This time the random walk loses

Michele Ca' Zorzi, Jakub Mućk, Michał Rubaszek

Notwithstanding the progress made in the field of exchange rate economics, we still know very little of what drives major currencies. This column argues that the best that one can do is to assume that currencies move to gradually restore (relative) purchasing power parity. Contrary to widely held beliefs, this is in general a much better strategy than to just assume that the exchange rate behaves like a random walk.

Forecasting nominal exchange rates

In their classic paper, Meese and Rogoff (1983) argued that forecasting nominal exchange rates is basically impossible, as all plausible macro-models fail to beat the random walk. For the last three decades, the lively debate in the literature has not changed the prevailing view that exchange rates are not forecastable. In this respect, a glimpse of light comes from the analysis of Engel et al. (2008) who argued that the dismal performance of exchange rates models arises from the pervasive impact of estimation error. This might explain why the performance of macro models tends to improve with large panels of data (e.g. Ince 2014) and be very poor for shorter samples.

Hope from the Purchasing Power Parity (PPP)?

There is, however, at least one exchange rate theory that is not totally discredited and has even enjoyed a positive reappraisal over recent years: PPP theory. It is indeed one of the oldest theories of economics, as it goes back to the Salamanca school in the 16th century and, in modern times, to Cassel (1918). But if we look back at the 20th century economic history, the consensus on the validity of PPP has shifted back and forward several times. As Dornbusch (1985) has put it “by different authors at different points in time has been considered an identity, a truism, an empirical regularity or a grossly misleading oversimplification”. According to Taylor and Taylor (2004), after a period when the real exchange rates were thought to behave as random walks, the literature has turned full circle to the pre-1970s view that PPP holds in the long run. If this is the case, real exchange rates should presumably be forecastable – at least over medium- to long-term horizons. Surprisingly, the articles that look at this issue directly can be counted on the fingers of one hand. Meese and Rogoff (1988) reached the conclusion that, like nominal exchange rates, real exchange rates are disconnected from economic fundamentals, whereas two studies from the mid-1990s argued that the random walk can be beaten for large datasets (Lothian and Taylor 1996, Jorion and Sweeney 1996). The question that we tackle in this paper is therefore the following:

Can the mean reverting property of real exchange rates be exploited to beat the random walk in relation to real and nominal exchange rate forecasting?

Forecast competition for real exchange rates

To beat the random walk one needs to consider both the crucial role of estimation forecast error and the strong persistence of real exchange rates. The strong persistence in the real exchange rate was affirmed in a series of studies conducted in the mid-1980s and early 1990s, which employed more than a hundred years of annual data. From an informal meta-analysis of these studies, Rogoff (1996) inferred that it takes between 3 and 5 years to halve real exchange rate deviations from the mean (HL, half-life). This has led to a broad, but not universal consensus, on the degree of persistence of real exchange rates, which is sometime dubbed the ‘Rogoff’s consensus’. In what follows, we shall evaluate the validity of this range exclusively from a forecasting perspective. As this range was determined only on the basis of pre-1990s data while our forecast evaluation sample starts in 1990, what we conduct is a true ‘out of sample’ forecasting exercise.

We are thus ready for undertaking a forecasting competition. At the start of the race, we have three equally plausible models:

One is based on the hypothesis that real exchange rates follow a random walk; and

Two on PPP, where real exchange rates are assumed to linearly adjust toward their mean.

The only difference between the two PPP models is that, in one case, we set a half-life adjustment of 5 years (HL), while in the second we estimate the duration of half-life adjustment using a simple autoregressive model (AR). We shall consider the real effective exchange rates of the euro (EUR) and the US dollar (USD) for the period between 1975:1 and 2012:3.

The forecasting scheme is a 15 year rolling window and the forecast horizon ranges between one and sixty months. To evaluate forecast accuracy we plot the mean squared forecast errors of the two PPP models divided by the mean squared forecast error of the random walk at different forecast horizons (Figure 1). Values below unity thus indicate that a given model forecasts better the real exchange rate than the random walk. The calibrated HL model (solid blue line) clearly beats the random walk both in the case of the euro and the dollar. Particularly persuasive is that the result holds also at short- term horizons and not just at longer horizons.

Figure 1. Mean squared forecast errors for real exchange rates (relative to the random walk).

Source: Author’s calculations.

Notes: The dashed red and solid blue lines stand for the AR and HL model, respectively. The forecast horizon, shown in the x-axis, is expressed in months.

The estimated AR model (dashed line) instead performs poorly, not only vis-à-vis the HL model but also vis-à-vis the random walk. In the extended version of this article (Ca’ Zorzi et al. 2015), we show analytically why for estimation windows of around 15 years of monthly data, the estimation error associated to the AR model is much more severe than the misspecification error associated to the random walk. This is exactly the reason why the random walk model appears to be successful, even if PPP holds. In the same paper we also provide several other robustness checks to confirm the validity of these results. In particular, it can be shown that the HL model beats the random walk for the entire 3 to 5 years half-life range proposed by Rogoff. The results hold also for several, albeit not all currencies in our sample.

Forecast competition for nominal effective exchange rates

The final step in our analysis consists in testing whether the mean reverting nature of the real exchange rate helps us to forecast nominal exchange rates. A simple approach is to assume that the real exchange rate adjustment predicted by our three alternative models, is entirely achieved via changes in nominal exchange rates, and that relative price changes play no role. Given the volatility of exchange rates, this is what one could expect in the case of flexible exchange rate regimes. The calibrated HL model (solid line) performs visibly better than the random walk while the estimated AR model (dashed line) underperforms both in the case of the euro and of the dollar (Figure 2). Our previous results hold also for the case of nominal exchange rate forecasting.

Figure 2. Mean squared forecast errors for nominal exchange rates (relative to the random walk).

Source: Author’s calculations.

Notes: as in Figure 1.

Currencies do not walk randomly

A long standing result of the academic literature is that exchange rates are not predictable as macroeconomic models cannot generally beat the random walk. The vast exchange rate literature provides, however, at least two reasons for being cautiously optimistic.

First, as discussed in Engel et al. (2008), estimation error is one of the root causes for the dismal forecasting performance of exchange rate models, which explains why the random walk is less competitive for larger datasets.

Second, the literature on PPP has shown that there is evidence of mean reversion in real exchange rates.

In this study we have illustrated how these findings can be exploited in exchange rate forecasting. In particular, we have proposed a simple model that just imposes a gradual return of the real exchange rate to its sample mean. From a theoretical perspective, this is a much more appealing alternative to the random walk for it takes into account that PPP holds over long-term horizons. The key finding of our study is that a calibrated half-life PPP model beats overwhelmingly the random walk in relation to real exchange rate forecasting. Our results are intuitive and not trivial:

The preferred forecasting model for real exchange rates resembles the random walk in the short-run while it gradually approaches PPP over long-term horizons.

A second key finding of our analysis is that, if the speed of mean reversion is estimated, rather than calibrated, the model performs significantly worse than the random walk due to estimation error.

Finally, we have shown that the mean reverting nature of real exchange rates can be exploited to outperform the random walk in relation to nominal exchange rate forecasting.

For both the case of the euro and the dollar we find that the nominal exchange rate has contributed to the mean reversion process of the real exchange rate rather than just followed a random walk.

Disclaimer: The views expressed are those of the authors and do not necessarily reflect those of the European Central Bank or the National Bank of Poland.

References

Ca’ Zorzi, M, J Muck and M Rubaszek (2015), “Real exchange rate forecasting and PPP: This time the random walk loses”, An earlier version was published as ECB Working Paper No. 1576, 2013,

Cassel, G (1918), “Abnormal deviations in international exchanges”, Economic Journal, 28 (112), 413-415.

Dornbusch, R (1985), “Purchasing Power Parity”, NBER Working Paper No. 1591.

Engel C, N C Mark and K D West (2008) “Exchange rate models are not as bad as you think”. In Acemoglu, D, K Rogoff and M Woodford (Eds), NBER Macroeconomics Annual 2007. Vol. 22 of NBER Chapters, National Bureau of Economic Research, Inc, pp. 381-441.

Ince, O (2014), “Forecasting exchange rates out-of-sample with panel methods and real-time data”, Journal of International Money and Finance 43 C, 1-43.

Jorion, P and R J Sweeney (1996), “Mean reversion in real exchange rates: evidence and implications for forecasting”, Journal of International Money and Finance 15 (4), 535-550.

Lothian, J R and M P Taylor (1996), “Real exchange rate behaviour: the recent float from the perspective of the past two centuries”, Journal of Political Economy 104 (3), 488-509.

Meese, R A and K Rogoff (1983), “Empirical exchange rate models of the seventies: Do they fit out of sample?”, Journal of International Economics 14 (1-2), 933-48.

Meese, R A and K Rogoff (1988) “Was it real? The exchange rate-interest differential relation over the modern floating-rate period”, Journal of Finance 43 (4), 933–48.

Rogoff, K (1996) “The PPP puzzle”, Journal of Economic Literature, 34, 647-668.

Taylor, A M and M P Taylor (2004) “The purchasing power parity debate”, Journal of Economic Perspectives 18 (4), 135-158.