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

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.