solution

 

In the National Footbal League, the philosophy for winning (rushing, passing, defense) seems to go through cycles. Consider a time series of the average number of rushing yards in the NFL per regular season from 1980 to 2008.28 DATADATA

DATADATADATA

DATADATADATADATADATA

 (a) Make a time series plot. Is there evidence that the average rushing yards is trending in one direction? Describe the general

movement of the series.

(b) Make a lagged time series plot and calculate the correlation between yt and yt−1.

(c) Fit an AR(1) model using yt as the response variable and yt−1 as the explanatory variable. Record the estimated regression

equation.

(d) Based on the AR(1) model, forecast the average number of rushing yards in the NFL for the 2009 regular season.

 

Q1501

 

Refer to the previous exercise. For each of the exponential smoothing models in part (a), calculate MAD, MSE, and MAPE (see Exercises 13.46, 13.47, and 13.48, pages 724–725). Which value of w provided the best fit?

 

Q1501

 

The 1993 average attendance per game (y78) was 32,363, and the 2005 average attendance per game (y90) was 38,272. Use only these facts and any information found in Example 13.18 (page 711) to calculate 2009 forecasts based on the 3-year moving-average model and the 15-year moving-average model.

 

Q1502

 

 

n Example 13.4 (page 696), an exponential trend model was fitted to the yearly time series of the number of passenger cars owned in China. In Exercise 13.16 (page 710), you were asked to investigate whether autocorrelation was present in the residuals associated with the trend fit.

(a) Fit an AR(1) model using yt as the response variable and yt−1 as the explanatory variable. Record the estimated regression equation.

(b) Based on the AR(1) model, forecast the number of passenger cars owned for 2008. Compare this forecast with the exponential trend forecast given in Example 13.4.

 

Q1503

 

Continue analyzing the Chinese car ownership data from the previous exercise.

 (a) Calculate the residuals for the exponential trend model fit of Example 13.4 (page 696), and calculate MAD, MSE, and MAPE (see Exercises 13.46, 13.47, and 13.48, pages 724–725).

(b) Calculate the residuals for the AR(1) fit of the previous exercise, and calculate MAD, MSE, and MAPE.

(c) Based on the results of parts (a) and (b), which model seems to provide a better fit?

 

Q1504

 

models for various financial time series are often mentioned in the business iterature. A simple random walk model specifies that one-period differences in the time series can be modeled as a constant term plus a random deviation term. In other words, period-to-period differences (or changes) will behave as a random process. The equation for this random walk model is

 

 yt − yt−1 = β0 +

 

t If we rewrite this equation solving for yt , we get

 

 yt = β0 + yt−1 + t

 

 which is our AR(1) model with β1 = 1. If we fit an AR(1) model and find that the β1 estimate is close to 1, then a simple random walk model for the time series is another modeling option. Consider the monthly closing prices of the Dow Jones Industrial Index beginning with January 2000 and ending with November 2008.29

(a) Fit an AR(1) model to the Dow Jones time series. Construct a 95% confidence interval for β1. Does the confidence interval suggest the possibility of a simple random walk model underlying the time series? Explain your answer.

(b) Obtain the one-period differences and plot them as a time series. Perform the runs test on the differences. Based on the plot of the differences and the runs test, can you justify the simple random walk model for the Dow Jones series? Provide appropriate justification.

(c) For a simple random walk model, the estimate of β0 is simply the average of all one-period differences yt − yt−1 (call this average ydiff), and the forecast equation is yt = yt−1 + ydiff. Calculate the one-period differences and their average, and use this to provide a “random walk forecast” for the monthly closing price of December 2008

 

 

Q1505

 

Refer to the time series of the monthly total revenue for the New York Times in Exercise 13.52 (page 730).

 (a) Fit a regression model based on a linear trend term and monthly seasonal indicator variables.

 (b) Make a time series plot of the residuals for the model in part (a). Is there evidence of nonlinearity in the residual plot? If so, describe the pattern.

(c) Fit a regression model based on a linear trend term, a quadratic trend term, and monthly seasonal indicator variables. What is the estimated coefficient for the month of January indicator variable? What is the P-value for the variable?

 

Q1506

 

ontinue the analysis of the New York Times revenue time series.

(a)    Make a time series plot of the residuals for the model in part

$1rn    (c) of Exercise 13.69. Identify the largest outlier in the residual series. In which month and year did the outlier occur?

 (b) When dealing with data sets that are not time series, tossing out an unusual observation from the data set poses no problems. However, with time series applications, deleting an observation can be problematic. In particular, observations that are normally two periods apart around the outlier become adjacent when the intermediate observation is deleted. Especially if lagged variables

re used, this can be a problem. As an alternative, we can create an indicator variable that has the value of 1 for the outlier time period and is 0 otherwise. Using such a variable for the identified outlier in the New York Times series, fit a regression model based on a linear trend term, a quadratic trend term, monthly seasonal indicator variables, and the special indicator variable. Report the estimated regression model.

 (c) For the model in part (b), what is the estimated coefficient for the month of January indicator variable? What is the P-value for the variable? Compare these results with part (c) of Exercise 13.69. Do you notice any other differences between the two fitted models? What does the comparison tell you about the effect of the outlier on the regression fit?

(d) Make a time series plot of the residuals for the model in part (b). What can you conclude about the fitted model?

(e) Based on the model in part (b), forecast monthly revenue for November 2008. (f) In Exercise 13.52 (page 730), you were asked to compute the 12-month moving averages for the monthly revenue series. You were also asked if there were any shifts in the moving-average values. Explain any shifts you observed in light of the present exercise.

 

Q1507

 

 In Example 13.13 (page 715) an AR(1) model was fitted to the corporate philanthropy series. Figure 13.24 (page 716) showed that the observation associated with the year 2005 was an outlier due to the unusual events of that period.

 (a) Refer to the discussion of Exercise 13.70 and create a special indicator variable for the year 2005. Fit the time series using a lag one variable and the special indicator variable. Report the estimated model. (b) What is the effect of the outlier on the estimated lag one coefficient?

 (c) Using the model fitted in part (a), forecast 2008’s corporate philanthropy percent. Compare this forecast with the forecast provided in Example 13.14 (page 717).

 

Q1508

 

Consider a time series of quarterly sales (in millions of dollars) for Best Buy, Inc. starting with the first quarter of fiscal year 2000 and ending with the second quarter of fiscal year 2009.30 FILE BESTBUY

 (a) Make a time series plot of quarterly sales. Describe the nature of the series in terms of trend and seasonality. Explain why an additive seasonal model is probably not applicable.

 (b) Make a time series plot of sales in logged units. Describe how this plot differs from the plot in part (a).

(c) Fit the logged sales series with a linear trend term and quarterly indicator variables. Report the estimated model and make a forecast for third-quarter sales of fiscal year 2009 in the original units.

 

Q1509

 

Continue the analysis of Best Buy sales in the previous exercise.

(a)    Obtain the residuals from the fit on logged sales from Exercise 13.72. Make a time series plot of the residuals. Do the residuals show any pattern? If so, describe the patter

Make a lagged residual plot and calculate the correlation between et and et−1. Is there evidence of autocorrelation in the residuals?

 (c) Fit the logged sales series with a linear trend term, quarterly indicator variables, and a lag one variable of logged sales.

(d) Obtain the residuals from the fitted model of part (c). Make a time series plot and a lagged residual plot. What do these plots indicate about the adequacy of the fitted model?

 (e) Based on the fitted model of part (c), make a forecast for thirdquarter sales of fiscal year 2009 in original units. How did the forecast change relative to the forecast made in Exercise 13.72, part (c)?

 

Q1510

 

In the previous exercises, seasonality was directly modeled along with trend in a regression equation. Since the seasonality does not appear to be additive in nature, the trend-and-season modeling was done on sales in logged units. Here you will be asked to separate the estimation of trend and seasonality.

 (a) Fit a linear trend model to the sales series in original units. Report the estimated model. Why does a linear trend model seem to be a more reasonable choice for the series than an exponential trend model?

 (b) Using the fitted trend model of part (a), calculate the seasonality factors for each quarter. Adjust the factors so that they average out to 1.

 (c) Make a forecast for third-quarter sales of fiscal year 2009

 

Q1511

 

sales. Walgreens is the nation’s largest drugstore chain, with approximately 7000 drugstores as of 2008. Consider a time series of its quarterly sales (in millions of dollars) starting with the first quarter of fiscal year 2000 and ending with the first quarter of fiscal year 2009.31 WALGREENS

 (a) Make a time series plot of quarterly sales. Describe the nature of the series in terms of trend and seasonality. Explain why an additive seasonal model is probably not applicable.

(b) Fit a linear trend model to the sales series. Report the estimated model.

 (c) Fit a quadratic trend model (using t and t2) to the sales series. Report the estimated model.

(d) Fit an exponential trend model to the sales series. Report the estimated model.

 (e) Calculate MAD, MSE, and MAPE (see Exercises 13.46, 13.47, and 13.48, pages 724–725) for each of the three trend fits. Which trend model provides the best fit?

 

Q1512

 

continue the analysis of Walgreen’s sales from the previous exercise.

(a) Using the fitted quadratic trend model from part (c) of Exercise 13.75, calculate the seasonality factors for each quarter. Adjust the factors so that they average out to 1.

(b) Make a forecast for second-quarter sales of fiscal year 2009.

 

Q15`13

 

how much more or less are Americans taking to the air? Consider a time series of monthly total number of passenger miles (in thousands) on U.S. domesticlights starting with January 2003 and ending with September 2008.32 AIRTRAFFIC

(a) Make a time series plot of monthly miles. Does the trend appear linear or curved? Describe the trend in the series over times.

(b) Relative to the general trend movement, does the seasonal variation appear to be additive or multiplicative in nature? Justify your answer.

 (c) Fit the time series to a quadratic trend (using t and t2) and monthly indicator variables. Report the estimated model and forecast the total number of passenger miles to be flown in October 2008.

 

Q1514

 

lights starting with January 2003 and ending with September 2008.32 AIRTRAFFIC

a) Make a time series plot of monthly miles. Does the trend appear linear or curved? Describe the trend in the series over times.

 (b) Relative to the general trend movement, does the seasonal variation appear to be additive or multiplicative in nature? Justify your answer.

(c) Fit the time series to a quadratic trend (using t and t2) and monthly indicator variables. Report the estimated model and forecast the total number of passenger miles to be flown in October 2008.

 

Q1515

 

ic. Continue the analysis of monthly total number of passenger miles on U.S. domestic flights from the previous exercise.

 (a) Obtain the residuals from the fitted model of Exercise 13.78, part (c). Make a time series plot of the residuals. To aid your visual inspection, superimpose a horizontal line at 0 (mean of the residuals). Do the residuals show any pattern? If so, describe the pattern.

(b) Make a lagged residual plot of et versus et−1. Also, make a lagged residual plot of et versus et−2. Calculate the correlation between et and et−1 and the correlation between et and et−2. Is there evidence of autocorrelation in the residuals that is not accounted for by the lag one variable in the fitted model?

(c) Fit the passenger miles series to a quadratic trend, monthly indicator variables, a lag one variable of miles flown, and a lag two variable of miles flown. Do the P-values for each of the lag variables indicate that the variables should be included in the model?

 (d) Forecast the total number of passenger miles to be flown in October 2008. How does this forecast differ from the forecasts from Exercises 13.77 and 13.78?

 

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