Stationary Gaussian AR1 likelihood
= Stationary Gaussian AR1 likelihood
{title2=$L(\mu,\phi,\sigma^2)$}
For $|\phi|<1$, independent $N(0,\sigma^2)$ innovations and stationary initial variance $\sigma^2/(1-\phi^2)$, the full <likelihood function> is
$$
L=(2\pi\sigma^2)^{-n/2}\sqrt{1-\phi^2}\exp\left[-\frac{(1-\phi^2)(x_1-\mu)^2+\sum_{t=2}^n\{(x_t-\mu)-\phi(x_{t-1}-\mu)\}^2}{2\sigma^2}\right].
$$
Conditioning on $X_1$ gives a different likelihood.