arXiv:2611.07219v1 Announce Type: new Abstract: We develop specification tests for parametric conditional tail index models of heavy-tailed distributions. The tests are based on a projected empirical process that removes the first-order effect of estimating the null parameter. Within the conditional Hall class, we identify conditions under which projection also eliminates the leading Pareto approximation bias in the test process, permitting lower thresholds while preserving asymptotically correct size. We establish the validity of the tests via multiplier bootstrap. For log-linear models, bootstrap validity extends to settings in which covariates among observations above a common threshold concentrate and the smallest eigenvalue of the information matrix tends to zero. We further propose a data-driven threshold selection rule based on an estimate of the Pareto approximation bias after projection. The rule allows lower thresholds to retain more observations for testing while limiting size distortions, with potential gains in power. The finite sample performance of the tests is examined through simulations and a real data application.