Conditional likelihood of an initialized Gaussian AR(2) process ($L\propto e^{-\frac{1}{2}{\sum }_t(x_{t+2}-a{x}_{t+1}-b{x}_t{)}^2}$)

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 $\exp [-\frac{1}{2}{\sum }_{t=0}^{n-2}(x_{t+2}-a{x}_{t+1}-b{x}_t{)}^2]$. The initial observation is a point mass and the first nonzero observation has a parameter-independent likelihood term.