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Testing Linearity in Cointegrating Relations with an Application to Purchasing Power Parity

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Abstract

This paper develops a linearity test that can be applied to cointegrating relations. We consider the widely used RESET specification test and show that when this test is applied to nonstationary time series its asymptotic distribution involves a mixture of noncentral chi^2 distributions, which leads to severe size distortions in conventional testing based on the central chi^2. Nonstationarity is shown to introduce two bias terms in the limit distribution, which are the source of the size distortion in testing. Appropriate corrections for this asymptotic bias leads to a modified version of the RESET test which has a central chi^2 limit distribution under linearity. The modified test has power not only against nonlinear cointegration but also against the absence of cointegration. Simulation results reveal that the modified test has good size infinite samples and reasonable power against many nonlinear models as well as models with no cointegration, confirming the analytic results. In an empirical illustration, the linear purchasing power parity (PPP) specification is tested using US, Japan, and Canada monthly data after Bretton Woods. While commonly used ADF and PP cointegration tests give mixed results on the presence of linear cointegration in the series, the modified test rejects the null of linear PPP cointegration.

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  • Seung Hyun Hong & Peter C. B. Phillips, 2005. "Testing Linearity in Cointegrating Relations with an Application to Purchasing Power Parity," Cowles Foundation Discussion Papers 1541, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1541
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    Cited by:

    1. Nidhi Choudhary & Girish K. Nair & Harsh Purohit, 2015. "Volatility In Copper Prices In India," Annals of Financial Economics (AFE), World Scientific Publishing Co. Pte. Ltd., vol. 10(02), pages 1-26, December.
    2. Alfred Haug & Syed Basher, 2011. "Linear or nonlinear cointegration in the purchasing power parity relationship?," Applied Economics, Taylor & Francis Journals, vol. 43(2), pages 185-196.
    3. Phillips, Peter C.B. & Li, Degui & Gao, Jiti, 2017. "Estimating smooth structural change in cointegration models," Journal of Econometrics, Elsevier, vol. 196(1), pages 180-195.
    4. Marmer, Vadim, 2008. "Nonlinearity, nonstationarity, and spurious forecasts," Journal of Econometrics, Elsevier, vol. 142(1), pages 1-27, January.
    5. Hongjun Li & Zhongjian Lin & Cheng Hsiao, 2015. "Testing purchasing power parity hypothesis: a semiparametric varying coefficient approach," Empirical Economics, Springer, vol. 48(1), pages 427-438, February.
    6. Wang, Qiying & Phillips, Peter C.B., 2009. "Asymptotic Theory For Local Time Density Estimation And Nonparametric Cointegrating Regression," Econometric Theory, Cambridge University Press, vol. 25(03), pages 710-738, June.
    7. Dong, Chaohua & Gao, Jiti & Tjøstheim, Dag & Yin, Jiying, 2017. "Specification testing for nonlinear multivariate cointegrating regressions," Journal of Econometrics, Elsevier, vol. 200(1), pages 104-117.
    8. Arai, Yoichi, 2016. "Testing For Linearity In Regressions With I(1) Processes," Hitotsubashi Journal of Economics, Hitotsubashi University, vol. 57(1), pages 111-138, June.
    9. Chen, Haiqiang & Fang, Ying & Li, Yingxing, 2015. "Estimation And Inference For Varying-Coefficient Models With Nonstationary Regressors Using Penalized Splines," Econometric Theory, Cambridge University Press, vol. 31(04), pages 753-777, August.
    10. Jia Chen & Jiti Gao & Degui Li & Zhengyan Lin, 2015. "Specification testing in nonstationary time series models," Econometrics Journal, Royal Economic Society, vol. 18(1), pages 117-136, February.
    11. Giovanni Millo, 2016. "The Income Elasticity of Nonlife Insurance: A Reassessment," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 83(2), pages 335-362, June.
    12. Stypka, Oliver & Wagner, Martin & Grabarczyk, Peter & Kawka, Rafael, 2017. "The Asymptotic Validity of "Standard" Fully Modified OLS Estimation and Inference in Cointegrating Polynomial Regressions," Economics Series 333, Institute for Advanced Studies.
    13. Li, Haiqi & Zheng, Chaowen & Guo, Yu, 2016. "Estimation and test for quantile nonlinear cointegrating regression," Economics Letters, Elsevier, vol. 148(C), pages 27-32.
    14. Chaohua Dong & Jiti Gao, 2014. "Specification Testing in Structural Nonparametric Cointegration," Monash Econometrics and Business Statistics Working Papers 2/14, Monash University, Department of Econometrics and Business Statistics.
    15. Derek Bond & Michael J. Harrison & Edward J. O'Brien, 2006. "Purchasing Power Parity: The Irish Experience Re-visited," Trinity Economics Papers tep200615, Trinity College Dublin, Department of Economics.
    16. repec:eee:reveco:v:49:y:2017:i:c:p:313-326 is not listed on IDEAS
    17. repec:wyi:journl:002195 is not listed on IDEAS
    18. Liew, Venus Khim-Sen & Ling, Tai-Hu & Chia, Ricky Chee-Jiun & Yoon, Gawon, 2012. "On the application of the rank tests for nonlinear cointegration to PPP: The case of Papua New Guinea," Economic Modelling, Elsevier, vol. 29(2), pages 326-332.

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    Keywords

    Nonlinear cointegration; Specification test; RESET test; Noncentral chi^2 distribution;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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