Relative performance of pretest estimators in the presence of misspecification and unit root

摘要

In this dissertation, some specific cases in linear statistical models are studied which are appropriate for general economic decision problems, with an objective to improve traditional estimation rules and develop inference procedures for a single data set. In order to arrive at the final model, researchers perform some preliminary statistical tests, e.g., specification tests. So the final model or estimators depend on these preliminary tests. The resulting estimators are known as the pre-test estimators.;I study two cases in which I have considered the performance of the pre-test estimators. In the first study, I pre-test for a unit root in an autoregressive model and study the sampling properties of certain estimators of the autoregressive coefficient. I compare the mean square errors of OLS, restricted and pre-test estimators and find that the pre-test estimator performs the best only when the true AR(1) coefficient is close to one. I also compare the weighted mean square errors of these estimators. I find that with or without weights, OLS estimator is the 'minimax' estimator. OLS also minimizes the 'average risk'. We show that the above results remain valid even when the null hypothesis and its alternative are interchanged. This is in Chapter II. I pre-test for unit root in AR(2) model in Chapter III and compare the performance of the pre-test estimators with those of OLS and RS (i.e. $rho=1)$ estimators. I find similar results like the AR(1) model. In Chapter IV, I consider weighted mean square errors with weights being the Sims' prior, reference prior, Jeffrey's prior, Leamer I and Leamer II priors on the AR(1) coefficient, and 'average risk' for the estimators are calculated. OLS minimizes the 'average risk' for almost all priors, except for Leamer II prior which resembles the real stock prices. RS estimator minimizes the 'average risk' with Leamer II prior. So pre-test estimator is never 'optimal'. In the second study, I have considered the pre-test estimation of the parameters of a linear regression model after a preliminary test for exact linear restrictions when the model is mis-specified through the omission of a relevant stochastic regressor. The predictive square error risk behavior of a restricted estimators (RS) and pre-test estimators (PT) are compared to the behavior of OLS estimators. This study is documented in Chapter V.