process consisting of a linear trend

2.6

w

Consider a process consisting of a linear trend with an additive noise term consisting of independent random variables wt with zero means and variances σ2 , that is,

xt = β0 + β1t + wt,

where β0, β1 are fixed constants.

(a) Prove xt is nonstationary.

(b) Prove that the first difference series ∇xt = xt − xt−1 is stationary by finding its mean and autocovariance function.

(c) Repeat part (b) if wt is replaced by a general stationary process, say yt, with mean function µy and autocovariance function γy (h).

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