The Long-Run U.S./U.K. Real Exchange Rate

Summary

Engel and Kim (1999) study the U.S./U.K. real exchange rate over 106 years (January 1890 – November 1995) using a state-space model that decomposes the series into a permanent random-walk component and a transitory first-order autoregressive (AR(1)) component, each with Markov-switching variance. The key finding is that the permanent component is large and dominates — contradicting long-run purchasing power parity (PPP). The transitory component's three variance regimes align closely with the international monetary regime (gold standard and Bretton Woods = low variance; modern float = high variance). The model is estimated by Gibbs sampling using the Kim (1994) smoother for the latent permanent component. Standard augmented Dickey-Fuller (ADF) unit root tests are shown to have grossly distorted size (19% actual vs. 5% nominal) under the estimated data-generating process (DGP), explaining why classical unit root tests fail to detect the permanent component.

Key Claims

• Model: $q_t = y_t + x_t$ where $y_t = y_{t-1} + \eta_t$ (permanent, random walk with switching variance $\sigma_\eta^2(s_t^P)$, $s_t^P \in \{1,2\}$) and $x_t = \phi x_{t-1} + \varepsilon_t$ (transitory, AR(1) with switching variance $\sigma_\varepsilon^2(s_t^T)$, $s_t^T \in \{1,2,3\}$). The two Markov chains evolve independently.
• Gibbs sampling estimation: Draw in turn from $p(\theta | s^P, s^T, y, q)$, $p(s^P | \theta, s^T, y, q)$, $p(s^T | \theta, s^P, y, q)$, and $p(y | \theta, s^P, s^T, q)$, using the Kim (1994) smoother for the latent permanent component. Full Bayesian treatment of the Markov-switching (MS) state-space model with 10,000-iteration chains.
• PPP rejection: The variance of innovations in the permanent component is much larger than the variance of innovations in the transitory component across all regimes. Since PPP requires the real exchange rate to be $I(0)$ (stationary, no permanent component), the finding strongly contradicts long-run PPP.
• Regime alignment with monetary history: The transitory component's three states correspond to: state 1 (very low variance) = gold standard 1890–1914 and Bretton Woods 1947–1971; state 2 (medium variance) = interwar fluctuations; state 3 (high variance) = post-1973 float. The permanent component's high-variance state corresponds to the two World Wars and the 1980s dollar episode.
• Cointegration test: A Johansen test with constrained cointegrating vector $(1,-1)$ on U.S. prices, U.K. prices, and the nominal exchange rate does not support cointegration — consistent with a non-stationary permanent component in the real exchange rate.
• ADF size distortion: A Monte Carlo exercise with 1,000 replications calibrated to the estimated model shows the ADF test has actual size of 19.08% at the nominal 5% critical value. Markov-switching heteroskedasticity in the DGP inflates the ADF rejection rate under the null of a unit root, making standard inference misleading.
• Balassa-Samuelson interpretation: Regressing the estimated permanent component on relative U.S./U.K. per capita output explains the bulk of its long-run variation — consistent with the Balassa-Samuelson effect: productivity differentials in traded vs. non-traded sectors create permanent real exchange rate deviations that PPP cannot account for.

Concepts Introduced or Extended

• Markov-Switching Model
• Purchasing Power Parity
• Cointegration

Entities Mentioned

• Charles Engel
• Chang-Jin Kim

Quotes

"The variance of innovations in the permanent component is much larger than the variance of innovations in the transitory component."

"We find that the real exchange rate is not cointegrated with U.S. and U.K. prices."

My Take

An elegant application of Bayesian Markov chain Monte Carlo (MCMC) to a long-horizon exchange rate question. The 106-year sample is the paper's key asset — it spans radically different monetary regimes, which the three-state transitory variance process captures naturally. The ADF size distortion finding (19% actual vs. 5% nominal) is methodologically important: it explains why the entire prior literature using ADF tests to test PPP may have been misled. The main limitation is that the permanent component is constrained to be a pure random walk — fractional integration or near-unit-root behavior is not accommodated, and the Balassa-Samuelson interpretation depends on the cleanness of the permanent/transitory split.