We study the optimal choice of quasi-likelihoods for nearly integrated,possibly non-normal, autoregressive models. It turns out that the two mostnatural candidate criteria, minimum Mean Squared Error (MSE) and maximumpower against the unit root null, give rise to different optimalquasi-likelihoods. In both cases, the functional specification of theoptimal quasi-likelihood is the same: it is a combination of the truelikelihood and the Gaussian quasi-likelihood. The optimal relativeweights, however, depend on the criterion chosen and are markedlydifferent. Throughout, we base our results on exact limiting distributiontheory. We derive a new explicit expression for the joint density of theminimal sufficient functionals of Ornstein-Uhlenbeck processes, which alsohas applications in other fields, and we characterize its behaviour forextreme values of its arguments. Using these results, we derive theasymptotic power functions of statistics which converge weakly tocombinations of these sufficient functionals. Finally, we evaluatenumerically our computationally-efficient formulae.