CHAPTER 13 Review Exercises

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Question 13.62

13.62 Just use last month’s figures!

Working with the financial analysts at your company, you discover that, when it comes to forecasting various time series, they often just use last period’s value as the forecast for the current period. As noted in the chapter, this is known as a naive forecast (page 660).

  1. If you could pick the estimates of and in the AR(1) model, could you pick values such that the AR(1) forecast equation would provide the same forecasts your company’s analysts use? If so, specify the values that accomplish this.
  2. What span in a moving-average forecast model would provide the same forecasts your company’s analysts use?
  3. What smoothing constant in a simple exponential smoothing model would provide the same forecasts your company’s analysts use?
  4. Under what circumstances is the naive forecast that your company’s analysts are using the most appropriate option? Explain your response.

Question 13.63

13.63 Egg shipments.

The U.S. Department of Agriculture tracks prices, sales, and movement of numerous food commodities. Consider the weekly number of eggs shipped in the Chicago retail market for the 52 weeks of 2012. Units are 30 dozen eggs in thousands.34

eggs

  1. Make a time plot of the egg shipment series.
  2. If software has the capability, produce an ACF for the series. If an ACF is not available with software, calculate the correlations between and and between and . Test these correlations against the null hypothesis that the underlying correlation .
  3. Based on parts (a) and (b), what do you conclude about the egg shipment process?

13.63

(a) The time plot of the egg shipment series looks random. (b) The ACF shows the egg shipment series is random. (c) The egg shipment series is random.

Question 13.64

13.64 Egg shipments.

Continue the analysis of the weekly egg shipments to Chicago.

eggs

  1. Make a histogram and Normal quantile plot of the egg data. What do you conclude from these plots?
  2. What is a 95% prediction interval for the weekly egg shipments?

Question 13.65

13.65 Annual precipitation.

Global temperatures are increasing. Great Lakes water levels meander up and down (see Figure 13.41, page 687). Do all environmental processes exhibit time series patterns? Consider a time series of the annual precipitation (inches) in New Jersey from 1895 through 2013.35

precip

  1. Make a time plot of the precipitation series.
  2. If software has the capability, produce an ACF for the series. If an ACF is not available with software, calculate the correlations between and and between and . Test these correlations against the null hypothesis that the underlying correlation .
  3. Based on parts (a) and (b), what do you conclude about the precipitation process?

13.65

(a) Precipitation over time looks random. (b) The ACF shows the annual precipitation series is random. (c) The annual precipitation series is random.

Question 13.66

13.66 Annual precipitation.

Continue the analysis of the annual precipitation time series.

precip

  1. Make a histogram and Normal quantile plot of the precipitation data. What do you conclude from these plots?
  2. What is a 90% prediction interval for the annual precipitation for 2014?

Question 13.67

13.67 NFL offense.

In the National Football League (NFL), many argue that rules changes over the years are favoring offenses. Consider a time series of the average number of offensive yards in the NFL per regular season from 1990 through 2013.36

nfloff

  1. Make a time plot. Is there evidence that the average number of offensive yards per game is trending in one direction? Describe the general movement of the series.
  2. Fit a trend model to the data and report the estimated model. What is the interpretation of the coefficient of the trend term in the context of this application?
  3. Based on the trend fit, forecast the average number of offensive yards per game for the 2014, 2015, and 2016 seasons.

13.67

(a) The offensive yards are increasing over time. (b) . The trend term means that the number of offensive yards increases by 1.54596 each year. (c) For 2014: 342.3953. For 2015: 343.9412. For 2016: 345.4872.

Question 13.68

13.68 Mexican population density.

Consider a time series on the annual population density (number of people per square kilometer) in Mexico from 2001 through 2013.37

mexico

  1. Make a time plot. Describe the movement of the data over time.
  2. Fit a linear trend model to the data series and report the estimated model.
  3. Obtain the residuals for the trend model fit and calculate MAD, MSE, and MAPE (see Exercises 13.47, 13.48, and 13.49, page 694).
  4. Make a time plot of the residuals. Are the residuals suggesting any concerns about the linear trend model? Explain.

Question 13.69

13.69 Mexican population density.

Continue the analysis of the annual population density in Mexico.

mexico

  1. Fit a trend model based on a linear term and a quadratic term . Report the estimated model.

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  2. Obtain the residuals for the quadratic trend model fit and calculate MAD, MSE, and MAPE (see Exercises 13.47, 13.48, and 13.49, page 694). How do these measures compare with the linear trend fit of the previous exercise?
  3. Forecast the population density of Mexico for year 2014.

13.69

(a) . (b) . All three measures are much smaller for the quadratic model than in the linear model. (c) 63.726.

Question 13.70

13.70 Facebook annual net income.

Consider a time series on the annual net income of Facebook (in millions of dollars) from 2007 through 20 1 3.38

fb

  1. Using Excel’s line plot option, make a time plot. Describe the movement of the data over time. Would a linear trend model be appropriate? Explain.
  2. Right click on any data point in the Excel plot and select the Add Trendline option. Fit the data to a quadratic model (that is, polynomial of order 2) and report the estimated model.
  3. Now use Excel to fit an exponential trend model. Report the estimated model.
  4. Based on Excel’s superimposed fits, which model visually appears to be a better fit? Explain.
  5. Obtain the residuals for each of the fitted models and calculate MAD, MSE, and MAPE (see Exercises 13.47, 13.48, and 13.49, page 694). Which model has better fit measures? Explain.

Question 13.71

13.71 U.S. poverty rate.

Consider a time series on the annual poverty rate of U.S. residents aged 18 to 64 from 1980 through 2012.39

poverty

  1. Make a time plot. Describe the movement of the data over time.
  2. Obtain the first differences for the series and test them for randomness. What do you conclude?
  3. Would you conclude that the poverty rate series behaves as a random walk? Explain.

13.71

(a) The data are not linear; they are somewhat seasonal through the first 20 observations then deviate afterward. (b) For the first differences of poverty, the Runs Test and the ACF show they are not random. (c) The first differences in a random walk are random; because the first differences for poverty are not random, it is unlikely the poverty series behaves like a random walk.

Question 13.72

13.72 U.S. poverty rate, continued.

Continue the analysis of the annual U.S. poverty rate.

poverty

  1. Set up an Excel spreadsheet to calculate forecasts for the time series using an exponential smoothing model with . Provide a forecast for the poverty rate for 2013.
  2. Set up an Excel spreadsheet to calculate forecasts for the time series using an exponential smoothing model with . Provide forecasts for the poverty rate in 2013 and 2014.

Question 13.73

13.73 U.S. poverty rate, continued.

Refer to Exercises 13.47, 13.48, and 13.49 (page 694) for explanation of the forecast accuracy measures MAD, MSE, and MAPE.

poverty

  1. Based on the forecasts calculated in part (a) of the previous exercise, calculate MAD, MSE, and MAPE.
  2. Based on the forecasts calculated in part (b) of the previous exercise, calculate MAD, MSE, and MAPE.

13.73

(a) . (b) .

Question 13.74

13.74 U.S. poverty rate, continued.

Continue the analysis of the annual U.S. poverty rate.

poverty

  1. Produce a PACF for the series. If a lag only based model were to be fit, how many lags does the PACF suggest?
  2. Based on the PACF from part (a), fit a lag only based model. Report the estimated model.
  3. Test the randomness of the residuals. What do you find? What is the implication on the fitted model?
  4. Based on the fitted model from part (b), provide a forecast for the poverty rate for 2013.

Question 13.75

13.75 Exponential smoothing for unemployment rate.

Consider the annual unemployment rate time series from Exercise 13.38 (page 690).

unempl

  1. Use statistical software to determine the optimal smoothing constant . Does this optimal fall in the traditional range for ? Explain.
  2. The series ended with the 2013 unemployment rate. Based on the reported optimal , calculate the forecasts for 2014 unemployment rate.

13.75

(a) . No, Normally should be between 0 and 1. (b) 7.3057.

Question 13.76

13.76 U.S. air carrier traffic.

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. domestic flights starting with January 2009 and ending with May 2014.40

airtrav

  1. Make a time plot of monthly miles. Describe the behavior of the series. Is there a trend? What months are consistently high versus low?
  2. Does the seasonal variation appear to be additive or multiplicative in nature? Justify your answer.
  3. Fit the time series to a trend and monthly indicator variables. Report the estimated model.
  4. Check the residuals from the trend-seasonal model for randomness. What do you conclude?
  5. Forecast the total number of passenger miles to be flown in June 2014.

Question 13.77

13.77 U.S. air carrier traffic.

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

airtrav

  1. Use the moving-average approach to compute seasonal ratios and report their values.
  2. Produce and plot the seasonally adjusted series. What are your impressions of this plot?
  3. Fit a trend model and report the -value of the trend coefficient. Is there enough evidence of the presence of trend?
  4. Given your conclusion of part (c), forecast total number of passenger miles to be flown in June 2014.

13.77

(a) 0.888, 0.833, 1.050, 0.995, 1.038, 1.098, 1.152, 1.108, 0.931, 0.995, 0.937, 0.966. (b) The seasonally adjusted series increases over time and is quite stable. (c) . The trend term is significant, . (d) For June 2014: 54,037,215.

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Question 13.78

13.78 Monthly warehouse club and superstore sales.

Consider the monthly warehouse club and superstore sales series discussed in Examples 13.15 and 13.16 (pages 671–674).

club

  1. Use the moving-average approach to compute seasonal ratios and report their values.
  2. Produce and plot the seasonally adjusted series. What are your impressions of this plot?
  3. Fit a regression model based on to the seasonally adjusted series. Report the -values for the linear trend term and the quadratic trend term. Are these terms significant?
  4. Provide a forecast for sales for June 2014.

Question 13.79

13.79 Daily trading volume of FedEx.

Refer to Example 13.22 (page 686) in which a prediction of logged trading volume of FedEx stock for period 150 was made.

  1. Exponentiate the log prediction value back to original units. What is the interpretation of the estimate?
  2. Refer now to Example 13.18 (pages 675–676) and make a similar adjustment to the untransformed value of part (a). (Note: You will need to refer to the regression output given in Figure 13.38, page 684.) What is the interpretation of the adjusted estimate?

13.79

(a) 1,641,137.24. (b) 1,721,060.624.