In the first chapter, We focus on the estimation of the ratio of trend slopes between two time series where is it reasonable to assume that the trending behavior of each series can be well approximated by a simple linear time trend. We obtain results under the assumption that the stochastic parts of the two time series comprise a zero mean time series vector that has sufficient stationarity and has dependence that is weak enough so that scaled partial sums of the vector satisfy a functional central limit theorem (FCLT). We compare two obvious estimators of the trend slope ratio and propose a third bias-corrected estimator. We show how to use these three estimators to carry out inference about the trend slope ratio. When trend slopes are small in magnitude relative to the variation in the stochastic components (the trend slopes are small relative to the noise), we find that inference using any of the three estimators is compromised and potentially misleading. We propose an alternative inference procedure that remains valid when trend slopes are small or even zero. We carry out an extensive theoretical analysis of the estimators and inference procedures with positive findings. First, the theory points to one of the three estimators as being preferred in terms of bias. Second, the theory unambiguously suggests that our alternative inference procedure is superior both under the null and under the alternative with respect to the magnitudes of the trend slopes. Finite sample simulations indicate that the predictions made by the asymptotic theory are relevant in practice. We give concrete and specific advice to empirical practitioners on how to estimate a ratio of trend slopes and how to carry out tests of hypotheses about the ratio. The second chapter is an extension of the first, where the stationarity assumption in the analysis is relaxed. It is assumed that the stochastic parts of the trending series follow an I(1) process. We consider the case when the individual series have unit roots in the stochastic parts and also the noise term in the IV regression equation has a unit root. We also consider the case of cointegration between the two series leading to I(0) error in the IV regression equation. The theory explicitly captures the impact of the magnitude of the trend slopes on the estimation and inference about the trend slopes ratio. If the trend slopes are relatively large in magnitude, the IV estimator is consistent for both I(1) and I(0) regression errors. For medium and small trend slopes, the IV estimator is inconsistent for I(1) case, but consistent for I(0) regression error. For inference, the test based on IV estimator has been compared with the alternative testing approach. Asymptotic theory and finite sample simulations suggest that the alternative testing approach is superior both under the null and under the alternative with respect to the magnitudes of the trend slopes. Whether the noise term in the IV regression equation is I(0) or I(1) has an impact on the power performance of the test for the trend slopes ratio. The third chapter is an empirical application of the methodology developed in the first and the second chapters. The empirical findings on convergence of per capita income across regions in convergence literature are mixed. There is evidence of convergence in a substantial number of cases, whereas evidence contrary to convergence has also been found. Where there is β-convergence found, it is interesting to come up with a measure of speed of convergence and estimate it. The speed of convergence has been shown to be proportional to a ratio of trend slopes, and using the methodology developed in the first and the second chapters, we estimate this ratio for all US regions which are converging. The higher the ratio, the greater is the speed of convergence.
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