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.
A semiparametric triangular systems approach shows how multicointegration can occur naturally in an I(1) cointegrated regression model. The framework reveals the source of multicointegration as singularity of the long run error covariance matrix in an I(1) system, a feature noted but little explored in earlier work. Under such singularity, cointegrated I(1) systems embody a multicointegrated structure and may be analyzed and estimated without appealing to the associated I(2) system but with consequential asymptotic properties that can introduce asymptotic bias into conventional methods of cointegrating regression. The present paper shows how estimation of such systems may be accomplished under multicointegration without losing the nice properties that hold under simple cointegration, including mixed normality and pivotal inference. The approach uses an extended version of high-dimensional trend IV estimation with deterministic orthonormal instruments that leads to mixed normal limit theory and pivotal inference in singular multicointegrated systems in addition to standard cointegrated I(1) systems. Wald tests of general linear restrictions are constructed using a fixed-b long run variance estimator that leads to robust pivotal HAR inference in both cointegrated and multicointegrated cases. Simulations show the properties of the estimation and inferential procedures in finite samples, contrasting the cointegration and multicointegration cases. An empirical illustration to housing stocks, starts and completions is provided.