= Conditional and stationary AR1 likelihood estimators
{title2=$\widehat\phi_{\rm cond}=\frac{\sum_{t=2}^TX_tX_{t-1}}{\sum_{t=2}^TX_{t-1}^2}$}
Conditioning on the first observation of a centered <autoregressive process of order one> makes the <maximum-likelihood estimator> of its coefficient the least-squares ratio above, when the optimum is interior. The stationary initial <normal distribution> contributes an additional factor $\sqrt{1-\phi^2}$ and quadratic term $(1-\phi^2)X_1^2$. These change the exact maximizer, but for a fixed interior stationary parameter their score is order one while the conditional score has order $T$, giving the same leading estimator.
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