= Solution
Use the chain rule for conditional densities. With $x_0=x_1=0$, the <likelihood function> of the nondegenerate observations is
$$
\boxed{L(a,b;x_1,\ldots,x_n)=(2\pi)^{-(n-1)/2}\exp\left[-\frac12\sum_{t=0}^{n-2}(x_{t+2}-ax_{t+1}-bx_t)^2\right].}
$$
The first residual is $x_2$ and carries no parameter information. There is no ordinary joint Lebesgue density for $X_1,\ldots,X_n$ because $X_1=0$ deterministically; this expression is the conditional <likelihood function> given the fixed initial values, or the <likelihood function> on $x_2,\ldots,x_n$. 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>.
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