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Q175Section III (Statistics)175 of 190

AR(1) with an intercept — the mean is not simply μ\mu

Let {Xt}\{X_t\} be a stationary AR(1) model with Xt=μ+ϕXt−1+ZtX_t=\mu+\phi X_{t-1} +Z_t, where Zt∼WN(0,σ2)Z_t\sim WN(0,\sigma^2). Then, which of the following statements is true? (i) E(Xt)=μE(X_t)=\mu. (ii) V(Xt)=σ2/(1−ϕ2)V(X_t)=\sigma^2/(1-\phi^2). (iii) Corr⁡(Xt,Xt+s)=ϕs\operatorname{Corr}(X_t,X_{t+s})=\phi^s. (iv) V(Xt)=1−ϕ2V(X_t)=1-\phi^2.

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