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Obviously if $X_t = \phi X_{t-1} + Z_t$, then the best linear predictor of $X_t$ given $X_{t-1}$ is $X_t = \phi X_{t-1}$.

But if $\phi$ is unknown, one may attempt to substitute $\phi$ by a Yule-Walker/MLE estimator which equals the sample autocorrelation at lag 1: $\hat{\phi} = p(1)$.

What is the mean squared error of $$E (X_t - \hat{\phi}X_{t-1})^2$$? If it is not known exactly, can we relate it to $E(X_t - \phi X_{t-1})^2$?

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  • $\begingroup$ Hi: usually, the first expectation is not attempted to be calculated because the variance of $\hat{\phi}$ is a function of the sample size. ( and also there are only numerical ways of estimating it since the likelihood of AR(1) has no closed form). So, the second expectation is the one used in general and the sampling error of the estimate is assumed negligible for large enough sample sizes. The second expectation is just the conditional variance one step ahead so it's the variance of $Z_{t}$ which is usually denoted $\sigma^2_{z_{t}}$. $\endgroup$ – mlofton Oct 7 '18 at 21:46
  • $\begingroup$ Note that above assumes that expectation of $Z_{t} = 0$. $\endgroup$ – mlofton Oct 7 '18 at 21:49

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