A Note on Nonlinear Cointegration, Misspecification, and Bimodality

Abstract

We derive the asymptotic distribution of the ordinary least squares estimator in a regression with cointegrated variables under misspecification and/or nonlinearity in the regressors. We show that, under some circumstances, the order of convergence of the estimator changes and the asymptotic distribution is non-standard. The t-statistic might also diverge. A simple case arises when the intercept is erroneously omitted from the estimated model or in nonlinear-in-variables models with endogenous regressors. In the latter case, a solution is to use an instrumental variable estimator. The core results in this paper also generalise to more complicated nonlinear models involving integrated time series.

Keywords:

• Asymptotic theory
• Bimodality
• Cointegration
• Misspecification
• Nonlinearity

JEL Classification:

• C22

ACKNOWLEDGMENTS

This work was carried out while the first author was visiting the Department of Economics at the University of Canterbury, and its kind hospitality is gratefully acknowledged. The authors are very thankful to Peter C. B. Phillips for many enlightening discussions. The suggestions of the Editor, Essie Maasoumi, and an Associate Editor are also gratefully acknowledged.

Notes

For references on classical results in cointegration theory see, e.g., Park and Phillips (1988) and Chapters 17, 18, and 19 in Hamilton (1994).

See Hamilton (1994, p. 486)

[X] denotes the integer part of X.

This last assumption excludes the case where the multivariate random walk is endogenous with respect to β. Generalizing our results to the case of endogenous x _t is considered in Section 4.

In order to simulate the distributions we consider that Ω in Assumption 1 is an identity matrix and α = 1. The Brownian motions are generated from 10,000 observations and the simulations repeated 10,000 times.

A second-order stationary random variable x _t has the following features: (a) 𝔼(x _t ) = μ, − ∞ < μ < ∞; (b) 𝔼[(x _t  − μ)²] = σ² < ∞; and 𝔼[(x _t  − μ)(x _t−j  − μ)] = γ _j , − ∞ < γ _j < ∞.

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lecr