This thesis investigates three topics in theoretical econometrics: goodness-of-fit tests for copulas,
copula density estimators which preserve the copula property, and bias-correction for the naive
kernel local linear estimators in the two-sample varying coefficient model with missing data.
In the first topic a family of goodness-of-fit tests for copulas is proposed. The tests use generalizations
of the information matrix equality of White (1982). The asymptotic distribution of the
generalized tests is derived. In Monte Carlo simulations, the behavior of the new tests is compared
with several Cramer-von Mises type tests and the desired properties of the new tests are confirmed
in high dimensions. In the second topic, a semi-parametric copula density estimation procedure
that guarantees that the estimator is a genuine copula density is outlined. A simulation-based study
is constructed to examine the performance of the proposed copula density estimation method and
compare it with the leading copula density estimators in the literature. The method is also applied to
estimate copula densities in two empirical cases. The third topic shows that the naive kernel estimator
using matching data is not consistent in the two-sample varying coefficient model with missing
data. A bias-corrected consistent estimator is proposed and the asymptotic theory is discussed. A
simulation study is conducted to support the theoretical results.