The Excess Volatility of Foreign Exchange Rates: Statistical Puzzle or Theoretical Artifact?∗

The inability to reconcile observed levels of foreign exchange rate volatility with predictions derived from rational expectations models represents one of the most persistent challenges in international finance. This paper shows that such excess volatility puzzles arise from informational assumptions by contrasting exchange rate equilibria under different expectational paradigms: rational expectations and their generalization, rational beliefs. Under the latter agents hold data rather than model consistent expectations requiring learning and inference. Uncertainty now arises endogenously as agents with diverse beliefs might trade even in the absence of new information. An analysis of currency volatility mechanisms now reveals that excess volatility is a theoretical consequence of rational expectations’ structural knowledge assumptions. Markets only transmit volatility from exogenous variables to exchange rates without any amplification mechanism. Hence, rational expectations equilibria provide a lower volatility bound on more general exchange rate processes solving the excess volatility puzzle in terms of endogenous volatility generation. Finally, the results are applied to explore the structure of currency crises as short-lived rational deviations from economic fundamentals. ∗This paper is based on earlier work circulated as ”Foreign Exchange Rates under Alternative Expectational Paradigms: A Resolution of the Excess Volatilty Puzzle.” †Stimulating discussions with and advice from Bob Flood, Thomas Gehrig, Peter Isard, Lutz Kilian, Mordecai Kurz, Lemma Senbet, Sunil Sharma, René Stulz, Jürgen von Hagen, Volker Wieland and Jeff Zwiebel are gratefully acknowledged. I also benefitted from comments and suggestions of seminar participants at the IMF, the Deutsche Bundesbank, Indiana, Freiburg, Bonn and the 1997 European Meetings of the Econometric Society, 1997 EFA meetings and 1997 EEA meetings where the earlier paper was presented. Special thanks go to Lars T. Nielsen, the EFA discussant. All errors, omissions or misinterpretations are solely my responsibility. The Excess Volatility of Foreign Exchange Rates: Statistical Puzzle or Theoretical Artifact? The statistical properties of foreign exchange rates continue to pose a serious theoretical challenge. [13, Flood and Taylor 1996], [15, Frankel and Rose 1994] and [26, Meese 1990] document the general failure of different model classes to generate sufficient amounts of exchange rate volatility in terms of economic fundamentals. Other statistical anomalies such as time varying risk premia or the forward rate bias surveyed in [17, Froot and Thaler 1990] cast further doubt on the validity of standard exchange rate models. While the former might explain the latter risk premia have to be disproportionately variable in comparison to the volatility of fundamentals as analyzed in [2, Bekaert 1995] and [3, Bekaert 1996]. In an attempt to resolve the tension between empirical facts and model predictions this paper examines the theoretical foundations of excessive exchange rate volatility by analyzing learning and diversity of beliefs in foreign currency markets. The one common feature that most exchange rate models share is their reliance on rational expectations. In particular, they all assume that agents know and agree on the structure of the true model. However, a growing body of empirical evidence suggests that such assumptions might be inappropriate for foreign currency markets. [24, Taylor 1995] makes the general case against homogeneous beliefs in exchange rate determination while [14, Frankel and Froot 1987] provide extensive evidence for heterogeneous expectations in foreign currency markets. Furthermore, the survey by [28, Takagi 1991] on market held exchange rate expectations raises serious doubts about the rationality of such heterogeneous beliefs. In his study on sterling-dollar exchange rates 1981-84 [10, Evans 1986] also identifies disequilibrium expectations whereas [17, Froot and Thaler 1990] report that the bias in interest rate differentials is almost entirely due to a bias in exchange rate expectations. While rational expectations insure model consistent beliefs and analytic tractability they also impose heavy statistical restrictions on equilibrium processes. The latter is a result of the severe informational assumption that agents have full knowledge of the economy and, more importantly, its underlying stochastic structure. Structural knowledge helps to solve for the endogenous variables in terms of exogenous ones so that the only uncertainty in equilibrium comes from the systematic variation of economic fundamentals. Hence, one can not increase the variability of endogenous variables (exchange rates) without first increasing the variability of exogenous ones. This observation explains why efforts to modify common rational expectations models of foreign currency markets such as [25, Manuelli and Peck 1990] and [3, Bekaert 1996] have been only partially successful. It also suggests a new line of attack on foreign currency puzzles. Instead of solving different models under the same beliefs assumption this paper compares the statistical properties of exchange rate equilibria in the same two-country [23, Lucas 1982] framework under alternative expectational paradigms. Rational expectations represent homogeneous beliefs while diversity in opinion and learning are modeled in terms of rational beliefs as developed in [19, Kurz 1991]. This approach lends itself to an investigation of the implied volatility mechanisms because rational beliefs encompass rational expectations as a special case. The former simply generalizes the latter’s rationality criterion of model consistent expectations to data consistent beliefs with learning . Consequently, rational beliefs lead to equilibria in which agents hold divergent, yet rational views. Equilibrium realizations then have differential information content so that contrary to rational expectations agents may trade even in the absence of new information. The interaction of learning, diversity in beliefs and trading now generates uncertainty endogenously thus resolving the excess volatility puzzles of exchange rates. Decomposing the rational beliefs variance into the rational expectations variance and an endogenous uncertainty term identifies the source of excess volatility: aggregated learning behavior. Learning behavior or its absence, in turn, determine the nature of volatility mechanisms. Rational expectations correspond to the deterministic transmission of exogenous uncertainty while rational beliefs endogenously amplify and generate uncertainty. Regarding first moments, exchange rate processes converge to the same asymptotic mean under both expectational hypotheses. However, endogenous uncertainty permits temporary deviations from fundamentals with correspondingly higher rational beliefs variance. The conclusion is immediate: the mystery of the missing exchange rate volatility is a theoretical artifact of rational expectations and unreasonable informational assumptions. This paper contributes to the growing literature that attempts to explain foreign exchange market anomalies in terms of expectational errors. Its main innovation lies in avoiding ad hoc beliefs specifications by casting rationality in terms of data consistent learning1. It turns out that this weakening of rationality is just sufficient to identify the volatility transmission mechanism as the deeper source of excess volatility puzzles. The next section presents an informal introduction to rational beliefs as a generalization of rational expectations. Section 2 develops a simple model of foreign currency markets, derives rational expectations and beliefs equilibria and compares their statistical properties. Section 3 analyzes exchange rate volatility in the light of the derived statistical properties of rational beliefs equilibria and introduces the concept of endogenous uncertainty. All the proofs and a summary of rational beliefs are relegated to the Appendix. 1. From Rational Expectations to Rational Beliefs Modeling decision making under uncertainty requires the judicious specification of beliefs about the stochastic environment. In economics, the paradigm of rational expectations (RE) has been so successful because it avoids logical contradictions and model inconsistencies by imposing This approach is similar in spirit to [22, Lewis 1989] where agents learn about regime changes one-time, unobservable shocks whose occurrence are only gradually accepted. The shock effects persist and excess returns are not traded away as one would expect in a learning environment.