Nearly Optimal Tests When a Nuisance Parameter Is Present Under the Null Hypothesis

Abstract

, and consider tests of the form

Measuring Uncertainty about Long-Run Predictions

“…Such a † is called the "least favourable distribution" for the problem. Elliott et al (2015) develop numerical methods for approximating least favourable distributions in related problems, and we use a variant of those methods here. See the Supplementary Appendix for details.…”

“…To make further progress, we follow Elliott et al (2015) (EMW in the following), and first formally state a lower bound on equation (14), and then define an ALFD * that solves (10) within a tolerance of . To ease notation, write…”

Pre and post break parameter inference

“…In particular, under the null hypothesis about the post break parameter value, the nuisance parameters are the break date and break magnitude. This non-standard testing problem falls into the class of problems considered by Elliott et al (2012) (EMW in the following), who derive a set of upper bounds on the weighted average power of any valid test, and suggest a numerical algorithm to determine both a low upper bound and a test with weighted average power close to the bound. We apply EMW's approach to the limit experiment corresponding to post break parameter inference.…”

Credibility of Confidence Sets in Nonstandard Econometric Problems

“…We thus apply Lemma 1 in Müller and Norets (2016) directly and numerically approximate K as a discrete measure on the grid θ = (γ 0) with γ ∈ {0 0 25 200} by iteratively adjusting the weight K j at θ j as a function of whether or not expected winnings at θ j are positive or negative under the optimal betting strategy based on the previous value of K. In this computation, the expected winnings are approximated by Monte Carlo integration using importance sampling over 200,000 draws of a stationary Gaussian AR(1) with 2,500 observations and γ drawn from the grid γ ∈ {0 0 25 200}. For a similar numerical approach, see Elliott, Müller, and Watson (2015).…”

“…For givenΛ, RPΛ(θ) can be approximated by Monte Carlo integration over X * . Furthermore, to approximate aΛ satisfying RPΛ(θ) dΛ(θ) = α, we posit a discrete grid Θ g on θ, and employ fixed-point iterations to adjust the mass points of a candidateΛ c on Θ g as a function of whether RPΛ c (θ) < α or RPΛ c (θ) > α, analogous to the algorithm suggested by Elliott, Müller, and Watson (2015). Specifically, Θ g in the weak instrument example is equal to θ = (0 ρ) with ρ j ∈ {0 0 05 0 01 10}, and it is equal to θ = (0 λ) with λ ∈ {0 1} and ∈ {0 0 05 0 01 15} in the Imbens-Manski example.…”