Comment on: Pseudo-True SDFs in Conditional Asset Pricing Models. Comparing Fixed- versus Vanishing-Bandwidth Estimators of Pseudo-True SDFs

The paper by Antoine, Proulx, and Renault (2018) (APR) deals with the econometric definition, economic interpretation, and statistical estimation of the pseudo-true stochastic discount factor (SDF) in misspecified conditional asset pricing models. The paper revolves around fundamental issues like the role of conditioning information and omitted risk factors, and has non-trivial interactions with the current debate in the literature on the impact of weak factors (weak identification) for assessing asset pricing models. Building on, and substantially extending, previous contributions in the literature, the approach of the authors to define a pseudo-true SDF relies on the minimizers of econometric criteria based on a conditional version of the Hansen–Jagannathan (HJ) distance, that is, an average across states of squared conditional pricing errors. The authors provide an insightful discussion of the economic interpretation of pseudo-true SDFs. APR advocate the use of a fixed bandwidth (i.e., independent of the sample size) when estimating the conditional pricing errors by kernel regression methods to facilitate statistical analysis. This route leads to bandwidth-dependent pseudo-true SDF parameters and estimators thereof. In our discussion, we investigate the different definitions of pseudo-true SDFs and interpret the fixed-bandwidth proposal as a model calibration which down-weights highfrequency Fourier components of the conditional pricing errors (Section 1). We compare the statistical properties of pseudo-true SDF parameters’ estimators relying on vanishing versus fixed bandwidth, and provide a condition under which the former have a smaller asymptotic variance than the latter (or viceversa). We look at these topics through the lens of misspecified conditional linear SDF models in which priced risk factors are omitted using both simulated and real data (Section 3). We skip regularity conditions and relegate some technical derivations in the Appendix of the paper.