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COINTEGRATING POLYNOMIAL REGRESSIONS: ROBUSTNESS OF FULLY MODIFIED OLS

Published online by Cambridge University Press:  15 February 2024

Oliver Stypka ,
Martin Wagner Open the ORCID record for Martin Wagner [Opens in a new window] ,
Peter Grabarczyk  and
Rafael Kawka
Oliver Stypka
Affiliation:
Flossbach von Storch
Martin Wagner*
Affiliation:
University of Klagenfurt, Bank of Slovenia, and Institute for Advanced Studies
Peter Grabarczyk
Affiliation:
TU Dortmund University
Rafael Kawka
Affiliation:
TU Dortmund University
*
Address correspondence to Martin Wagner, Department of Economics, University of Klagenfurt, Klagenfurt, Austria; e-mail: [email protected].
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Abstract

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Cointegrating polynomial regressions (CPRs) include deterministic variables, integrated variables, and their powers as explanatory variables. Based on a novel kernel-weighted limit result and a novel functional central limit theorem, this paper shows that the fully modified ordinary least squares (FM-OLS) estimator of Phillips and Hansen (1990, Review of Economic Studies 57, 99–125) is robust to being used in CPRs. Being used in CPRs refers to a widespread empirical practice that treats the integrated variables and their powers, incorrectly, as a vector of integrated variables and uses textbook FM-OLS. Robustness means that this “formal” FM-OLS practice leads to a zero mean Gaussian mixture limiting distribution that coincides with the limiting distribution of the Wagner and Hong (2016, Econometric Theory 32, 1289–1315) application of the FM estimation principle to the CPR case. The only restriction for this result to hold is that all integrated variables to power one are included as regressors. Even though simulation results indicate performance advantages of the Wagner and Hong (2016, Econometric Theory 32, 1289–1315) estimator, partly even in large samples, the results of the paper give an asymptotic foundation to “formal” FM-OLS and thus enlarge the usability of the Phillips and Hansen (1990, Review of Economic Studies 57, 99–125) estimator implemented in many software packages.

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 21
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Footnotes

All authors acknowledge partial financial support from the Collaborative Research Center 823: Statistical Modelling of Nonlinear Dynamic Processes supported by the Deutsche Forschungsgemeinschaft (DFG). Martin Wagner furthermore acknowledges research support from the Jubilaumsfonds of the Oesterreichische Nationalbank via several grants. We gratefully acknowledge the helpful comments of, in particular, the editor Peter C. B. Phillips, as well as of a co-editor and four reviewers. We furthermore are grateful for the many comments received from participants in numerous conferences and seminars since 2016. This paper solely reflects the views of the authors and not necessarily those of Flossbach von Storch, the Bank of Slovenia, or the European System of Central Banks. On top of this, the usual disclaimer applies.

References

REFERENCES

Chan, N., & Wang, Q. (2015). Nonlinear regressions with nonstationary time series. Journal of Econometrics , 185, 182–195.CrossRefGoogle Scholar
Chang, Y., Park, J. Y., & Phillips, P. C. B. (2001). Nonlinear econometric models with cointegrated and deterministically trending regressors. Econometrics Journal , 4, 1–36.CrossRefGoogle Scholar
Darvas, Z. (2008). Estimation bias and inference in overlapping autoregressions: Implications for the target-zone literature. Oxford Bulletin of Economics and Statistics , 70, 1–22.CrossRefGoogle Scholar
Di Iorio, F., & Fachin, S. (2022). Fiscal reaction functions for the advanced economies revisited. Empirical Economics , 62, 2865–2891.CrossRefGoogle Scholar
Grabarczyk, P., Wagner, M., Frondel, M., & Sommer, S. (2018). A cointegrating polynomial regression analysis of the material Kuznets curve hypothesis. Resources Policy , 57, 236–245.CrossRefGoogle Scholar
Grossman, G. M., & Krueger, A. B. (1995). Economic growth and the environment. Quarterly Journal of Economics , 110, 353–377.CrossRefGoogle Scholar
Ibragimov, R., & Phillips, P. C. B. (2008). Regression asymptotics using martingale convergence methods. Econometric Theory , 24, 888–947.CrossRefGoogle Scholar
Jansson, M. (2002). Consistent covariance matrix estimation for linear processes. Econometric Theory , 18, 1449–1459.CrossRefGoogle Scholar
Kasparis, I. (2008). Detection of functional form misspecification in cointegrating relations. Econometric Theory , 24, 1373–1403.CrossRefGoogle Scholar
Kuznets, S. (1955). Economic growth and income inequality. American Economic Review , 45, 1–28.Google Scholar
Labson, B. S., & Crompton, P. L. (1993). Common trends in economic activity and metals demand: Cointegration and the intensity of use debate. Journal of Environmental Economics and Management , 25, 147–161.CrossRefGoogle Scholar
Le Gall, J. F. (2016). Brownian motion, martingales, and stochastic calculus . Springer International Publishing.CrossRefGoogle Scholar
Liang, H., Phillips, P. C. B., Wang, H., & Wang, Q. (2016). Weak convergence to stochastic integrals for econometric applications. Econometric Theory , 32, 1349–1375.CrossRefGoogle Scholar
Malenbaum, W. (1978). World demand for raw materials in 1985 and 2000 . McGraw-Hill, EMJ Mining Information Services.Google 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. (1987). Time series regression with a unit root. Econometrica , 55, 277–301.CrossRefGoogle Scholar
Phillips, P. C. B. (1989). Partially identified econometric models. Econometric Theory , 5, 181–240.CrossRefGoogle Scholar
Phillips, P. C. B. (1991a). Optimal inference in cointegrated systems. Econometrica , 59, 283–306.CrossRefGoogle Scholar
Phillips, P. C. B. (1991b). Spectral regression for cointegrated time series. In Barnett, W., Powell, J., & Tauchen, G. (Eds.), Nonparametric and semiparametric methods in econometrics and statistics (pp. 413–436). Cambridge University Press.Google Scholar
Phillips, P. C. B. (1995). Fully modified least squares and vector autoregression. Econometrica , 63, 1023–1078.CrossRefGoogle Scholar
Phillips, P. C. B., & Hansen, B. E. (1990). Statistical inference in instrumental variables regression with $I(1)$ processes. Review of Economic Studies , 57, 99–125.CrossRefGoogle Scholar
Shin, Y. (1994). A residual-based test for the null of cointegration against the alternative of no cointegration. Econometric Theory , 10, 91–115.CrossRefGoogle Scholar
Stypka, O., & Wagner, M. (2019). The Phillips unit root tests for polynomials of integrated processes revisited. Economics Letters , 176, 109–113.CrossRefGoogle Scholar
Svensson, L. E. O. (1992). An interpretation of recent research on exchange rate target zones. Journal of Economic Perspectives , 6, 119–144.CrossRefGoogle Scholar
Wagner, M. (2015). The environmental Kuznets curve, cointegration and nonlinearity. Journal of Applied Econometrics , 30, 948–967.CrossRefGoogle Scholar
Wagner, M. (2023). Residual-based cointegration and non-cointegration tests for cointegrating polynomial regressions. Empirical Economics , 65, 1–31.CrossRefGoogle Scholar
Wagner, M., & Hong, S. H. (2016). Cointegrating polynomial regressions: Fully modified OLS estimation and inference. Econometric Theory , 32, 1289–1315.CrossRefGoogle Scholar
Yandle, B., Bhattarai, M., & Vijayaraghavan, M. (2004). Environmental Kuznets curves: A review of findings, methods, and policy implications. PERC Research Study , 2(1), 1–38.Google Scholar
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