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14.2 Introduction to Time Series Data and Serial Correlation 575

autocorrelation coefficient) is the correlation between Yt and Yt - 1—that is, the
correlation between values of Y at two adjacent dates. The second autocorrelation
is the correlation between Yt and Yt - 2, and the jth autocorrelation is the correla-
tion between Yt and Yt - j. Similarly, the jth autocovariance is the covariance
between Yt and Yt - j. Autocorrelation and autocovariance are summarized in Key
Concept 14.2.

The jth population autocovariances and autocorrelations in Key Concept 14.2
can be estimated by the jth sample autocovariances and autocorrelations,

cov(Yt, Yt - j) and rnj:

cov(Yt, Yt - j) = 1 T (Yt - Yj + 1: T)(Yt - j - Y1:T - j) (14.5)
T
t a 1

=j+

rn j = cov(Yt, Yt - j), (14.6)
var(Yt)

where Yj + 1: T denotes the sample average of Yt computed using the observations
t = j + 1, c, T and where var(Yt) is the sample variance of Y. 2

The first four sample autocorrelations of GDPGR, the growth rate of GDP,

are rn1 = 0.34, rn2 = 0.27, rn3 = 0.13, and rn4 = 0.14. These values suggest that
GDP growth rates are mildly positively autocorrelated; if GDP grows faster than

average in one period, it tends to also grow faster than average in the following period.

Other Examples of Economic Time Series

Economic time series differ greatly. Four examples of economic time series are
plotted in Figure 14.2: the U.S. unemployment rate; the rate of exchange between
the dollar and the British pound; the logarithm of an index of industrial produc-
tion in Japan; and the daily return on the Wilshire 5000 stock price index.

The U.S. unemployment rate (Figure 14.2a) is the fraction of the labor force
out of work, as measured in the Current Population Survey (see Appendix 3.1).
Figure 14.2a shows that the unemployment rate increases by large amounts
during recessions (the shaded areas in Figure 14.1) and falls during recoveries
and expansions.

2The summation in Equation (14.5) is divided by T, whereas in the usual formula for the sample
covariance [see Equation (3.24)], the summation is divided by the number of observations in the sum-
mation, minus a degrees-of-freedom adjustment. The formula in Equation (14.5) is conventional for
the purpose of computing the autocovariance. Equation (14.6) uses the assumption that var(Yt) and
var(Yt - j) are the same—an implication of the assumption that Y is stationary, which is discussed in
Section 14.4.