Conditional and stationary AR1 likelihood estimators (${\hat{\varphi }}_{\mathrm{cond}}=\frac{{\sum }_{t=2}^T{X}_t{X}_{t-1}}{{\sum }_{t=2}^T{X}_{t-1}^2}$)

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 $\sqrt{1-{\varphi }^2}$ and quadratic term $(1-{\varphi }^2)X_1^2$. 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.