Conditional likelihood of an initialized Gaussian AR(2) process
= Conditional likelihood of an initialized Gaussian AR(2) process
{title2=$L\propto e^{-\frac12\sum_t(x_{t+2}-ax_{t+1}-bx_t)^2}$}
With two fixed initial states and iid N(0,1) errors, the parameter likelihood is the product of conditional transition densities. For $X_0=X_1=0$, it is proportional to $\exp[-\tfrac12\sum_{t=0}^{n-2}(x_{t+2}-ax_{t+1}-bx_t)^2]$. The initial observation is a point mass and the first nonzero observation has a parameter-independent likelihood term.