## Abstract

Traded good prices affect the real exchange rate first through their effect on the overall price level and second through their effect on the nominal exchange rate. Whereas the price level effect, which is positive in sign, is universally recognized, the nominal exchange rate effect, which is negative in sign, is routinely ignored. We calculate to which extent real exchange rate changes are accounted for by traded good prices and other components of the real exchange rate. We find that the nominal exchange rate effect neutralizes the price level effect entirely, suggesting that, contrary to popular belief, good market arbitrage is not conducive to purchasing power parity (the purchasing power parity fallacy). Rather than traded or non-traded good prices, the main driving force behind the real exchange rate is currency market pressure, a variable that, as we argue, is largely determined by the cumulative trade and capital flows of a country.

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## Notes

- 1.
The only exception is the study by Eleftheriou and Müller-Plantenberg (forthcoming), who run nonparametric regressions on various data sets to examine how the real exchange rate, the price differential and the nominal exchange rate react to an overvalued real exchange rate over time. In line with the hypothesis of this paper, their results show that when price adjustment is fast (which it is for traded good real exchange rate data), the nominal exchange rate tends to diverge in response to deviations from purchasing power parity.

## References

Alexius A, Nilsson J (2000) Real exchange rates and fundamentals: evidence from 15 OECD countries. Open Econ Rev 11:383–397

Bache IW, Sveen T, Torstensen KN (2013) Revisiting the importance of non-tradable goods’ prices in cyclical real exchange rate fluctuations. Eur Econ Rev 57:98–107

Baum CF, Barkoulas JT, Caglayan M (2001) Nonlinear adjustment to purchasing power parity in the post-Bretton Woods era. J Int Money Financ 20 (3):379–399

Beckmann J (2013) Nonlinear adjustment, purchasing power parity and the role of nominal exchange rates and prices. North American Journal of Economics and Finance 24:176–190

Betts CM, Kehoe TJ (2006) U.S. real exchange rate fluctuations and relative price fluctuations. J Monet Econ 53(7):1297–1326

Betts CM, Kehoe TJ (2008) Real exchange rate movements and the relative price of non-traded goods. Federal Reserve Bank of Minneapolis Research Department Staff Reports (415)

Brooks R, Edison HJ, Kumar M, Sløk T (2001) Exchange rates and capital flows. International Monetary Fund Working Paper Series (190)

Burstein A, Eichenbaum M, Rebelo S (2006) The importance of nontradable goods’ prices in cyclical real exchange rate fluctuations. Jpn World Econ 18(3):247–253

Ca’ Zorzi M, Muck J, Rubaszek M (2016) Real exchange rate forecasting and PPP: this time the random walk looses. Open Econ Rev 27(3):585–609

Chen L-L, Choi S, Devereux J (2006) Accounting for US regional real exchange rates. J Money, Credit, Bank 38(1):229–244

Cheung Y-W, Lai KS, Bergman M (2004) Dissecting the PPP puzzle: the unconventional roles of nominal exchange rate and price adjustments. J Int Econ 64 (1):135–150

Cheung Y-W, Chinn MD, García Pascual A, Zhang Y (2017) Exchange rate prediction redux: new models, new data, new currencies. National Bureau of Economic Research Working Paper Series (23267)

Choi K, Kapetanios G, Shin Y (2002) Nonlinear mean reversion in real exchange rates. Econ Lett 77(3):411–417

Combes J-L, Kinda T, Plane P (2012) Capital flows, exchange rate flexibility, and the real exchange rate. J Macroecon 34(4):1034–1043

Drozd LA, Nosal JB (2010) The nontradable goods’ real exchange rate puzzle. In: Reichlin L, West KD (eds) National Bureau of Economic Research International Seminar on Macroeconomics 2009, vol 6, pp 227–249

Eleftheriou M, Müller-Plantenberg NA (forthcoming) Price level convergence and purchasing power divergence. International Finance

Enders W (1988) ARIMA and cointegration tests of purchasing power parity. Rev Econ Stat 70(3):504–508

Engel C (1999) Accounting for US real exchange rate changes. J Polit Econ 107(3):507–538

Engel C, Morley JC (2001) The adjustment of prices and the adjustment of the exchange rate. National Bureau of Economic Research Working Paper Series (8550)

Gabaix X, Maggiori M (2015) International liquidity and exchange rate adjustment. Q J Econ 130(3):1369–1420

Hall SG, Hondroyiannis G, Kenjegaliev A, Swamy PAVB, Tavlas GS (2013) Is the relationship between prices and exchange rates homogeneous? J Int Money Financ 37:411–438

Heimonen K (2006) Nonlinear adjustment in PPP: evidence from threshold cointegration. Empir Econ 31(2):479–495

Ho CSF, Ariff M (2011) Sticky prices and time to equilibrium: evidence from Asia-Pacific trade-related economies. Appl Econ 43(21):2851–2861

Imbs J, Mumtaz H, Ravn MO, Rey H (2003) Non-linearities and real exchange rate dynamics. J Eur Econ Assoc 1(2):639–649

Imbs J, Mumtaz H, Ravn MO, Rey H (2005) PPP strikes back: aggregation and the real exchange rate. Q J Econ 120(1):1–43

Juvenal L, Taylor MP (2008) Threshold adjustment of deviations from the law of one price. Studies in Nonlinear Dynamics and Econometrics 12(3):1–44

Lo MC, Zivot E (2001) Threshold cointegration and nonlinear adjustment to the law of one price. Macroecon Dyn 5(4):533–576

Manzur M, Ariff M (1995) Purchasing power parity: new methods and extensions. Appl Financ Econ 5(1):19–26

Mendoza EG (2000) On the instability of variance decompositions of the real exchange rate across exchange-rate regimes: evidence from Mexico and the United States. National Bureau of Economic Research Working Paper Series (7768)

Michael P, Nobay AR, Peel DA (1997) Transaction costs and nonlinear adjustment in real exchange rates: an empirical investigation. J Polit Econ 105(4):862–879

Müller-Plantenberg NA (2006) Japan’s imbalance of payments. In: Hutchison MM, Westermann F (eds) Japan’s Great Stagnation: financial and monetary policy lessons for advanced economies. MIT Press, Cambridge and London, pp 239–266

Müller-Plantenberg NA (2010) Balance of payments accounting and exchange rate dynamics. Int Rev Econ Financ 19(1):46–63

Müller-Plantenberg NA (2017a) Boom-and-bust cycles, external imbalances and the real exchange rate. World Econ 40(1):56–87

Müller-Plantenberg NA (2017b) Currency flows and currency crises. CESifo Econ Stud 63(2):182–209

Nakagawa H (2010) Investigating nonlinearities in real exchange rate adjustment: threshold cointegration and the dynamics of exchange rates and relative prices. J Int Money Financ 29(5):770–790

Obstfeld M, Taylor AM (1997) Nonlinear aspects of goods-market arbitrage and adjustment: Heckscher’s commodity points revisited. J Jpn Int Econ 11:441–479

Pavlidis EG, Paya I, Peel DA (2011) Real exchange rates and time-varying trade costs. J Int Money Financ 30(6):1157–1179

Paya I, Venetis IA, Peel DA (2003) Further evidence on PPP adjustment speeds: the case of effective real exchange rates and the EMS. Oxf Bull Econ Stat 65 (4):421–437

Rogoff KS (1996) The purchasing power parity puzzle. J Econ Lit 34(2):647–668

Sarno L, Taylor MP (2002) Purchasing power parity and the real exchange rate. International Monetary Fund Staff Papers 49(1):65–105

Taylor AM, Taylor MP (2004) The purchasing power parity debate. J Econ Perspect 18(4):135–158

Taylor MP, Peel DA, Sarno L (2001) Nonlinear mean-reversion in real exchange rates: toward a solution to the purchasing power parity puzzles. Int Econ Rev 42(4):1015–1042

Vataja J (2000) Should the law of one price be pushed away? Evidence from international commodity markets. Open Econ Rev 11(4):399–415

Woo K-Y, Lee S-K, Chan A (2014) Non-linear adjustments to intranational PPP. J Macroecon 40:360–371

Yoon G (2010) Nonlinear mean-reversion to purchasing power parity: exponential smooth transition autoregressive models and stochastic unit root processes. Appl Econ 42(4):489–496

## Acknowledgments

We thank two anonymous reviewers for very helpful comments.

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## Appendices

### Appendix A: The Unbounded and Bounded Contribution Measures versus Alternative Contribution Measures

In this appendix, the performance of the UCM and the BCM is compared to that of various contribution measures proposed in the literature. We start by introducing those alternative contribution measures in Appendix A.1. The comparison is then carried out in Appendix A.2 using a Monte Carlo approach.

### A.1 Alternative contribution measures

#### A.1.1 Engel’s (1999) Contribution Measures

Engel (1999) uses contribution measures that build on the mean squared error criterion (MSE). Although the mean squared error is a concept used in statistics to quantify the difference between values implied by an estimator and the true values of the quantity being estimated, we follow Engel and adopt this concept analogously in the context of exchange rate accounting. Suppose that *x*_{
t
} is the sum of only two components, *x*_{1,t} and *x*_{2,t}, so that *x*_{
t
} = *x*_{1,t} + *x*_{2,t}. From the definition of the mean squared error, we can deduce the following:

The mean squared error of the composite series Δ_{
h
}*x*_{
t
} is thus equal to the sum of the mean squared errors of its two components, Δ_{
h
}*x*_{1,t} and Δ_{
h
}*x*_{2,t}, plus an additional term involving the product of the “biases” and the covariance of both components.

It is natural to use the mean squared errors of both components to measure their respective contribution to the mean squared error of the composite series. However, it is less clear what should be done with the additional term. Two possibilities that suggest themselves are to either ignore this term or to attribute it equally to both component series. Engel applies both alternatives and finds that they lead to similar results in his context. Specifically, the two contribution measures he considers are the following:

Note that the “bias” is measured around zero and is thus equal to the arithmetic mean of the variable in question. For the variance, Engel uses a small-sample correction:

#### A.1.2 Other Contribution Measures

One contribution measure used in the literature is based on sample variances:

Similar to the contribution measure CM^{MSE(1)}, the measure CM^{Var(1)} can only take values between zero and one.

An alternative measure is also based on variances, but takes the correlation between Δ_{
h
}*x*_{1,t} and Δ_{
h
}*x*_{2,t} into account:

Yet another contribution measure proposed in the literature is based on the sample standard deviations:

Finally, authors have looked at the sample correlation between the component series and the composite series:

Note that − 1 ≤CM^{Corr} ≤ 1. There is no obvious relationship with the UCM.

The contribution measures CM^{Var(1)}, CM^{Var(2)}, CM^{Std} and CM^{Corr} have been proposed by Betts and Kehoe (2006, 2008). These authors apply those measures to the levels of the real exchange rate and its components and to the linearly detrended levels. They further use the measures CM^{MSE(1)}, CM^{Std} and CM^{Corr} for the one-year and four-year differences of the real exchange rate and its components. Burstein et al. (2006) adopt CM^{Var(2)} with a slight modification: by attributing the covariance term of the variance of the composite series, 2Cov(*x*_{1},*x*_{2}), either fully to one component series or to the other one, they obtain lower and upper bounds for the contribution of the component series to the composite series. Moreover, Burstein et al. actually do not apply CM^{Var(2)} to the differences of the series but to their levels, where the levels have previously been detrended using a Hodrick-Prescott filter with a smoothing parameter of 1600. Drozd and Nosal (2010) use the measures CM^{Std} and CM^{Corr}, Bache et al. (2013) the measure CM^{Var(1)}. However, Chen et al. (2006) use Engel’s (1999) original measures, CM^{MSE(1)} and CM^{MSE(2)}.

### A.2 Comparison of Contribution Measures

In order to compare the properties of the UCM and the BCM with those of the contribution measures used in the literature, we rely on Monte Carlo simulations. We look at three different models, in all of which *x*_{1,t} and *x*_{2,t} are modeled as random walks:

where *t* = 2, 3,…,*T*, *T* = 100 and *x*_{i,1} = 0. The stochastic processes in models 1 to 3 differ only with respect to their innovations, Δ_{
h
}*x*_{1,t} and Δ_{
h
}*x*_{2,t}, as will be shown below. We also experimented with stochastic processes with some kind of mean reversion or error correction mechanism, but the results we obtained were similar; this is why we only report the results for the random walk processes, which are easier to interpret. The sample size of the Monte Carlo runs is 1000 in each case. The results are shown in Table 5. Differences refer to first-order differences (*h* = 1).

### A.2.1 Model 1

Model 1 takes the following form:

where *u*_{
t
} ∼U(0, 1) and *ε*_{
t
} ∼N(0, 1). Note that in this model, it will always be the case that Δ*x*_{1,t}/Δ*x*_{
t
} < 0, so that necessarily UCM < 0 and BCM = 0. In fact, since Δ*x*_{1,t} = −*u*_{
t
} × *ε*_{
t
}, Δ*x*_{
t
} = (1 − *u*_{
t
}) × *ε*_{
t
} and \(\text {E}(u_{t}) = \text {E}(1-u_{t}) = \frac {1}{2}\), the UCM should take on a value near minus one in the simulated sample:

Turning now to the contribution measures used in the literature, it is easy to see that the standard deviation of the component series Δ*x*_{1,t} is equal in expectation to that of the composite series Δ*x*_{
t
}. This implies that CM^{Std} should be positive and near one in the simulated sample, no matter whether we look at the series in differences or in levels, even though *x*_{1,t} clearly does not contribute at all to the movements of *x*_{
t
}. Therefore, CM^{Std} is not useful as a measure of co-movement between the component and composite series.

Another insight to take away from model 1 is that the contribution measures CM^{MSE(1)} and CM^{Var(1)} ignore the negative correlation between Δ*x*_{1,t} and Δ*x*_{
t
} and between *x*_{1,t} and *x*_{
t
} and thus take on positive values, whereas the UCM and the BCM suggest that they should be negative or zero. Both CM^{MSE(1)} and CM^{Var(1)} thus appear inadequate as contribution measures, too. While the measures CM^{MSE(2)} and CM^{Var(2)} do take on negative values in this model, the values lie somewhere between the UCM and the BCM and are arguably more difficult to interpret than the latter.

### A.2.2 Model 2

Model 2 takes the following form:

where *μ*_{1} = 1, *μ*_{2} = 0.2, *𝜃* = 1.78 and *ε*_{
t
} ∼N(0, 1). The parameter *𝜃* is chosen so that the UCM is about one half (with the BCM near 0.4). However, note that the correlation between the differences of the component series and the composite series is perfectly negative, whereas the correlation between the corresponding levels is almost perfectly positive. This demonstrates that CM^{Corr}, whether measured for differences or levels, is a misleading measure of the contribution of *x*_{1,t} and *x*_{2,t} to the movements of *x*_{
t
}.

As mentioned, the contribution of *x*_{1,t} to the movements of *x*_{
t
} is roughly one half on average. The contribution measure that comes closest to this value is CM^{MSE(1)} when applied to differences; it takes a value 0.382, or about 76% of the UCM. However, this measure did badly in model 1, where it took a positive sign, rather than a negative one as the UCM. The values of the measures CM^{MSE(2)} and CM^{Var(1)} for differences are unreasonably low (around 0.2), whereas those of the measures CM^{MSE(1)}, CM^{MSE(2)}, CM^{Var(1)} and CM^{Var(2)} for levels are all too high (around 0.8–0.9). Due to the negative correlation between Δ*x*_{1,t} and Δ*x*_{2,t}, the measure CM^{Var(2)} takes on a negative value when applied to the differences:

This example shows why it is problematic to use a contribution measure that ignores the drifts of *x*_{1,t} and *x*_{2,t} if they exist.

### A.2.3 Model 3

Model 3 takes the following form:

where *ε*_{1,t} ∼N(0, 1), *ε*_{2,t} ∼N(0, 1) and *ε*_{1,t} and *ε*_{2,t} are drawn independently. This example demonstrates that if *x*_{1,t} and *x*_{2,t} are driven by shocks from different distributions, CM^{MSE(1)}, CM^{MSE(2)}, CM^{Var(1)} and CM^{Var(2)} can all be significantly biased compared to the UCM and the BCM. The only measures that come close to the UCM and the BCM in this particular example are CM^{Std} and CM^{Corr}. However, those latter measures proved clearly inadequate in the other two experiments.

### A.2.4 Conclusions from the Comparison of Contribution Measures

When a variable is the exact sum of other variables, the contribution of the latter variables to the composite series can be calculated precisely using the UCM and the BCM. These two contribution measures are simple and intuitive. As the Monte Carlo simulations have shown, all the measures proposed in the literature deviate from the UCM and the BCM for even quite simple stochastic processes. In many cases, they have the wrong sign or are significantly biased. Even if those measures do well in one model, they do badly in others. Our impression is that for very simple models, the measure CM^{MSE(2)} based on differences is the one that comes closest to the UCM. However, as the models 1 to 3 show, CM^{MSE(2)} can also be quite off the target.

### Appendix B: Data

### B.1 Food and Construction Price Data Set

The food and construction price data set used in this paper has already been described at the beginning of Section 4. Here we add information on the country coverage and length of the time series data that we took from the Main Economic Indicators of the OECD. The data set covers the following countries:

Austria, Brazil, Canada, Denmark, Finland, France, Germany, Greece, Iceland, Ireland, Italy, Mexico, Norway, Portugal, Spain and Sweden.

The available series vary in length, with the longest price and nominal exchange rate series spanning the period from 1960Q1 to 2017Q2. Note that with 16 countries, we can compute nominal and real exchange rates as well as price differentials for a total of 120 country pairs (= (16 × 15) / 2).

We thought that the fact that some country pairs had fixed bilateral exchange rates during parts of the sample could be of relevance, yet found that taking out those pairs hardly affected our results and hence left them in. It should be noted that using data on countries that temporarily fix their exchange rates vis-à-vis other countries will overstate the effect of traded good prices on the real exchange rate as the nominal exchange rate effect is reduced. This adds, of course, confidence to our finding that traded good prices account for almost none of the movements of real exchange rates. However, as already said, our results were practically identical for the full sample and the sub-sample.

### B.2 PPI and CPI Data Set

The PPI and CPI data set that we use for our robustness analysis in Section 5.3 was taken from the International Financial Statistics of the IMF. In this database, nominal exchange rate as well as PPI and CPI data are available for the following set of countries:

Albania, Algeria, Argentina, Armenia, Austria, Belgium, Brazil, Bulgaria, Canada, Central African Republic, Chile, China, Colombia, Costa Rica, Croatia, Czech Republic, Denmark, Ecuador, Egypt, El Salvador, Estonia, Ethiopia, Finland, France, Georgia, Germany, Greece, Hong Kong, Hungary, India, Indonesia, Iran, Israel, Italy, Japan, Kazakhstan, Kuwait, Kyrgyzstan, Latvia, Lithuania, Macedonia, Malaysia, Mexico, Morocco, Netherlands, Norway, Pakistan, Panama, Paraguay, Peru, Philippines, Poland, Portugal, Republic of the Congo, Romania, Saudi Arabia, Senegal, Singapore, Slovakia, Slovenia, South Africa, South Korea, Spain, Sri Lanka, Sweden, Switzerland, Syria, Thailand, Trinidad and Tobago, Tunisia, Turkey, Ukraine, United Kingdom, United States, Uruguay, Venezuela and Zambia.

The available series vary in length, with the longest price and nominal exchange rate series spanning the period from 1957Q1 to 2016Q2. Note that with 77 countries, we can compute nominal and real exchange rates as well as price differentials for a total of 2926 country pairs (= (77 × 76) / 2).

### Appendix C: Determining Inflation-Offsetting Currency Market Pressure

In this paper, inflation-offsetting currency market pressure, \(\tilde {s}_{t}^{\,\text {inflation-offsetting}}\), is defined as \(\gamma p_{t}^{\textsc {t}}\), where *γ* is chosen such that \(\text {UCM}(p_{t}^{\textsc {t}}, \tilde {s}_{t}^{\,\text {core}}) = 0\). Denoting \(\text {UCM}(p_{t}^{\textsc {t}}, \tilde {s}_{t}^{\,\text {core}})\) as Φ(*γ*), we see that:

where

It is easily verified that Φ_{2}(*γ*) is weakly decreasing in *γ*, that \({\Phi }_{2}(\gamma ) \rightarrow \bar {\Phi }_{2} > 0\) as *γ* →−*∞* and that \({\Phi }_{2}(\gamma ) \rightarrow \underline {\Phi }_{2} < 0\) as *γ* → + *∞*. Hence there exists exactly one interval of values of *γ* for which Φ_{2}(*γ*) = 0. Since Φ_{1}(*γ*) > 0 always, it follows that Φ(*γ*) = 0 for the same interval of values of *γ* for which Φ_{2}(*γ*) = 0. Computationally, this interval can be estimated using a grid search with an arbitrarily fine, possibly adaptive grid. As for the value of *γ*, we used the midpoint of the estimated interval. It should be noted, however, that the estimated interval is of zero length most of the time so that a point estimate can be used for *γ*.

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### Cite this article

Eleftheriou, M., Müller-Plantenberg, N.A. The Purchasing Power Parity Fallacy: Time to Reconsider the PPP Hypothesis.
*Open Econ Rev* **29, **481–515 (2018). https://doi.org/10.1007/s11079-017-9473-9

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### Keywords

- Real exchange rate accounting
- Good market arbitrage
- Purchasing power parity fallacy
- Currency market pressure

### JEL Classification

- F31