Exam 16: Time-Series Forecasting and Index Numbers

To assess the adequacy of a forecasting model, one measure that is often used is

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 Y^{^} = 3.37 + 0.117 X - 0.083 Q₁ + 1.28 Q₂ + 0.617 Q₃ where Y^{^} 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. -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?

TABLE 16-14 Given below are the average prices for three types of energy products in the United States from 1992 to 1995. Year Electricity N atural Gas Fuel Oil 1992 43.205 25.893 0.892 1993 16.959 28.749 0.969 1994 47.202 28.933 1.034 1995 48.874 29.872 0.913 1996 48.693 28.384 0.983 -Referring to Table 16-14, what is the Paasche price index for the group of three energy items in 1995 for a family that consumed 13 units of electricity, 26 units of natural gas and 235 units of fuel oil in 1995 using 1992 as the base year?

TABLE 16-6 The number of cases of merlot wine sold by a Paso Robles winery in an 8-year period follows. 1991 270 1992 356 1993 398 1994 456 1995 358 1996 500 1997 410 1998 376 -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____

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

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. -Referring to Table 16-6, the Holt-Winters method for forecasting with 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.

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_____ .

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 MultipleR 0.604 R Square 0.365 Adjusted R Square 0.316 Standard Error 4.800 Observations 17 Coefficients Intercept 31.2 Coded Year 0.78 -Referring to Table 16-8, the forecast for sales (in millions of dollars) in 2000 is ______.

TABLE 16-12 The manager of a health club has recorded average 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 XVariable 2 0.85 SUMMARY OUTPUT - 1st 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 $\underline{\text {X Variable 1 \quad 1.10}}$ -Referring to Table 16-12, based on the parsimony principle, the second-order model is the better model for making forecasts.

TABLE 16-6 The number of cases of merlot wine sold by a Paso Robles winery in an 8-year period follows 1991 270 1992 356 1993 398 1994 456 1995 358 1996 500 1997 410 1998 376 -Referring to Table 16-6, plot both the wine sales series and the series of Holt-Winters forecast for 1999 to 2002 using a smoothing constant of 0.2 for both level and trend.

TABLE 16-6 $\text { The number of cases of merlot wine sold by a Paso Robles winery in an 8-year period follows. }$ 1991 270 1992 356 1993 398 1994 456 1995 358 1996 500 1997 410 1998 376 -Referring to Table 16-6, the Holt-Winters method for forecasting with 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.

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. -Referring to Table 16-7, the number of arrivals will be smoothed with a 5-term moving average. The last smoothed value will be ______.

TABLE 16-6 The number of cases of merlot wine sold by a Paso Robles winery in an 8-year period follows. 1991 270 1992 356 1993 398 1994 456 1995 358 1996 500 1997 410 1998 376 -Referring to Table 16-6, the Holt-Winters method for forecasting with 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.

TABLE 16-15 Given below are the prices of a basket of four food items from 1996 to 2000. Year Wheat (\ / Bushel ) Corn (\ / Bushel ) Soybeans (\ / Bushel ) Milk (\ / hundredweight) 1996 4.25 3.71 7.41 15.03 1997 3.43 27 7.55 13.63 1998 2.63 23 6.05 15.18 1999 2.11 1.97 4.68 14.72 2000 2.16 1.9 4.81 12.32 -Referring to Table 16-15, what is the unweighted aggregate price index for the basket of four food items in 1999 using 1996 as the base year?

When using the exponentially weighted moving average for purposes of forecasting rather than smoothing,

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 0.996 R Square 0.992 Adjusted R Square 0.991 Standard Error 0.02831 Observations 12 Coefficients Intercept 1.44 Coded Year 0.068 -Referring to Table 16-9, the forecast for the demand in 2002 is______ .

TABLE 16-14 Given below are the average prices for three types of energy products in the United States from 1992 to 1995. Year Electricity N atural Gas Fuel Oil 1992 43.205 25.893 0.892 1993 16.959 28.749 0.969 1994 47.202 28.933 1.034 1995 48.874 29.872 0.913 1996 48.693 28.384 0.983 -Referring to Table 16-14, what are the simple price indexes for electricity, natural gas and fuel oil, respectively, in 1992 using 1996 as the base year?

The fairly regular fluctuations that occur within each year would be contained in the component.

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: log₁₀Y^{^} = 6.102 + 0.012 X - 0.129 Q₁ - 0.054 Q₂ + 0.098 Q₃ where Y^{^} 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. -Referring to Table 16-13, the best interpretation of the constant 6.102 in the regression equation is

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 -Referring to Table 16-1, does there appear to be a relationship between year and the number of cases of wine sold?