This thesis investigates three topics in theoretical and applied econometrics:
two sample nonparametric estimation of intergenerational income mobility, sparse sieve maximum likelihood estimation, and asymptotic efficiency of Improved QMLE and Sieve MLE.

The first essay proposes a two sample nonparametric GMM estimator, which
extends the local linear GMM estimator to two sample settings, and applies it to
estimate the intergnerational income mobility in the U.S and Sweden. The second
essay proposes an estimator that uses the Dantzig Selector to improve the finite
sample performance of Sieve MLE in a panel data setting. We show that in
simulations the sparsity imposed by the Dantzig Selector is innocuous with
respect to the sieve MLE, and substantially improves its computational
efficiency. The third essay compares  an optimal GMM estimator, known as
Improved QMLE, with sieve MLE in a panel data setting. We derive a condition
when these two estimators are equally efficient asymptotically and provide
simulation results to illustrate the extent of efficiency loss.