Purchasing Power Parity

Definition

Purchasing power parity (PPP) states that the real exchange rate — the nominal exchange rate adjusted for relative price levels — should be constant (absolute PPP) or stationary (relative PPP) in the long run. If PPP holds, the log real exchange rate $q_t = s_t + p_t^* - p_t$ is $I(0)$ with a fixed mean, and the nominal rate adjusts to offset price differentials.

Key Ideas

Absolute PPP: The law of one price holds for all goods; the real exchange rate equals a constant (normalized to 1). Violated in practice by trade costs, non-traded goods, and product differentiation.
Relative PPP: Changes in the nominal exchange rate equal changes in relative price levels; $\Delta s_t = \pi_t - \pi_t^*$ (inflation differential). Implies $q_t$ is stationary around a constant mean — the weaker, more empirically testable form.
The PPP puzzle (Rogoff 1996): Real exchange rates are highly volatile in the short run (close to nominal rate volatility) but mean-revert very slowly. Estimated half-lives of PPP deviations are 3–5 years even when stationarity can be formally established — far too slow for goods-market arbitrage to explain.
Engel-Kim (1999): 106 years of U.S./U.K. monthly data, modeled as a permanent/transitory state-space decomposition with Markov-switching variance, yields a large permanent component — directly rejecting long-run PPP. Johansen cointegration test on $(s_t, p_t, p_t^*)$ with constrained vector $(1,-1)$ fails to support cointegration.
Balassa-Samuelson effect (Samuelson 1964): Countries with faster productivity growth in traded goods pay higher wages economy-wide, raising the relative price of non-traded goods. This creates a permanent appreciation of the real exchange rate relative to PPP — the equilibrium real exchange rate drifts with productivity differentials.

How It Works

The Real Exchange Rate

Let $s_t$ = log nominal exchange rate (domestic price of foreign currency), $p_t$ = log domestic consumer price index (CPI), $p_t^*$ = log foreign CPI. The log real exchange rate is: $q_t = s_t + p_t^* - p_t$

PPP requires $q_t \sim I(0)$ . Equivalently, $(s_t, p_t, p_t^*)$ must be cointegrated with vector $(1,-1,1)$ — the nominal rate and relative price level are bound in a long-run relationship.

Testing PPP: Three Approaches

Unit root tests on $q_t$ : Augmented Dickey-Fuller (ADF)/Phillips-Perron (PP) tests of $H_0: q_t \sim I(1)$ . Typically fail to reject even over long samples. Power is low under near-unit-root alternatives, and size is distorted under heteroskedasticity — Engel-Kim (1999) show the ADF test has actual size of 19% at the nominal 5% critical value when data are generated by their Markov-switching (MS) state-space model.
Johansen cointegration test on $(s_t, p_t, p_t^*)$ : Tests whether the cointegrating space contains the PPP vector $(1,-1,1)$ . Engel-Kim (1999) fail to reject no cointegration for U.S./U.K. over 1890–1995.
Panel unit root tests (Frankel-Rose 1996): Pool data across countries to boost power against slow mean-reversion. Panel tests find more support for PPP at long horizons, but cross-sectional dependence can inflate rejection rates spuriously.

Permanent/Transitory Decomposition (Engel-Kim 1999)

Rather than testing stationarity directly, Engel and Kim decompose $q_t$ into: $q_t = \underbrace{y_t}_{\text{permanent}} + \underbrace{x_t}_{\text{transitory}}, \quad y_t = y_{t-1} + \eta_t, \quad x_t = \phi x_{t-1} + \varepsilon_t$ with Markov-switching variance for both. The large estimated variance of permanent innovations $\eta_t$ relative to transitory innovations $\varepsilon_t$ is direct evidence that the exchange rate is not stationary — PPP fails even over 106 years.

Time-Varying Mean Decomposition (Kleijn-Van Dijk 2002)

Kleijn and Van Dijk (2002) implement Engel's (2000) suggestion that the long-run PPP equilibrium drifts slowly over time. The log real exchange rate is decomposed as: $q_t = \mu_t + \psi_{1,t} + \psi_{2,t}$ where $\mu_t$ is a smooth I(2) trend (local linear, second-difference disturbance $\zeta_t \sim N(0,\sigma_\zeta^2)$ ) and $\psi_{1,t}$ , $\psi_{2,t}$ are damped stochastic sinusoids (cycles) with damping factors $\rho_i$ and frequencies $\lambda_i$ . The I(2) specification is preferred over I(1) because its variance $\sigma_\zeta^2$ is estimated more precisely, avoiding the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) low-power problem that afflicts random-walk trends.

Applied to DEM/USD and FF/DEM monthly series (Jan 1973–Dec 1998, $T=312$ ), Bayes factors decisively reject the constant-mean null ( $BF \approx 0.0004$ ), confirming time variation in the PPP equilibrium. The key finding: once the flexible trend absorbs low-frequency variation, the remaining transitory component reverts fast — posterior half-life medians fall to 12 months (DEM/USD) and 11 months (FF/DEM), versus the canonical 3–5 years. The mechanism is decomposition rather than faster reversion: standard estimates inflate persistence by attributing trend drift to the stationary component. Impulse responses are non-monotonic (initially amplified before decaying) due to the oscillatory cycle structure. The DEM/USD trend variance $\hat\sigma_\zeta$ is ~19 $\times$ larger than FF/DEM, consistent with the tighter French-German monetary coordination under the European Monetary System (EMS).

Why It Matters

PPP is the foundational long-run equilibrium condition in open-economy macroeconomics; its failure means real exchange rates are not anchored by goods arbitrage alone.
The PPP puzzle motivates models incorporating non-traded goods (Balassa-Samuelson), pricing-to-market, and international risk sharing.
Exchange rate models that impose PPP as a long-run constraint are mis-specified if the real exchange rate has a permanent component.

Open Questions

Whether PPP failure reflects genuine non-stationarity or extremely slow mean-reversion ( $t_{1/2} > 10$ years) that is observationally equivalent to a unit root in century-long samples.
Goods-level data show stronger PPP support than aggregate CPI data; the puzzle may be primarily a non-traded goods phenomenon.
New-Keynesian models can generate slow mean-reversion under nominal rigidities and local-currency pricing, but require implausibly large persistent real shocks to match the volatility (Rogoff 1996).
What drives the time-varying equilibrium? Kleijn-Van Dijk (2002) establish that the PPP equilibrium drifts but remain agnostic on the source — Balassa-Samuelson productivity differentials, pricing-to-market, or other structural factors all remain candidate explanations.