Question

15) 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 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

Week	1	2	3	4	5	6	7	8
Buildup Factor	2.3	4.6	6.2	7.5	8.0	9.4	10.6	12.1

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

x	2.3	4.6	6.2	7.5	8.0	9.4	10.6
y	4.6	6.2	7.5	8.0	9.4	10.6	12.1

For convenience, we are given the following sums.

Σx = 48.6,

Σy = 58.4,

Σx² = 385.86,

Σy² = 526.98,

Σxy = 448.7

(a) Use the sums provided (or a calculator with least-squares regression) to compute the equation of the sample least-squares line,

ŷ = a + bx.

(Use 4 decimal places.)

a
b

If the buildup factor was

x = 5.4

one week, what would you predict the buildup factor to be the next week? (Use 2 decimal places.)


(b) Compute the sample correlation coefficient r and the coefficient of determination

r².

(Use 4 decimal places.)

r
r2

Test

ρ > 0

at the 1% level of significance. (Use 2 decimal places.)

t
critical t

Answer 1

(a)

the equation of the sample least-squares line,

= a + bx

b = 0.8926

the equation of the sample least-squares line,

= 2.1457 + 0.8926x

If the buildup factor was  x = 5.4 one week, predict the buildup factor to be the next week

predicted buildup factor to be the next week : = 2.1457 +0.8926 x 5.4 = 6.96574 6.97

(b)

correlation coefficient r

r = 0.9853

coefficient of determination r² = 0.9853² = 0.9708

Null hypothesis : H_o : =0

Alternate Hypothesis : H_a: > 0

Right tailed test:

Test Statistic :

Degrees of freedom = 8-2 =6

For right tailed test : Critical value of t at 1% level of significance (=0.01) for 6 degrees of freedom = 3.143

As Value of the test statistic : 14.128 > Critical value of t : 3.143; Reject the null hypothesis.

There is sufficient evidence to conclude that > 0