SECTION 13.2 Exercises

For Exercises 13.9 and 13.10, see pages 660–661; for 13.11, see page 663.

Question 13.12

13.12 Consumer sentiment index.

Each month, the University of Michigan and Thomson Reuters conduct a survey of consumer attitudes concerning both the present situation as well as expectations regarding economic conditions. The results of the survey are used to construct a Consumer Sentiment Index. The index has been normalized to have a value of 100 in December 1964. Consider the monthly values for the index from January 2000 to January 2014.7

umcsent

664

  1. Test the randomness of the index series. What do you conclude?
  2. Obtain the first differences for the series and test them for randomness. What do you conclude?
  3. Would you conclude that the consumer index series behaves as a random walk? Explain.
  4. Make a forecast for the value of the index for February 2014.

Question 13.13

13.13 Gold prices.

Consider the monthly data on the price of gold ($ per troy ounce) from January 2000 to July 2014.8

gprice

  1. Test the randomness of the index series. What do you conclude?
  2. Obtain the first differences for the series and plot these differences over time. What do you observe?
  3. Take the log of the gold prices and then take their first differences. What do these differences approximate?
  4. Plot the differences of the logged prices over time. Describe what you observe with this plot in comparison to the plot from part (b).

13.13

(a) The Runs Test and the ACF show the price of gold series is not random. (b) The variation of the price changes for gold is increasing with time. (c) These differences approximate the percent change in the price of gold. (d) The first differences of the log price data show much more constant variance than the original differences.

Question 13.14

13.14 Gold prices.

Continue the previous exercise.

gprice

  1. Using the Normal distribution, provide an approximate 95% prediction interval for future monthly returns on gold prices.
  2. Use the prediction interval on monthly returns found in part (a) to obtain an approximate 95% prediction interval for the price of gold for August 2014.

Question 13.15

13.15 U.S.-Canadian exchange rates.

Consider the daily U.S.-Canadian exchange rates (Canadian dollars to one U.S. dollar) from the beginning of 2013 through the first week of August (bank holidays excluded).9

canrate

  1. Test the randomness of the exchange rate series. What do you conclude?
  2. Obtain the first differences for the series and test them for randomness. What do you conclude?
  3. Would you conclude that the exchange rate series behaves as a random walk? Explain.

13.15

(a) The Runs Test and the ACF show the exchange rate is not random. (b) For the first differences of rate, the Runs Test and the ACF show they are random. (c) Yes, because the exchange rates are not random but their first differences are random.

Question 13.16

13.16 Honda returns.

Consider the approximated returns plot in Figure 13.18 from Example 13.11. image

  1. Obtain a histogram and Normal quantile plot of the returns data. What do you conclude about the distribution of the returns?
  2. In the previous chapter on control charts, limits were placed plus and minus three standard deviations around the sample mean to identify unusual observations. What are the plus and minus three standard deviation limits for the return data?
  3. There are returns in the series shown in Figure 13.18. If the data were consistent with the Normal distribution, how many returns would we expect to fall outside the limits computed in part (b)? Is the actual number of returns falling outside of the limits close to this expected number?
  4. Eyeballing Figure 13.18, where would you roughly place the limits so that the expected number of observations falling outside of the limits is close to the expected number found in part (c)?

honda2

Question 13.17

13.17 U.S.-Canadian exchange rates.

Refer to Exercise 13.15.

canrate

  1. Obtain a histogram and Normal quantile plot of the daily changes in exchange rates. What do you conclude?
  2. Test the daily changes against the null hypothesis that the underlying mean change is 0. What do you conclude? What is the implication of your conclusion in terms of the exchange rate series having drift or not?
  3. Given your conclusion in part (b), what would be your forecast in general for the next day’s exchange rate?
  4. If today’s exchange rate is equal to 1, what would be the 95% prediction interval for tomorrow’s exchange rate?

13.17

(a) The histogram and Normal quantile plot show a Normal distribution with one high potential outlier. (b) . The underlying mean is not significantly different from 0. We have no evidence of a drift in the exchange rate series. (c) The best forecast is a naive forecast, or today’s rate. (d) (0.9927, 1.0078).