I study the role of heterogeneity and idiosyncratic risk in Macroeconomics, and their implications on problems of income taxation. In the first chapter, I study the effects of redistributive taxation in an incomplete market economy with heterogeneous agents and idiosyncratic risk. I focus on the role of distortions in labor supply decisions and the interplay of heterogeneity and uninsurable idiosyncratic shocks, conducting the first general equilibrium analysis of a Negative Income Tax (NIT). I show that a NIT is a serious candidate to replace the current income tax in the United States. I find that the optimal NIT has a marginal tax rate of 28% and a transfer of 10% of per capita GDP, roughly $4600. The welfare gains of replacing the current US income tax with a NIT are equivalent to a 6.3% increase in annual consumption in every state of the world. Low-ability agents, in the bottom quintile of the productivity distribution, benefit the most, while high-ability agents are worse off. A consequence of the reform is that the composition of the labor force changes, with high-productivity agents working more, in relative terms, than low-productivity agents. Finally, I find that the riskier the economy, the higher the welfare gains of the NIT as a provider of public insurance. In the second chapter, I study labor income dynamics over the life cycle and introduce a novel methodology that can detect the presence of patterns in the idiosyncratic earnings shocks and recognize economic forces in action. Using a sample from the Panel Study of Income Dynamics (PSID), I estimate a Bayesian Logistic Smoothed Transition Autoregressive model of order 1 (LSTAR(1)) with a rich level of heterogeneity in the innovations. I find that there is a life-cycle pattern in the earning shocks: before the age 29, young workers experience shocks with higher variance and a positive probability of lower persistence than older workers. A comparison with conventional models shows that an incorrect model specification introduces bias in the estimates. The proposed model can be easily approximated with a discrete Markov process. This means that this model can be used by macroeconomists to calibrate income processes.

Abstract This thesis considers some linear and nonlinear time series models. In the linear case, the analysis of a large number of short time series generated by a first order autoregressive type model is considered. The conditional and exact maximum likelihood procedures are developed to estimate parameters. Simulation results are presented and compare the bias and the mean square errors of the parameter estimates. In Chapter 3, five important nonlinear models are considered and their time series properties are discussed. The estimating function approach for nonlinear models is developed in detail in Chapter 4 and examples are added to illustrate the theory. A simulation study is carried out to examine the finite sample behavior of these proposed estimates based on the estimating functions.

Abstract This thesis considers some linear and nonlinear time series models. In the linear case, the analysis of a large number of short time series generated by a first order autoregressive type model is considered. The conditional and exact maximum likelihood procedures are developed to estimate parameters. Simulation results are presented and compare the bias and the mean square errors of the parameter estimates. In Chapter 3, five important nonlinear models are considered and their time series properties are discussed. The estimating function approach for nonlinear models is developed in detail in Chapter 4 and examples are added to illustrate the theory. A simulation study is carried out to examine the finite sample behavior of these proposed estimates based on the estimating functions.

The estimation and analysis of signals that have polynomial phase and constant or time-varying amplitudes with the addititve noise is considered in this dissertation.Much work has been undertaken on this problem over the last decade or so, and there are a number of estimation schemes available. The fundamental problem when trying to estimate the parameters of these type of signals is the nonlinear characterstics of the signal, which lead to computationally difficulties when applying standard techniques such as maximum likelihood and least squares. When considering only the phase data, we also encounter the well known problem of the unobservability of the true noise phase curve. The methods that are currently most popular involve differencing in phase followed by regression, or nonlinear transformations. Although these methods perform quite well at high signal to noise ratios, their performance worsens at low signal to noise, and there may be significant bias. One of the biggest problems to efficient estimation of these models is that the majority of methods rely on sequential estimation of the phase coefficients, in that the highest-order parameter is estimated first, its contribution removed via demodulation, and the same procedure applied to estimation of the next parameter and so on. This is clearly an issue in that errors in estimation of high order parameters affect the ability to estimate the lower order parameters correctly. As a result, stastical analysis of the parameters is also difficult. In thie dissertation, we aim to circumvent the issues of bias and sequential estiamtion by considering the issue of full parameter iterative refinement techniques. ie. given a possibly biased initial estimate of the phase coefficients, we aim to create computationally efficient iterative refinement techniques to produce stastically efficient estimators at low signal to noise ratios. Updating will be done in a multivariable manner to remove inaccuracies and biases due to sequential procedures. Stastical analysis and extensive simulations attest to the performance of the schemes that are presented, which include likelihood, least squares and bayesian estimation schemes. Other results of importance to the full estimatin problem, namely when there is error in the time variable, the amplitude is not constant, and when the model order is not known, are also condsidered.