A note on mean-squared prediction errors of the least squares predictors in random walk models (Q2784956)

An AR(1) model \(x_t=\beta x_{t-1}+\varepsilon_t\) is considered, where \(\varepsilon_t\) are i.i.d. with \(E\varepsilon_t=0\) and \(E\varepsilon^2_t=\sigma^2\). Let \(\hat\beta_n\) be the least squares estimator for \(\beta\) by \(x_i\), \(i=1,\dots,n\), \(\hat x_{n+1}=\hat\beta_n x_n\) is the predictor of \(x_{n+1}\). The author derives an asymptotic expansion NEWLINE\[NEWLINEE(x_{n+1}-\hat x_{n+1})^2 =\sigma^2+c/n +o(n^{-1})NEWLINE\]NEWLINE with \(c=2\sigma^2\) in the case \(\beta=1\) (the random walk case). It is shown that in this case the use of \(\hat\beta_n\) estimated by the same realization yields better prediction than an independent realization \(\hat\beta_n.\)