The Purchasing Power Parity Fallacy: Time to Reconsider the PPP Hypothesis

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.

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.

1.1 A.1 Alternative contribution measures

1.1.1 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:

$$\begin{array}{@{}rcl@{}} \text{MSE}({\Delta}_{h} x_{t}) &=& \text{MSE}({\Delta}_{h} x_{1, t} + {\Delta}_{h} x_{2, t})\\ &=& \text{Var}({\Delta}_{h} x_{1, t} + {\Delta}_{h} x_{2, t}) + [\text{Bias}({\Delta}_{h} x_{1, t} + {\Delta}_{h} x_{2, t})]^{2}\\ &=& \text{Var}({\Delta}_{h} x_{1, t}) + \text{Var}({\Delta}_{h} x_{2, t}) + 2 \text{Cov}({\Delta}_{h} x_{1, t}, {\Delta}_{h} x_{2, t})\\ && + [\text{Bias}({\Delta}_{h} x_{1, t})]^{2} + [\text{Bias}({\Delta}_{h} x_{2, t})]^{2} + 2 \text{Bias}({\Delta}_{h} x_{1, t}) \text{Bias}({\Delta}_{h} x_{2, t})\\ &=& \text{MSE}({\Delta}_{h} x_{1, t}) + \text{MSE}({\Delta}_{h} x_{2, t}) + 2 \text{Bias}({\Delta}_{h} x_{1, t}) \text{Bias}({\Delta}_{h} x_{2, t})\\ && + 2 \text{Cov}({\Delta}_{h} x_{1, t}, {\Delta}_{h} x_{2, t}). \end{array} $$

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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:

$$ \text{CM}^{\text{MSE}(1)}(x_{1, t}, x_{t}) = \frac{\text{MSE}({\Delta}_{h} x_{1, t})}{\text{MSE}({\Delta}_{h} x_{1, t}) + \text{MSE}({\Delta}_{h} x_{2, t})}, $$

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$$\begin{array}{@{}rcl@{}} && \text{CM}^{\text{MSE}(2)}(x_{1, t}, x_{t})\\ &=& \frac{\text{MSE}({\Delta}_{h} x_{1, t}) + \text{Bias}({\Delta}_{h} x_{1, t})\text{Bias}({\Delta}_{h} x_{2, t}) + \text{Cov}({\Delta}_{h} x_{1, t}, {\Delta}_{h} x_{2, t})}{\text{MSE}({\Delta}_{h} x_{t})}. \end{array} $$

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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:

$$\begin{array}{@{}rcl@{}} \text{Bias}({\Delta}_{h} x_{\cdot, t}) &=& \frac{h}{T-1} (x_{\cdot, T} - x_{\cdot, 1}), \end{array} $$

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$$\begin{array}{@{}rcl@{}} \text{Var}({\Delta}_{h} x_{\cdot, t}) &=& \frac{T}{(T-h-1)(T-h)} \sum\limits_{t=h + 1}^{T} [{\Delta}_{h} x_{\cdot, t} - \text{Bias}({\Delta}_{h} x_{\cdot, t})]^{2}. \end{array} $$

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1.1.2 A.1.2 Other Contribution Measures

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

$$ \text{CM}^{\text{Var}(1)}(x_{1, t}, x_{t}) = \frac{\text{Var}({\Delta}_{h} x_{1, t})}{\text{Var}({\Delta}_{h} x_{1, t}) + \text{Var}({\Delta}_{h} x_{2, t})}. $$

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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:

$$\begin{array}{@{}rcl@{}} \text{CM}^{\text{Var}(2)}(x_{1, t}, x_{t}) &\,=\,& \frac{\text{Var}({\Delta}_{h} x_{1, t}) + \text{Cov}({\Delta}_{h} x_{1, t}, {\Delta}_{h} x_{2, t})}{\text{Var}({\Delta}_{h} x_{t})}\\ &\,=\,& \frac{\text{Var}({\Delta}_{h} x_{1, t}) + \text{Cov}({\Delta}_{h} x_{1, t}, {\Delta}_{h} x_{2, t})}{\text{Var}({\Delta}_{h}x_{1, t}) + \text{Var}({\Delta}_{h} x_{2, t}) + 2 \text{Cov}({\Delta}_{h} x_{1, t}, {\Delta}_{h} x_{2, t})}. \end{array} $$

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

$$ \text{CM}^{\text{Std}}(x_{1, t}, x_{t}) = \frac{ \text{Std}({\Delta}_{h} x_{1, t}) }{ \text{Std}({\Delta}_{h} x_{t}) }. $$

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

$$ \text{CM}^{\text{Corr}}(x_{1, t}, x_{t}) = \frac{\text{Cov}({\Delta}_{h} x_{1, t}, {\Delta}_{h} x_{t})}{\text{Std}({\Delta}_{h} x_{1, t}) \text{Std}({\Delta}_{h} x_{t})}. $$

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

Table 5 Comparison of contribution measures

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(x₁,x₂), 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).

1.2 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:

$$ x_{i, t} = x_{i, t-1} + {\Delta} x_{i, t}, \qquad \text{for } i = 1, 2, $$

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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).

1.3 A.2.1 Model 1

Model 1 takes the following form:

$$\begin{array}{@{}rcl@{}} {\Delta} x_{1, t} &=& - u_{t} \times \varepsilon_{t}, \end{array} $$

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$$\begin{array}{@{}rcl@{}} {\Delta} x_{2, t} &=& \varepsilon_{t}, \end{array} $$

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where u _t ∼U(0, 1) and ε _t ∼N(0, 1). Note that in this model, it will always be the case that Δx_1,t/Δx _t < 0, so that necessarily UCM < 0 and BCM = 0. In fact, since Δx_1,t = −u _t × ε _t , Δx _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:

$$\begin{array}{@{}rcl@{}} \lim\limits_{T \rightarrow \infty} \text{UCM}(x_{1, t}, x_{t}) &=& \lim\limits_{T \rightarrow \infty} \sum\limits_{t = 1}^{T} \frac{|(1-u_{t}) \varepsilon_{t}|}{{\sum}_{\tau= 1}^{T} |(1-u_{\tau}) \varepsilon_{\tau}|} \times \frac{-u_{t} \varepsilon_{t}}{(1-u_{t}) \varepsilon_{t}}\\ &=& \lim\limits_{T \rightarrow \infty} \sum\limits_{t = 1}^{T} \frac{- u_{t} |\varepsilon_{t}|}{{\sum}_{\tau= 1}^{T}(1-u_{\tau}) |\varepsilon_{\tau}|} = \lim\limits_{T \rightarrow \infty} \frac{- \frac{1}{2} {\sum}_{t = 1}^{T} |\varepsilon_{t}|}{\frac{1}{2} {\sum}_{t = 1}^{T} |\varepsilon_{t}|} = -1.\\ \end{array} $$

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Turning now to the contribution measures used in the literature, it is easy to see that the standard deviation of the component series Δx_1,t is equal in expectation to that of the composite series Δx _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 Δx_1,t and Δx _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.

1.4 A.2.2 Model 2

Model 2 takes the following form:

$$\begin{array}{@{}rcl@{}} {\Delta} x_{1, t} &=& \mu_{1} + \varepsilon_{t}, \end{array} $$

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$$\begin{array}{@{}rcl@{}} {\Delta} x_{2, t} &=& \mu_{2} - \theta \varepsilon_{t}, \end{array} $$

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where μ₁ = 1, μ₂ = 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 Δx_1,t and Δx_2,t, the measure CM^Var(2) takes on a negative value when applied to the differences:

$$ \text{CM}^{\text{Var}(2)}(x_{1, t}, x_{t}) = (1 - 1.78)/(1 + 1.78^{2} - 2 \times 1.78) = -1.282. $$

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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.

1.5 A.2.3 Model 3

Model 3 takes the following form:

$$\begin{array}{@{}rcl@{}} {\Delta} x_{1, t} = \varepsilon_{1, t}, \end{array} $$

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$$\begin{array}{@{}rcl@{}} {\Delta} x_{2, t} = \varepsilon_{2, t}^ 3, \end{array} $$

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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.

1.6 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

1.1 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.

1.2 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:

$$\begin{array}{@{}rcl@{}} {\Phi}(\gamma) &=& \text{UCM}(p_{t}^{\textsc{t}}, \tilde{s}_{t}^{\,\text{core}})\\ &=& \text{UCM}(p_{t}^{\textsc{t}}, \tilde{s}_{t} - \gamma p_{t}^{\textsc{t}})\\ &=& \sum\limits_{t=h + 1}^{T} \frac{|{\Delta}_{h} \tilde{s}_{t} - \gamma {\Delta}_{h} p_{t}^{\textsc{t}}|}{{\sum}_{\tau=h + 1}^{T}|{\Delta}_{h} \tilde{s}_{\tau} - \gamma {\Delta}_{h} p_{\tau}^{\textsc{t}}|} \times \frac{{\Delta}_{h} p_{t}^{\textsc{t}}}{{\Delta}_{h} \tilde{s}_{t} - \gamma {\Delta}_{h} p_{t}^{\textsc{t}}}\\ &=& {\Phi}_{1}(\gamma) \times {\Phi}_{2}(\gamma), \end{array} $$

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where

$$\begin{array}{@{}rcl@{}} {\Phi}_{1}(\gamma) &=& \frac{1}{{\sum}_{\tau=h + 1}^{T}|{\Delta}_{h} \tilde{s}_{\tau} - \gamma {\Delta}_{h} p_{\tau}^{\textsc{t}}|},\\ {\Phi}_{2}(\gamma) &=& \sum\limits_{t=h + 1}^{T} \text{sgn}({\Delta}_{h} \tilde{s}_{t} - \gamma {\Delta}_{h} p_{t}^{\textsc{t}}) \times {\Delta}_{h} p_{t}^{\textsc{t}}. \end{array} $$

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It is easily verified that Φ₂(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 Φ₂(γ) = 0. Since Φ₁(γ) > 0 always, it follows that Φ(γ) = 0 for the same interval of values of γ for which Φ₂(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.