53
where L is the number of lags, 9 (L) is the autocorrelation, Y-(L) is
the covariance between xt and xt+L, and Yx(0) is the variance of the
stochastic process under the assumption of stationarity. The covariance
of the series is given as
Vl> W Et(*t E
where t 0,1,2,...T. The variance is given as
Yx(0) COV(xt, xt+0) COV(xt, xt) =* VAR(xt)
Thus, 9x(L) is defined as the autocorrelation at lag L.
The very strict assumption of stationarity of a time series implies
that Yx(L) and Yx(0) are the same for all values of t. In fact, sta
tionarity implies that the joint and conditional probability functions
are invariant with respect to time. In particular, a stationary time
series will be characterized by -1 < 9X(L) < 1 for L > 0. In addition,
a time series characterized by
10, where L 0
1, L 0
is called a white noise process. A white noise process is not autocor-
related and, thus, exhibits no interdependency (the series is serially
uncorrelated). White noise is that part of a time series that cannot be
explained by its own past.
As Pindyck and Rubinfeld (1981) note, most time series encountered
in economic studies are not white noise processes and are non-station-
ary. However, these series can usually be differenced one or more times
to obtain stationarity. The number of differences taken, d, is known as
the order of homogeneity. A differenced series wt ig gven aa