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.
New time and frequency domain tests for the presence of a unit root are developed. The tests are based on generalized least squares (GLS) methods in both the time and the frequency domains. For the time domain tests, moving average processes are assumed for the error terms on the autoregression. For the frequency domain tests, general assumptions are made which allow for stationary and weakly dependent error processes. The limiting distributions of feasible GLS tests are derived under MA(1) errors in the time domain. This theory is extended to higher order moving average processes under an invertibility condition. The limiting distributions of both full and band spectrum tests in the frequency domain are also derived.
All of these limiting distributions are shown to be free of nuisance parameters. Some results on test consistency are also reported. Extensive Monte Carlo simulations are performed to study the size and power of the proposed tests in finite samples.
JEL Classifications: 211
Keywords: Unit root, spectral methods, generalized least squares, asymptotic theory, Monte Carlo