Regressions for Partially Identified, Cointegrated Simultaneous Equations

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 identifiability 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.