In this dissertation we proposed two generalizations of the Second-Order Least Squares (SLS) approach in two popular dynamic econometrics models. The first one is the regression model with time varying nonlinear mean function and autoregressive conditionally heteroskedastic (ARCH) disturbances. The second one is a linear dynamic panel data model. We used a semiparametric framework in both models where the SLS approach is based only on the first two conditional moments of response variable given the explanatory variables. There is no need to specify the distribution of the error components in both models. For the ARCH model under the assumption of strong-mixing process with finite moments of some order, we established the strong consistency and asymptotic normality of the SLS estimator. It is shown that the optimal SLS estimator, which makes use of the additional information inherent in the conditional skewness and kurtosis of the process, is superior to the commonly used quasi-MLE, and the efficiency gain is significant when the underlying distribution is asymmetric. Moreover, our large scale simulation studies showed that the optimal SLSE behaves better than the corresponding estimating function estimator in finite sample situation. The practical usefulness of the optimal SLSE was tested by an empirical example on the U.K. Inflation. For the linear dynamic panel data model, we showed that the SLS estimator is consistent and asymptotically normal for large N and finite T under fairly general regularity conditions. Moreover, we showed that the optimal SLS estimator reaches a semiparametric efficiency bound. A specification test was developed for the first time to be used whenever the SLS is applied to real data. Our Monte Carlo simulations showed that the optimal SLS estimator performs satisfactorily in finite sample situations compared to the first-differenced GMM and the random effects pseudo ML estimators. The results apply under stationary/nonstationary process and wih/out exogenous regressors. The performance of the optimal SLS is robust under near-unit root case. Finally, the practical usefulness of the optimal SLSE was examined by an empirical study on the U.S. airfares.