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Published online by Cambridge University Press:  14 August 2017

Offer Lieberman*
Bar-Ilan University
Peter C.B. Phillips
Yale University, University of Auckland, Southampton University, and Singapore Management University
*Address correspondence to Offer Lieberman, Department of Economics and Research Institute for Econometrics (RIE), Bar-Ilan University, Ramat Gan 52900, Israel; e-mail:


Lieberman and Phillips (2017; LP) introduced a multivariate stochastic unit root (STUR) model, which allows for random, time varying local departures from a unit root (UR) model, where nonlinear least squares (NLLS) may be used for estimation and inference on the STUR coefficient. In a structural version of this model where the driver variables of the STUR coefficient are endogenous, the NLLS estimate of the STUR parameter is inconsistent, as are the corresponding estimates of the associated covariance parameters. This paper develops a nonlinear instrumental variable (NLIV) as well as GMM estimators of the STUR parameter which conveniently addresses endogeneity. We derive the asymptotic distributions of the NLIV and GMM estimators and establish consistency under similar orthogonality and relevance conditions to those used in the linear model. An overidentification test and its asymptotic distribution are also developed. The results enable inference about structural STUR models and a mechanism for testing the local STUR model against a simple UR null, which complements usual UR tests. Simulations reveal that the asymptotic distributions of the NLIV and GMM estimators of the STUR parameter as well as the test for overidentifying restrictions perform well in small samples and that the distribution of the NLIV estimator is heavily leptokurtic with a limit theory which has Cauchy-like tails. Comparisons of STUR coefficient and standard UR coefficient tests show that the one-sided UR test performs poorly against the one-sided STUR coefficient test both as the sample size and departures from the null rise. The results are applied to study the relationships between stock returns and bond spread changes.

Copyright © Cambridge University Press 2017 

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This paper is a revised version of an earlier paper entitled “IV and GMM Estimation and Testing of Multivariate Stochastic Unit Root Models.” We thank the CoEditor, Pentti Saikkonen and two referees for helpful comments and Tim Ginker for research assistance. Support from the NSF under Grant No. SES 1258258 is acknowledged by Peter C.B. Phillips. Support from the Israel Science Foundation grant No. 1082-14 and from The Pinhas Sapir Center for Development at Tel Aviv University is gratefully acknowledged by Offer Lieberman.



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