Testing Generalized Regression Monotonicity

Abstract

We propose a test for a generalized regression monotonicity (GRM) hypothesis. The GRM hypothesis is the sharp testable implication of the monotonicity of certain latent structures, as we show in this article. Examples include the monotonicity of the conditional mean function when only interval data are available for the dependent variable and the monotone instrumental variable assumption of Manski and Pepper (2000). These instances of latent monotonicity can be tested using our test. Moreover, the GRM hypothesis includes regression monotonicity and stochastic monotonicity as special cases. Thus, our test also serves as an alternative to existing tests for those hypotheses. We show that our test controls the size uniformly over a broad set of data generating processes asymptotically, is consistent against fixed alternatives, and has nontrivial power against some ${n^{ - 1/2}}$ local alternatives.

An adaptive test of stochastic monotonicity

“…Remark 3.3 (Comparison with Hsu, Liu, and Shi (2018)). On can show that the test by Hsu, Liu, and Shi (2018) also has better power relative to our test against sequences of wide alternatives, as discussed below Theorem 3.3.…”

“…Remark 3.3 (Comparison with Hsu, Liu, and Shi (2018)). On can show that the test by Hsu, Liu, and Shi (2018) also has better power relative to our test against sequences of wide alternatives, as discussed below Theorem 3.3. On the other hand, since it is based on inference techniques by Andrews and Shi (2013), we conjecture that similar calculations as those in Appendix K of Chernozhukov, Lee, and Rosen (2013) can be used to show that Hsu, Liu, and Shi (2018)'s test is not rate-optimal against sequences of alternatives in M β .…”

“…Remark 3.5 (Higher-order stochastic monotonicity). It is straightforward to adapt our test and subsequent formal results for testing the null of higher-order stochastic monotonicity as in Example 2.5 of Hsu, Liu, and Shi (2018).…”

“…Fan (1996), Fan and Li (2000), Horowitz and Spokoiny (2001)). Hsu, Liu, and Shi (2018) and Lee, Song, and Whang (2018) propose tests of functional inequalities of which testing the null of stochastic monotonicity is a special case.…”

An Adaptive Test of Stochastic Monotonicity

“…Fan (1996), Fan and Li (2000), Horowitz and Spokoiny (2001)). Hsu, Liu, and Shi (2019) and Lee, Song, and Whang (2018) propose tests of functional inequalities of which testing the null of stochastic monotonicity is a special case.…”