Three essays in econometrics: Idempotent matrix, adaptive regression, and unit root test

This dissertation consists of three distinctly different essays on econometric subjects. Essay I deals with an idempotent matrix which is often referred to as the 'residual maker.' The applicability of the idempotent matrix is much broader than we are aware of in econometrics. In this essay, two fundamental properties of the idempotent matrix are formally established: order-invariability (well known but not formally established) and decomposability (hitherto unknown). Then, the idempotent matrix is applied to derive an analytic expression for each element in the OLS estimator vector, which is otherwise possible but very cumbersome. Using the idempotent matrix, we also easily verify the Frisch-Waugh-Lovell Theorem. More importantly, the matrix is used to derive a new result, i.e., the exact analytical expression for a correlation coefficient between any pair of elements in the OLS estimator vector.;Essay II investigates the distribution of the statistic for testing parameter instability in the adaptive regression model. In a number of empirical studies, the test statistic has been assumed to be asymptotically normally distributed. However, we show experimentally that the asymptotic distribution of the test statistic is indeed nonnormal. Based on the Monte Carlo results for various sample sizes, a statistical table has been produced which practitioners of the adaptive regression model may find useful for significance tests of the instability parameter estimate.;Essay III addresses the problem of the unit root test associated with measurement errors contained in the observed unit root process. Though virtually all observed economic variables are likely to be subject to measurement errors to an extent, the possibility of serious ramifications on unit root test have been almost completely ignored in practice. Augmented Dickey-Fuller tests were carried out for various sample sizes, with data generated by a random walk process contaminated by stationary random measurement errors. As conjectured, for small sample sizes the test statistic substantially deviates from its nominal distributions tabulated by Dickey and Fuller, with the deviation depending on the degree of the measurement errors. The deviations, however, taper off as the sample size increases.