Corrigendum to ‘A Gaussian approach for continuous time models of short‐term interest rates’

Abstract

This note corrects an error in Yu and Phillips (2001, hereafter YP) where a time transformation was used to induce Gaussian disturbances in the discrete time version of a continuous time model. The error occurs in equations (3.7)-(3.10) of YP where the Dambis, Dubins-Schwarz (hereafter DDS) theorem was applied to the quadratic variation of the error term in Equation (3.6), [M ]h, in order to induce a sequence of stopping time points {tj} for which the disturbance term in (3.10) follows a normal distribution, facilitating Gaussian estimation. To apply the DDS theorem, the original error process, M(h) needs to be a continuous martingale with finite quadratic variation. In YP, it was assumed that M(h) was a continuous martingale. This note shows that the assumption is generally not warranted and so the DDS theorem does not induce a Brownian motion. However, a simple decomposition splits the error process into a trend component and a continuous martingale process. The DDS theorem can then be applied to the detrended error process, generating a Brownian motion residual. With the presence of the time varying trend component, the discrete time model is heteroskedastic and the regressor is endogenous. The endogeneity is addressed using an instrumental variable We thank Joon Park for bringing to our attention an error in Yu and Phillips (2001) and for helpful discussion on the same issue, and the editor and a referee for helpful comments. Phillips gratefully acknowledges support from the NSF under Grant Nos. SES 06-47086 and SES 09-56687. Yale University, University of Auckland, University of Southampton, Singapore Management University. Cowles Foundation for Research in Economics, Yale University, Box 208281, Yale Station, New Haven, Connecticut 06520-8281. Email: peter.phillips@yale.edu. School of Economics and Sim Kee Boon Institute for Financial Economics, Singapore Management University; email: yujun@smu.edu.sg. URL: http://www.mysmu.edu/faculty/yujun/default.htm.