Many estimators and tests are of the form of a ratio of quadratic forms in normal variables+ Excepting a few very special cases little is known about the density or distribution of these ratios, particularly if we allow for noncentrality in the quadratic forms+ This paper assumes this generality and derives saddlepoint approximations for this class of statistics+ We first derive and prove the existence of an exact inversion based on the joint characteristic function+ Then the saddlepoint algorithm is applied and the leading term found, and analytic justification of the asymptotic nature of the approximation is given+ As an illustration we consider the calculation of sizes and powers of F-tests, where a new exact result is found+

This paper considers the information available to invariant unit root tests at and near the unit root. Since all invariant tests will be functions of the maximal invariant, the Fisher information in this statistic will be the available information. The main finding of the paper is that the available information for all tests invariant to a linear trend is zero at the unit root. This result applies for any sample size, over a variety of distributions and correlation structures and is robust to the inclusion of any other deterministic component. In addition, an explicit bound upon the power of all invariant unit root tests is shown to depend solely upon the information. This bound is illustrated via comparison with the local-to-unity power envelope and a brief simulation study illustrates the impact that the requirements of invariance have on power.

No abstract

The fundamental contributions made by Paul Newbold have highlighted how crucial it is to detect when economic time series have unit roots. This paper explores the effects that model specification has on our ability to do that. Asymptotic power, a natural choice to quantify these effects, does not accurately predict finite-sample power. Instead, here the Kullback-Leibler divergence between the unit root null and any alternative is used and its numeric and analytic properties detailed. Numerically it behaves in a similar way to finite-sample power. However, because it is analytically available we are able to prove that it is a minimizable function of the degree of trending in any included deterministic component and of the correlation of the underlying innovations. It is explicitly confirmed, therefore, that it is approximately linear trends and negative unit root moving average innovations that minimize the efficacy of unit root inferential tools. Applied to the Nelson and Plosser macroeconomic series the effect that different types of trends included in the model have on unit root inference is clearly revealed.Thanks are due to

This paper derives transformations for multivariate statistics that eliminate asymptotic skewness, extending the results of Niki and Konishi~1986, Annals of the Institute of Statistical Mathematics 38, 371-383!+ Within the context of valid Edgeworth expansions for such statistics we first derive the set of equations that such a transformation must satisfy and second propose a local solution that is sufficient up to the desired order+ Application of these results yields two useful corol-laries+ First, it is possible to eliminate the first correction term in an Edgeworth expansion, thereby accelerating convergence to the leading term normal approx-imation+ Second, bootstrapping the transformed statistic can yield the same rate of convergence of the double, or prepivoted, bootstrap of Beran~1988, Journal of the American Statistical Association 83, 687-697!, applied to the original statistic, implying a significant computational saving+ The analytic results are illustrated by application to the family of exponential models, in which the transformation is seen to depend only upon the properties of the likelihood+ The numerical properties are examined within a class of nonlinear regression models~logit, probit, Poisson, and exponential regressions!, where the adequacy of the limiting normal and of the bootstrap~utilizing the k-step procedure of Andrews, 2002, Econometrica 70, 119-162! as distributional approximations is assessed+ This paper is derived from my Ph+D+ thesis, "Higher-Order Asymptotics for Econometric Estimators and Tests," for which thanks for patient and helpful supervision go to Grant Hillier+ Comments by