Question: Consider the random walk with drift model xt = + xt1 + wt, for t = 1, 2, . . . , with x0

Consider the random walk with drift model xt = δ + xt−1 + wt, for t = 1, 2, . . . , with x0 = 0, where wt is white noise with variance σ2 w.

(a) Show that the model can be written as xt = δt + Pt k=1 wk.

(b) Find the mean function and the autocovariance function of xt.

(c) Argue that xt is not stationary.

(d) Show ρx(t − 1, t) = qt−1 t → 1 as t → ∞. What is the implication of this result?

(e) Suggest a transformation to make the series stationary, and prove that the transformed series is stationary. (Hint: See Problem 1.6b.)

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