This paper studies regressions for partially identiﬁed equations in simultaneous equations models (SEMs) where all the variables are I(l) and cointegrating relations are present. Asymptotic properties of OLS and 2SLS estimators under partial identiﬁcation are derived. The results show that the identiﬁabilitv condition is important for consistency of estimates in nonstationary SEMs as it is for stationary SEMS. Also, OLS and 2SLS estimators are shown to have diﬀerent rates of convergence and divergence under partial identiﬁcation, though they have the same rates of convergence and divergence for the two polar cases of full identiﬁcation and total lack of identiﬁability. Even in the case of full identiﬁcation. however, the OLS and 2SLS estimators have diﬀerent distributions in the limit. Fully modiﬁed OLS regression and leads-and-lags regression methods are also studied. The results show that these two estimators have nuisance parameters in the limit under general assumptions on the regression errors and are not suitable for structural inference. The paper proposes 2SLS versions of these two nonstationary regression estimators that have mixture normal distributions in the limit under general assumptions on the regression errors, that are more eﬀicient than the unmodiﬁed estimators, and that are suited to statistical inference using asymptotic chi-squared distributions. Some simulation results are also reported.
Choi, In and Phillips, Peter C.B., "Regressions for Partially Identified, Cointegrated Simultaneous Equations" (1997). Cowles Foundation Discussion Papers. 1410.