Are OLS or MLE estimators of autoregressive model asymptotically efficient if errors are i.i.d?
Consider the case of an AR(1) model $$x_t=\alpha x_{t-1} + \epsilon_t$$ with $\epsilon_t$ ~ $i.i.d. N(0,\sigma^2)$ and $abs(\alpha) < 1$. We know that the estimator for $\alpha$, $\hat{\alpha}$, is biased, although consistent (c.f. e.g. this StackExchange answer).
Whether or not we derive this estimator through OLS or conditional Maximum Likelihood, since $\hat{\alpha}$ is biased, we can appeal neither to Gauss-Markov nor the Cramer-Rao Lower Bound for efficiency.
Does OLS or maximum likelihood yield efficient estimates, or are other estimators like LASSO or Ridge more efficient? Or is the efficient estimator not yet worked out?
