This dissertation consists of two essays on maximum likelihood estimation of Dynamic Stochastic General Equilibrium (DSGE) models. The first essay focuses on a monetary DSGE model of term structure, while the second essay explores and compares three different versions of New Keynesian DSGE models. In Chapter 1, a general background is given for the DSGE models, and their estimation techniques along with a review of the term structure models and New Keynesian models. The first essay, which is a joint work with Hwagyun Kim, empirically evaluates the relationships between money, inflation, output growth, and the interest rates of different maturities using a monetary DSGE model of term structure, featuring inflation targeting behavior, asset market segmentation, and external habit extended for nominal economy. This model can generate liquidity effect, average upward sloping yield curve, and time-varying bond risk premia for bearing inflation and real shocks. By exploiting the term structure equations derived from the model, the deep parameters of the model describing risk preference, inflation targeting behavior, and market segmentation between bond traders and non-traders are estimated. The model is estimated under alternative specifications: latent factors; macroeconomic factors; and both latent and macroeconomic factors. The empirical findings show that all the methods give consistent estimates of the parameters, and conclude that asset market segmentation, inflation targeting, and time-varying risk aversion are significant to account for the term structure dynamics. They also suggest that monetary factors and monetary policy are important to understand both short-run and long-run behaviors of bond prices. In the second essay, three different versions of New Keynesian DSGE models are developed, and their structural parameters are estimated by maximum likelihood estimation. Specifically, the role of velocity of money on the dynamics of real variables is empirically examined by constructing a money in the utility model and two special cases of transactions cost model. Wealth effects, previously ignored in many transactions cost models, are taken into consideration in one of the cases examined here, and comparisons are made between the transactions cost model that includes the wealth effects and the transactions cost model that ignores the wealth effects entirely. The equivalence of money in the utility model and transactions cost model with wealth effects is also quantitatively examined. The results show that there is no evidence of quantitative equivalence between these two models. Although the magnitude of impulse responses are different among the models studied here, all three models give consistent estimates for the structural parameters. The empirical findings from the maximum likelihood estimates of all three models' parameters also suggest that the velocity of money is a very important part of the IS and Phillips curves of all three models developed here, and should be included in IS and Phillips curves when examining the inflation and output dynamics.