In: Statistics and Probability
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 company that produces and markets video games wants to estimate
the predictability of per capita consumer spending on video games
in a particular country. For the most recent 7 years, the amount of
annual spending per person per year is shown here.
Original Time Series
Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
$ per capita | 32.26 | 34.01 | 37.83 | 43.32 | 44.68 | 49.65 | 51.88 |
(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. Construct the data set
(x, y)
for serial correlation by filling in the following table.
x | 32.26 | 34.01 | 37.83 | 43.32 | 44.68 | 49.65 |
y |
(b) For the
(x, y)
data set of part (a), compute the equation of the sample least-squares line
ŷ = a + bx.
(Use 4 decimal places.)
a | |
b |
If the per capita spending was
x = $43
one year, what do you predict for the spending the next year? (Use 2 decimal places.)
$ per capita
(c) 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 |