This dissertation uses innovative methods to study different perspectives of macroeconomics. In particular, topics in this dissertation including macroeconomic modeling, computational technics, and empirical analysis.In the first chapter, I propose a numerical routine to approximate the radius of convergence (ROC) for perturbation methods. The classical issue of solving dynamic stochastic models with perturbation methods is that the solution only converges (to the true solution) locally over the state space. Moreover, such a convergence range is usually undetectable under current existing numerical methods. The proposed algorithm, in the limit, can approximate this range both necessarily and suciently. It resolves the diculty of assessing the appropriateness of perturbation solutions. This chapter makes two types of contributions. First, method-wise, the proposed algorithm is the first numerical routine on approximating the ROC of perturbation solutions. In addition, as shown in this chapter, the approximated ROC converges to the true value (no matter if it is analytically obtainable or not) as the order of the approximation increases. Second, modeling-wise, this algorithm provides two main insights on the real business cycle (RBC) model: (i) the perturbation method is capable of solving the standard RBC model; the ROC is large enough to cover most commonly used calibrations; (ii) models with recursive utility or stochastic volatility can not be solved appropriately by the perturbation method as the standard calibration for TFP volatility exceeds the ROC of value function on this dimension. In the second chapter, I try to shed light on the frontier research discussions around declining trends in business dynamism using a general equilibrium model structure with heterogeneous individuals. The key mechanism of the framework is the strategic technological innovation process of firms in response to an individual’s wealth holding and lifetime value of being in different occupations. Such an endogenous TFP process accompanies by wealth inequality reflects a firm’s relative technological position among other producers. The resulting ”winner takes all” dynamics help the model jointly account for several observed empirical trends of the U.S. economy. To accomplish this analysis, I adept a two-stage approach in my structural model. In the first stage, I run a comparative analysis to numerically show that multiple structural shocks are unnecessary to create stylized facts of declining business dynamism. As a consequence, for the second stage, I introduce the ”winner takes all” dynamics to the model and provide intuitions on the mechanism. I try to use both stages to emphasize the key role of the interactions between wealth inequality and firm-specific productivity growth in explaining the DBD facts during transitional dynamics. In the third chapter, I investigate whether the series of President Trump’s tweets had any effect on the stock market. In order to accomplish this, I divide this project into two separate parts. For the first part, I collect all president Donald Trump’s tweets from his Twitter database. I then run sentiment analysis on those tweets which contain content related to a particular company that is in the S&P 500 index. For the other part, I create a new routine for obtaining the abnormal daily return of stock indexes by time series forecasting. Combining both parts together, I test the main hypothesis that a tweet from President Donald Trump will significantly affect the stock price of the mentioned company. This chapter suggests only tweets with positive sentiment have significant effects on stock price movement. Moreover, such an ffect only lasts for about one trading day.