Consider a causal AR(1) process $X_t = \phi X_{t−1} + Z_t$ with $(Z_t)$ iid with mean 0 and finite variance. I am reading in a book, that $\phi X_n$ is the best predictor for $X_{n+1}$ because it minimizes $E[(X_{n+1}-f(X_1,\ldots,X_n))^2]$.

But then it says that if $\phi$ is unknown, then $\hat{\phi} X_n$ (with $\hat{\phi}$ being the Yule-Walker estimate) is not the best predictor for $X_{n+1}$ in the same sense. I can't really see why this is, could someone explain?


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