Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T17:56:25.007Z Has data issue: false hasContentIssue false
  • English
  • Français

NONLINEAR COINTEGRATING REGRESSION UNDER WEAK IDENTIFICATION

Published online by Cambridge University Press:  25 November 2011

Xiaoxia Shi  and
Peter C.B. Phillips
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Research Article
Information
Econometric Theory , Volume 28 , Issue 3 , June 2012 , pp. 509 - 547
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Our thanks to two referees and the co-editor, Pentti Saikkonen, for helpful comments on the original version. The paper originated in a 2008 Yale take-home examination. The first complete draft was circulated in December 2009. Xiaoxia Shi gratefully acknowledges support from the Cowles Foundation via a Carl Arvid Anderson Fellowship at Yale University. Peter Phillips thanks the NSF for support under grant SES 06-47086 and 09-56687.

References

REFERENCES

Andrews, D.W.K. & Soares, G. (2010) Inference for parameters defined by moment inequalities using generalized moment selection. Econometrica 78, 119–157.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Campbell, J.Y. & Yogo, M. (2006) Efficient tests of stock return predictability. Journal of Financial Economics 8, 27–60.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2011) Endogeneity in nonlinear regressions with integrated time series. Econometric Reviews 30, 51–87.CrossRefGoogle Scholar
Cheng, X. (2008) Robust Confidence Intervals in Nonlinear Regression under Weak Identification. Manuscript, Department of Economics, Yale University.Google Scholar
Cheng, X. (2010) Essays on weak identification and cointegrating rank selection. Ph.D. Dissertation, Yale University.Google Scholar
de Jong, R. (2002) Nonlinear Estimators with Integrated Regressors but without Exogeneity. Working paper, Michigan State University.Google Scholar
Giacomini, R. & Granger, C.W.J. (2004) Aggregation of space-time processes. Journal of Econometrics 118, 7–26.CrossRefGoogle Scholar
Granger, C.W.J. (1987) Implications of aggregation withc common factors. Econometric Theory 3, 208–222.CrossRefGoogle Scholar
Hansen, B.E. (1996) Stochastic equicontinuity for unbounded dependent heterogeneous arrays. Econometric Theory 12, 347–359.CrossRefGoogle Scholar
Ibragimov, R. & Phillips, P.C.B. (2008) Regression asymptotics using martingale convergence methods. Econometric Theory 24, 888–947.CrossRefGoogle Scholar
Jeganathan, P. (2008) Limit Theorems for Functionals of Sums That Converge to Fractional Brownian and Stable Motions. Cowles Foundation Discussion paper 1649, Yale University.Google Scholar
Kallenberg, O. (2001) Foundations of Modern Probability, 2nd ed. Springer-Verlag.Google Scholar
Kim, J. & Pollard, D. (1990) Cube root asymptotics. Annals of Statistics 18, 191–219.CrossRefGoogle Scholar
Kurtz, T.G. & Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 1035–1070.CrossRefGoogle Scholar
Marmer, V. (2008) Nonlinearity, nonstationarity, and spurious forecasts. Journal of Econometrics 142, 1–27.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (1999) Asymptotics for nonlinear transformations of integrated time series. Econometric Theory 15, 269–298.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (2001) Nonlinear regressions with integrated time series. Econometrica 69, 117–161.CrossRefGoogle Scholar
Phillips, P.C.B. (1989) Partially identified econometric models. Econometric Theory 5, 181–240.CrossRefGoogle Scholar
Phillips, P.C.B. & Magdalinos, T. (2009) Unit root and cointegrating limit theory when initialization is in the infinite past. Econometric Theory 25, 1682–1715.CrossRefGoogle Scholar
Pollard, D. (1990) Empirical Process Theory and Applications. Institute of Mathematical Statistics.CrossRefGoogle Scholar
Stock, J.H. & Wright, J. (2000) GMM with weak identification. Econometrica 68, 1055–1096.CrossRefGoogle Scholar
van der Vaart, A. & Wellner, J. (1996) Weak Convergence and Empirical Processes: With Applications to Statistics. Springer-Verlag.CrossRefGoogle Scholar
Wang, Q. & Phillips, P.C.B. (2009) Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25, 710–738.CrossRefGoogle Scholar