 Solution

Use the chain rule for conditional densities. With $x_0=x_1=0$, the likelihood function of the nondegenerate observations isThe first residual is $x_2$ and carries no parameter information. There is no ordinary joint Lebesgue density for $X_1,\dots ,X_n$ because $X_1=0$ deterministically; this expression is the conditional likelihood function given the fixed initial values, or the likelihood function on $x_2,\dots ,x_n$. The initial point mass is parameter independent and has no effect on inference. This is the conditional likelihood of an initialized Gaussian AR(2) process.