This dissertation presents two essays on macroeconometrics. In the second chapter, I empirically compare alternative specifications of time-varying volatility in the context of linearized dynamic stochastic general equilibrium models. I consider time variation in the volatility of structural innovations in two ways: one in which the logarithm of the volatility is assumed to follow a simple autoregressive process (stochastic volatility) and the other in which the volatility follows a Markov-switching process. A comprehensive simulation study is presented to assess the fit and performance of two specifications. I show that modeling heteroscedasticity in a highly synchronized fashion across shocks may lead to distorted estimation of the volatility. In the empirical application to the United States data, stochastic volatility model delivers the best-fit and accounts for the heteroscedasticity present in the data well. In the third chapter, I conduct a quantitative evaluation of the potential role of adaptive expectations in a two-country dynamic stochastic general equilibrium model. Under the learning mechanism economic agents are assumed to form their expectations of forward-looking variables using a simple vector autoregressive forecasting model. The agents estimate their vector autoregression based on past model variables and update the estimates every period via a constant gain learning algorithm. I show in a simulation study that the learning mechanism increases the volatility and persistence of the endogenous variables and that as the constant gain parameter grows larger, so do these increases. The two-country DSGE model is then estimated with data from the United States and Euro area. A comparison based on log marginal data densities favors the learning over the rational expectations specification. The learning mechanism generates more persistent responses of variables to the monetary shocks. The improvement in terms of fitting the observed Dollar-Euro exchange rate dynamics is limited.