The first difference of a random walk

Question

A random walk is Xt = X{t-1} + \varepsilont, where \varepsilont is white noise with variance \sigma^2 = 4. Define the differenced series Yt = Xt - X{t-1}. What is \operatorname{Var}(Yt), and is \{Yt\} stationary?

Accepted Answer

Differencing collapses the recursion directly: Yt = Xt - X{t-1} = \varepsilont. So the differenced series is nothing but the white-noise shocks themselves. Its variance is \operatorname{Var}(Yt) = \operatorname{Var}(\varepsilont) = \sigma^2 = 4, constant for every t. Its mean is 0 and successive values are uncorrelated, so \{Yt\} is stationary. This is the whole idea behind differencing nonstationary series: the random walk \{Xt\} has variance that grows as t\sigma^2, but one difference removes the accumulation and returns you to flat, stationary white noise. Takeaway. The first difference of…

The first difference of a random walk

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