Short Memory and the PPP-puzzle

The so-called purchasing-power parity (PPP) puzzle takes one of two forms. In its first form, early tests of the PPP-hypothesis failed to reject unit roots in real exchange rates, thus rejecting the hypothesis of PPP holding in the long term. In the more recent literature, the literature on the PPP puzzle focused on stochastic real exchange rate models that allow long term PPP to hold. However, early estimates of such models suggested that the real exchange rates converge to their long-term PPP levels at a very slow rate (usually expressed as a half-life of 3~5 years), c.f. Rogoff (1996). Attempts to solve the second form of the PPP-puzzle have been forthcoming in recent years, mostly by using non-linear stochastic models of real exchange rates. However, despite the introduction of non-linearities, the literature has continued to focus on the notion of "half-life" as a measure of persistence. We argue the half-life measure is only appropriate in linear settings, failing to capture the richness of non-linear dynamics introduced in the more recent literature. We propose operational measures of persistence in such non-linear models, which we label short-memory in distribution (SMD, which is similar to the notion of -mixing), and short-memory in mean (SMM, c.f. Granger (1995)). We show that focusing on a simple notion such as "half-life" can be very misleading. In particular, we show that it is possible to match desired "half-lives" for any of the most popular non-linear models recently proposed in the literature. However, we show that those models imply very different persistent properties for the same half-life. We conclude that depending on the models and criteria selected for investigating the PPP puzzle, the puzzle may be in the eye of the beholder.