Deviations from PPP: a micro-macro disconnect?

Purchasing power parity (PPP) implies that adjusted for exchange rates, similar goods should have the same price across countries. However, this does not hold true, and for extended periods. Why this happens is a puzzle, which is even more confounding considering that deviations from aggregate PPP have a longer half-life than deviations from good-level PPP. Jean Imbs, Haroon Mumtaz, Morten Ravn and Hélène Rey have claimed that this apparent micro-macro disconnect can be explained by a composition effect.

Not so fast, say Paul Bergin, Reuven Glick and Jyh-Lin Wu. Their point is that macro- and micro-PPP follow different processes, the first following aggregate shocks and the second mostly relative-price shocks. Using a vector error correction model, which controls for these shocks, they show then that the remaining deviations are consistent between macro and micro data and have very similar half-lives. This means that the previous explanation based on the heterogeneity of goods and different price stickiness does not hold water. Back to the drawing board...