PR DIFFICULTY: Difficult KEYWORDS: Holt-Winters

Time-Series Forcasting and Index Numbers

247

115. Referring to Table 16-7, plot both the number of arrivals series and the series of HoltWinters forecasts for the seventh through twelfth Mondays using a smoothing constant of 0.9 for both level and trend. ANSWER: 180 160 140 120 100

Prediction

80

Arrivals

60 40 20 0 1

2

3

4

5

6

7

8

9

10 11 12

TYPE: PR DIFFICULTY: Difficult KEYWORDS: Holt-Winters, forecasting, scatter plot TABLE 16-8 The president of a chain of department stores believes that her stores' total sales have been showing a linear trend since 1980. She uses Microsoft Excel to obtain the partial output below. The dependent variable is sales (in millions of dollars), while the independent variable is coded years, where 1980 is coded as 0, 1981 is coded as 1, etc. SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations Intercept Coded Year

0.604 0.365 0.316 4.800 17

Coefficients 31.2 0.78

116. Referring to Table 16-8, the fitted trend value for 1980 is __________. ANSWER: 31.2 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, fitted value

248

Time-Series Forecasting and Index Numbers

117. Referring to Table 16-8, the fitted trend value for 1985 is __________. ANSWER: 35.1 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, fitted value 118. Referring to Table 16-8, the estimate of the rate at which sales are increasing each year is __________. ANSWER: 0.78 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, slope, interpretation 119. Referring to Table 16-8, the forecast for sales in 2000 is __________. ANSWER: 46.8 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, forecasting 120. Referring to Table 16-8, the forecast for sales in 2005 is __________. ANSWER: 50.7 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, forecasting

Time-Series Forcasting and Index Numbers

249

TABLE 16-9 The executive vice-president of a drug manufacturing firm believes that the demand for the firm's most popular drug has been evidencing an exponential trend since 1985. She uses Microsoft Excel to obtain the partial output below. The dependent variable is the log base 10 of the demand for the drug, while the independent variable is years, where 1985 is coded as 0, 1986 is coded as 1, etc. SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

Intercept Coded Year

0.996 0.992 0.991 0.02831 12

Coefficients 1.44 0.068

121. Referring to Table 16-9, the fitted trend value for 1985 is __________. ANSWER: 27.54 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, fitted value 122. Referring to Table 16-9, the fitted trend value for 1990 is __________. ANSWER: 60.26 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, fitted value

123. Referring to Table 16-9, the fitted exponential trend equation to predict Y is __________. ANSWER: 27.54(1.17)X TYPE: FI DIFFICULTY: Difficult KEYWORDS: least-squares trend fitting, fitted value 124. Referring to Table 16-9, the forecast for the demand in 1999 is __________. ANSWER: 246.6 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, forecasting

250

Time-Series Forecasting and Index Numbers 125. Referring to Table 16-9, the forecast for the demand in 2002 is __________. ANSWER: 394.46 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, forecasting TABLE 16-10 The manager of a marketing consulting firm has been examining his company's yearly profits. He believes that these profits have been showing a quadratic trend since 1980. He uses Microsoft Excel to obtain the partial output below. The dependent variable is profit (in thousands of dollars), while the independent variables are coded years and squares of coded years, where 1980 is coded as 0, 1981 is coded as 1, etc. SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

Intercept Coded Year Year Squared

0.998 0.996 0.996 4.996 17

Coefficients 35.5 0.45 1.00

126. Referring to Table 16-10, the fitted value for 1980 is __________. ANSWER: 35.5 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, fitted value 127. Referring to Table 16-10, the fitted value for 1985 is __________. ANSWER: 62.75 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, fitted value 128. Referring to Table 16-10, the forecast for profits in 2000 is __________. ANSWER: 444.5 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, forecasting 129. Referring to Table 16-10, the forecast for profits in 2005 is __________. ANSWER:

Time-Series Forcasting and Index Numbers

251

671.75 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, forecasting 130. Microsoft Excel was used to obtain the following quadratic trend equation: Sales = 100 – 10X + 15X2. The data used was from 1989 through 1998, coded 0 to 9. The forecast for 1999 is __________. ANSWER: 1,500 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, forecasting 131. The manager of a company believed that her company's profits were following an exponential trend. She used Microsoft Excel to obtain a prediction equation for the logarithm (base 10) of profits: log10(Profits) = 2 + 0.3X. The data she used were from 1993 through 1998, coded 0 to 5. The forecast for 1999 profits is __________. ANSWER: 6,309.57 TYPE: FI DIFFICULTY: Moderate KEYWORDS: least-squares trend fitting, forecasting 132. A first-order autoregressive model for stock sales is: Salesi = 800 + 1.2(Sales)i-1. If sales in 1998 are 6000, the forecast of sales for 1999 is __________. ANSWER: 8,000 TYPE: FI DIFFICULTY: Moderate KEYWORDS: autoregressive model, forecasting 133. A second-order autoregressive model for average mortgage rate is: Ratei = – 2.0 + 1.8(Rate)i-1 – 0.5 (Rate)i-2. If the average mortgage rate in 1998 was 7.0, and in 1997 was 6.4, the forecast for 1999 is __________. ANSWER: 7.4 TYPE: FI DIFFICULTY: Moderate KEYWORDS: autoregressive model, forecasting

252

Time-Series Forecasting and Index Numbers

134. A second-order autoregressive model for average mortgage rate is: Ratei = – 2.0 + 1.8(Rate)i-1 – 0.5 (Rate)i-2. If the average mortgage rate in 1998 was 7.0, and in 1997 was 6.4, the forecast for 2000 is __________. ANSWER: 7.82 TYPE: FI DIFFICULTY: Difficult KEYWORDS: autoregressive model, forecasting TABLE 16-11 Business closures in Laramie, Wyoming from 1989 to 1994 were: 1993 10 1994 11 1995 13 1996 19 1997 24 1998 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 Intercept X Variable 1 X Variable 2

Coefficients -5.77 0.80 1.14

SUMMARY OUTPUT – 1st Order Model Intercept X Variable 1

Coefficients -4.16 1.59

135. Referring to Table 16-11, the fitted values for the first-order autoregressive model are ________, ________, ________, ________, and ________. ANSWER: 11.74, 13.33, 16.51, 26.05, 34.00 TYPE: FI DIFFICULTY: Moderate KEYWORDS: autoregressive model, fitted value

Time-Series Forcasting and Index Numbers

253

136. Referring to Table 16-11, the residuals for the first-order autoregressive model are ________, ________, ________, ________, and ________. ANSWER: -.74, -.33, 2.49, -2.05, 1.00 TYPE: FI DIFFICULTY: Moderate KEYWORDS: autoregressive model, residual

137. Referring to Table 16-11, the value of the MAD for the first-order autoregressive model is ________. ANSWER: 1.322 TYPE: FI DIFFICULTY: Moderate KEYWORDS: autoregressive model, model selection 138. Referring to Table 16-11, the fitted values for the second-order autoregressive model are ________, ________, ________, and ________. ANSWER: 14.43, 17.17, 24.25, 35.09 TYPE: FI DIFFICULTY: Moderate KEYWORDS: autoregressive model, fitted value 139. Referring to Table 16-11, the residuals for the second-order autoregressive model are ________, ________, ________, and ________. ANSWER: – 1.43, 1.83, – 0.25, – 0.09 TYPE: FI DIFFICULTY: Moderate KEYWORDS: autoregressive model, residual

140. Referring to Table 16-11, the value of the MAD for the second-order autoregressive model is ________. ANSWER: 0.90 TYPE: FI DIFFICULTY: Moderate KEYWORDS: autoregressive model, model selection

141. True or False: Referring to Table 16-11, the values of the MAD for the two models indicate that the first-order model should be used for forecasting. ANSWER: False TYPE: TF DIFFICULTY: Moderate KEYWORDS: autoregressive model, model selection

254

Time-Series Forecasting and Index Numbers TABLE 16-12 The manager of a health club has recorded average attendance in the club’s 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 R Square Adjusted R Square Standard Error Observations

Intercept X Variable 1 X Variable 2

0.993 0.987 0.985 9.276 15

Coefficients 5.86 0.37 0.85

SUMMARY OUTPUT – 1st Order Model Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

Intercept X Variable 1

0.993 0.987 0.985 9.150 15

Coefficients 5.66 1.10

142. Referring to Table 16-12, using the first-order model, the forecast of average attendance for month 16 is __________. ANSWER: 295.51 TYPE: FI DIFFICULTY: Moderate KEYWORDS: autoregressive model, forecasting

Time-Series Forcasting and Index Numbers

255

143. Referring to Table 16-12, using the first-order model, the forecast of average attendance for month 17 is __________. ANSWER: 330.72 TYPE: FI DIFFICULTY: Moderate KEYWORDS: autoregressive model, forecasting 144. Referring to Table 16-12, using the second-order model, the forecast of average attendance for month 16 is __________. ANSWER: 315.09 TYPE: FI DIFFICULTY: Moderate KEYWORDS: autoregressive model, forecasting 145. Referring to Table 16-12, using the second-order model, the forecast of average attendance for month 17 is __________. ANSWER: 346.42 TYPE: FI DIFFICULTY: Moderate KEYWORDS: autoregressive model, forecasting 146. True or False: Referring to Table 16-12, based on the parsimony principle, the secondorder model is the better model for making forecasts. ANSWER: False TYPE: TF DIFFICULTY: Easy KEYWORDS: autoregressive model, model selection

256

Time-Series Forecasting and Index Numbers TABLE 16-13 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 1998 to 2002. The following is the resulting regression equation: log10 Yˆ = 6.102 + 0.012 X – 0.129 Q1 – 0.054 Q2 + 0.098 Q3 where Yˆ is the estimated number of contracts in a quarter X is the coded quarterly value with X = 0 in the first quarter of 1998. 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.

147. Referring to Table 16-13, the best interpretation of the constant 6.102 in the regression equation is: a) the fitted value for the first quarter of 1998, prior to seasonal adjustment, is log10(6.102). b) the fitted value for the first quarter of 1998, after seasonal adjustment, is log10(6.102). c) the fitted value for the first quarter of 1998, prior to seasonal adjustment, is 106.102. d) the fitted value for the first quarter of 1998, after seasonal adjustment, is 106.102. ANSWER: c TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, intercept, interpretation

148. Referring to Table 16-13, the best interpretation of the coefficient of X (0.012) in the regression equation is: a) the quarterly growth rate in revenues is around 1.2%. b) the annual growth rate in revenues is around 1.2%. c) the quarterly growth rate in revenues is around 12%. d) the annual growth rate in revenues is around 12%. ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, slope, interpretation 149. Referring to Table 16-13, the estimated quarterly compound growth rate in revenues is: a) the quarterly growth rate in revenues is around 1.2%. b) the annual growth rate in revenues is around 2.8%. c) the quarterly growth rate in revenues is around 12%. d) the annual growth rate in revenues is around 28%. ANSWER: b TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, slope, interpretation

150. Referring to Table 16-13, the best interpretation of the coefficient of Q2 (–0.054) in the regression equation is:

Time-Series Forcasting and Index Numbers

257

a) the revenues in the second quarter of a year are approximately 5.4% lower than the average over all 4 quarters. b) the revenues in the second quarter of a year are approximately 5.4% lower than they would be during the fourth quarter. c) the revenues in the second quarter of a year are approximately 11.69% lower than the average over all 4 quarters. d) the revenues in the second quarter of a year are approximately 11.69% lower than they would be during the fourth quarter. ANSWER: d TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, slope, interpretation

151. Referring to Table 16-13, the best interpretation of the coefficient of Q3 (0.098) in the regression equation is: a) the revenues in the third quarter of a year are approximately 9.8% higher than the average over all 4 quarters. b) the revenues in the third quarter of a year are approximately 9.8% higher than they would be during the fourth quarter. c) the revenues in the third quarter of a year are approximately 25.31% higher than the average over all 4 quarters. d) the revenues in the third quarter of a year are approximately 25.31% higher than they would be during the fourth quarter. ANSWER: d TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, slope, interpretation 152. Referring to Table 16-13, to obtain a forecast for the first quarter of 2002 using the model, which of the following sets of values should be used in the regression equation? a) X = 16, Q1 = 1, Q2 = 0, Q3 = 0 b) X = 16, Q1 = 0, Q2 = 1, Q3 = 0 c) X = 17, Q1 = 1, Q2 = 0, Q3 = 0 d) X = 17, Q1 = 0, Q2 = 1, Q3 = 0 ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, forecasting

258

Time-Series Forecasting and Index Numbers 153. Referring to Table 16-13, to obtain a forecast for the fourth quarter of 1999 using the model, which of the following sets of values should be used in the regression equation? a) X = 7, Q1 = 0, Q2 = 0, Q3 = 0 b) X = 7, Q1 = 1, Q2 = 0, Q3 = 0 c) X = 8, Q1 = 0, Q2 = 0, Q3 = 0 d) X = 8, Q1 = 1, Q2 = 0, Q3 = 0 ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, forecasting 154. Referring to Table 16-13, to obtain a forecast for the third quarter of 2003 using the model, which of the following sets of values should be used in the regression equation? a) X = 22, Q1 = 0, Q2 = 0, Q3 = 0 b) X = 22, Q1 = 0, Q2 = 0, Q3 = 1 c) X = 23, Q1 = 0, Q2 = 0, Q3 = 0 d) X = 23, Q1 = 0, Q2 = 0, Q3 = 1 ANSWER: b TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, forecasting 155. Referring to Table 16-13, using the regression equation, what is the forecast for revenues in the third quarter of 2003? ANSWER: 2910717.12 TYPE: PR DIFFICULTY: Moderate KEYWORDS: exponential model, forecasting 156. Referring to Table 16-13, using the regression equation, what is the forecast for revenues in the first quarter of 2005? ANSWER: 2037042.08 TYPE: PR DIFFICULTY: Moderate KEYWORDS: exponential model, forecasting 157. Referring to Table 16-13, using the regression equation, what is the forecast for revenues in the fourth quarter of 2004? ANSWER: 2666858.67 TYPE: PR DIFFICULTY: Moderate KEYWORDS: exponential model, forecasting

Time-Series Forcasting and Index Numbers

259

158. Referring to Table 16-13, in testing the significance of the coefficient of X in the regression equation (0.012), which has a p-value of 0.0000, which of the following is the best interpretation of this result? a) The quarterly growth rate in revenues is significantly different than 0% ( = 0.05). b) The quarterly growth rate in revenues is not significantly different than 0% ( = 0.05). c) The quarterly growth rate in revenues is significantly different than 1.2% ( = 0.05). d) The quarterly growth rate in revenues is not significantly different than 1.2% ( = 0.05).

α

α

α

α

ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, t test on slope, decision, conclusion, interpretation

159. Referring to Table 16-13, in testing the significance of the coefficient for Q1 in the regression equation (– 0.129), which has a p-value of 0.492, which of the following is the best interpretation of this result? a) The revenues in the first quarter of the year are significantly different than the revenues in an average quarter ( = 0.05). b) The revenues in the first quarter of the year are not significantly different than the revenues in an average quarter ( = 0.05). c) The revenues in the first quarter of the year are significantly different than the revenues in the fourth quarter ( = 0.05). d) The revenues in the first quarter of the year are not significantly different than the revenues in the fourth quarter ( = 0.05).

α

α

α

α

ANSWER: d TYPE: MC DIFFICULTY: Moderate KEYWORDS: exponential model, t test on slope, decision, conclusion, interpretation