An Alternative Sense of Asymptotic Efficiency

The paper studies the asymptotic efficiency and robustness of hypothesis tests when models of interest are defined in terms of a weak convergence property. The null and local alternatives induce different limiting distributions for a random element, and a test is considered robust if it controls asymptotic size for all data generating processes for which the random element has the null limiting distribution. Under weak regularity conditions, asymptotically robust and efficient tests are then simply given by efficient tests of the limiting problem--that is, with the limiting random element assumed observed--evaluated at sample analogues. These tests typically coincide with suitably robustified versions of optimal tests in canonical parametric versions of the model. This paper thus establishes an alternative and broader sense of asymptotic efficiency for many previously derived tests in econometrics, such as tests for unit roots, parameter stability tests and tests about regression coefficients under weak instruments, and it provides a concrete limit on the potential for more powerful tests in less parametric set-ups.hypothesis tests, optimality, robustness, weak convergence

Hodges-Lehmann Optimality for Testing Moment

This paper studies the Hodges and Lehmann (1956) optimality of tests in a general setup. The tests are compared by the exponential rates of growth to one of the power functions evaluated at a fixed alternative while keeping the asymptotic sizes bounded by some constant. We present two sets of sufficient conditions for a test to be Hodges-Lehmann optimal. These new conditions extend the scope of the Hodges-Lehmann optimality analysis to setups that cannot be covered by other conditions in the literature. The general result is illustrated by our applications of interest: testing for moment conditions and overidentifying restrictions. In particular, we show that (i) the empirical likelihood test does not necessarily satisfy existing conditions for optimality but does satisfy our new conditions; and (ii) the generalized method of moments (GMM) test and the generalized empirical likelihood (GEL) tests are Hodges-Lehmann optimal under mild primitive conditions. These results support the belief that the Hodges-Lehmann optimality is a weak asymptotic requirement.Asymptotic optimality, Large deviations, Moment condition, Generalized method of moments, Generalized empirical likelihood

Inference on the mode of weak directional signals: a Le Cam perspective on hypothesis testing near singularities

We revisit, in an original and challenging perspective, the problem of testing the null hypothesis that the mode of a directional signal is equal to a given value. Motivated by a real data example where the signal is weak, we consider this problem under asymptotic scenarios for which the signal strength goes to zero at an arbitrary rate~$\eta_n$. Both under the null and the alternative, we focus on rotationally symmetric distributions. We show that, while they are asymptotically equivalent under fixed signal strength, the classical Wald and Watson tests exhibit very different (null and non-null) behaviours when the signal becomes arbitrarily weak. To fully characterize how challenging the problem is as a function of~$\eta_n$, we adopt a Le Cam, convergence-of-statistical-experiments, point of view and show that the resulting limiting experiments crucially depend on~$\eta_n$. In the light of these results, the Watson test is shown to be \emph{adaptively} rate-consistent and essentially adaptively Le Cam optimal. Throughout, our theoretical findings are illustrated via Monte-Carlo simulations. The practical relevance of our results is also shown on the real data example that motivated the present work

Inference with Weak Instruments

This paper reviews recent developments in methods for dealing with weak instruments (IVs) in IV regression models. The focus is more on tests (and confidence intervals derived from tests) than estimators. The paper also presents new testing results under "many weak IV asymptotics," which are relevant when the number of IVs is large and the coefficients on the IVs are relatively small. Asymptotic power envelopes for invariant tests are established. Power comparisons of the conditional likelihood ratio (CLR), Anderson-Rubin, and Lagrange multiplier tests are made. Numerical results show that the CLR test is on the asymptotic power envelope. This holds no matter what the relative magnitude of the IV strength to the number of IVs.Conditional likelihood ratio test, instrumental variables, many instrumental variables, power envelope, weak instruments

A simple and general test for white noise

This article considers testing that a time series is uncorrelated when it possibly exhibits some form of dependence. Contrary to the currently employed tests that require selecting arbitrary user-chosen numbers to compute the associated tests statistics, we consider a test statistic that is very simple to use because it does not require any user chosen number and because its asymptotic null distribution is standard under general weak dependent conditions, and hence, asymptotic critical values are readily available. We consider the case of testing that the raw data is white noise, and also consider the case of applying the test to the residuals of an ARMA model. Finally, we also study finite sample performance

Model Checks Using Residual Marked Empirical Processes

This paper proposes omnibus and directional tests for testing the goodness-of-fit of a parametric regression time series model. We use a general class of residual marked empirical processes as the building-blocks for estimation and testing of such models. First, we establish a weak convergence theorem under mild assumptions, which allows us to study in a unified way the asymptotic null distribution of the test statistics and their asymptotic behavior against Pitman's local alternatives. To approximate the asymptotic null distribution of test statistics we justify theoretically a bootstrap procedure. Also, some asymptotic theory for the estimation of the principal components of the residual marked processes is considered. This asymptotic theory is used to derive optimal directional tests and efficient estimation of regression parameters. Finally, a Monte Carlo study shows that the bootstrap and the asymptotic results provide good approximations for small sample sizes and an empirical application to the Canadian lynx data set is considered.

Semiparametrically efficient rank-based inference for shape I. optimal rank-based tests for sphericity

We propose a class of rank-based procedures for testing that the shape matrix $\mathbf{V}$ of an elliptical distribution (with unspecified center of symmetry, scale and radial density) has some fixed value ${\mathbf{V}}_0$; this includes, for ${\mathbf{V}}_0={\mathbf{I}}_k$, the problem of testing for sphericity as an important particular case. The proposed tests are invariant under translations, monotone radial transformations, rotations and reflections with respect to the estimated center of symmetry. They are valid without any moment assumption. For adequately chosen scores, they are locally asymptotically maximin (in the Le Cam sense) at given radial densities. They are strictly distribution-free when the center of symmetry is specified, and asymptotically so when it must be estimated. The multivariate ranks used throughout are those of the distances--in the metric associated with the null value ${\mathbf{V}}_0$ of the shape matrix--between the observations and the (estimated) center of the distribution. Local powers (against elliptical alternatives) and asymptotic relative efficiencies (AREs) are derived with respect to the adjusted Mauchly test (a modified version of the Gaussian likelihood ratio procedure proposed by Muirhead and Waternaux [Biometrika 67 (1980) 31--43]) or, equivalently, with respect to (an extension of) the test for sphericity introduced by John [Biometrika 58 (1971) 169--174]. For Gaussian scores, these AREs are uniformly larger than one, irrespective of the actual radial density. Necessary and/or sufficient conditions for consistency under nonlocal, possibly nonelliptical alternatives are given. Finite sample performances are investigated via a Monte Carlo study.Comment: Published at http://dx.doi.org/10.1214/009053606000000731 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org