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as a confirmation that we have eliminated any remaining serial correlation from the residuals
and thus have a good model fit.
Now that we have examined DWN we are going to move on to a famous model for some
financial time series, which is known as the random walk model.
9.4 Random Walk
A random walk is a time series model where the current observation is equal to the previous
observation with a random step up or down. It is formally defined below:
Definition 9.4.1. Random Walk. A random walk is a time series model x t such that x t =
x t−1 + w t , where w t is a discrete white noise series.
Recall above that we defined the backward shift operator B. We can apply the BSO to the
random walk:
x t = Bx t + w t = x t−1 + w t (9.2)
And stepping back further:
x t−1 = Bx t−1 + w t−1 = x t−2 + w t−1 (9.3)
If we repeat this process until the end of the time series we get:
x t = (1 + B + B 2 + . . .)w t =⇒ x t = w t + w t−1 + w t−2 + . . . (9.4)
Hence it is clear to see how the random walk is simply the sum of the elements from a discrete
white noise series.
9.4.1 Second-Order Properties
The second-order properties of a random walk are a little more interesting than that of discrete
white noise. While the mean of a random walk is still zero, the covariance is actually timedependent.
Hence a random walk is non-stationary:
µ x = 0 (9.5)
γ k (t) = Cov(x t , x t+k ) = tσ 2 (9.6)
In particular, the covariance is equal to the variance multiplied by the time. Hence, as time
increases, so does the variance.
What does this mean for random walks? Put simply, it means there is very little point in
extrapolating "trends" in them over the long term, as they are literally random walks.