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A note on tests of efficient market hypotheses: the case of the forward exchange rate.

I. Introduction

According to the definition of market efficiency [Fama, 1970], market prices reflect all available information instantly: 1) in the absence of transaction costs, 2) given costless information, and 3) for homogeneous expectations. As a consequence, the market is always perfectly arbitraged, with no opportunity for above normal profits [Jensen, 1978]. The last 20 years have seen a significant amount of research into the theoretical problems that arise when one or more of the assumptions underlying Fama's [19701 definition are violated.(1) Grossman and Stiglitz [1980], for example, have pointed out the problems that are associated with the existence of information costs. If market prices did indeed reflect all information instantly, there would be no incentive for market participants to gather and process information, because they could not get compensated for the costs incurred. Andersen [1983-84] demonstrates that the equilibrium asset prices implied by the efficient market hypothesis may be indeterminate, even if one assumes that all three assumptions given by Fama [19701 hold.

The purpose of this paper is to point out that the definition of market efficiency as put forth by Fama [1970] also presents serious problems for measurement and empirical hypothesis testing. These problems derive from the fact that one concept of the passage of time is used for theorizing on market efficiency, whereas an entirely different concept is employed in empirical tests. This idea is developed with respect to the application of the efficient market hypothesis to the foreign exchange market, i.e., the notion that the forward exchange rate is an unbiased and efficient predictor of future spot exchange rates. This topic has captured intense interest since the mid-1970s, not only because of the switch from fixed to floating exchange rates but also because of the increasing internationalization of industrialized economies.(2)

The paper is organized as follows. The following section will explain the theoretical concept of market efficiency in the context of the forward market for foreign exchange. Next, it is shown why empirical tests have so far fallen short of truly testing the theory of market efficiency for the forward exchange rate. The concluding section outlines how an adequate empirical test may have to be constructed.

II. The Forward Market Efficiency Hypothesis

As with any efficient market hypothesis, the forward exchange rate efficiency hypothesis (FMEH) consists of two components: a postulated equilibrium relationship between the model's variables and the assumption of rational expectations by market participants. The simplest equilibrium relationship assumes that speculators act to equilibrate the forward rate with the expected spot rate:

(1) [Mathematical Expression Omitted]

where: [E.sub.t-1] is the expectations operator at time t-1; [s.sub.t.] is the logarithm of the spot exchange rate; [Mathematical Expression Omitted] is the logarithm of the spot exchange rate at time t that is expected at time t-1; [I.sub.t-1] is the information set available at time t- 1; and [f.sub.t-1] is the logarithm of the forward rate quoted at time t- 1 on a contract maturing at time t. The hypothesis that expectations of future spot rates are rational in the sense of Muth [1961] can be formalized as:

(2) [Mathematical Expression Omitted]

where [[espilon].sub.t] is a white-noise error process.

The hypothesis summarized in (1)--which states that the forward rate is equal to the expected future spot rate--relies on the assumption that speculators will buy or sell unlimited amounts of foreign exchange in the forward market if this condition does not hold. Equation (1) does not contain a risk premium and, therefore, implies that speculators are risk neutral. To allow for risk aversion on the part of speculators, a risk premium ([ohm]) can be added to (1) to obtain:(4)

(3) [Mathematical Expression Omitted]

The rational expectations hypothesis of (2) assumes that speculators either know or act as if they know the equations and parameters of the true underlying model for exchange rate formation. Based on this knowledge, market participants do not make systematic forecasting mistakes. Their forecasts deviate from the actually observed exchange rate only insofar as unexpected and unpredictable shocks occur in period t, i.e., insofar as the error term is not equal to zero.

Combining (1) with (2) or, alternatively, (3) with (2) leads to equations:

(4) [s.sub.t] = [f.sub.t-1] + [[epsilon].sub.t],

(5) [s.sub.t] = [f.sub.t-1] - [ohm] + [[epsilon].sub.t],

respectively. The empirical equivalent of the more general (5) can be written as:

(6) [s.sub.t] = [a.sub.0] + [a.sub.1] [f.sub.t-1],

where the joint restriction [a.sub.0] = 0 and [a.sub.1] = 1 represents the FMEH in the absence of a risk premium (4), and where the restriction [a.sub.1] = 1 characterizes the FMEH with a risk premium (5).

Equation (6) represents a long-run relationship between actual spot and associated forward rate because dependent and independent variables have a unit root and are cointegrated [Baillie and Bollerslev, 1989; Hakkio and Rush, 1989; Barnhart and Szakmary, 1991].(5) The parameters of (6), therefore, provide information only on the question whether the FMEH holds as a long-run equilibrium relationship, not whether it is valid continuously, at every point in time.

The FMEH has been widely tested on data with sampling intervals ranging from one day to one quarter.(6) All empirical tests are based on an estimating equation of the form:

(7) [Mathematical Expression Omitted]

The tests differ in the number and types of restrictions that are imposed a priori on the parameters of this equation. The simplest and earliest tests (e.g., Cornell [1977]) assume that all [[theta].sub.i] and [[delta].sub.j] are zero. This set of restrictions makes estimating (7) equivalent to the long-run equilibrium (6). Thus, these early tests only look at the long-run equilibrium implications of the FMEH. The overwhelming majority of these tests finds that the data conform to the FMEH without risk premium, i.e., the joint restriction ao = 0 and al 1 in (6) cannot be rejected.

Later work, such as Frankel [1979], Fama [1984], and most studies since the mids, 1980s, do not use the a priori assumption that [[theta].sub.i] = [[delta].sub.j] = 0 for all i and j. Hence, this latter segment of the literature does provide a meaningful test of the view that the FMEH holds continuously, either in its form with or without a risk premium. Essentially, all of these studies find that some [[theta].sub.i] and [[delta].sub.j] are always different from zero. These results imply that the FMEH does not hold continuously or at every instant in time.(7) Yet the same estimates also confirm the earlier result that the FMEH holds as a long-run equilibrium relationship. In particular, one can show for essentially all reported studies that the estimated short-run parameters of (7) imply long-run parameters for the relationship between actual spot and associated forward rate that conform to the restrictions of (4). It should be noted in this context that the long-run equilibrium intercept term implied by (7) is given as:

[[alpha]/(1-[[sigma].sub.j][[delta].sub.j])] = 0,

and the long-run equilibrium slope parameter as:

[([beta] + [[sigma].sub.j][[theta].sub.j])/(1-[[sigma].sub.jj][[delta].sub.j ])] = 1.

III. Empirical Tests in a Rational Expectations Framework

According to Fama [1991], the most significant problem encountered in tests for market efficiency is the so-called "joint hypothesis" problem. The expectations mechanism, (2) in this case, is always necessarily tested jointly with a particular assumption about the equilibrium relationship among the involved variables, (1) or (3) in this case. Hence, if the parameter restrictions relating to continuous market efficiency are rejected, i.e., [[theta].sub.i] = [[delta].sub.i] = 0 for all i and j in (7) for the case of the FMEH, it is never clear whether that result derives from a breakdown of the expectation mechanism or from a misspecified equilibrium condition.

Since it has become apparent that the FMEH does not hold continuously, research has concentrated on finding the possible reasons. Most of the work has concentrated on amending the equilibrium relationship (1) by adding a risk premium, as in (3), or by adding other variables. This line of work continues to assume speculators in the forward market form expectations according to (2). Empirical results of this type of work are ambiguous.(8) In general, however, there is no evidence that the addition of a risk premium causes the FMEH to hold continuously.

As an alternative to allowing for risk premia, a number of recent studies have tried to check the assumption of rational expectations in the forward market by survey methods [Goodhart, 1988; Froot and Frankel, 1989; Ito, 1990]. Interestingly, all find that the rational expectations assumption, as incorporated in (2), is clearly rejected on a continual basis. Participants in the forward market do not appear to hold expectations of future exchange rates that are consistent with (2) at every point in time.(9) This finding may explain why the burgeoning work on risk premia has not been conclusive, because all models on risk premia continue to rely on the assumption of rational expectations. It could also explain why the FMEH does not hold continuously in typical empirical tests but only as a long-run equilibrium relationship.

The central question that arises in this context is whether the strong empirical findings against rational expectations in the forward market are sufficient evidence against the FMEH. Should the FMEH be discarded and replaced with an alternative theory? Clearly, evidence from ex ante survey data cannot be dismissed with the standard argument that may be leveled against econometric tests on ex post data: even if ex ante expectations are fully rational, expectations calculated from ex post data may look irrational when compared to the information set that is available ex post. But there is an alternative defense of the assumption of rational expectations: the tests of rationality conducted so far, including those on survey data, do not capture the essence of the theory to be tested. In particular, the empirical tests use a concept of time that is different from the one that is at the heart of the theory of rational expectations. Hence, the tests are, in a strict sense, inadequate. They build up and shoot down an artificial construct rather than providing sufficient attention to all aspects of the theory.

The concept of time is one aspect of the theory of rational expectations as set out by Muth [1961] that is easily overlooked by empirical researchers. It is crucial to remember that the theory of rational expectations does not specify the time horizon over which a rational expectations equilibrium, as described by (2), holds. This is because the rational expectations theory does not contain an explicit model of learning,'o but concentrates rather on the outcome of this learning process and its economic consequences [Hahn, 1982, p. 3]. To put it differently, the theory of rational expectations relies on the construct of logical time, rather than the historical time that empirical researchers are dealing with [Hoover, 1988]. The time subscript t in (2), therefore, refers to logical time, whereas the time subscripts elsewhere in the model relate to historical time.

In logical time, a time period is as long as it takes to generate a rational expectations equilibrium. Historical time, by contrast, is divided into standard increments (hours, days, months, and so on) that are determined by a system of measurement, not by the economic phenomenon under investigation. Clearly, there would be little to worry about the difference between logical time and historical time if the two were always equal. In reality, however, this is most likely not the case. In general, it is impossible to say how many standard increments of historical time it takes to attain a rational expectations equilibrium. The number of historical time periods depends not only on the economic environment, its complexity and tendency for change, but also on the learning ability of the economic actors. Learning, however, is hardly ever instantaneous, unless information and calculation costs are negligible, which is a rather unlikely scenario in reality. I For empirical work on equally spaced observations, however, just this unrealistic assumption has to be made: (2) has to be assumed to hold for all t in historical time. Otherwise, one cannot combine (2) with (1) or (3) because it would mean combining a time subscript defined in logical time (2) with one defined in historical time, (1) or (3).

It is apparent that ceteris paribus, the assumption that a rational expectations equilibrium is attained within a given increment of historical time, becomes ever less likely the shorter the sampling interval is and vice versa. It is no surprise, then, that the survey studies [Goodhart, 1988; Froot and Frankel, 1989; Ito, 1990] tend to reject the hypothesis of rational expectations and, therefore, the FMEH for short-time increments but appear to be unable to reject it as a long-run equilibrium proposition [Liu and Maddala, 1992]. As mentioned earlier, this result is consistent with most econometric tests of the FMEH based on ex post data.

One has to conclude, then, the evidence collected so far from econometric studies on both ex post and ex ante survey data has conclusively rejected neither the hypothesis of rational expectations nor the FMEH because no test has been conducted that is adequate relative to the theory. The question arises as to what would be a true or adequate test of the FMEH. The most important aspect of such a true test would be a sampling methodology that is radically different from those used in the past. In particular, rather than mechanically sampling observations at equally spaced intervals, one would want to include only those observations in the sample for which (2) holds, i.e., for which market participants have reached a rational expectations equilibrium. A sample of equally spaced historical data points will most likely contain an unspecified number of observations for which market participants have not yet attained a rational expectations equilibrium but are, instead, on the adjustment path toward one. This is clearly all the more likely if the sample includes periods of significant economic change, such as 1978, 1981, and 1985, for tests of the FMEH for the U.S. dollar.

However, selectively choosing observations for which a rational expectations equilibrium has been reached is a tall order and, for all practical purposes, impossible for the type of data typically employed in studies of the FMEH. There is little knowledge of when a rational expectations equilibrium has been reached in the foreign exchange market. Hence, there is little chance that one can test empirically whether the FMEH holds in the short run. The best one can probably do for the data commonly used is to artificially simulate a rational expectations equilibrium by looking only at long-run equilibrium parameters. These long-run parameters can be obtained either from the corresponding cointegrating regression (6) or, as outlined earlier, from the estimated short-run parameters of (7).

IV. Conclusion

The paper has suggested that no true or adequate empirical test of continuous market efficiency has so far been conducted for the forward exchange rate market or, for that matter, for other efficient market hypotheses. In fact, one may doubt whether a true test is even possible. Such a test would have to be conducted in logical time because the assumption of rational expectations, which underlies the concept of efficient markets, is itself defined in logical time. The difficulty with such a test is that continuous market efficiency in logical time has no easily discernible counterpart in historical time. Sampling data at equally spaced time periods is certainly much too mechanistic an approach. It would be an acceptable strategy only for a world subject to no change in the underlying fundamentals that drive prices and expectations of prices. For a world subject to change, however, sampling has to occur only at those points that can be considered rational expectations equilibria. Yet these are essentially impossible to identify in historical time.

Because all tests of the FMEH or other efficient market hypotheses utilize equally spaced observations, one wonders what they can contribute to the literature. Since the equilibrium concept implied by rational expectations is best approximated by the empirical concept of a long-run or steady state equilibrium, they can, at best, provide meaningful empirical tests on long-run equilibrium propositions. In the context of the FMEH, for example, this would include a test whether a risk premium exists in the long run or whether the forward rate is a biased predictor in steady state.

By contrast, it makes little economic sense to test on equally spaced observations of whether the FMEH or other efficient market hypotheses hold continuously in historical time. Since the theory relates to logical time, these tests in historical time cannot reject the theory. They are, in a true sense, meaningless or inadequate for the theory at hand. Clearly, they also cannot provide any guidance on how to modify the theory. In this context, one may ask to what extent a theory is useful if its implications are essentially impossible to test empirically. Certainly, one has to agree with Goodhart et al. [19921 that the FMEH provides no basis for short-run forecasting of future exchange rate changes.

(1) An exhaustive summary of research on efficient markets in a closed economy context is provided by Fama [1991].

(2) A recent summary of the issues surrounding the testing of the efficiency of the forward market for foreign exchange is provided by Froot and Thaler [1990].

(3) More complex equilibrium relationships have also been developed. The most popular adds a risk premium to the determining equation of the forward rate. This is taken up later in the text.

(4) Compare Grauer et al. [1976] for a formal derivation of the economic rationale for a risk premium.

(5) However, compare the work by Sephton and Larsen [1991] on the fragility of the cointegration results for conclusions on long-run relationships.

(6) Surveys of the empirical work are provided in Fama [1984], Boothe and Longworth [1986], Froot and Frankel [1989], and Barnhart and Szakmary [1991].

(7) The result does not depend on whether a risk premium is allowed for or not.

(8) The literature on risk premia in the forward market for foreign exchange is immense. Some selected articles include Frankel [1982], Fama [1984], Hodrick and Srivastava [1984], Domowitz and Hakkio [19851, Baillie and Bollersiev [1990], Kaminsky and Peruga [1990], and Engel [1992].

(9) In a more recent study, Liu and Maddala [19921 suggest that the rational expectations assumption does hold for the forward rate. However, the authors solely rely on cointegration tests and, therefore, only test for a long-run relationship. They do not explicitly address the question whether expectations are rational at every instant in time. Hence, their conclusions do not contradict those of the other survey studies.

(10) In fact, according to recent work by Evans and Ramey [19921, there is nothing rational about rational expectations because the theory of rational expectations totally ignores "expectational preferences and technology."

(11) Lewis [1989] provides some empirical evidence on the importance of learning in the forward exchange market of the 1980s.

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JOACHIM ZIETZ, Middle Tennessee State University.
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