Kleijn-van-Dijk (2002) A Bayesian Analysis of the PPP Puzzle Using an Unobserved Components Model

Summary

Kleijn and Van Dijk implement Engel's (2000) suggestion that the long-run equilibrium real exchange rate varies slowly over time, using a Bayesian unobserved components (UC) model of the purchasing power parity (PPP) real exchange rate with an I(2) smooth trend and two stochastic cycles. Applied to DEM/USD and FF/DEM monthly rates (1973:01–1998:12, $T=312$), Bayes factors decisively reject a constant mean ($BF \approx 0.0004$), confirming time-variation. Once the flexible mean is accounted for, the half-life of PPP deviations drops from the canonical 3–5 years to approximately one year — providing a resolution to the PPP puzzle without abandoning mean reversion.

Key Claims

• Standard unit root tests assume a fixed long-run mean; if the equilibrium itself varies slowly (as Engel 2000 argues), tests are misspecified — inflating apparent persistence.
• The preferred model decomposes $q_t = \mu_t + \psi_{1,t} + \psi_{2,t}$, where $\mu_t$ follows a smooth I(2) trend and $\psi_{i,t}$ are damped stochastic sinusoids (cycles). The I(2) trend is preferred over an I(1) random walk because its variance is estimated more precisely, avoiding the low-power problem that afflicts Kwiatkowski-Phillips-Schmidt-Shin (KPSS) testing.
• Bayes factors for $H_0: \sigma_\zeta^2 = 0$ (constant mean = strict PPP): 0.000357 (DEM/USD) and 0.000553 (FF/DEM) — overwhelming evidence of time-varying equilibrium.
• Posterior half-life medians: 12 months (DEM/USD) and 11 months (FF/DEM), vs. 3–5 years in the standard literature. The mechanism: the trend absorbs the slow low-frequency variation, leaving only fast-reverting deviations in $\psi_{1,t} + \psi_{2,t}$.
• Impulse responses are non-monotonic: shocks are initially amplified before decaying (oscillatory behavior due to cycle structure), consistent with Cheung-Lai (2000).
• DEM/USD has trend variance $\sigma_\zeta \approx 19\times$ larger than FF/DEM, reflecting tighter French-German monetary linkage under the European Monetary System.
• Posterior cycle period medians: 12.3 and 71.6 months (DEM/USD); 9.6 and 57.1 months (FF/DEM). The longer cycle (near-unit-root: $\hat\rho_2 \approx 0.97$) is the main source of PPP persistence.
• Bayes factors computed via Laplace approximation (Kass et al. 1990) rather than Chib's (1995) method, because integration constants are unknown for some conditional densities in this model.

Concepts Introduced or Extended

• Purchasing Power Parity
• Unit Root Tests
• Kalman Filter

Entities Mentioned

• Richard Kleijn
• Herman K. van Dijk

Quotes

"Including a flexible mean is the main reason that we find half-lives around one year for both real exchange rates. This is much shorter than the half-lives reported in previous studies which are in the range of 3 to 5 years."

"The Bayes factors in Table 5.3 provide very strong evidence against $\sigma_\zeta^2 = 0$."

My Take

The paper is a clean Bayesian implementation of a compelling economic idea: if the PPP equilibrium drifts, standard tests are measuring the wrong thing. The UC + Gibbs sampler approach is well-suited to the problem, and the Bayesian posterior for the half-life is considerably more informative than classical point estimates. The main limitation is that the I(2) trend is a flexible curve-fitting device without structural interpretation — the authors acknowledge the approach is "pragmatic in nature." Whether the time-varying mean reflects Balassa-Samuelson effects, pricing-to-market, or something else is not addressed. The working paper dates from 2002; the final journal version may differ slightly. Computations in Ox + SsfPack.