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This paper studies exchange rate volatility within the context of the monetary model of exchange rates. We assume that agents regard this model as merely a benchmark, or reference model, and attempt to construct forecasts that are robust to model misspecification. We show that revisions of robust forecasts are more volatile than revisions of nonrobust forecasts, and that empirically plausible concerns for model misspecification can explain observed exchange rate volatility. We also briefly discuss the implications of robust forecasts for a number of other exchange rate puzzles.

Exchange rate volatility remains a mystery. Over the years, many explanations have been offered — bubbles, sunspots, ‘unobserved fundamentals’, noise traders, etcetera. Our paper offers a new explanation. Our explanation is based on a disciplined retreat from the Rational Expectations Hypothesis. The Rational Expectations Hypothesis involves two assumptions: (1) agents know the correct model of the economy (at least up to a small handful of unknown parameters, which can be learned about using Bayes' rule), and (2) given their knowledge of the model, agents make statistically optimal forecasts. In this paper, we try to retain the idea that agents process information efficiently, while at the same time relaxing what we view as the more controversial assumption, namely, that agents know the correct model up to a finite dimensional parameterization. Of course, if agents don't know the model, and do not have conventional finite-dimensional priors about it, the obvious question becomes — how are they supposed to forecast the future? Our answer is to suppose that agents possess a simple benchmark model of the economy, containing a few key macroeconomic variables. We further suppose that agents are aware of their own ignorance, and respond to it strategically by constructing forecasts from the benchmark model that are robust to a wide spectrum of potential misspecifications. We show that revisions of robust forecasts are quite sensitive to new information, and in the case of exchange rates, can easily account for observed exchange rate volatility. Our paper is closely related to prior work by Hansen and Sargent (2008), Kasa (2001), and Lewis and Whiteman (2008). Hansen and Sargent have pioneered the application of robust control methods in economics. This literature formalizes the idea of a robust policy or forecast by viewing agents as solving dynamic zero sum games, in which a so-called ‘evil agent’ attempts to subvert the control or forecasting efforts of the decision maker. Hansen and Sargent show that concerns for robustness and model misspecification shed light on a wide variety of asset market phenomena, although they do not focus on exchange rate volatility. Kasa (2001) used frequency domain methods to derive a robust version of the well-known Hansen and Sargent (1980) prediction formula. This formula is a key input to all present value asset pricing models. Lewis and Whiteman (2008) use this formula to study stock market volatility. They show that concerns for model misspecification can explain observed violations of Shiller's variance bound. They also apply a version of Hansen and Sargent's detection error probabilities to gauge the empirical plausibility of the agent's fear of model misspecification. Since robust forecasts are the outcome of a minmax control problem, one needs to make sure that agents are not being excessively pessimistic, by hedging against models that could have been easily rejected on the basis of observed historical time series. Lewis and Whiteman's results suggest that explaining stock market volatility solely on the basis of a concern for robustness requires an excessive degree of pessimism on the part of market participants. Interestingly, when we modify their detection error calculations slightly, we find that robust forecasts can explain observed exchange rate volatility.1 Since there are already many explanations of exchange rate volatility, a fair question at this point is — why do we need another one? We claim that our approach enjoys several advantages compared to existing explanations. Although bubbles and sunspots can obviously generate a lot of volatility, these models require an extreme degree of expectation coordination. So far, no one has provided a convincing story for how bubbles or sunspots emerge in the first place. Our approach requires a more modest degree of coordination. Agents must merely agree on a simple benchmark model, and be aware of the fact that this model may be misspecified.2 It is also clear that noise traders can generate a lot of volatility. However, as with bubbles and sunspots, there is not yet a convincing story for where these noise traders come from, and why they aren't driven from the market. An attractive feature of our approach is that, if anything, agents in our model are smarter than usual, since they are aware of their own lack of knowledge about the economy.3 Our approach is perhaps most closely related to the ‘unobserved fundamentals’ arguments in West (1987), Engel and West (2004), and Engel et al. (2007). These papers all point out that volatility tests aren't very informative unless one is confident that the full array of macroeconomic fundamentals are captured by a model.4 As a result, they argue that rather than test whether markets are ‘excessively volatile’, it is more informative to simply compute the fraction of observed exchange rate volatility that can be accounted for by innovations in observed fundamentals. Our perspective is similar, yet subtlely different. In West, Engel–West, and Engel–Mark–West, fundamentals are only unobserved by the outside econometrician. Agents within the (Rational Expectations) model are presumed to observe them. In contrast, in our model it is the agents themselves who suspect that there might be missing fundamentals, in the form of unobserved shocks that are correlated both over time and with the observed fundamentals. In fact, however, their benchmark model could be perfectly well specified. (In the words of Hansen and Sargent, their doubts are only ‘in their heads’). It is simply the prudent belief that they could be wrong that makes agents aggressively revise forecasts in response to new information. In contrast to ‘unobserved fundamentals’ explanations, which are obviously untestable, there is a sense in which our model is testable. Since specification doubts are only ‘in their heads’, we can ask whether an empirically plausible degree of doubt can rationalize observed exchange rate volatility. That is, we only permit agents to worry about alternative models that could have plausibly generated the observed time series of exchange rates and fundamentals, where plausible is defined as an acceptable detection error probability, in close analogy to a significance level in a traditional hypothesis test. We find that given a sample size in the range of 100–150 quarterly observations, detection error probabilities in the range of 10–20% can explain observed exchange rate volatility. The remainder of the paper is organized as follows. Section 2 briefly outlines the monetary model of exchange rates. We assume that agents regard this model as merely a benchmark, and so construct forecasts that are robust to a diffuse array of unstructured alternatives. Section 3 briefly summarizes the data. We examine quarterly data from 1973:1–2011:3 on six US dollar bilateral exchange rates: the Australian dollar, the Canadian dollar, the Danish kroner, the Japanese yen, the Swiss franc, and the British pound. Section 4 contains the results of a battery of traditional excess volatility tests: Shiller's original bound applied to linearly detrended data, the bounds of West (1988) and Campbell and Shiller (1987), which are robust to inside information and unit roots, and finally, a couple of more recent tests proposed by Engel and West (2004) and Engel (2005). Although the results differ somewhat by test and currency, a fairly consistent picture of excess volatility emerges. Section 5 contains the results of our robust volatility bounds. We first apply Kasa's (2001) robust Hansen–Sargent prediction formula, based on a so-called H∞ approach to robustness, and show that in this case the model actually predicts that exchange rates should be far more volatile than the observed exchange rate volatility. We then follow Lewis and Whiteman (2008) and solve a frequency domain version of Hansen and Sargent's evil agent game, which allows us to calibrate the degree of robustness to detection error probabilities. This is accomplished by assigning a penalty parameter to the evil agent's actions. We find that observed exchange rate volatility can be explained if agents are hedging against models that have a 10–20% chance of being the true data-generating process. Section 6 relates robustness to other puzzles in the foreign exchange market. In particular, we show that robust forecasts can explain the forward premium puzzle. In fact, explaining the forward premium puzzle is easier than explaining the volatility puzzle, since the associated detection error probabilities are larger. Section 7 contains a few concluding remarks.

This paper has proposed a solution to the excess volatility puzzle in foreign exchange markets. Our solution is based on a disciplined retreat from the Rational Expectations Hypothesis. We abandon the assumption that agents know the correct model of the economy, while retaining a revised notion of statistically optimal forecasts. We show that an empirically plausible concern for robustness can explain observed exchange rate volatility, even in a relatively simple environment like the constant discount rate/flexible-price monetary model. Of course, there are many competing explanations already out there, so why is ours better? We think our approach represents a nice compromise between the two usual routes taken toward explaining exchange rate volatility. One obvious way to generate volatility is to assume the existence of bubbles, herds, or sunspots. Although these models retain the idea that agents make rational (self-fulfilling) forecasts, in our opinion they rely on an implausible degree of expectation coordination. Moreover, they are often not robust to minor changes in market structure or information. At the other end of the spectrum, many so-called ‘behavioral’ explanations have the virtue of not relying on strong coordination assumptions, but only resolve the puzzle by introducing rather drastic departures from conventional notions of optimality. As noted at the outset, our paper is closely related to Lewis and Whiteman (2008). They argue that robustness can explain observed US stock market volatility. However, they also find that if detection errors are based only on the agent's ability to discriminate between alternative models for the economy's exogenous dividend process, then implausibly small detection error probabilities are required. If instead detection errors are based on the agent's ability to discriminate between bivariate models of dividends and prices, then stock market volatility can be accounted for with reasonable detection errors. This is not at all surprising, since robustness delivers a substantially improved fit for prices. Interestingly, we find that even if detection errors are only based on the exogenous fundamental process, exchange rate volatility can be accounted for with reasonable detection error probabilities. Still, one could argue that they are a bit on the low side, so it might be a useful extension to apply the bivariate Lewis–Whiteman approach to computing detection error probabilities. We conjecture that this would only strengthen our results. A second useful extension would be to consider in more detail the links between robustness and other exchange rate puzzles. For example, while we have shown that robust forecasts can also explain the forward premium puzzle, it would be interesting to see to what extent both puzzles can be explained simultaneously.