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Question

Problem 1. Consider a random walk with drift model, xt = δt + ∑ti=1 wi, where wt is white noise as usual. You have already shown that this model is not stationary. (a) In class, we showed that the autocovariance function is γx(s, t) = min(s, t)σw. Use this to show that the correlation between xt−1 and xt is ρx(t−1,t) = very large? Explain why this is to be expected. (b) Let xt be the random walk model from above. Consider the time series yt formed by taking first differences of the random walk model, yt = xt − xt−1. Show that yt is stationary.

          Problem 1. Consider a random walk with drift model, xt = δt + ∑ti=1 wi, where wt is white noise as usual.

You have already shown that this model is not stationary.

(a) In class, we showed that the autocovariance function is γx(s, t) = min(s, t)σw. Use this to show that the correlation between xt−1 and xt is ρx(t−1,t) = very large? Explain why this is to be expected.

(b) Let xt be the random walk model from above. Consider the time series yt formed by taking first differences of the random walk model, yt = xt − xt−1. Show that yt is stationary.

Added by Jeffery G.

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