Improved asymptotics for econometric estimators and tests

Chapter two derives saddlepoint approximations for the density and distribution of a ratio of non-central quadratic forms in normal variables. Excepting a few special cases, little is known of the exact, finite sample properties of these statistics. Hence we derive and prove the existence of an exact inversion based upon the joint characteristic function. Thence the saddlepoint algorithm is applied and the leading term approximation is found. An illustration accuracy of the approximation.Chapter three derives higher-order, leating term approximations for the density of the MLE in three classes of models. It is found, by exploiting the properties of the likelihood directly, that approximations so constructed often coincide with the saddlepoint approximation. Importantly though we circumvent the necessity of calculating the characteristic function and of solving some saddlepoint defining equation. For each class of models: linear exponential, curved exponential and a class of non-exponential models, the central results are given along with a simple illustrative example.Chapter four investigates the application of Edgeworth series as symptotic expansions for the densities of minimal sufficient statistics. Since much inference is constructed through functions of this statistic validity for both the statistic itself and arbitrary functions of it is proved. Further, numerical accuracy of such approximations is shown to be influenced by transformation. Hence, conditions for an optimal transformation are derived, that is one which minimises some criterion of numerical error, given an asymptotic order of error.Chapter five applies this 'optimal' transformation in the context of first-order regression. We find that asymptotic inference in this model can be significantly improved, i.e. made more accurate, if inference is made via this transformed statistic.