This dissertation consists of three essays on time series and panel data econometrics. The first essay considers the bootstrap method for the covariates augmented Dickey-Fuller (CADF) unit root test suggested by Hansen (1995). It is known that the CADF test is very powerful. However, its limit distribution depends on the nuisance parameter, and thus inference is not possible. To solve this problem, we propose to use the bootstrap method, and establish the asymptotic validity of the bootstrap CADF test. Simulations show that the bootstrap method provides correct sizes, with drastic power gains over the conventional ADF test. Our testing procedures are applied to the Nelson-Plosser data set and annual real exchange rates. In the second essay, we extend Chang's (2001) IV methodology for panel unit root test in three important directions. First, we allow for dependencies across cross sections at both short-run and long-run levels. Second, our theory permits the use of covariates to increase power as well as to control idiosyncrasies of individual units. Third, we re-examine the formulation of the unit root hypothesis in panels, and propose to analyze the hypotheses that only a fraction of cross-sectional units have unit roots. The resulting test statistics are all Gaussian and easy to implement. Simulations are performed, and the new tests are applied to quarterly and monthly real exchange rates. The third essay introduces a new estimation method for time-varying individual effects in a panel data model. An important application is the estimation of time-varying technical inefficiencies of individual firms using the fixed effects model. Most previous models require rather strong distributional assumptions about inefficiency and random noise, and/or impose explicit restrictions on the temporal pattern of inefficiency. This essay drops such assumptions, and provides a semiparametric method for estimation of the time-varying effects. The methods are related to principal component analysis, and estimate the time-varying effects using a small number of common functions. Simulations are performed, and the methods are applied to the U.S. banks data.

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Roughly speaking, a separating algebra is a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this thesis, we introduce the notion of a geometric separating algebra, a more geometric notion of a separating algebra. We find two geometric formulations for the notion of separating algebra which allow us to prove, for geometric separating algebras, the results found in the literature for separating algebras, generally removing the hypothesis that the base field be algebraically closed. Using results from algebraic geometry allows us to prove that, for finite groups, when a polynomial separating algebra exists, the group is generated by reflections, and when a complete intersection separating algebra exists, the group is generated by bireflections. We also consider geometric separating algebras having a small number of generators, giving an upper bound on the number of generators required for a geometric separating algebra. We end with a discussion of two methods for obtaining new separating sets from old. Interesting, and relevant examples are presented throughout the text. Some of these examples provide answers to questions which previously appeared in print. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2008-08-28 14:14:04.138