COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1768 Nonlinear Cointegrating Regression under Weak Identification Xiaoxia Shi and Peter C. B. Phillips September 2010 An asymptotic theory is developed for a weakly identified cointegrating
regression model in which the regressor is a nonlinear transformation of an integrated
process. Weak identification arises from the presence of a loading coefficient for the
nonlinear function that may be close to zero. In that case, standard nonlinear
cointegrating limit theory does not provide good approximations to the finite sample
distributions of nonlinear least squares estimators, resulting in potentially misleading
inference. A new local limit theory is developed that approximates the finite sample
distributions of the estimators uniformly well irrespective of the strength of the
identification. An important technical component of this theory involves new results
showing the uniform weak convergence of sample covariances involving nonlinear functions
to mixed normal and stochastic integral limits. Based on these asymptotics, we construct
confidence intervals for the loading coefficient and the nonlinear transformation
parameter and show that these confidence intervals have correct asymptotic size. As in
other cases of nonlinear estimation with integrated processes and unlike stationary
process asymptotics, the properties of the nonlinear transformations affect the
asymptotics and, in particular, give rise to parameter dependent rates of convergence and
differences between the limit results for integrable and asymptotically homogeneous
functions. |
