This paper proposes a procedure for the estimation of discrete Markov decision models and studies its statistical and computational properties. Our Nested Pseudo-Likelihood method (NPL) is similar to Rust's Nested Fixed Point algorithm (NFXP), but the order of the two nested algorithms is swapped. First, we prove that NPL produces the Maximum Likelihood Estimator under the same conditions as NFXP. Our procedure requires fewer policy iterations at the expense of more likelihood-climbing iterations. We focus on a class of infinite-horizon, partial likelihood problems for which NPL results in large computational gains. Second, based on this algorithm we define a class of consistent and asymptotically equivalent Sequential Policy Iteration (PI) estimators, which encompasses both Hotz-Miller's CCP estimator and the partial Maximum Likekihood estimator. This presents the researcher with a ''menu'' of sequential estimators reflecting a trade-off between finite-sample precision and computational cost. Using actual and simulated data we compare the relative performance of these estimators. In all our experiments the benefits in terms of precision of using a 2-stage PI estimator instead of 1-stage (i.e., Hotz-Miller) are very significant. More interestingly, the benefits of MLE relative to 2-stage PI are small.