Stationary Gaussian AR1 likelihood ($L(\mu ,\varphi ,{\sigma }^2)$)

For $|\varphi |<1$, independent $N(0,{\sigma }^2)$ innovations and stationary initial variance ${\sigma }^2/(1-{\varphi }^2)$, the full likelihood function isConditioning on $X_1$ gives a different likelihood.

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.