Deck 14: Time Series: Descriptive Analyses, Models, and Forecasting Available on CD

Question 1

(Situation M) Fast food chains are closely watching what proposed legislation will do to consumption of "huge meals" in the United States. Researchers have accumulated statistics on the annual consumption of "huge for the past 25 years. The goal of the analysis is to use the past data to predict future consumption and then compare the predicted consumption to the actual consumption in those years.

-Propose a straight-line model that includes both a long-term trend and a seasonal component for the time series. Let $t =$ the year in which the data was collected $( t = 1,2 , \ldots , 25 )$ and let $Q _ { 1 } , Q _ { 2 }$ , and $Q _ { 3 }$ be dummy variables used to model a seasonal effect.

A) $\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { Q } _ { 1 } + \beta _ { 3 } \mathrm { Q } _ { 2 } + \beta _ { 4 } \mathrm { Q } _ { 3 }$
B) $\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 1 \mathrm { t } }$
C) $\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t } + \beta _ { 2 } \mathrm { Q } _ { 1 } + \beta _ { 3 } \mathrm { Q } _ { 2 } + \beta _ { 4 } \mathrm { Q } _ { 3 }$
D) $\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t }$

Answer 1

$\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t } + \beta _ { 2 } \mathrm { Q } _ { 1 } + \beta _ { 3 } \mathrm { Q } _ { 2 } + \beta _ { 4 } \mathrm { Q } _ { 3 }$

Question 2

(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table. $$\begin{array} { l c c c r c } \hline \text { Year } & \begin{array} { l } \text { Commercial } \\\text { Service }\end{array} & \text { Construction } & \begin{array} { c } \text { Manufacturing } \\\text { and Mining }\end{array} & \begin{array} { r } \text { Retail } \\\text { Trade }\end{array} & \begin{array} { l } \text { Wholesale } \\\text { Trade }\end{array} \\\hline 1985 & 1637 & 2262 & 1645 & 4799 & 1089 \\1986 & 1331 & 1770 & 1360 & 4139 & 1028 \\1987 & 1041 & 1463 & 1122 & 3406 & 887 \\1988 & 773 & 1204 & 1013 & 2889 & 740 \\1989 & 930 & 1378 & 1165 & 3183 & 908 \\1990 & 1594 & 2355 & 1599 & 4910 & 1284 \\1991 & 2366 & 3614 & 2223 & 6882 & 1709 \\1992 & 3840 & 4872 & 3683 & 9730 & 2783 \\1993 & 8627 & 5247 & 4433 & 11,429 & 3598 \\1994 & 12,787 & 6936 & 5759 & 13,787 & 4882 \\1995 & 16,647 & 7004 & 5662 & 13,501 & 4835 \\1996 & 20,911 & 7035 & 5641 & 13,509 & 4808 \\\hline\end{array}$$

-Using 1985 as the base period and using just construction failures, calculate the simple index for 1992.

A) 217.88
B) 46.43
C) 215.38
D) 487.20

Answer 2

215.38

Question 3

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.
$$\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}$$

-Using the exponential smoothing technique to the data from 1980 to 1985, forecast the number of Chevrolet passenger cars sold to U.S. and Canadian dealers in 1986 using $\mathrm { w } = 0.3$

A) 2.448 million cars
B) 3.427 million cars
C) 4.882 million cars
D) 3.834 million cars

Answer 3

2.448 million cars

Question 4

(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991 The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is
shown in the table. $$\begin{array} { l c c } \hline & \text { Wheat Harvested by Coop. Member } \\\text { Year } & \text { Time } & ( \mathrm { y } , \text { in thousands of bushels) } \\\hline 1991 & 1 & 75 \\1992 & 2 & 78 \\1993 & 3 & 82 \\1994 & 4 & 82 \\1995 & 5 & 84 \\1996 & 6 & 85 \\1997 & 7 & 87 \\1998 & 8 & 91 \\1999 & 9 & 92 \\2000 & 10 & 92 \\2001 & 11 & 93 \\2002 & 12 & 96 \\2003 & 13 & 101 \\2004 & 14 & 102 \\\hline\end{array}$$

-A forecast was obtained for the year 2005 and the corresponding 95% prediction interval was found to be (103, 107). Interpret this interval.

A) We are 95% confident that the volume of wheat harvested in 2005 will be between 103,000 and 107,000 bushels.
B) We are 95% confident that the mean volume of wheat harvested in all years will be between 103,000 and 107,000 bushels.
C) We expect the volume of wheat harvested to increase between 103,000 and 107,000 bushels from one year to the next.
D) We are 95% confident that the 2005 harvest will be between 103,000 and 107,000 bushels larger than the harvest in 2004.

Question 5

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985. $$\begin{array} { c l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}$$

-Using a smoothing constant of $\mathrm { w } = 0.80$ ate the value of the exponentially smoothed series in 1983.

A) 1.289
B) 1.340
C) 1.228
D) 1.235

Question 6

The _______ generally describes fluctuations of the time series that are attributable to business and economic conditions.

A) seasonal variation
B) secular trend
C) cyclical effect
D) residual effect

Question 7

(Situation O) Using data from the post-Korean war period, an economist modeled annual consumption, $y _ { t }$ , as a function of total labor income, $\mathrm { x } _ { 1 \mathrm { t } }$ , and total property income, $\mathrm { x } _ { 2 \mathrm { t } }$ , with the following results. Assume data for $\mathrm { n } = 40$ years were used in the analysis.
$$\hat { y } _ { t } = 7.81 + 0.91 x _ { 1 t } + 0.57 x _ { 2 t } \quad s = 1.29 \quad \text { Durbin-Watson } d = 2.09$$

-For the situation above, give the rejection region for the Durbin-Watson test for autocorrelation of residuals. Use ? = 0.10.

A) $\mathrm { d } < 1.39$
B) $d > 1.60$ or $4 - d > 1.60$
C) $1.39 < \mathrm { d } < 1.60$
D) $\mathrm { d } < 1.39$ or $4 - \mathrm { d } < 1.39$

Question 8

Using just the wool prices and a smoothing constant w = 0.8, find the exponentially smoothed value for May.

A) 272.0
B) 275.0
C) 287.0
D) 276.9

Question 9

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985. $$\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}$$

-Using the exponential smoothing technique to the data from 1980 to 1985, forecast the number of Chevrolet passenger cars to be sold to U.S. and Canadian dealers in 1986 using $w = 0.7$

A) 4.882 million cars
B) 3.427 million cars
C) 3.834 million cars
D) 2.448 million cars

Question 10

(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986 through 1992 are shown in the table below.
$$\begin{array} { l r c r } \hline \text { Year } & \text { G.M. } & \text { Ford } & \text { Chrysler } \\\hline 1986 & 8993 & 5810 & 1796 \\1987 & 7101 & 4328 & 1225 \\1988 & 6762 & 4313 & 1283 \\1989 & 6244 & 4255 & 1182 \\1990 & 7769 & 4934 & 1494 \\1991 & 8256 & 5585 & 2034 \\1992 & 9305 & 5551 & 2157 \\\hline\end{array}$$

-Using 1986 as the base year, find the simple composite index for 1990.

A) 85.53
B) 116.92
C) 65.81
D) 151.95

Question 11

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.
$$\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}$$

-The _______ is what remains of a time series value after the secular, cyclical, and seasonal components have been removed.

A) exponential effect
B) additive effect
C) residual effect
D) error effect

Question 12

The tendency of a series of values to increase or decrease over a long period of time is known as the _______ of a time series.

A) seasonal variation
B) cyclical fluctuation
C) secular trend
D) residual effect

Question 13

(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table. $$\begin{array} { l r c c r c } \hline \text { Year } & \begin{array} { l } \text { Commercial } \\\text { Service }\end{array} & \text { Construction } & \begin{array} { c } \text { Manufacturing } \\\text { and Mining }\end{array} & \begin{array} { l } \text { Retail } \\\text { Trade }\end{array} & \begin{array} { l } \text { Wholesale } \\\text { Trade }\end{array} \\\hline 1985 & 1637 & 2262 & 1645 & 4799 & 1089 \\1986 & 1331 & 1770 & 1360 & 4139 & 1028 \\1987 & 1041 & 1463 & 1122 & 3406 & 887 \\1988 & 773 & 1204 & 1013 & 2889 & 740 \\1989 & 930 & 1378 & 1165 & 3183 & 908 \\1990 & 1594 & 2355 & 1599 & 4910 & 1284 \\1991 & 2366 & 3614 & 2223 & 6882 & 1709 \\1992 & 3840 & 4872 & 3683 & 9730 & 2783 \\1993 & 8627 & 5247 & 4433 & 11,429 & 3598 \\1994 & 12,787 & 6936 & 5759 & 13,787 & 4882 \\1995 & 16,647 & 7004 & 5662 & 13,501 & 4835 \\1996 & 20,911 & 7035 & 5641 & 13,509 & 4808 \\\hline\end{array}$$

-Using 1985 as the base year and using all five types of firms, calculate the simple composite index for 1995.

A) 217.88
B) 23.99
C) 476.49
D) 416.80

Question 14

(Situation O) Using data from the post-Korean war period, an economist modeled annual consumption, $y _ { t }$ as a function of total labor income, $\mathrm { x } _ { 1 } \mathrm { t }$ , and total property income, $\mathrm { x } _ { 2 } \mathrm { t }$ , with the following results. Assume data for $\mathrm { n } = 40 \mathrm { years }$ were used in the analysis.
$$\hat { y } _ { t } = 7.81 + 0.91 x _ { 1 t } + 0.57 x _ { 2 t } \quad s = 1.29 \quad \text { Durbin-Watson } d = 2.09$$

-For the situation above, set up the null and alternative hypotheses for testing for the presence of autocorrelation of residuals.

A) $\mathrm { H } _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = 0$
$\mathrm { H } _ { \mathrm { a } }$ : At least one $\beta \neq 0$

B) $\mathrm { H } _ { 0 }$ : No first-order autocorrelation
$\mathrm { H } _ { \mathrm { a } }$ : Negative first-order autocorrelation

C) $\mathrm { H } _ { 0 }$ : No first-order autocorrelation
$\mathrm { H } _ { \mathrm { a } }$ : Positive or Negative first-order autocorrelation

D) $\mathrm { H } _ { 0 }$ : No first-order autocorrelation
$\mathrm { H } _ { \mathrm { a } }$ : Positive first-order autocorrelation

Question 15

(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991-2004.
The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is shown in the table.
$$\begin{array} { l c c } \hline \text { Year } & \text { Time } & \begin{array} { c } \text { Wheat Harvested by Coop. Member } \\\text { (y, in thousands of bushels) }\end{array} \\\hline 1991 & 1 & 75 \\1992 & 2 & 78 \\1993 & 3 & 82 \\1994 & 4 & 82 \\1995 & 5 & 84 \\1996 & 6 & 85 \\1997 & 7 & 87 \\1998 & 8 & 91 \\1999 & 9 & 92 \\2000 & 10 & 92 \\2001 & 11 & 93 \\2002 & 12 & 96 \\2003 & 13 & 101 \\2004 & 14 & 102 \\\hline\end{array}$$

-Suppose the least squares regression equation is $\hat { y } _ { t } = 75 + 2 t$ Interpret the estimate of $\beta _ { 1 }$ in terms of the problem.

A) We expect the mean volume of wheat harvested to increase 2000 bushels from one year to the next.
B) We expect the volume of wheat harvested to increase 2000 bushels for each additional corporate member.
C) We expect the volume of wheat harvested to be 2000 bushels in any given year.
D) We expect to harvest 2000 bushels of wheat in 2005.

Question 16

(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991 The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is
shown in the table. $$\begin{array} { l c c } \hline \text { Year } & \text { Time } & \begin{array} { c } \text { Wheat Harvested by Coop. Member } \\\text { (y, in thousands of bushels) }\end{array} \\\hline 1991 & 1 & 75 \\1992 & 2 & 78 \\1993 & 3 & 82 \\1994 & 4 & 82 \\1995 & 5 & 84 \\1996 & 6 & 85 \\1997 & 7 & 87 \\1998 & 8 & 91 \\1999 & 9 & 92 \\2000 & 10 & 92 \\2001 & 11 & 93 \\2002 & 12 & 96 \\2003 & 13 & 101 \\2004 & 14 & 102\end{array}$$

-Suppose the least squares regression equation is $\hat{ y } _ { \mathrm { t } } = 75 + 2 \mathrm { t }$ . Use the regression model to forecast the harvest in 2005.

A) 110,000 bushels
B) 102,000 bushels
C) 105,000 bushels
D) 103,000 bushels

Question 17

(Situation M) Fast food chains are closely watching what proposed legislation will do to consumption of "huge meals" in the United States. Researchers have accumulated statistics on the annual consumption of "huge for the past 25 years. The goal of the analysis is to use the past data to predict future consumption and then compare the predicted consumption to the actual consumption in those years.

-To test for first-order autocorrelation, we use the _______ test.

A) Paasche
B) Laspeyres
C) Durbin-Watson
D) Wilcoxon

Question 18

(Situation O) Using data from the post-Korean war period, an economist modeled annual consumption, $y _ { t }$ as a function of total labor income, $\mathrm { x } _ { 1 } \mathrm { t }$ , and total property income, $\mathrm { x } _ { 2 } \mathrm { t }$ , with the following results. Assume data for $\mathrm { n } = 40 \mathrm { years }$ were used in the analysis.
$$\hat { y } _ { t } = 7.81 + 0.91 x _ { 1 t } + 0.57 x _ { 2 t } \quad s = 1.29 \quad \text { Durbin-Watson } d = 2.09$$

-Is there evidence of positive autocorrelation of residuals in the consumption model presented above? Test using $\alpha = 0.10$ .

A) Yes, since the standard deviation $\mathrm { s } = 1.29$ is small.
B) Yes, since the Durbin-Watson statistic $d = 2.09$ falls in the rejection region.
C) No, since the standard deviation $\mathrm { s } = 1.29$ is small.
D) No, since the Durbin-Watson statistic $\mathrm { d } = 2.09$ falls in the nonrejection region.

Question 19

(Situation M) Fast food chains are closely watching what proposed legislation will do to consumption of "huge meals" in the United States. Researchers have accumulated statistics on the annual consumption of "huge for the past 25 years. The goal of the analysis is to use the past data to predict future consumption and then compare the predicted consumption to the actual consumption in those years.

-Propose a straight-line model for the long-term trend of the time series. Do not include a seasonal component. Let $t =$ the year in which the data was collected $( t = 1,2 , \ldots , 25 )$ .

A) $E \left( Y _ { t } \right) = \beta _ { 0 } + \beta _ { 1 } t + \beta _ { 2 } Q _ { 1 } + \beta _ { 3 } Q _ { 2 } + \beta _ { 4 } Q _ { 3 }$
B) $\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 1 } \mathrm { t }$
C) $\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { Q } _ { 1 } + \beta _ { 3 } \mathrm { Q } _ { 2 } + \beta _ { 4 } \mathrm { Q } _ { 3 }$
D) $\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t }$

Question 20

(Situation N) An economist wishes to study the monthly trend in the Dow Jones Industrial Average (DJIA). Data collected over the past 40 months were used to fit the model $E \left( Y _ { t } \right) = \beta _ { 0 } + \beta _ { 1 } t _ { \text {, where } }$ , monthly close of the DJIA and $t =$ month $( 1,2,3 , \ldots , 40 )$ . The regression results appear below:
$$\hat { y } = 88 + 0.25 t \quad R ^ { 2 } = 0.37 \quad \text { MSE } = 144 \quad F = 4.25 \quad \text { Durbin-Watson } \mathrm { d } = 0.96$$

-Since the data are recorded over time (months), there is a strong possibility that the residuals are positively correlated. How could you check for residual correlation using a graphical technique?

A) Plot the residuals against $\hat { y }$ and look for a funnel shape.
B) Plot the residuals against $\hat { y }$ and look for outliers.
C) Plot the residuals against $t$ and look for long runs of positive and negative residuals.
D) Plot the residuals against $\hat { y }$ and look for a linear trend.

Question 21

Question 22

(Situation K) Foreign Exchange rates, the values of foreign currency in U.S. dollars, are important to investors and international travelers. The table lists the monthly foreign exchange rates of the British pound (in U.S. dollars per pound) for a certain year.
$$\begin{array} { l l } \underline {\text { Month }} & \underline { \text { Exchange Rate } } \\\text { January } & 1.13 \\\text { February } & 1.10 \\\text { March } & 1.13 \\\text { April } & 1.23 \\\text { May } & 1.25 \\\text { June } & 1.28 \\\text { July } & 1.38 \\\text { August } & 1.39 \\\text { September } & 1.36 \\\text { October } & 1.42 \\\text { November } & 1.44 \\\text { December } & 1.44\end{array}$$

-Calculate the value of the exponentially smoothed series for April using a smoothing constant of $w = 0.7$

Question 23

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985. $$\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}$$

-Using a smoothing constant of $w = 0.30$ 0.30, calculate the value of the exponentially smoothed series in 1982.

A) 1.150
B) 1.289
C) 1.301
D) 1.425

Question 24

The _______ of a time series can account for fluctuations that recur during specific time periods.

A) secular trend
B) cyclical fluctuation
C) seasonal effect
D) residual effect

Question 25

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985. $$\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}$$

-Use the Holt forecasting model with trend to forecast the number of Chevrolet passenger cars sold to U.S. and Canadian dealers in 1990 using $w = 0.6 \text { and } v = 0.5$

A) 6.068 million cars
B) 8.72 million cars
C) 8.952 million cars
D) 6.39 million cars

Question 26

Which of the following statements about the Durbin-Watson d-statistic is true?

A) It can assume any value between 0 and 2 .
B) It can assume any value between 0 and $4 .$
C) It can assume any value between $- 4$ and 0 .
D) It can assume any value between $- 4$ and 4 .

Question 27

(Situation H) The prices of coffee, gasoline, and sugar for each month of 1983 are shown below in the table.
Using just the price of gasoline and a smoothing constant of lculate the exponentially smoothed value for March.

A) $1.102
B) $1.12
C) $1.076
D) $1.084

Question 28

Consider the monthly time series shown in the table. $\begin{array}{l|c|c}\hline \text { Month } & \mathrm{t} & \mathrm{Y} \\\hline \text { January } & 1 & 185 \\\hline \text { February } & 2 & 192 \\\hline \text { March } & 3 & 189 \\\hline \text { April } & 4 & 201 \\\hline \text { May } & 5 & 195 \\\hline \text { June } & 6 & 199 \\\hline \text { July } & 7 & 206 \\\hline \text { August } & 8 & 203 \\\hline \text { September } & 9 & 208 \\\hline \text { October } & 10 & 209 \\\hline \text { November } & 11 & 218 \\\hline \text { December } & 12 & 216 \\\hline\end{array}$
a. Use the values of $Y$ in the table to forecast the values of $Y$ for the next two months, using simple exponential smoothing with $\mathrm { w } = 0.7$ .
b. Use the Holt model with $w = 0.7$ and $v = 0.7$ to forecast the values of $Y$ for the next two months.

Question 29

(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table. $$\begin{array} { r r r r r r } \hline \text { Year } & \text { Commercial } & \text { Manufacturing } & \begin{array} { r } \text { Retail } \\\text { Trade }\end{array} & \begin{array} { l } \text { Wholesale } \\\text { Trade }\end{array} \\\hline 1985 & 1637 & 2262 & 1645 & 4799 & 1089 \\1986 & 1331 & 1770 & 1360 & 4139 & 1028 \\1987 & 1041 & 1463 & 1122 & 3406 & 887 \\1988 & 773 & 1204 & 1013 & 2889 & 740 \\1989 & 930 & 1378 & 1165 & 3183 & 908 \\1990 & 1594 & 2355 & 1599 & 4910 & 1284 \\1991 & 2366 & 3614 & 2223 & 6882 & 1709 \\1992 & 3840 & 4872 & 3683 & 9730 & 2783 \\1993 & 8627 & 5247 & 4433 & 11,429 & 3598 \\1994 & 12,787 & 6936 & 5759 & 13,787 & 4882 \\1995 & 16,647 & 7004 & 5662 & 13,501 & 4835 \\1996 & 20,911 & 7035 & 5641 & 13,509 & 4808 \\\hline\end{array}$$

-Using just the wholesale trade failures and a smoothing constant $\mathrm { w } = 0.7$ , calculate the exponentially smoothed value for 1988.

A) 1015.6
B) 798.4
C) 932.9
D) 845

Question 30

Consider the table below which displays the price of a commodity for six consecutive
years. $\begin{array}{c|c}\hline \text { Year } & \text { Price (dollars) } \\\hline 1 & 250 \\\hline 2 & 255 \\\hline 3 & 253 \\\hline 4 & 255 \\\hline 5 & 259 \\\hline 6 & 261 \\\hline\end{array}$
a. Calculate the values in the exponentially smoothed series using $\mathrm { w } = 0.6$ .
b. Calculate the forecast errors for Years 7-10 if the actual values in those years are 262, 264, 263, 266 respectively.
c. Calculate MAD, MAPE, and RMSE, using the forecast errors for Years 7-10.

Question 31

Consider the actual values Y and forecast values F given in the table below. $$\begin{array} { | c | c | c | } \hline \text { Time Period } & \mathrm { Y } & \mathrm { F } \\\hline 1 & 19.5 & 19.3 \\\hline 2 & 21.5 & 20.9 \\\hline 3 & 22.6 & 22.5 \\\hline\end{array}$$ Calculate the root mean squared error (RMSE) of the forecasts.

A) 0.90
B) 0.30
C) 0.37
D) 1.42

Question 32

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985. $$\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}$$

-Use the Holt forecasting model with trend to forecast the number of Chevrolet passenger cars sold to U.S. and Canadian dealers in 1990 using $\mathrm { w } = 0.4 \text { and } \mathrm { v } = 0.5$

A) 8.952 million cars
B) 8.72 million cars
C) 6.068 million cars
D) 6.39 million cars

Question 33

Consider the monthly time series shown in the table. $\begin{array}{l|c|c}\hline {\text { Month }} & \mathrm{t} & \mathrm{Y} \\\hline \text { January } & 1 & 185 \\\hline \text { February } & 2 & 192 \\\hline \text { March } & 3 & 189 \\\hline \text { April } & 4 & 201 \\\hline \text { May } & 5 & 195 \\\hline \text { June } & 6 & 199 \\\hline \text { July } & 7 & 206 \\\hline \text { August } & 8 & 203 \\\hline \text { September } & 9 & 208 \\\hline \text { October } & 10 & 209 \\\hline \text { November } & 11 & 218 \\\hline \text { December } & 12 & 216 \\\hline\end{array}$

a. Use the method of least squares to fit the model $\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t }$ to the data. Write the prediction equation.
b. Construct a residual plot for the model.
c. Is there evidence of a cyclical component? Explain.

Question 34

Consider the actual values Y and forecast values F given in the table below. $$\begin{array} { | c | c | c | } \hline \text { Time Period } & \mathrm { Y } & \mathrm { F } \\\hline 1 & 19.5 & 19.3 \\\hline 2 & 21.5 & 20.9 \\\hline 3 & 22.6 & 22.5 \\\hline\end{array}$$ Calculate the mean absolute deviation (MAD) of the forecasts.

A) 1.42
B) 0.90
C) 0.30
D) 0.37

Question 35

A(n) _______ is a number that measures the change in a variable over time relative to the value of the variable during a base period.

A) index number
B) exponential smoothing constant
C) residual value
D) time series

Question 36

(Situation K) Foreign Exchange rates, the values of foreign currency in U.S. dollars, are important to investors and international travelers. The table lists the monthly foreign exchange rates of the British pound (in U.S. dollars per pound) for a certain year.
$$\begin{array} { l l } \underline {\text { Month }} & \underline { \text { Exchange Rate } } \\\text { January } & 1.13 \\\text { February } & 1.10 \\\text { March } & 1.13 \\\text { April } & 1.23 \\\text { May } & 1.25 \\\text { June } & 1.28 \\\text { July } & 1.38 \\\text { August } & 1.39 \\\text { September } & 1.36 \\\text { October } & 1.42 \\\text { November } & 1.44 \\\text { December } & 1.44\end{array}$$

-Consider the table below which displays the price of a commodity for six consecutive
years.

$\begin{array}{c|c}\hline \text { Year } & \text { Price } \\\hline 1 & 250 \\\hline 2 & 255 \\\hline 3 & 253 \\\hline 4 & 255 \\\hline 5 & 259 \\\hline 6 & 261 \\\hline\end{array}$

a. $\text { Use the method of least squares to fit the model } E\left(Y_{t}\right)=\beta_{0}+\beta_{1} t \text { to the data. Write the }$ prediction equation.

b. Calculate the residuals and construct a residual plot.
c. Calculate the Durbin Watson d statistic.

Question 37

Consider the actual values Y and forecast values F given in the table below. $$\begin{array} { | c | c | c | } \hline \text { Time Period } & \text { Y } & \text { F } \\\hline 1 & 19.5 & 19.3 \\\hline 2 & 21.5 & 20.9 \\\hline 3 & 22.6 & 22.5 \\\hline\end{array}$$ Calculate the mean absolute percentage error (MAPE) of the forecasts.

A) 0.37
B) 0.30
C) 1.42
D) 0.90

Question 38

(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991
The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is shown in the table.
$$\begin{array} { l c c } \hline \text { Year } & \text { Time } & \begin{array} { c } \text { Wheat Harvested by Coop. Member } \\\text { (y, in thousands of bushels) }\end{array} \\\hline 1991 & 1 & 75 \\1992 & 2 & 78 \\1993 & 3 & 82 \\1994 & 4 & 82 \\1995 & 5 & 84 \\1996 & 6 & 85 \\1997 & 7 & 87 \\1998 & 8 & 91 \\1999 & 9 & 92 \\2000 & 10 & 92 \\2001 & 11 & 93 \\2002 & 12 & 96 \\2003 & 13 & 101 \\2004 & 14 & 102 \\\hline\end{array}$$

-Find the least squares prediction equation for the model $y _ { t } = \beta _ { 0 } + \beta _ { 1 } t + \varepsilon$ .

A) $\hat { y } _ { t } = - 74.2 - 1.9165 \mathrm { t }$
B) $\hat { y } _ { \mathrm { t } } = 1.9165 - 74.2 \mathrm { t }$
C) $\hat { y } _ { t } = 74.2 - 1.9165 \mathrm { t }$
D) $\hat { y } _ { \mathrm { t } } = 74.2 + 1.9165 \mathrm { t }$

Question 39

(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986
through 1992 are shown in the table below. $$\begin{array} { l r r r } \hline \text { Year } & \text { G.M. } & \text { Ford } & \text { Chrysler } \\\hline 1986 & 8993 & 5810 & 1796 \\1987 & 7101 & 4328 & 1225 \\1988 & 6762 & 4313 & 1283 \\1989 & 6244 & 4255 & 1182 \\1990 & 7769 & 4934 & 1494 \\1991 & 8256 & 5585 & 2034 \\1992 & 9305 & 5551 & 2157\end{array}$$

-Using 1986 as the base year, and only using the Chrysler sales data, find the simple index for 1992.

A) 102.49
B) 83.26
C) 97.57
D) 120.10

Question 40

(Situation N) An economist wishes to study the monthly trend in the Dow Jones Industrial Average (DJIA). Data collected over the past 40 months were used to fit the model $E \left( Y _ { t } \right) = \beta _ { 0 } + \beta _ { 1 } t$ , where $y =$ monthly close of the DJIA and $t =$ month $( 1,2,3 , \ldots , 40 )$ . The regression results appear below:
$$\hat { y } = 88 + 0.25 t \quad R ^ { 2 } = 0.37 \quad \text { MSE } = 144 \quad F = 4.25 \quad \text { Durbin-Watson } \mathrm { d } = 0.96$$

-What is the value of the test statistic for testing whether autocorrelation exists in the data?

A) 4.25
B) 0.25
C) 0.37
D) 0.96

Question 41

It is common to use dummy variables to describe seasonal differences in a time series.

Question 42

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985. $$\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}$$

-Using a smoothing constant of $\mathrm { w } = 0.70 ,$ 0.70, calculate the value of the exponentially smoothed series in 1985.

Question 43

The printout below shows a regression analysis for a time series that included 20 observations.

Regression Analysis: C2 versus C1

The regression equation is
$\mathrm { C } 2 = 1.20 + 0.0362 \mathrm { C } 1$

$\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \mathrm { T } & \mathrm { P } \\ \text { Constant } & 1.19947 & 0.08471 & 14.16 & 0.000 \\ \mathrm { C } 1 & 0.036241 & 0.007072 & 5.12 & 0.000 \\\end{array}$

$\begin{array} { lll}\mathrm {~S} = 0.182361 & \mathrm { R } - \mathrm { Sq } = 59.3 \% \quad \mathrm { R } - \mathrm { Sq } ( \text { adj } ) = 57.1 \% \end{array}$

Analysis of Variance
$\begin{array}{lcrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\\text { Regression } & 1 & 0.87340 & 0.87340 & 26.26 & 0.000 \\\text { Residual Error } & 18 & 0.59860 & 0.03326 & & \\\text { Total } & 19 & 1.47200 & & &\end{array}$

Locate the Durbin-Watson d-statistic and test the null hypothesis that there is no autocorrelation of residuals. Use $\alpha = 0.10$ .

Question 44

Since the theoretical distributional properties of the forecast errors with smoothing methods are unknown, many analysts regard smoothing methods as descriptive procedures rather than inferential procedures.

Question 45

The Laspeyres index is a weighted index while the Paasche index is not weighted.

Question 46

A composite index number represents combinations of the prices or quantities of several commodities.

Question 47

Fourth-order autocorrelation in a quarterly time series may indicate seasonality.

Question 48

The table below shows the price of a commodity for each of ten consecutive years.

Question 49

The d-test requires that the residuals be normally distributed.

Question 50

The table below shows the price of a commodity for each of ten consecutive years. $\begin{array}{l}\begin{array} { l l l l l l l l l l l } \hline \text { Year } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\hline\text { Price } & \$ 1.19 & \$ 1.22 & \$ 1.23 & \$ 1.45 & \$ 1.39 & \$ 1.42 & \$ 1.47 & \$ 1.55 & \$ 1.62 & \$ 1.65 \\\hline\end{array}\end{array}$

a. Using Year 1 as the base period, calculate the simple index for the price of the commodity for each year.
b. Plot the simple indexes for years 1-10.
c. Use the simple index to interpret the trend in the price of the commodity.

Question 51

The table below shows the prices and quantities of three commodities for six consecutive
years.
$\quad$$\quad$$\quad$$\quad$$\text {Commodity A}$$\quad$ $\quad$$\text {Commodity B}$$\quad$$\text {Commodity C}$
$\begin{array} { c c c c c c c } \hline \text { Year } & \text { Price } & \text { Quantity } & \text { Price } & \text { Quantity } & \text { Price } & \text { Quantity } \\\hline 1 & 250 & 1200 & 121 & 3200 & 675 & 1800 \\2 & 255 & 1500 & 115 & 3500 & 700 & 1900 \\3 & 253 & 2700 & 128 & 2400 & 714 & 2100 \\4 & 255 & 1800 & 126 & 2800 & 721 & 2500 \\5 & 259 & 2100 & 129 & 2700 & 725 & 3100 \\6 & 261 & 2000 & 135 & 2500 & 734 & 3900 \\\hline\end{array}$ a. Compute the Laspeyres price index for the six-year period, using Year 1 as the base period.
b. Compute the Paasche price index for the six-year period, using Year 1 as the base period.
c. Plot the Laspeyres and Paasche indexes on the same graph. Comment on the
differences.

Question 52

Consider the monthly time series shown in the table. $$\begin{array} { l | c | c } \hline \text { Month } & \text { t } & \mathrm { Y } \\\hline \text { January } & 1 & 185 \\\hline \text { February } & 2 & 192 \\\hline \text { March } & 3 & 189 \\\hline \text { April } & 4 & 201 \\\hline \text { May } & 5 & 195 \\\hline \text { June } & 6 & 199 \\\hline \text { July } & 7 & 206 \\\hline \text { August } & 8 & 203 \\\hline \text { September } & 9 & 208 \\\hline \text { October } & 10 & 209 \\\hline \text { November } & 11 & 218 \\\hline \text { December } & 12 & 216 \\\hline\end{array}$$ a. Calculate the values in the exponentially smoothed series using w $w = 0.6$ b. Graph the time series and the exponentially smoothed series on the same graph.

Question 53

Consider the monthly time series shown in the table. $$\begin{array} { l | c | c } \hline \text { Month } & \mathrm { t } & \mathrm { Y } \\\hline \text { January } & 1 & 185 \\\hline \text { February } & 2 & 192 \\\hline \text { March } & 3 & 189 \\\hline \text { April } & 4 & 201 \\\hline \text { May } & 5 & 195 \\\hline \text { June } & 6 & 199 \\\hline \text { July } & 7 & 206 \\\hline \text { August } & 8 & 203 \\\hline \text { September } & 9 & 208 \\\hline \text { October } & 10 & 209 \\\hline \text { November } & 11 & 218 \\\hline \text { December } & 12 & 216 \\\hline\end{array}$$ a. Use the method of least squares to fit the mo $\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 }$ t to the data. Write the prediction equation.
b. Use the prediction equation to obtain forecasts for the next two months.
c. Find 95% forecast intervals for the next two months.

Question 54

The exponential smoothing constant can be any number between 0 and 100.

Question 55

We plot time series residuals against observed values of Y to determine whether a cyclical component is apparent.

Question 56

Consider the table below which displays the price of a commodity for six consecutive years. $$\begin{array} { c | c } \hline \text { Year } & \text { Price (dollars) } \\\hline 1 & 250 \\\hline 2 & 255 \\\hline 3 & 253 \\\hline 4 & 255 \\\hline 5 & 259 \\\hline 6 & 261 \\\hline\end{array}$$ a. Use the Holt model to forecast values for Years 7-10 using $\mathrm { w } = 0.6$ and $\mathrm { v } = 0.5$ .
b. Calculate the forecast errors for Years 7-10 if the actual values in those years are 263, 267, 269, 268 respectively.
c. Calculate MAD, MAPE, and RMSE, using the forecast errors for Years 7-10.

Question 57

(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986 through 1992 are shown in the table below. $$\begin{array} { l r c r } \hline \text { Year } & \text { G.M. } & \text { Ford } & \text { Chrysler } \\\hline 1986 & 8993 & 5810 & 1796 \\1987 & 7101 & 4328 & 1225 \\1988 & 6762 & 4313 & 1283 \\1989 & 6244 & 4255 & 1182 \\1990 & 7769 & 4934 & 1494 \\1991 & 8256 & 5585 & 2034 \\1992 & 9305 & 5551 & 2157 \\\hline\end{array}$$

-Using 1986 as the base year, find the simple composite index for 1992.

Question 58

Price indexes measure changes in the price of a commodity or group of commodities over time.

Question 59

Consider the table below which displays the price of a commodity for six consecutive years.
$\begin{array}{c|c}\hline \text { Year } & \text { Price (dollars) } \\\hline 1 & 250 \\\hline 2 & 255 \\\hline 3 & 253 \\\hline 4 & 255 \\\hline 5 & 259 \\\hline 6 & 261 \\\hline\end{array}$

a. Use the method of least squares to fit the model $E \left( Y _ { t } \right) = \beta _ { 0 } + \beta _ { 1 } t$ to the data. Write the prediction equation.
b. Use the prediction equation to obtain forecasts of the prices in years 7 and 8 .
c. Find $95 \%$ prediction intervals for years 7 and 8 .

Question 60

Retail sales for a home improvement store in quarters 1-4 over a five-year period are shown (in millions of dollars) in the table below. $$\begin{array} { c | c | c | c | c } \hline { \text { Quarter } } \\\hline \text { Year } & 1 & 2 & 3 & 4 \\\hline 1 & 1.2 & 1.4 & 1.5 & 1.1 \\\hline 2 & 1.3 & 1.6 & 1.5 & 1.2 \\\hline 3 & 1.4 & 1.8 & 1.8 & 1.6 \\\hline 4 & 1.4 & 1.7 & 1.9 & 1.6 \\\hline 5 & 1.6 & 2.0 & 2.1 & 1.9 \\\hline\end{array}$$ a. Write a regression model that contains trend and seasonal components to describe the sales data.
b. Use least squares regression to fit the model.
c. Use the regression model to forecast the quarterly sales during Year 6. Give 95% prediction intervals for the forecasts.

Question 61

The least squares model is an excellent choice for forecasting time series since it works particularly well outside the region of known observations.

Question 62

Smaller values of the trend smoothing constant v assign more weight to the most recent trend of the series and less to past trends.

Question 63

One of the major weaknesses of exponential smoothing is that it is not easily adapted to forecasting.

Question 64

Smoothing techniques are used to remove rapid fluctuations in a time series so the general trend can be seen.

Question 65

With N time periods in your data, a good rule of thumb is to forecast ahead no more than 2N time periods.

Question 66

A major advantage of forecasting with smoothing techniques is that the standard deviation of the forecast errors is known prior to observing the future values.

Question 67

The Holt forecasting model consists of both an exponentially smoothed component and a trend component.

Question 68

The straight-line regression model accounts for both the secular trend and the cyclical effect in a time series.

Question 69

The exponentially smoothed forecast takes into account both changes in trend and seasonality.

Question 70

The value of the Durbin-Watson d-statistic always falls in the interval from 0 to 1.

Question 71

The choice of exponential smoothing constant w has little or no effect on forecast values found using exponential smoothing.

Question 72

The Laspeyres index uses the purchase quantities of the period as weights.

Question 73

Smaller choices of the exponential smoothing constant w assign more weight to the current value of the series and yield a smoother series.