Econometric issues with Laubach and Williams' estimates of the natural rate of interest

Holston, Laubach and Williams' (2017) estimates of the natural rate of interest are driven by the downward trending behaviour of 'other factor' $z_{t}$. I show that their implementation of Stock and Watson's (1998) Median Unbiased Estimation (MUE) to determine the size of the $\lambda _{z}$ parameter which drives this downward trend in $z_{t}$ is unsound. It cannot recover the ratio of interest $\lambda _{z}=a_{r}\sigma _{z}/\sigma _{\tilde{y}}$ from MUE required for the estimation of the full structural model. This failure is due to an 'unnecessary' misspecification in Holston et al.'s (2017) formulation of the Stage 2 model. More importantly, their implementation of MUE on this misspecified Stage 2 model spuriously amplifies the point estimate of $\lambda _{z}$. Using a simulation experiment, I show that their procedure generates excessively large estimates of $\lambda _{z}$ when applied to data generated from a model where the true $\lambda _{z}$ is equal to zero. Correcting the misspecification in their Stage 2 model and the implementation of MUE leads to a substantially smaller $\lambda _{z}$ estimate, and with this, a more subdued downward trending influence of 'other factor' $z_{t}$ on the natural rate. Moreover, the $\lambda _{z}$ point estimate is statistically highly insignificant, suggesting that there is no role for 'other factor' $z_{t}$ in this model. I also discuss various other estimation issues that arise in Holston et al.'s (2017) model of the natural rate that make it unsuitable for policy analysis.