## Abstract

This note shows that the mixed normal asymptotic limit of the trend IV estimator with a fixed number of deterministic instruments (fTIV) holds in both singular (multicointegrated) and nonsingular cointegration systems, thereby relaxing the exogeneity condition in (Phillips and Kheifets, 2024, Theorem 1(ii)). The mixed normality of the limiting distribution of fTIV allows for asymptotically pivotal F tests about the cointegration parameters and for simple efficiency comparisons of the estimators for different numbers K of instruments, as well as comparisons with the trend IV estimator when K → ∞ with the sample size.