Chapter 9.3, Problem 16P

Expand Your Knowledge: Time Series and Serial Correlation An Internet advertising agency is studying the number of “hits” on a certain web site during an advertising campaign. It is hoped that as the campaign progresses, the number of hits on the web site will also increase in a predictable way from one day to the next. For 10 days of the campaign, the number of hits × 10⁵ is shown:

Original Time Series

1. (a) To construct a serial correlation, we use data pairs (x, y) where x = original data and y = original data shifted ahead by one time period. Verify that the data set (x, y) for serial correlation is shown here. (For discussion of serial correlation, see Problem 15.)

1. (b) For the (x, y) data set of part (a), compute the equation of the sample least-squares line $\hat{y}=a+bx$. If the number of hits was $9.3\left(\times {10}^5\right)$ one day, what do you predict for the number of hits the next day?
2. (c) Compute the sample correlation coefficient r and the coefficient of determination r². Test ρ > 0 at the 1 % level of significance. Would you say the time series of web site hits is relatively predictable from one day to the next? Explain.

   15. Expand Your Knowledge: Time Series and Serial Correlation Serial correlation, also known as autocorrelation, describes the extent to which the result in one period of a time series is related to the result in the next period. A time series with high serial correlation is said to be very predictable from one period to the next. If the serial correlation is low (or near zero), the time series is considered to be much less predictable. For more information about serial correlation, see the book Ibbotson SBBI published by Morningstar.

A research veterinarian at a major university has developed a new vaccine to protect horses from West Nile virus. An important question is: How predictable is the buildup of antibodies in the horse’s blood after the vaccination is given? A large random sample of horses from Wyoming were given the vaccination. The average antibody buildup factor (as determined from blood samples) was measured each week after the vaccination for 8 weeks. Results are shown in the following time series:

Original Time Series

To construct a serial correlation, we simply use data pairs (x, y) where x = original buildup factor data and y = original data shifted ahead by 1 week. This gives us the following data set. Since we are shifting 1 week ahead, we now have 7 data pairs (not 8).

Data for Serial Correlation

For convenience, we are given the following sums:

∑x = 48.6 ∑y = 58.5 ∑x² = 383.84 ∑y² = 529.37 ∑xy = 448.7

1. (a) Use the sums provided (or a calculator with least-squares regression) to compute the equation of the sample least-squares line, $\hat{y}=a+bx$. If the buildup factor was x = 5.8 one week, what would you predict the buildup factor to be the next week?
2. (b) Compute the sample correlation coefficient r and the coefficient of determination r². Test ρ > 0 at the 1% level of significance. Would you say the time series of antibody buildup factor is relatively predictable from one week to the next? Explain.