To derive the log-likelihood function of an MA(1) we condition on [TEX] \epsilon_0 [/TEX].
But when deriving the log-likelihood function function for an AR(1) we don't.
Why not? Is it just because [TEX] \epsilon_0 [/TEX] isn't found in the AR(1) equation?
That's right. In the AR(1) setting the log-likelihood function can be written completely in terms of the observable time series X_t and the parameters of the model. This is not the case for MA(1). There we need to know the initial shock Epsilon_0 to write the likelihood-terms for subsequent X_t.