Size Distortion of the Nested Clark–West Test under Constant-Gain Learning, and a Monte-Carlo Remedy

The general principle that parameter-estimation error distorts out-of-sample tests of equal predictive accuracy is well established; it is the motivation for the West (1996), Clark–McCracken (2001, 2005), Clark–West (2007), and Giacomini–White (2006) literature. This working paper documents a specific, practically important manifestation: when a frozen, fixed-coefficient forecast is compared against a constant-gain adaptive-learning forecast, the nested Clark–West test is severely anti-conservative, rejecting a true no-adaptation null roughly 60% of the time at the nominal 5% level. An exact one-step algebraic identity shows the Clark–West summand equals twice the product of the restricted forecast error and the learner's forecast revision; under constant gain that revision is a noise-chasing average of recent errors and stays positively correlated with the error even under the null, so the statistic is positively centered. With a decreasing (1/t) gain the revision vanishes and the statistic recenters — the distortion is a violation of Clark–West's vanishing-estimation-error premise, not a flaw in the test. The paper recommends Monte-Carlo size-calibration against a fixed-coefficient null (and treats the Giacomini–White fixed-window design as the asymptotically motivated alternative). Applied to a three-equation New Keynesian forecasting exercise on US data (1960–2019), the size gate converts two spurious p = 0.000 results into honest non-results, and finds no robust evidence that belief adaptation improves New Keynesian inflation forecasting. Code and data are fully reproducible.