The predictive power of equilibrium exchange rate models

Prepared by Michele Ca' Zorzi, Pablo Anaya Longaric and Michał Rubaszek^[1]

Published as part of the ECB Economic Bulletin, Issue 7/2021.

1 Introduction

Central banks carefully monitor the evolution of exchange rates. In the case of the European Central Bank (ECB) and other major central banks, the exchange rate is not a policy target. But the “market value” of the euro is highly relevant for understanding the medium-term inflation outlook via its impact – through import prices and through general equilibrium effects – on the real economy.

It would therefore be very useful to be able to anticipate future exchange rate movements, but this has proven rather elusive, especially at short horizons. The view that exchange rates are largely disconnected from economic fundamentals at short horizons has mainly predominated since the seminal paper by Meese and Rogoff (1983), which showed that exchange rate models were unable to deliver more accurate nominal exchange rate forecasts than the simple prediction of “no change” associated with the random walk (RW) model.^[2] The difficulty of predicting exchange rates with systematically better results than by using the RW model, especially at short-horizons, was reiterated by Rogoff (2008)^[3] and, more recently, in the influential articles by Rossi (2013)^[4] and Cheung et al. (2019)^[5].

Exchange rates have an important role beyond their key contribution to understanding the inflation outlook. While, in line with their monetary policy strategies, the ECB, the US Federal Reserve System and other major central banks do not treat the exchange rate as a target variable,^[6] they are mindful that if large and persistent nominal exchange rate fluctuations occur, real exchange rate misalignments may develop over time that could have significant implications for the economic outlook. This is because over- and undervalued currencies, in an environment of price rigidity, could lead to competitiveness imbalances, excessive real exchange rate volatility and, potentially, sharp economic adjustments with adverse effects on consumption and production.^[7] This is the context in which currencies’ “fair value” is often discussed. In exceptional instances, major central banks have intervened directly or only verbally in foreign exchange markets in a concerted manner to influence exchange rate dynamics. It is hence not surprising that academics and policymakers have continued to strive in recent decades to improve their methodological frameworks for assessing equilibrium exchange rates (e.g. Bussière et al., 2010; Phillips et al., 2013; Fidora et al., 2017; Couharde et al., 2018; and Cubeddu et al., 2019).^[8]

Recent papers have argued that concepts of equilibrium exchange rates, besides their intrinsic interest, could be helpful for understanding and predicting exchange rate movements. These papers suggest that, even if the dynamic adjustment of exchange rates cannot be fully anticipated, it is known that they should eventually adjust to their equilibrium, i.e. a terminal condition defined by economic theory. For example, Ca’ Zorzi et al. (2016) and Ca’ Zorzi and Rubaszek (2020)^[9] show that it is sufficient to assume that the real exchange rate gradually converges to the simplest definition of the equilibrium exchange rate, i.e. relative purchasing power parity (PPP), to produce surprisingly accurate real and nominal exchange rate forecasts. In theory, this approach is best suited to predicting real exchange rates, as the concept of the equilibrium exchange rate is defined in real terms. But empirical evidence shows that real exchange rate adjustments occur primarily through currency movements rather than via relative price changes. Thus, at least for countries with moderate inflation rates, measures of real equilibrium exchange rates can also be employed to forecast nominal exchange rates.^[10] From a different starting point, Ca’ Zorzi et al. (2017)^[11] and Eichenbaum et al. (2020)^[12] suggest that more advanced macroeconomic models, known as dynamic stochastic general equilibrium (DSGE) models – which assume that real and nominal exchange rate fluctuations are driven by differences in the monetary policy stance adjusted for risk premia – also offer a fairly good description of exchange rate dynamics and perform well overall in forecasting real and, to a lesser extent, nominal exchange rates. The explanation for this is that, like simpler approaches, such models imply a gradual return of the real exchange rate toward its equilibrium PPP value, but they tend to underestimate the empirical regularity of a strong co-movement between real and nominal exchange rates.

The key question addressed in this article is whether concepts of equilibrium exchange rates other than PPP might strengthen the predictability of exchange rates. To the extent that long-term drivers of exchange rates can be understood, and hence the equilibrium exchange rate can be estimated more precisely, it should in theory be possible to better forecast the future trajectory of the real and nominal exchange rate. For that purpose, this article evaluates the predictive power of three popular equilibrium exchange rate models. Besides the PPP model, we also investigate simplified versions of the behavioural equilibrium exchange rate (BEER) and the macroeconomic balance (MB) approaches along the lines of Ca’ Zorzi et al. (2022).^[13]

2 Three methods for assessing equilibrium exchange rates

Equilibrium exchange rate models are employed to decompose the real exchange rate ( r e r) into its equilibrium ( r e r e q) and misalignment ( r e r m i s) components:

The split between the two components depends in part on the time horizon that is used. As discussed by Driver and Westaway (2005), exchange rate movements are driven by long, medium and short-term economic fundamentals and by an unexplained component.^[14] The approach taken in this article is to distinguish between movements of the equilibrium exchange rates, and movements of the exchange rate around the equilibrium. The former are driven by long and medium-term economic fundamentals, while the latter are driven by short-term fundamentals and an unexplained component. This is consistent with theoretical general equilibrium models,^[15] in which over the business cycle fluctuations in exchange rates around their equilibria are partly driven by central banks’ relative monetary policy stance, adjusted for risk premia.^[16] The definition of equilibrium exchange rates is hence important from a monetary policy perspective.

The first equilibrium exchange rate model considered here is the PPP model, i.e. the oldest theory of real exchange rate determination, which was restored to prominence in modern times by Gustav Cassel and is still today a key benchmark for determining exchange rate parities in fixed exchange rate regimes.^[17] In a nutshell, the PPP model starts from the law of one price, which states that international arbitrage helps to equalise the price of any tradable product denominated in a common currency. The concept of strong PPP emerges from applying this law to consumption baskets, i.e. the same basket of goods should cost the same across countries when denominated in a common currency. In contrast, the weak version of PPP theory states that, in equilibrium, the relative cost of the same basket of goods across countries is constant over time but might deviate from unity owing to factors such as taxes and/or transportation costs. The weak version of PPP is empirically more relevant and appealing from a practical perspective, as it implies that a long-run sample mean of the real exchange rate is a good proxy for the PPP-based equilibrium real exchange rate.

The BEER model generalises PPP theory by assuming that persistent PPP deviations should not be treated as “disequilibria” if they are driven by economic fundamentals. In other words, while the PPP model suggests that the long-term equilibrium exchange rate is a constant, in the BEER model the medium-term equilibrium fluctuates over time in line with changes in a set of economic fundamentals. In practice, the level of the BEER is estimated from a regression linking the real exchange rate to a set of economic variables indicated by economic theory considerations (Fell, 1996; MacDonald, 1998; Maeso-Fernandez et al., 2001; and Lee et al., 2008).^[18] The literature has discussed at length the best choice of fundamentals for the BEER and the expected sign and magnitude of the parameters (for a comprehensive literature review, see Fidora et al., 2017).^[19] In this analysis the choice of economic fundamentals is limited to three key ones: relative per capita GDP, net foreign assets and the terms of trade. All three variables aim to explain medium and long-term exchange rate movements while, as discussed above, the analysis does not include real interest rate differentials, fiscal variables or other short-term factors. In most of the literature it is taken for granted that a rise in relative per capita GDP ( g d p) leads to an appreciation of the real exchange rate. The explanation is twofold. From a demand perspective, an increase in relative income should lead to stronger demand for domestic non-traded goods and hence to an increase in their price relative to traded goods, i.e. a real exchange rate appreciation. The supply perspective is based on the Balassa-Samuelson effect, which suggests that a relative increase in productivity in the production of tradable goods leads to an increase in g d p and a real exchange rate appreciation (Lee et al., 2013; and Zhang, 2017).^[20] Another frequently cited explanation for long-run trends in equilibrium real exchange rates emphasises the role of net foreign assets ( n f a). The rationale is that a rise in n f a increases the interest income on the current account that must be counterbalanced by a trade balance deterioration, requiring in turn a real exchange rate appreciation (Lane and Milesi-Ferretti, 2002).^[21] The terms of trade ( t o t) is the third most used explanatory variable in BEER regressions. A rise in t o t should lead to an improved trade balance, and therefore a real exchange rate appreciation is needed to restore the trade balance to its initial level. For the above reasons we estimate the level of the BEER with the specification used by Faruqee (1995) and Lane and Milesi-Ferretti (2004),^[22] so that the value of the BEER for country i in period t is given by:

where all explanatory variables are expressed relative to a trade-weighted average of foreign values and the estimated signs of the regression match the theoretical predictions.

The MB approach defines equilibrium exchange rates from a current account sustainability perspective. The methodology of the MB approach differs substantially from that of the PPP and BEER models. Instead of estimating an equilibrium exchange rate, the MB approach estimates or calibrates/postulates an equilibrium current account (or norm) and the current account gap, i.e. the distance of the current account from this norm.^[23] It then seeks to estimate or calibrate a relationship between the current account balance and the real exchange rate in order to establish to what degree the real exchange rate needs to adjust so as to close the current account gap. For that reason, this definition is closely connected with the debate on global (current account) imbalances and the role played by the exchange rate in unwinding them. Technically, to calculate the equilibrium exchange rate with the MB framework, three issues must be addressed. The first consists of projecting the level at which the current account balance would stabilise if exchange rates remained unchanged and output gaps were closed ( c a ~). The second relates to setting the current account norm ( c a n o r m), which is typically estimated using a panel data regression with similar fundamentals to those chosen for the above-mentioned BEER regressions. The third is estimating how changes in the real exchange rate affect the current account (elasticity η). Addressing these issues as in Ca’ Zorzi et al. (2022),^[24] the equilibrium exchange rate ( r e r i t M B) can then be simply computed as:

Equation (2) reveals the critical role played by the current account elasticity η, which measures the current account adjustment in response to a 1% exchange rate appreciation. For example, doubling the value of this elasticity would halve the exchange rate adjustment required to bring back the long-run value of the current account ( c a ~) to its norm ( c a n o r m). The academic literature has shown that the value of the current account elasticity is uncertain and dependent on some critical assumptions about the extent to which exchange rate changes pass through to export and import prices.^[25]

In terms of country coverage, the analysis in this article includes the group of advanced economies that issue so-called G10 currencies, namely Australia (AU), Canada (CA), Switzerland (CH), the euro area (EA),the United Kingdom (UK), Japan (JP), Norway (NO), New Zealand (NZ), Sweden (SE) and the United States (US) and our quarterly dataset spans the period between the first quarter of 1975 and the fourth quarter of 2020. Such a sample is comparable to those used in studies evaluating exchange rate trading strategies for the ten globally most traded currencies (e.g. Opie and Riddiough, 2020), as well as recent studies on exchange rate forecasting (e.g. Engel and Wu, 2021).^[26] There are two additional criteria for the country selection, namely the presence of flexible exchange rate regimes for most of the sample and the availability of sufficient macroeconomic data for a meaningful forecast evaluation exercise.^[27] A full description of the data sources, estimated regressions and equilibrium exchange rate estimates is available in Ca’ Zorzi et al. (2022), which for the purpose of this article was updated to the fourth quarter of 2020.

3 In-sample adjustment

A key question considered in our analysis is to what extent the euro, the US dollar and other major currencies tend to converge to their long-run exchange rate equilibria. For each equilibrium exchange rate model, currency and horizon h we estimate the parameters of the local projection regression:

in which the parameter β h measures the average fraction of misalignment eliminated at horizon h. If the real exchange rate reverts to its equilibrium over time, the value of β h should converge to 1 as h increases. In parallel, we estimate the local projection:

which enables the extent to which the nominal exchange rate accomplishes the required adjustment to be evaluated.

The evidence points to a clear tendency for exchange rates to converge to their equilibrium and for the pace of the convergence to be model-dependent. In Table 1, for each country the first line reports the fraction of the real exchange rate misalignment absorbed by the real exchange rate change ( β h ) and the second line reports the fraction absorbed by the nominal exchange rate change ( δ h )at each horizon h.

Four interesting findings are evident from Table 1:

First, the adjustment of the real exchange rate starts almost immediately. In the case of the PPP model, for example, in one quarter the real exchange rate moves to absorb 7% of the real exchange rate misalignment for the euro and 5% for both the US dollar and the full panel of G10 currencies.^[28] The adjustment is similar and marginally stronger in the case of the BEER model. It is remarkable that within the short horizon of one quarter a small fraction of the required exchange rate adjustment is already accomplished. It is hence not the case – despite being often stated – that short-run exchange rate movements are entirely unpredictable. The adjustment toward equilibrium is very slow but starts well before the one-year horizon.

Second, nominal exchange rate changes are at least in part predictable. The reason for this is that the required real exchange rate adjustment takes place via currency movements and not via an adjustment in relative prices across countries. This is apparent from the observation that the fraction of adjustment completed by the nominal exchange rate δ h is almost as sizeable as the fraction achieved by β h.^[29] In the case of the PPP model, the numbers are only slightly lower than for the real exchange rate: the nominal exchange rate absorbs 5% of the real exchange rate misalignment in the case of the euro and 4% for both the US dollar and on average for the panel. This is true across different models and horizons irrespective of the degree of exchange rate predictability, which differs across models.

Third, the pace of the adjustment toward the equilibrium varies considerably across models. With the PPP model, within three years the real exchange rate absorbs 72% of the required adjustment for the euro, 74% for the US dollar and 56% for the full panel of G10 currencies. However, the pace of adjustment is not the same for different equilibrium exchange rate concepts. It is somewhat stronger for the BEER model: within three years 79% of the adjustment is completed for the euro, 90% for the US dollar and 69% for the full panel.^[30]

Fourth, the pace of the real exchange rate adjustment is much weaker with the MB model. After three years it is basically nil for the euro, 48% for the US dollar and only 24% for the full panel. This suggests that the in-sample explanatory power of equilibrium exchange rates estimated with the current account-based model is overall very poor. A similar pattern emerges if we focus on the adjustment of the nominal exchange rate (see Table 1).

Finally, there is an elegant graphical equivalent way to show that nominal exchange rates over time absorb past exchange rate misalignments. Below we produce a set of scatter plots showing nominal exchange rate changes at different horizons against the initial real exchange rate misalignment, i.e. the deviation of the real exchange rate from its model-consistent equilibrium value. To the extent that there is an adjustment, this relationship should be negative, and, in the case of full adjustment accomplished by the nominal exchange rate and no movements in the equilibrium exchange rate, the slope of the regression should be equal to minus one. Chart 1 shows the results for the three models at different time horizons, indicating with yellow circles the observations for the US dollar, with blue circles those for the euro and with red circles those for all other G10 currencies.

The chart confirms visually that for both the PPP and BEER models there is an adjustment mechanism at play, which ensures that the initial real exchange rate misalignment is absorbed through a nominal exchange rate movement. This is already visible at the one-year horizon and becomes particularly evident at longer horizons. For the MB model the adjustment is again much weaker and visible only for the US dollar at the five-year horizon.

4 Out-of-sample evidence

The evidence that exchange rates converge to their long-term equilibria begs the key question: can this information be exploited to forecast real and nominal exchange rates? To find the answer, in the next step we evaluate the out-of-sample forecast accuracy for horizons ranging from one to 20 quarters ahead in the period from the first quarter of 1995 to the fourth quarter of 2020 on the basis of our full panel of data. In particular, for each vintage period s, the local projection regressions (5) and (6) are employed to calculate forecasts for real and nominal exchange rates:

where α h | s, β h | s, γ h | s, δ h | s, and r e r s | s e q are the parameters and equilibrium exchange rate estimates based on a sample ending in period s.

The chart showing sequential forecasts illustrates the predictive content of three equilibrium exchange rate models and immediately indicates that the PPP and BEER models perform relatively well. Chart 2 presents sequential forecasts for the US dollar and the euro. It shows that by predicting a gradual reversion of the real exchange rate to the corresponding equilibrium, the PPP and BEER models both usually deliver relatively accurate forecasts. By contrast, the MB model performs poorly^[31] as forecasts obtained from this model are only weakly correlated with subsequent outcomes.^[32]

The above anecdotal evidence on the relative performance of the three models can also be shown more formally vis-à-vis the RW benchmark. To this end, in Chart 3 we plot an indicator of the performance of each model (i.e. the root mean squared forecast error – RMSFE) measured relative to the performance of the RW model. If this indicator is below unity, the model concerned delivers more accurate forecasts than those obtained using the simple assumption of an unchanged exchange rate.

The findings confirm that the PPP and BEER models perform well in relative terms while the MB model performs poorly for the euro area. In the cases of the euro and the US dollar, the PPP model (yellow line) and the BEER model (red line) clearly outperform the RW model (blue line) in terms of forecasting both real (Chart 3, panel a) and nominal (Chart 3, panel b) exchange rates, as this indicator of forecasting performance is always below the threshold of 1. Interestingly, for the PPP model the result is already statistically significant at the one-year horizon. The MB model (green line) conversely fails to outperform the RW model in the case of the euro, while it outperforms the RW model for the US dollar at horizons longer than one year.

More mixed results are obtained for other G10 currencies. Table 2 presents the results for all countries that issue G10 currencies, with nominal and real exchange rates at the four-quarter and 20-quarter horizons. The results at short horizons show that equilibrium-based model forecasts beat the RW model forecasts approximately half of the time, albeit in only in a few instances in a significant way. However, at the longer, 20-quarter horizon, the PPP and BEER models show a stronger forecast performance than the RW model, while the MB model is generally inaccurate.^[33] The PPP model is the best performing among the three models participating in the forecasting competition.

The analysis highlights that the RW model still remains a strong contender, which is most likely due to the role of estimation error. Indeed, the effort made to estimate the pace of the adjustment to equilibrium with the local projection regressions (5) and (6) has not been very helpful in terms of forecasting. To show this, the previous exercise can be repeated with the only differences being that (i) this time the nominal exchange rate moves to ensure a movement of the real exchange rate toward its equilibrium value and that (ii) such adjustment is calibrated to be rather slow in line with the real exchange rate literature. Table 3 reports the results for a half-life adjustment of three years, but the results are valid for the wider range of between two and a half years and five years, while for half-lives longer than five years the forecasts are almost the same as those given by the RW model.

The forecasting power of the PPP equilibrium exchange rate (i.e. the sample mean) becomes fully evident in this setting. The PPP model beats the RW benchmark seven times out of ten in real exchange rate forecasting and eight times out of ten in nominal exchange rate forecasting. The performance of the BEER model is almost as good as that of the PPP model, but not better as it is also affected by other sources of estimation error. The predictive power of the MB model is generally considerably worse than that of the other two models.

An additional insight gained from this result is that most of the forecasting power that the models possess relative to the RW benchmark arises directly from the mean-reverting properties of the real exchange rate.^[34] It is enough to set up a simple “rule of thumb”, whereby the nominal exchange rate gradually moves to restore the real exchange rate to its mean, in order to outperform the RW model and most other models. In summary, PPP is both a competitive model in terms of equilibrium exchange rate estimation and an accurate benchmark for real and nominal exchange rate forecasting. The BEER model has good in-sample properties and a similar out-of-sample performance. Conversely, the MB model is highly unreliable in predicting future exchange rate changes.

5 Concluding remarks

Exchange rates are not a policy target for major central banks but are an important variable in their information set. They are important because anticipating their future movements could help to gauge the transmission mechanism of monetary policy. And they are important because sharp, large deviations from fair values could presage a large adjustment with potentially adverse repercussions for the outlook for inflation and for output. This article has presented an analysis bringing together the literature on equilibrium exchange rates and exchange rate forecasting. Several important, unexpected lessons can be drawn from an analysis of the predictive power of equilibrium exchange rate models.

The first is that in-sample, on average, real exchange rates tend to slowly converge to their equilibria if the latter are defined by the PPP or BEER models. This is not the case for equilibrium exchange rate models based on the current account. The equilibrium exchange rate estimates based on such models tend to be unstable and unreliable in explaining real exchange rate movements.

The second is that this convergence property can be exploited to forecast not only real exchange rates but also nominal exchange rates. The reason is intuitive as most of the adjustment toward equilibrium is achieved by a currency movement and not through relative price changes across countries. While the RW model remains, in some cases, a formidable competitor, the PPP and BEER models can help to forecast future exchange rate movements. The MB model estimates conversely are not indicative of future exchange rate movements.

The third is that most of the forecasting power comes from the mean-reverting properties of real exchange rates rather than from exploiting the relationship between exchange rates and economic fundamentals. While the notion of PPP is much more compatible with economic theory and hence offers reassurance as to the usefulness of macroeconomic theory and the role of the exchange rate in the transmission of monetary policy, the original insight of Meese and Rogoff (1983) – that the future path of exchange rates cannot easily be better extrapolated from economic models than by using a “rule of thumb” – remains largely true. In this article we have qualified this result by suggesting that the most competitive “rule of thumb” is often one which foresees a gradual adjustment of the exchange rate to restore the real exchange rate to its mean value rather than one that assumes “no change”. The key fundamentals that need to be known for out-of-sample forecasting are the real and nominal exchange rates, while there is relatively little information to be extracted from other economic fundamentals in normal times. Large current account surpluses or deficits may influence the path of adjustment of the exchange rate but, in line with theory, they have a limited influence on the long-term value of the exchange rate.