Variance growth in a random walk

Question

A random walk is Xt = X{t-1} + \varepsilont with X0 = 0, where \varepsilont is white noise with variance \sigma^2 = 4. Find \operatorname{Var}(X{10}), and state whether the series is stationary and what first-differencing produces.

Accepted Answer

Unroll the recursion into a running sum of shocks: Xt = \varepsilon1 + \varepsilon2 + \dots + \varepsilont. The shocks are uncorrelated, so the variances simply add: \operatorname{Var}(Xt) = t\,\sigma^2. At t = 10 with \sigma^2 = 4: \operatorname{Var}(X{10}) = 10 	imes 4 = 40. Because this variance grows without bound as t increases, the random walk is not stationary. First-differencing fixes that: Xt - X{t-1} = \varepsilont, which is plain white noise, stationary with constant variance \sigma^2 = 4. A series that becomes stationary after one difference is called integrated of order 1, or I(…

Variance growth in a random walk

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