Jonathan B. Hill

Professor of Economics and Director of Graduate Studies, University of North Carolina-Chapel Hill

Recent Activity

A Smoothed P-Value Test When There is a Nuisance Parameter under the Alternative: submitted, paper (updated Nov. 2018), pdf (arXiv), appendix

We present a new test when there is a nuisance parameter λ under the alternative hypothesis. The test exploits the p-value occupation time [PVOT], the measure of the subset of a nuisance parameter on which a p-value test based on a test statistic rejects the null hypothesis. Key contributions are: (i) An asymptotic critical value upper bound for our test is the significance level α, making inference easy. Conversely, test statistic functionals need a bootstrap or simulation step which can still lead to size and power distortions, and bootstrapped or simulated critical values are not asymptotically valid under weak or non-identification. (ii) We only require the test statistic to have a known or bootstrappable limit distribution, hence we do not require √n-Gaussian asymptotics, and weak or non-identification is allowed. Finally, (iii) a test based on the sup-p-value may be conservative and in some cases have nearly trivial power, while the PVOT naturally controls for this by smoothing over the nuisance parameter space. We give examples and related controlled experiments concerning PVOT tests of: omitted nonlinearity; GARCH effects; and a one time structural break. Across cases, the PVOT test variously matches, dominates or strongly dominates standard tests based on the supremum p-value, or supremum or average test statistic (with wild bootstrapped p-value).


A Max-Correlation White Noise Test for Weakly Dependent Time Series (2017, with K. Motegi: revised and resubmitted to Econometric Theory): paper (2018), pdf (arXiv), appendix

This paper presents a bootstrapped p-value white noise test based on the maximum correlation, for a time series that may be weakly dependent under the null hypothesis. The time series may be prefiltered residuals. The test statistic is a normalized weighted maximum sample correlation coefficient max{1≤h≤Ln}: root(n)|ω{n,h}ρ(n,h)|, where ω{n,h} are weights and the maximum lag Ln increases at a rate slower than the sample size n. We only require uncorrelatedness under the null hypothesis, along with a moment contraction dependence property that includes mixing and non-mixing sequences. We show Shao’s (2011) dependent wild bootstrap is valid for a much larger class of processes than originally considered. It is also valid for residuals from a general class of parametric models as long as the bootstrap is applied to a first order expansion of the sample correlation. We prove the bootstrap validity without exploiting extreme value theory (standard in the literature) or recent Gaussian approximation theory. Finally, we extend Escanciano and Lobato’s (2009) automatic maximum lag selection to our setting with an unbounded choice set, and find it works strikingly well in controlled experiments. Our proposed test achieves accurate size under various white noise null hypotheses and high power under various alternative hypotheses including distant serial dependence.

Inference When There is a Nuisance Parameter under the Alternative and Some Parameters are Possibly Weakly Identified (2018) pdf (arXiv)

We present a new robust bootstrap method for a test when there is a nuisance parameter under the alternative, and some parameters are possibly weakly or non-identified. We focus on a Bierens (1990)-type conditional moment test of omitted nonlinearity for convenience, and because of difficulties that have been ignored to date. Existing methods include the supremum p-value which promotes a conservative test that is generally not consistent, and test statistic transforms like the supremum and average for which bootstrap methods are not valid under weak identification. We propose a new wild bootstrap method for p-value computation by targeting specific identification cases. We then combine bootstrapped p-values across polar identification cases to form an asymptotically valid p-value approximation that is robust to any identification case. The wild bootstrap does not require knowledge of the covariance structure of the bootstrapped processes, whereas Andrews and Cheng’s (2012, 2013, 2014) simulation approach generally does. Our method allows for robust bootstrap critical value computation as well. Our bootstrap method (like conventional ones) does not lead to a consistent p-value approximation for test statistic functions like the supremum and average. We therefore smooth over the robust bootstrapped p-value as the basis for several tests which achieve the correct asymptotic level, and are consistent, for any degree of identification. A simulation study reveals possibly large empirical size distortions in non-robust tests when weak or non-identification arises. One of our smoothed p-value tests, however, dominates all other tests by delivering accurate empirical size and comparatively high power.

© 2019 Jonathan B. Hill

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