SECTION 13.5 Exercises

For Exercises 13.46 to 13.49, see pages 693–694; for 13.50, see page 699; and for 13.51 and 13.52, see pages 705–706.

Question 13.53

13.53 It’s exponential.

Exponential smoothing models are so named because the weights

decrease in value exponentially. For this exercise, take . Use software to do the calculations.

  1. Calculate the weights for a smoothing constant of .
  2. Calculate the weights for a smoothing constant of .
  3. Calculate the weights for a smoothing constant of .
  4. Plot each set of weights from parts (a), (b), and (c). The weight values should be measured on the vertical axis, while the horizontal axis can simply be numbered 1, 2, …, 9, 10 for the 10 coefficients from each part. Be sure to use a different plotting symbol and/or color to distinguish the three sets of weights and connect the points for each set. Also, label the plot so that it is clear which curve corresponds to each value of used.
  5. Describe each curve in part (d). Which curve puts more weight on the most recent value of the time series when calculating a forecast?
  6. The weight of in the exponential smoothing model is . Calculate the weight of for each of the values of in parts (a), (b), and (c). How do these values compare to the first 10 weights you calculated for each value of ? Which value of puts the greatest weight on when calculating a forecast?

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(a)–(c)

0.1 0.5 0.9
1 0.1000 0.5000 0.9000
2 0.0900 0.2500 0.0900
3 0.0810 0.1250 0.0090
4 0.0729 0.0625 0.0009
5 0.0656 0.0313 9.E-05
6 0.0590 0.0156 9.E-06
7 0.0531 0.0078 9.E-07
8 0.0478 0.0039 9.E-08
9 0.0430 0.0020 9.E-09
10 0.0387 0.0010 9.E-10

(e) The higher values put more weight on the current observation, so the curve with . (f) 0.3487, 0.000977, 1E-10. puts the most weight on .

Question 13.54

13.54 It’s exponential.

In the previous exercise, you explored the behavior of the exponential smoothing model weights when the smoothing constant is between 0 and 1. We noted in the section that software can actually report an optimal smoothing constant greater than 1. Suppose software reports an optimal of 1.6.

  1. What is the value of the damping factor?
  2. Starting with the current observation and going back to , calculate the weights for each of these observations.
  3. Describe how the weights behave moving back in time.

Question 13.55

13.55 Number of iPhones sold globally.

Consider data on the quarterly global sales (in millions of dollars) of iPhones from the first quarter of 2012 to the third quarter of 2014. In Exercise 13.22 (page 677), you were asked to fit a trend-and-season model using regression.

iphone

  1. Use the moving-average approach to compute seasonal ratios and report their values.

    707

  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 iPhone sales for the fourth quarter of 2014 and for the first quarter of 2015.

13.55

(a) 1.310, 1.020, 0.811, 0.824. (b) The seasonally adjusted sales data is increasing over time. (c) . The trend term is significant, . (d) For the fourth quarter of 2014: 37.085. For the first quarter of 2015: 60.693.

Question 13.56

13.56 H&R Block quarterly tax services revenue.

H&R Block is the world’s largest consumer tax services provider. Consider a time series of its quarterly tax services revenues (in thousands of $) starting with the first quarter of fiscal year 2010 and ending on the second quarter of fiscal year 2014.31

hrblock

  1. Make a time plot of the revenue series. Describe any important features of the time series. Which fiscal quarter is associated with the April 15 tax season?
  2. Use the moving-average approach to compute seasonal ratios and report their values.
  3. Produce and plot the seasonally adjusted series. What are your impressions of this plot?
  4. Fit a trend model and report the -value of the trend coefficient? Is there enough evidence of the presence of trend?
  5. Given your conclusion of part (d), forecast tax services revenues for the third and fourth quarter of fiscal year 2014.

Question 13.57

13.57 Moving averages and linear trend.

The moving-average model provides reasonable predictions only under certain scenarios. Consider monthly seasonally adjusted Consumer Price Index (CPI) data, starting with January 1990 and ending in July 2014.32

cpi

  1. Make a time plot of the CPI series. Describe its movement over time.
  2. Using software, calculate moving average forecasts for spans of and 100. Superimpose the moving averages (on a single time plot) on a plot of the original time series.
  3. As the span increases, what do you observe about the plot of the moving averages?
  4. At the stock market analysis website stockcharts.com, it is stated that moving averages “are best suited for trend identification and trend following purposes, not for prediction.” Explain whether or not your results from part (a) are consistent with this claim.

13.57

(a) The CPI series is steadily increasing over time. (c) The plot gets smoother with a larger span. (d) The results are consistent with the quote. Both moving average predictions would grossly underestimate the CPI values. However, they do show the general pattern, or upward trend, of the CPI data.

Question 13.58

13.58 CTA commuters.

The Chicago rapid transit rail system is well known as the “L” (abbreviation for “elevated”). It is the third busiest system after the New York City Subway and the Washington Metro. Consider the daily count of commuters going through a particular station. The count is based on how many commuters went through all the turnstiles at the station. In particular, the data are for the downtown station of Randolph/ Wabash from April 7, 2014 (Monday), to May 11, 2014 (Sunday).33

cta

  1. Make a time plot of the commuter series. Describe any important features of the time series.
  2. Use the moving-average approach to compute seasonal ratios and report their values. Be careful to recognize that the number of seasons (days) is odd.
  3. Produce and plot the seasonally adjusted series. What are your impressions of this plot?
  4. Fit a trend model and report the -value of the trend coefficient? Is there enough evidence of the presence of trend?
  5. Given your conclusion of part (d), forecast the number of daily commuters going through the Randolph/ Wabash for each of the seven days of the week.

Question 13.59

13.59 Exponential smoothing for information services hires rate.

Consider the monthly information services sector hires rate time series from Exercise 13.46 (pages 693–694).

hires

  1. Calculate and plot (on a single time plot) exponential smoothing models using smoothing constants of , , and .
  2. Comment on the smoothness of each exponential smoothing model in part (a).
  3. The series ended with the hires rate of June 2014. For each model in part (a), calculate the forecasts for the hires rate of July 2014.

13.59

(b) The smaller the smoothing constant is, the smoother the model is. Or alternatively, a higher damping factor, , provides a smoother model. (c) 2.587, 2.874, 3.061.

Question 13.60

13.60 Exponential smoothing for information services hires rate.

Continue the analysis of monthly information services sector hires rate time series.

hires

  1. Use statistical software to determine the optimal smoothing constant .
  2. The series ended with the hires rate of June 2014. Based on the reported optimal , calculate the forecast for the hires rate of July 2014.

Question 13.61

13.61 Exponential smoothing forecast equation.

We have learned that the exponential smoothing forecast equation is written as

  1. Show that the equation can be written as

    where is the residual, or prediction error, for period .

  2. Explain in words how the reexpressed equation can be interpreted.

13.61

(a) . (b) The forecasted value is equal to the previous predicted value plus a percentage of the residual of the previous value.