standard deviation increases linearly with ). In this case, unlike all the
previous cases, the variance increases monotonically with time; that is, the
values are capable of moving to extremes with the passage of time. Also, the
statistical parameters like the unconditional mean and variance are not time
invariant, or stationary. The series is therefore called a nonstationarytime
series.
The correlation between a value and its immediate lagging value is 1.
Our prediction for the next time step would then be a value with mean
equal to the current time step; that is,. The variance, of course,
is the variance of the white noise realizations. As a matter of fact, our pre-
diction for any number of time steps would be a distribution whose mean is
the current value of the series. However, because the variance increases lin-
early with time, the error in our prediction progressively increases with the
number of time steps.
Of the different time series reviewed so far, the random walk is the only
series in which the prediction of the mean value for the next time step is the
current value. Such series where the expected value at the next time step is
the value at the current time step are known as martingales. The random
walk qualifies as a martingale.
The random walk also exhibits a strong trending behavior. Let us ex-
amine that statement by contrasting the behavior of the random walk with
yyttpred= − 1tTime Series 23
FIGURE 2.4 Random Walk Series.10 30 50 70 90–5
0510
15