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Spurious OLS Estimators of Detrending Method by Adding a Linear Trend in Difference-Stationary Processes - A Mathematical Proof and its Verification by Simulation

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  • Rémy Herrera

    (CNRS - Centre National de la Recherche Scientifique, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Long Zhiming

    (THU - Tsinghua University [Beijing])

Abstract

first_page settings Order Article Reprints Open AccessArticle Spurious OLS Estimators of Detrending Method by Adding a Linear Trend in Difference-Stationary Processes—A Mathematical Proof and Its Verification by Simulation by Zhiming LONG 1,* and Rémy HERRERA 2 1 Research Center for College Moral Education, Tsinghua University, 307C Shanzhai Building, Tsinghua University, Beijing 100084, China 2 CNRS (National Center for Scientific Research)—UMR 8174 Centre d'Économie de la, Maison des Sciences Economiques de l'Université de Paris 1 Panthéon-Sorbonne 106-112 boulevard de l'Hôpital, 75013 Paris, France * Author to whom correspondence should be addressed. Mathematics 2020, 8(11), 1931; https://doi.org/10.3390/math8111931 Received: 2 September 2020 / Revised: 23 September 2020 / Accepted: 25 September 2020 / Published: 2 November 2020 (This article belongs to the Section Dynamical Systems) Download Browse Figures Versions Notes Abstract Adding a linear trend in regressions is a frequent detrending method in economic literatures. The traditional literatures pointed out that if the variable considered is a difference-stationary process, then it will artificially create pseudo-periodicity in the residuals. In this paper, we further show that the real problem might be more serious. As the Ordinary Least Squares (OLS) estimators themselves are of such a detrending method is spurious. The first part provides a mathematical proof with Chebyshev's inequality and Sims–Stock–Watson's algorithm to show that the OLS estimator of trend converges toward zero in probability, and the other OLS estimator diverges when the sample size tends to infinity. The second part designs Monte Carlo simulations with a sample size of 1,000,000 as an approximation of infinity. The seed values used are the true random numbers generated by a hardware random number generator in order to avoid the pseudo-randomness of random numbers given by software. This paper repeats the experiment 100 times, and gets consistent results with mathematical proof. The last part provides a brief discussion of detrending strategies.

Suggested Citation

  • Rémy Herrera & Long Zhiming, 2020. "Spurious OLS Estimators of Detrending Method by Adding a Linear Trend in Difference-Stationary Processes - A Mathematical Proof and its Verification by Simulation," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-03083782, HAL.
    • Handle: RePEc:hal:cesptp:hal-03083782
    DOI: 10.3390/math8111931
    Note: View the original document on HAL open archive server: https://hal.science/hal-03083782
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