Dynamic stochastic general equilibrium (DSGE) models generally do not admit analytic solutions. Although DSGE models are widely used in macroeconomics and finance, no statistically sound estimation methods for policy functions such as the price-dividend ratio function have been developed. Because they rely on a fully specified data generating process (DGP) of state variables, numerical solution methods, extensively adopted in the literature, may discredit model evaluation due to model misspecification of state variables and poor approximations of unknown functions. In the second chapter, I propose a convenient nonparametric 2SLS series regression method that is built on consumption based asset pricing models (CAPM), and we investigate its performance in comparison with analytic and existing numerical solutions. The new method proposes to estimate a recursively specified function embedded in Euler equations always admits a data-based closed-form solution, and it is not only easy to implement but also asymptotically free of endogeneity biases and approximation errors, even when the CAPM becomes complex. This new method does not require specifying a DGP of state variables, which avoids model misspecification of state variables and enables us to connect the solutions of DSGE models to empirical data. Our method always provides a consistent estimation of the price-dividend ratio function for a broad class of stationary Markov state variables. The newly proposed 2SLS series regression method will become a pivotal approach for obtaining a consistent estimation of the price-dividend ratio function in the presence of a misspecified or unknown DGP of state variables, and it can help construct the most reliable and accurate model implications. In the third chapter, I consider dynamic stochastic general equilibrium models (DSGE) with recursive preferences, which provide powerful means for investigating the connection between economic fundamentals, asset returns and agent preferences. A system of Euler equations is derived as a pivotal tool to obtain model implications. It often involves multiple recursively specified unknown functions of state variables over different time periods, such as the price-dividend ratio function and the wealth-consumption ratio function. Given the fact that analytic solutions are extremely difficult, if not impossible, numerical solution methods for such functions are extensively adopted in the literature. Because cross dependence exists among unknown functions, all existing numerical solution methods can only provide function approximations sequentially. Therefore, approximation errors from one solution may accumulate and contaminate the others, thereby resulting in conflicting model conclusions. Despite this importance, no statistically sound methods that provide estimation and inference on this class of multiple unknown functions have been developed. Built upon the Epstein and Zin's (1989) consumption based asset pricing model (CAPM), we propose a new nonparametric generalized method of moments (GMM) series procedure and investigate its performance in comparison with existing numerical solution methods. Instead of approximating unknown functions sequentially, our method can consistently estimate all unknown functions simultaneously, while capturing their interactions using the variance-covariance of the derived estimators. Moreover, our GMM series approach is asymptotically free of simultaneous equation biases, endogeneity biases and functional form misspecification as the sample size increases, no matter how complex the DSGE model is. In addition, compared to all existing numerical solution methods which can only provide function approximations given a fully specified dynamics of state variables, our nonparametric GMM series procedure does not require any specification for the dynamics of state variables, thus avoiding potential misspecification for the data generating process (DGP) of state variables. To incorporate a wide variety of empirically relevant setups, this paper discusses two types of the GMM series estimators, namely the two-stage and continuously updating efficient (CUE) GMM series estimators. Our nonparametric CUE GMM series estimator will improve accuracy of inference when instruments are weakly correlated with Euler equation errors. Because there is an infinite number of moments due to series approximations, our nonparametric GMM series method contributes to the GMM literature by establishing a new result on consistency and asymptotic normality, which further helps facilitate rigorous inference on the DSGE model implications. Three simulation studies are considered, and our new method has been proven to perform reasonably well in the finite sample in comparison with popular numerical solution methods such as the log linearization, discretization and projection methods. In the fourth chapter, investor extrapolation biases in the dynamics of economic fundamental variables are introduced into the traditional Lucas Jr (1978) consumption-based asset pricing models (CAPM). Given the involvement of subjective expectations in the estimation procedure, this paper proposes a feasible generalized method of moments (GMM) approach to provide consistent estimation of model parameters. Using this new estimation method, we discover different patterns of investor extrapolation biases for local investors in China, the United States, Japan and the United Kingdom. Investors in U.S., Japan and U.K. tend to react to changes in the mean levels of economic fundamentals, whereas investors in China only pay extra attention to the overall volatile levels of the aggregate economic background. Once equipped with their specific estimated extrapolation biases, models for all these four countries show good performance in explaining well-documented economic anomalies, such as the equity premium puzzle and accumulative equity returns for the aggregate stock markets. Different types of distorted investor beliefs identified in this paper help understand why China's stock market has been deviating from economic fundamentals in recent years. These distorted investor beliefs also shed light on how the regulation of China's stock market can be improved.