Analysis Of Bootstrap Methods And Its Applications In A Few Linear Model Classes

This paper focuses on analyzing problems in a few linear-model classes, mainly concerning about the test of equality of a group of means and significant test of estimated parameters in linear regression model. Firstly, we study the equality of two normal means under different and unknown variances, called Behrens-Fisher problem. We combine bootstrap idea and the existed score test and get a better test method for Behrens-Fisher problem. Based on this work, we further consider the equality test of means in the case which involves more than two normal populations. Drawing on Max Likelihood idea, we derive a parametric bootstrap method to solve the problem of testing a groups of normal means under heteroscedasticity. Finally, we manage to incorporate bootstrap method into traditional quadratic test variable to avoid solving the inverse of singular covariance in testing the significance of estimated parameter in a multivariable linear model-CAPM, especially when the number of the excess returns we study is much bigger than sample size.Through Monte Carlo simulation, the parametric bootstrap tests we proposed perform better than traditional welch t-test and generalized F test in terms of type I error and power both in comparing two normal means and more than two ones, especially when the sizes of all groups is small and variant. As for the efficiency test of CAPM in high dimension case, the standard inverse of symmetric matrix is replaced by a special generalized counterpart, the Moore-Penrose Inverse and we proposed parametric bootstrap test, which could be applied to more general situations. Furthermore, the simulation results show that the existed test designed for high dimension cases are susceptible to the size of non-diagonal elements in the covariance matrix of random term. Specifically, prior test methods perform well under negligee cross-sectional correlation. While the type I error of these tests will increase when cross-sectional correlation is getting stronger. However, the bootstrap test could suit various situation of cross-sectional correlation and performs much more stably.