CHAPTER 17TIME SERIES ANALYSIS AND FORECASTING MULTIPLE CHOICE 1. Which of the following is not present in a time series? a. seasonality b. operational variations c. trend d. cycles ANS: B PTS: 1 TOP: Time Series Analysis and Forecasting 2. Given an actual demand of 61, forecast of 58, and an period be using simple exponential smoothing? a. 57.1 b. 58.9 c. 61.0 d. 65.5 ANS: B PTS: 1 of .3, what would the forecast for the next TOP: Time Series Analysis and Forecasting 3. Which of the following smoothing constants would make an exponential smoothing forecast equivalent to a naive forecast? a. 0 b. 1 divided by the number of periods c. 0.5 d. 1.0 ANS: D PTS: 1 TOP: Time Series Analysis and Forecasting 4. The time series component which reflects a regular, multi-year pattern of being above and below the trend line is a. a trend b. seasonal c. cyclical d. irregular ANS: C PTS: 1 TOP: Time Series Analysis and Forecasting 5. The time series component that reflects variability during a single year is called a. a trend b. seasonal c. cyclical d. irregular ANS: B PTS: 1 TOP: Time Series Analysis and Forecasting 6. The time series component that reflects variability due to natural disasters is called a. a trend b. seasonal c. cyclical d. irregular ANS: D PTS: 1 TOP: Time Series Analysis and Forecasting 7. The time series component that reflects gradual variability over a long time period is called a. a trend b. seasonal c. cyclical d. irregular ANS: A PTS: 1 TOP: Time Series Analysis and Forecasting 8. The trend component is easy to identify by using a. moving averages b. exponential smoothing c. regression analysis d. the Delphi approach ANS: C PTS: 1 TOP: Time Series Analysis and Forecasting 9. The forecasting method that is appropriate when the time series has no significant trend, cyclical, or seasonal effect is a. moving averages b. mean squared error c. mean average deviation d. qualitative forecasting methods ANS: A PTS: 1 TOP: Time Series Analysis and Forecasting 10. If data for a time series analysis is collected on an annual basis only, which component may be ignored? a. trend b. seasonal c. cyclical d. irregular ANS: B PTS: 1 TOP: Time Series Analysis and Forecasting 11. For the following time series, you are given the moving average forecast. Time Period 1 2 3 4 5 6 7 Time Series Value 23 17 17 26 11 23 17 Moving Average Forecast 19 20 18 20 The mean squared error equals a. 0 b. 6 c. 41 d. 164 ANS: C PTS: 1 TOP: Time Series Analysis and Forecasting 12. If the estimate of the trend component is 158.2, the estimate of the seasonal component is 94%, the estimate of the cyclical component is 105%, and the estimate of the irregular component is 98%, then the multiplicative model will produce a forecast of a. 1.53 b. 1.53% c. 153.02 d. 153,020,532 ANS: C PTS: 1 TOP: Time Series Analysis and Forecasting 13. Below you are given the first four values of a time series. Time Period 1 2 3 4 Time Series Value 18 20 25 17 Using a 4-period moving average, the forecasted value for period 5 is a. 2.5 b. 17 c. 20 d. 10 ANS: C PTS: 1 TOP: Time Series Analysis and Forecasting 14. Below you are given the first two values of a time series. You are also given the first two values of the exponential smoothing forecast. Time Period (t) 1 2 Time Series Value (Y t) 18 22 Exponential Smoothing Forecast (F t) 18 18 If the smoothing constant equals .3, then the exponential smoothing forecast for time period three is a. 18 b. 19.2 c. 20 d. 40 ANS: B PTS: 1 TOP: Time Series Analysis and Forecasting 15. The following linear trend expression was estimated using a time series with 17 time periods. Tt = 129.2 + 3.8t The trend projection for time period 18 is a. 68.4 b. 193.8 c. 197.6 d. 6.84 ANS: C Exhibit 18-1 PTS: 1 TOP: Time Series Analysis and Forecasting Below you are given the first five values of a quarterly time series. The multiplicative model is appropriate and a four-quarter moving average will be used. Year 1 Quarter 1 2 3 4 1 2 Time Series Value Yt 36 24 16 20 44 16. Refer to Exhibit 18-1. An estimate of the trend component times the cyclical component (T 2Ct) for Quarter 3 of Year 1, when a four-quarter moving average is used, is a. 24 b. 25 c. 26 d. 28 ANS: B PTS: 1 TOP: Time Series Analysis and Forecasting 17. Refer to Exhibit 18-1. An estimate of the seasonal-irregular component for Quarter 3 of Year 1 is a. .64 b. 1.5625 c. 5.333 d. 30 ANS: A PTS: 1 TOP: Time Series Analysis and Forecasting 18. You are given the following information on the seasonal-irregular component values for a quarterly time series: Seasonal-Irregular Component Values (StIt) 1.23, 1.15, 1.16 .86, .89, .83 .77, .72, .79 1.20, 1.13, 1.17 Quarter 1 2 3 4 The seasonal index for Quarter 1 is a. .997 b. 1.18 c. 4 d. 3 ANS: B PTS: 1 TOP: Time Series Analysis and Forecasting 19. Below you are given some values of a time series consisting of 26 time periods. Time Period 1 2 3 4 . . . Time Series Value 37 48 50 63 23 24 25 26 105 107 112 114 The estimated regression equation for these data is Yt = 16.23 + .52Yt-1 + .37Yt-2 The forecasted value for time period 27 is a. 53.23 b. 109.5 c. 116.65 d. 116.95 ANS: D PTS: 1 TOP: Time Series Analysis and Forecasting 20. A group of observations measured at successive time intervals is known as a. a trend component b. a time series c. a forecast d. an additive time series model ANS: B PTS: 1 TOP: Time Series Analysis and Forecasting 21. A component of the time series model that results in the multi-period above-trend and below-trend behavior of a time series is a. a trend component b. a cyclical component c. a seasonal component d. an irregular component ANS: B PTS: 1 TOP: Time Series Analysis and Forecasting 22. The model that assumes that the actual time series value is the product of its components is the a. forecast time series model b. multiplicative time series model c. additive time series model d. None of these alternatives is correct. ANS: B PTS: 1 TOP: Time Series Analysis and Forecasting 23. A method of smoothing a time series that can be used to identify the combined trend/cyclical component is a. the moving average b. the percent of trend c. exponential smoothing d. the trend/cyclical index ANS: A PTS: 1 TOP: Time Series Analysis and Forecasting 24. A method that uses a weighted average of past values for arriving at smoothed time series values is known as a. a smoothing average b. a moving average c. an exponential average d. an exponential smoothing ANS: D PTS: 1 TOP: Time Series Analysis and Forecasting 25. In the linear trend equation, T = b0 + b1t, b1 represents the a. trend value in period t b. intercept of the trend line c. slope of the trend line d. point in time ANS: C PTS: 1 TOP: Time Series Analysis and Forecasting 26. In the linear trend equation, T = b0 + b1t, b0 represents the a. time b. slope of the trend line c. trend value in period 1 d. the Y intercept ANS: D PTS: 1 TOP: Time Series Analysis and Forecasting 27. A parameter of the exponential smoothing model which provides the weight given to the most recent time series value in the calculation of the forecast value is known as the a. mean square error b. mean absolute deviation c. smoothing constant d. None of these alternatives is correct. ANS: C PTS: 1 TOP: Time Series Analysis and Forecasting 28. One measure of the accuracy of a forecasting model is a. the smoothing constant b. a deseasonalized time series c. the mean square error d. None of these alternatives is correct. ANS: C PTS: 1 TOP: Time Series Analysis and Forecasting 29. A qualitative forecasting method that obtains forecasts through "group consensus" is known as the a. Autoregressive model b. Delphi approach c. mean absolute deviation d. None of these alternatives is correct. ANS: B PTS: 1 TOP: Time Series Analysis and Forecasting Exhibit 18-2 Consider the following time series. t 1 2 3 4 Yi 4 7 9 10 30. Refer to Exhibit 18-2. The slope of linear trend equation, b 1, is a. 2.5 b. 2.0 c. 1.0 d. 1.25 ANS: B PTS: 1 TOP: Time Series Analysis and Forecasting 31. Refer to Exhibit 18-2. The intercept, b0, is a. 2.5 b. 2.0 c. 1.0 d. 1.25 ANS: A PTS: 1 TOP: Time Series Analysis and Forecasting 32. Refer to Exhibit 18-2. The forecast for period 5 is a. 10.0 b. 2.5 c. 12.5 d. 4.5 ANS: C PTS: 1 TOP: Time Series Analysis and Forecasting 33. Refer to Exhibit 18-2. The forecast for period 10 is a. 10.0 b. 25.0 c. 30.0 d. 22.5 ANS: D PTS: 1 TOP: Time Series Analysis and Forecasting Exhibit 18-3 Consider the following time series. Year (t) 1 2 3 4 5 Yi 7 5 4 2 1 34. Refer to Exhibit 18-3. The slope of linear trend equation, b 1, is a. -1.5 b. +1.5 c. 8.3 d. -8.3 ANS: A PTS: 1 TOP: Time Series Analysis and Forecasting 35. Refer to Exhibit 18-3. The intercept, b0, is a. -1.5 b. +1.5 c. 8.3 d. -8.3 ANS: C PTS: 1 TOP: Time Series Analysis and Forecasting 36. Refer to Exhibit 18-3. In which time period does the value of Y i reach zero? a. 0.000 b. 0.181 c. 5.53 d. 4.21 ANS: C PTS: 1 TOP: Time Series Analysis and Forecasting 37. Refer to Exhibit 18-3. The forecast for period 10 is a. 6.7 b. -6.7 c. 23.3 d. 15 ANS: B PTS: 1 TOP: Time Series Analysis and Forecasting PROBLEM 1. Consider the following time series. Time 1 2 3 4 5 6 a. b. Yt 18 20 22 24 26 28 Develop a linear trend equation for this time series. What is the forecast for t = 17? ANS: a. Yt = 16+2t b. 30 PTS: 1 TOP: Time Series Analysis and Forecasting 2. What is the forecast for July based on a three-month weighted moving average applied to the following past demand data and using the weights: 5, 3,and 2 (largest weight is for most recent data)? Show all of your computations for April through July. Month January February March April May June July ANS: Month Demand 40 45 57 60 75 87 Demand Forecast Forecast January February March April May June July 40 45 57 60 75 87 50.00 56.10 66.90 78.00 PTS: 1 TOP: Time Series Analysis and Forecasting 3. What is the forecast for July based on a three-month weighted moving average applied to the following past demand data and using the weights: 6, 4, and 2 (largest weight is for most recent data)? Show all of your computations for April through July. Month Actual Demand (A) January 40 February 45 March 57 April 60 May 75 June 87 July ANS: Month April May June July Demand 60 75 87 PTS: 1 Forecast 50.17 56.50 67.00 78.50 TOP: Time Series Analysis and Forecasting 4. What is the forecast for July based on a three-month weighted moving average applied to the following past demand data and using the weights: 5, 4, and 3 (largest weight is for most recent data)? Show all of your computations for April through July. Month January February March April May June July ANS: Month April May June Demand 80 83 87 90 95 98 Demand 90 95 98 Forecast 83.92 87.25 91.33 July 95.00 PTS: 1 TOP: Time Series Analysis and Forecasting 5. Actual sales for January through April are shown below. Observation 1 2 3 4 5 Month January February March April May Actual Sales (A) 18 23 20 16 Forecasted Sales (F) Use exponential smoothing with = 0.2 to calculate smoothed averages and forecast sales for May from the above data. Assume the forecast for the initial period (January) is 18. Show all of your computations. ANS: Observation 1 2 3 4 5 Month January February March April May PTS: 1 Actual Sales (A) 18 23 20 16 Forecasted Sales (F) 18.00 18.00 19.00 19.20 18.56 TOP: Time Series Analysis and Forecasting 6. Actual sales for January through April are shown below. Month Actual Sales (A) January 18 February 25 March 30 April 40 May Use exponential smoothing with = 0.3 to calculate smoothed values and forecast sales for May from the above data. Assume the forecast for the initial period (January) is 18. Show all of your computations from February through May. ANS: Month January February Actual Sales (A) 18 25 Smoothed Value F 18.00 March April May 30 40 20.10 23.07 28.15 PTS: 1 TOP: Time Series Analysis and Forecasting 7. Actual sales of a company (in millions of dollars) for January through April are shown below. Month January February March April May a. b. c. Sales 18 25 30 40 Use = 0.3 to compute the exponential smoothing values for sales. Compute MSE and forecast sales for May. Show all of your computations from February through May. Use = 0.1 to compute the exponential smoothing values for sales. Compute MSE and forecast sales for May. Show all of your computations from February through May. Based on MSE, which provides a better forecast? Explain why? ANS: a. When = 0.30: Month January February March April May b. When Sales 18 25 30 40 Error 2 18.00 20.10 23.07 28.15 49.00 98.01 286.63 144.55 MSE = = 0.10: Month January February March April May c. Smoothed Values Sales 18 25 30 40 Smoothed Values Error 2 18.00 18.70 19.83 21.85 49.00 127.69 406.83 194.51 = 0.30 yields a smaller MSE, therefore, it is a better PTS: 1 MSE = to use. TOP: Time Series Analysis and Forecasting 8. The actual demand for a product and the forecast for the product are shown below. Calculate MAE and MSE. Show all of your computations. Observation 1 2 3 Actual Demand (A) 35 30 26 Forecast (F) --35 30 34 28 38 4 5 6 26 34 28 ANS: Observation 1 2 3 4 5 6 Actual Sales (A) 35 30 26 34 28 38 Forecast (F) --35 30 26 34 28 Error = A-F ---5.00 -4.00 8.00 -6.00 10.00 Total Average PTS: 1 Error squared --25.00 16.00 64.00 36.00 100.00 Absolute Error --5.00 4.00 8.00 6.00 10.00 241.00 48.20 MSE 33.00 6.6 MAE TOP: Time Series Analysis and Forecasting 9. Demand for a product and the forecasting department's forecast (nave model) for a product are shown below. a. b. Compute the mean absolute error. Compute the mean squared error. Actual Forecasted Demand (A) Demand (F) 45 --- Period 1 2 48 45 3 42 48 4 44 42 5 50 44 6 60 50 ANS: a. 5.4 b. 37.0 PTS: 1 TOP: Time Series Analysis and Forecasting 10. For the following time series data, using the nave method (the most recent value as the forecast for the next period), compute the following measures of forecast accuracy. Month Value 1 2 3 4 5 6 a b. c. 12 14 10 16 29 22 Mean absolute error (MAE) Mean squared error (MSE) What is the forecast for period 7? ANS: Observation Actual (A) Forecast (F) Error = A-F Error 2 1 2 3 4 5 6 12 14 10 16 29 22 --12 14 10 16 29 --2.00 -4.00 6.00 13.00 -7.00 --4.00 16.00 36.00 169.00 49.00 Absolute Error --2.00 4.00 6.00 13.00 7.00 274.00 32.00 Total a. b. c. MAE = 6.40 MSE = 54.80 22 PTS: 1 TOP: Time Series Analysis and Forecasting 11. The quarterly sales of a company (in millions of dollars) over the past three years are given in the following table. 2007 Quarter 1 Quarter 2 Quarter 3 Quarter 4 a. b. 2008 2009 170 111 270 250 180 96 280 220 190 120 290 223 Compute the four seasonal factors (Seasonal Indexes). Show all of your computations. The trend for these data is Trend = 174 + 4 t (t represents time, where t=1 for Quarter 1 of 2007 and t=12 for Quarter 4 of 2009). Forecast sales for the first quarter of 2010 using the trend and seasonal indexes. Show all of your computations. ANS: a. 2007 2008 2009 Quarter Total Quarter Average Seasonal Index Quarter 1 Quarter 2 Quarter 3 Quarter 4 170 111 270 250 180 96 280 220 190 120 290 223 Overall average = b. 540 327 840 693 180 109 280 231 0.900 0.545 1.400 1.155 200 Trend = 174 + 4 t = 174 +4 (13) = $226 Millions Forecast = Trend*(SI for Quarter 1) = (226)*(0.9) = $203.40 Millions PTS: 1 TOP: Time Series Analysis and Forecasting 12. The quarterly sales of a company (in millions of dollars) over the past three years are given in the following table. 2007 106 135 149 Quarter 2 256 280 292 Quarter 3 273 280 290 Quarter 4 c. 2009 Quarter 1 a. b. 2008 190 180 209 Compute the four seasonal factors (Seasonal Indexes). Show all of your computations. The trend for these data is Trend = 185.86 + 5.25 t (t represents time, where t=1 for Quarter 1 of 2007 and t=12 for Quarter 4 of 2009). Forecast sales for the first quarter of 2010 using the trend only. Show all of your computations. Forecast sales for the first quarter of 2010 using the trend and seasonal indexes and write your answer below. Show all of your computations. ANS: a. b. c. Quarter 1 Quarter 2 Quarter 3 Quarter 4 254.14 million dollars 150.17 million dollars PTS: 1 0.591 1.255 1.277 0.877 TOP: Time Series Analysis and Forecasting 13. The quarterly sales of a company (in millions of dollars) over the past three years are given in the following table. Quarter Quarter 1 Quarter 2 Quarter 3 Quarter 4 2007 14 20 36 10 2008 28 16 40 14 2009 30 18 38 12 a. b. c. Compute the four seasonal factors (Seasonal Indexes). Show all of your computations. The trend for these data is Trend = 20.82 + 0.336 t (t represents time, where t=1 for Quarter 1 of 2007 and t=12 for Quarter 4 of 2009). Forecast sales for the first quarter of 2010 using the trend only. Show all of your computations. Forecast sales for the first quarter of 2010 using the trend and seasonal indexes and write your answer below. Show all of your computations. ANS: a. Overall average = 23 Quarter Quarter 1 Quarter 2 Quarter 3 Quarter 4 b. c. Quarter Total 72 54 114 36 Quarter Average 24.00 18.00 38.00 12.00 $25.188 Millions $26.711 Millions PTS: 1 TOP: Time Series Analysis and Forecasting 14. The sales records of a company over a period of seven years are shown below. Year (t) 1 2 3 4 5 6 7 a. b. Sales (In Millions of Dollars) 12 16 17 19 18 21 22 Develop a linear trend expression for the above time series. Forecast sales for period 10. ANS: a. b. Tt = 12 + 1.464t $26,640,000 PTS: 1 TOP: Time Series Analysis and Forecasting 15. Student enrollment at a university over the past six years is given below. Year (t) 1 2 3 4 5 6 Enrollment (In 1,000s) 6.30 7.70 8.00 8.20 8.80 8.00 Seasonal Index 1.043 0.783 1.652 0.522 a. b. Develop a linear trend expression for the above time series. Forecast enrollment for year 10. ANS: a. b. Tt = 6.633 + 0.343t 10,063 PTS: 1 TOP: Time Series Analysis and Forecasting 16. The following time series shows the sales of a clothing store over a 10-week period. Week 1 2 3 4 5 6 7 8 9 10 a. b. c. d. Sales ($1,000s) 15 16 19 18 19 20 19 22 15 21 Compute a 4-week moving average for the above time series. Compute the mean square error (MSE) for the 4-week moving average forecast. Use = 0.3 to compute the exponential smoothing values for the time series. Forecast sales for week 11. ANS: a. b. c. d. 17, 18, 19, 19, 20, 19 7.67 15.00, 15.00, 15.30, 16.40, 16.89, 17.52, 18.26, 19.38, 18.07, 18.95 $19,560 PTS: 1 TOP: Time Series Analysis and Forecasting 17. The following time series shows the number of units of a particular product sold over the past six months. Month 1 2 3 4 5 6 a. Units Sold (Thousands) 8 3 4 5 12 10 Compute a 3-month moving average (centered) for the above time series. b. c. d. Compute the mean square error (MSE) for the 3-month moving average. Use = 0.2 to compute the exponential smoothing values for the time series. Forecast the sales volume for month 7. ANS: a. b. c. d. 5, 4, 7 MSE = 73/3 = 24.33 8, 8, 7, 6.4, 6.12, 7.296 F7 = 7.836 PTS: 1 TOP: Time Series Analysis and Forecasting 18. The sales volumes of CMM, Inc., a computer firm, for the past 8 years is given below. Year (t) 1 2 3 4 5 6 7 8 a. b. Sales (In Millions of Dollars) 2 3 5 4 6 8 9 9 Develop a linear trend expression for the above time series. Forecast sales for period 9. ANS: a. b. Tt = 0.929 + 1.071t $10,568,000 PTS: 1 TOP: Time Series Analysis and Forecasting 19. The sales records of a major auto manufacturer over the past ten years are shown below. Year (t) 1 2 3 4 5 6 7 8 9 10 Number of Cars Sold (In thousands of Units) 195 200 250 270 320 380 440 460 500 500 Develop a linear trend expression and project the sales (the number of cars sold) for time period t = 11. ANS: Tt = 136 + 39.182t T11 = 567 PTS: 1 TOP: Time Series Analysis and Forecasting 20. The following data show the quarterly sales of Amazing Graphics, Inc. for the years 6 through 8. Year 6 Quarter 1 2 3 4 1 2 3 4 1 2 3 4 7 8 a. b. c. Sales 2.5 1.5 2.4 1.6 2.0 1.4 1.7 1.9 2.5 2.0 2.4 2.1 Compute the four-quarter moving average values for the above time series. Compute the seasonal factors for the four quarters. Use the seasonal factors developed in Part b to adjust the forecast for the effect of season for year 6. ANS: a. b. c. Centered Moving Averages: 1.94; 1.87; 1.77; 1.72; 1.82; 1.96; 2.12; 2.26 Seasonal Factors: 1.16; 0.85; 1.09; 0.92 Deseasonalized Sales (Year 6): 2.16; 1.76; 2.20; 1.74 PTS: 1 TOP: Time Series Analysis and Forecasting 21. John has collected the following information on the amount of tips he has collected from parking cars the last seven nights. Day 1 2 3 4 5 6 7 a. b. c. d. Tips 18 22 17 18 28 20 12 Compute the 3-day moving averages for the time series. Compute the mean square error for the forecasts. Compute the mean absolute deviation for the forecasts. Forecast John's tips for day 7. ANS: a. b. c. d. 19, 19, 21, 22, 20 45.75 5.25 22 PTS: 1 TOP: Time Series Analysis and Forecasting 22. The following information has been collected on the sales of greeting cards for the past 6 weeks. Week 1 2 3 4 5 6 a. b. c. d. Sales 105 90 95 110 105 100 Produce exponential smoothing forecasts for the series using a smoothing constant of .2. Compute the mean square error for the forecasts produced with a smoothing constant of .2. What is the forecast of sales for week 7? Is a smoothing constant of .2 or .3 better for the sales data? Explain. ANS: a. b. c. d. 105, 105, 102, 100.6, 102.48, 102.984 75.523 102.39 0.2 is better since the MSE is smaller PTS: 1 TOP: Time Series Analysis and Forecasting 23. Consider the following annual series on the number of people assisted by a county human resources department. Year 1 2 3 4 5 6 7 8 9 10 11 a. b. People (in 100s) 22 24 28 24 22 24 20 26 24 28 26 Prepare 3-year moving average values to be used as forecasts for periods 4 through 11. Calculate the mean squared error (MSE) measure of forecast accuracy for periods 4 through 11. Use a smoothing constant of .4 to compute exponential smoothing values to be used as forecasts for periods 2 through 11. Calculate the MSE. c. Compare the results in Parts a and b. ANS: a. b. c. 24.667, 25.333, 24.667, 23.333, 22, 23.333, 23.333, 26, MSE = 7.667 22, 22.8, 24.88, 24.528, 23.5168, 23.71, 22.226, 23.7356, 23.8414, 25.505, MSE = 8.405 The forecasts produced in Part a are better than those produced in Part b. PTS: 1 TOP: Time Series Analysis and Forecasting 24. The temperature in Chicago has been recorded for the past seven days. You are given the information below. Day 1 2 3 4 5 6 7 a. b. c. d. Temperature 82 80 84 83 80 79 82 Produce exponential smoothing forecasts for the series using a smoothing constant of .2. Compute the mean square error for the forecasts produced with a smoothing constant of .2. What is the forecasted temperature for day 8? Is a smoothing constant of .2 or .3 better for the temperature data? Explain. ANS: a. b. c. d. 82, 81.6, 82.08, 82.264, 81.8112, 81.249 4.033 81.399 A smoothing constant of 0.2 is better because the MSE is lower when 0.2 is used. PTS: 1 TOP: Time Series Analysis and Forecasting 25. The yearly series below exhibits a long-term trend. Use the appropriate forecasting technique to produce forecasts for years 11 and 12. Year 1 2 3 4 5 6 7 8 9 10 ANS: Time Series Value 120 132 148 152 160 175 182 190 195 205 T = 115.2 + 9.218182t T11 = 216.6 T12 = 225.82 PTS: 1 TOP: Time Series Analysis and Forecasting 26. The following time series gives the number of units sold during 5 years at a boat dealership. Year 1 2 3 4 5 a. b. c. d. e. f. g. Quarter 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Number of Units 300 240 240 290 350 300 280 320 410 400 390 410 490 450 440 510 540 530 520 540 Find the four-quarter centered moving averages. Plot the series and the moving averages on a graph. Compute the seasonal-irregular component. Compute the seasonal factors for all four quarters. Compute the deseasonalized time series for sales. Calculate the linear trend from the deseasonalized sales. Forecast the number of units sold in each quarter of year 6. ANS: a. 273.75, 287.5, 300, 308.75, 320, 340, 366.25, 391.25, 412.5, 428.75, 441.25, 460, 478.75, 495, 515, 528.75 b. c. d. e. f. g. 0.8767, 1.0087, 1.1667, 0.9717, 0.875, 0.9412, 1.1195, 1.0224, 0.9455, 0.9563, 1.1105, 0.9783, 0.9191, 1.0303, 1.0485, 1.0024 1.1132, 0.9954, 0.9056, 0.9858 269.498, 241.109, 265.018, 294.177, 314.409, 301.386, 309.187, 324.609, 368.308, 401.849, 430.654, 415.906, 440.172, 452.08, 485.866, 517.346, 485.088, 532.449, 574.205, 547.778 T = 216.2993 + 17.35763t 646.56, 595.42, 557.42, 623.90 PTS: 1 TOP: Time Series Analysis and Forecasting 27. Below you are given information on John's income for the past 7 years. Year 1 2 3 4 5 6 7 a. b. Income (In Thousands) 15.0 16.2 17.1 18.1 18.8 19.2 20.5 Use regression analysis to obtain an expression for the linear trend component. Forecast John's income for the next 5 years. ANS: a. b. T = 14.3857 + 0.86429t 21.3, 22.2, 23.0, 23.9, 24.8 PTS: 1 TOP: Time Series Analysis and Forecasting 28. You are given the following information on the quarterly profits for Ajax Corporation. Year 1 2 Quarter 1 2 3 4 1 2 Quarterly Profits Yt 150 120 160 150 150 130 3 4 1 2 3 4 1 2 3 4 3 4 a. b. c. d. 180 160 170 140 200 180 200 150 230 200 Find the four-quarter centered moving averages. Compute the seasonal-irregular component. Compute the seasonal factors for all four quarters. Represent the deseasonalized series. ANS: a. b. c. d. 145, 146.25, 150, 153.75, 157.5, 161.25, 165, 170, 176.25, 181.25, 186.25, 192.5 1.103, 1.026, 1, 0.846, 1.143, 0.992, 1.03, 0.824, 1.135, 0.993, 1.074, 0.779 1.04, 0.82, 1.132, 1.008 144.23, 146.34, 141.34, 148.81, 144.23, 158.54, 159.01, 161.28, 163.46, 170.73, 176.68, 178.57, 192.31, 182.93, 203.18, 198.41 PTS: 1 TOP: Time Series Analysis and Forecasting 29. Below you are given information on crime statistics for Middletown. Year 1 2 3 4 Quarter 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Number of Crimes Committed Yt 10 20 25 5 10 30 35 25 20 40 35 15 20 50 45 35 The seasonal factors for these data are Quarter 1 2 3 4 Seasonal Factor St .589 1.351 1.335 .726 a. b. c. Deseasonalize the series. Obtain an estimate of the linear trend for this series. Use the seasonal and trend components to forecast the number of crimes for each quarter of Year 5. ANS: a. b. c. 16.98, 14.8, 18.78, 20.66, 16.98, 22.21, 26.22, 34.44, 33.96, 29.61, 26.22, 20.66, 33.96, 37.01, 33.71, 48.21 T = 13.5155 + 1.603765t 24.02, 57.26, 58.72, 33.1 PTS: 1 TOP: Time Series Analysis and Forecasting 30. Below you are given the seasonal factors and the estimated trend equation for a time series. These values were computed on the basis of 5 years of quarterly data. Quarter 1 2 3 4 Seasonal Factor St 1.2 .9 .8 1.1 T = 126.23 - 1.6t Produce forecasts for all four quarters of year 6 by using the seasonal and trend components. ANS: 111.156, 81.927, 71.544, 96.613 PTS: 1 TOP: Time Series Analysis and Forecasting 31. The following data show the quarterly sales of a major auto manufacturer (introduced in exercise 4) for the years 8 through 10. Year 8 9 10 a. b. c. Quarter 1 2 3 4 1 2 3 4 1 2 3 4 Sales 160 180 190 170 200 210 260 230 210 240 290 260 Compute the four-quarter moving average values for the above time series. Compute the seasonal factors for the four quarters. Use the seasonal factors developed in Part b to adjust the forecast for the effect of season for year 9. ANS: a. b. c. 180.00, 188.75, 201.25, 217.50, 226.25, 231.25, 238.75, 245.25 0.935, 0.975, 1.1, 0.945 213.90, 215.38, 236.36, 243.39 PTS: 1 TOP: Time Series Analysis and Forecasting