On the Statistical Properties of Floating Exchange Rates
A Reassessment of Recent Experience and Literature
  • 1 https://isni.org/isni/0000000404811396, International Monetary Fund

The paper reviews the statistical behavior of major currency exchange rates during 1975-86. A close inspection indicates small deviations of recent exchange rate behavior from random walks and some systematic movements in monthly data, possibly corresponding to the relatively infrequent arrivals of information concerning major macroeconomic variables. The distributional characteristics of exchange rate changes differ between daily and monthly data and thus imply the possible presence of heterogeneity in underlying factors. These and other observations suggest care in the use of daily data in empirical work and the usefulness of explicit modeling of heterogeneity among market participants and in information structure.

Abstract

The paper reviews the statistical behavior of major currency exchange rates during 1975-86. A close inspection indicates small deviations of recent exchange rate behavior from random walks and some systematic movements in monthly data, possibly corresponding to the relatively infrequent arrivals of information concerning major macroeconomic variables. The distributional characteristics of exchange rate changes differ between daily and monthly data and thus imply the possible presence of heterogeneity in underlying factors. These and other observations suggest care in the use of daily data in empirical work and the usefulness of explicit modeling of heterogeneity among market participants and in information structure.

I. Introduction

The paper will review the statistical properties of nominal exchange rates between major currencies in the light of some 15 years of experience with floating exchange rates and the relevant literature that has emerged in recent years. The paper is intended both as a summary description of empirical nominal exchange rate behavior and as a review of the existing literature on this subject. As a summary description, it will present basic statistical measures of the spot and forward exchange rates between four major currencies during 1975-86; as a review, it will trace recent theoretical and empirical developments that are important to our understanding of the nature of exchange rate behavior.

Basic understanding of the statistical behavior of exchange rates is important because it forms the basis for discussion and analysis of other pertinent issues involving exchange rates. In particular, the empirical behavior of exchange rates is important for at least three principal reasons. First, the time-series behavior of exchange rates has implications for the question of market efficiency that has received far more attention in the literature. Second, the distributional property of exchange rates in part determines the riskiness of the foreign exchange market and the validity of statistical inference in empirical work. Third, the statistical behavior of exchange rates provides insight into the nature of the process governing exchange rate determination. A systematic review of empirical exchange rate behavior in these three principal areas might also provide useful implications for the direction of future research.

The paper will first clarify the often confused relationship between market efficiency and exchange rate behavior; it will show that market efficiency imposes certain constraints on the relationship among endogenous variables but not necessarily on the stochastic process governing the time-series behavior of exchange rates. It will then discuss the statistical properties of nominal exchange rates, which are divided for analytical convenience into time-series and distributional properties. This division corresponds to the two important elements of the traditional random walk theory of asset prices (Fama, 1965): (1) the serial correlation of successive price changes and (2) the type of probability distribution to which those price changes conform. For each type of statistical behavior, the paper will review the literature and draw implications for the nature of forces that underlie the exchange rate generating process.

The paper will show that, although the level of the empirical exchange rates of major currencies followed a process that is closely approximated by a random walk, some serial dependence in successive exchange rate changes was almost always present on a closer examination. Moreover, monthly data showed generally greater serial dependence than daily data, possibly suggesting the presence of systematic information in low frequency data corresponding to macroeconomic variables. Regarding the distributional properties, the paper confirmed that the distribution of daily exchange rate changes was in general too “peaked” and “fat-tailed” to be normal (i.e., leptokurtic); in contrast, the distribution of monthly changes could be characterized as approximately normal. The normal with an autoregressive conditional heteroskedasticity (ARCH) process for innovations has shown some promise as a model of the empirical distribution of exchange rate changes. However, the most one can say with confidence is that the distributional characteristics of daily and monthly exchange rate changes point to the presence of heterogeneity among market participants as well as changing parameters over time.

The paper is organized as follows. Section II clarifies the concept of efficiency in foreign exchange markets and presents an example of the way market efficiency is related to the time-series behavior of nominal exchange rates using a simple monetary-type model. Section III discusses the time-series properties of exchange rates, including random walk tests of the levels of exchange rates and various serial correlation tests of the first differences of their logarithms. Section IV then discusses the distributional properties of the first logarithmic differences of exchange rates and critically analyzes some of the possible hypotheses that have been proposed to explain the observed behavior. Finally, Section V presents concluding remarks.

II. Market Efficiency and Exchange Rate Behavior

1. Efficiency in foreign exchange markets

It is sometimes thought that market efficiency necessarily specifies the type of time-series behavior of exchange rates, such as a random walk. In reality, however, market efficiency can be consistent with many types of statistical behavior. What market efficiency does specify is the kind of relationship that must exist between certain endogenous variables. The time-series behavior of exchange rates in an efficient market would be in turn determined by the nature of the time-series behavior of the exogenous variables underlying that relationship.

In general, an efficient capital market is defined as a market in which prices fully reflect all available information and, consequently, investors cannot systematically earn an “unusual” profit on the basis of information available in the market (Fama, 1976). This definition of market efficiency, however, lacks testable content in and of itself unless an operational definition is given for what constitutes an “unusual” profit and “available” information. The first operational definition requires a formal economic model of asset price determination; otherwise, one cannot determine what is “normal,” hence what is “unusual.” The second operational definition, on the other hand, simply requires a characterization of the information set against which the question of market efficiency is tested.

In the empirical literature on stock markets, broadly two types of characterization have been made, corresponding to what is called weak form efficiency and semi-strong form efficiency. 1/ Weak form efficiency requires that an asset’s price should not be predicted on the basis of the past history of its own prices. Popular forms of weak form efficiency tests have involved testing either for the existence of a trading rule based on observed prices that would yield a higher return than a simple buy-and-hold strategy, or for the existence of serial correlation in securities returns on the assumption that the returns are constant.

Semi-strong form efficiency requires that the asset price should be the market’s best prediction based on all available information. Popular empirical tests have involved analysis of the residuals calculated as the difference between actual prices and the prices predicted by a model of market equilibrium (e.g., the Market Model or the Capital Asset Pricing Model). Others have tested the profitability of different strategies to trade on published information.

In the context of foreign exchange markets, the term “efficiency” has come to acquire the two additional meanings of covered interest parity and forward market efficiency. First, covered interest parity is a condition that riskless arbitrage yields no profit, i.e.:

tft+1et=Rt+ut,(1)

where tft+1 is the one-period ahead forward exchange rate and et is the spot exchange rate, both expressed in logarithm as the domestic currency price of the foreign currency at time t (i.e., an increase in f or e means a depreciation of the domestic currency); Rt is a differential between the domestic and foreign one-period interest rates formed at t, both expressed as the log of one plus the nominal interest rate; and ut is a random deviation at t. 2/ In general, no restrictions need to be placed on the distribution of ut; it may or may not be white noise. For example, in the presence of capital controls, deviations from covered interest parity can be in one direction, corresponding to one-sided restrictions on either outflows or inflows; thus, ut can be serially correlated.

Equation (1) states that, in a state where no exploitable profit opportunity exists, a forward premium must be offset by the corresponding nominal interest rate differential, except for a deviation (u) whose magnitude must be within the cost of arbitrage. In this case, an “unusual” profit means a profit in excess of transactions costs provided that domestic and foreign assets are perfect substitutes. There is, however, a conceptual difficulty regarding the interpretation of the transactions cost arising from capital controls. For example, the deviation from covered interest parity can be large in the presence of capital controls; however, the market can be considered efficient in the sense of equation (1) as long as there is no exploitable profit opportunity within that given regulatory framework. Available evidence suggests that, at least for the 1980s, deviations from covered interest parity between similar short-term instruments in major industrial countries have been extremely small and serially uncorrelated, suggesting that the market has been unambiguously efficient with the recent elimination of capital control measures in these countries. 3/

Second, forward market efficiency expresses the idea that the forward exchange rate incorporates all available information about the expected future spot rate, i.e.:

tft+1=Etet+1+rt(2)

where Et is a mathematical expectations operator based on the set of information available at t, and r is a deviation term that reflects, at most, the possible presence of a risk premium plus random mean-zero error. In the empirical literature, most studies have made the operational assumption of risk neutrality and tested the hypothesis that the forward exchange rate is an unbiased predictor of the future spot rate. Although not conclusive, empirical evidence generally points to the rejection of the joint hypothesis of risk neutrality and forward market efficiency, suggesting either market inefficiency or the existence of a risk premium (for a survey of the empirical literature, see Levich, 1985; Boothe and Longworth, 1986; and Isard, 1987). 4/

Covered interest parity and forward market efficiency jointly imply a more familiar condition of market efficiency,

Etet+1et=Rt+utrt.(3)

This condition amounts to uncovered interest parity if ut is white noise and rt (the risk premium) is zero. Equation (3) corresponds to the usual formulation of market efficiency in stock markets, because the left-hand-side variable in (3) can be interpreted as the expected rate of return from speculation in the spot exchange market. That is to say, covered interest parity and forward market efficiency are components of the more traditional concept of market efficiency. One can interpret condition (3) as weak form efficiency if one assumes risk neutrality and a constant nominal interest differential; in this case, the presence of serial dependence in an exchange rate series will reject the efficiency hypothesis. 5/ More generally, one can also interpret (3) as semi-strong form efficiency; in this case, empirical testing of market efficiency requires a knowledge of how the risk premium and the nominal interest differential are determined.

Compared with tests of market efficiency in stock markets, empirical testing of (3) in foreign exchange markets involves two types of limitations (Levich, 1985). First, in testing weak form efficiency, one has less justification for assuming that the nominal interest differential plus the risk premium are constant particularly for monthly or quarterly data. Second, as a test of semi-strong form efficiency, one has no satisfactory model of market equilibrium in the foreign exchange market comparable to the Capital Asset Pricing Model. In the absence of a satisfactory model of market equilibrium, foreign exchange market efficiency possesses little testable content. 6/

2. Examples of time-series behavior

The preceding discussion suggests that the behavior of expected changes in exchange rates has little to do with market efficiency; rather, it depends on the structure of underlying economic variables that influence the risk premium (r) and the joint determination of e and R. This idea can be made explicit by specifying general forms of market equilibrium in the goods and money markets and imposing them on equation (3). As a simple illustration, let us assume the following form of international price linkage as the goods market equilibrium condition:

et=βpt+ηt,(4)

where the exchange rate is influenced by both a systematic relative price factor (p) and a non-systematic factor (η); if β = 1 and η = 0, equation (4) corresponds to purchasing power parity.

For the money market, let us assume the following equilibrium relationship between the differential in domestic and foreign money supplies (m) and the relative price in the consolidated world market:

mt=ptγRt,(5)

where γ, interpreted as the global interest elasticity of money demand, assumes that the country elasticities are identical for simplicity. Alternatively, equation (5) can be interpreted as a characterization of a less specific equilibrium condition in which the relative price is related to other real and nominal variables; in this case, m can be more broadly interpreted as a fundamental economic variable.

Substituting (4) and (5) into (3), we obtain:

et=[β/(1+φ)]mt+[φ/(1+φ)]Etet+1+[φ/(1+φ)](rtut)+ηt,(6)

where ϕ ≡ βγ. This intermediate solution, which relates the current exchange rate to the current value of the underlying fundamental variable, the expected value of the future exchange rate and random errors, is a general solution form obtained from most models of exchange rate determination in which an intertemporal variable (such as R) plays a role (Genberg, 1984; and Frenkel and Mussa, 1985). 7/ A solution of equation (6) can be obtained by forward iteration:

et=[β/(1+φ)]Σj=0[φ/(1+φ)]jEtmt+j+Σj=0[φ/(1+φ)]j+1Etrt+j+zt,(7)

where z ≡ η - (ϕ1/+ϕ)u. This solution assumes that all expected future values of z are zero and rules out the existence of a rational bubble.

Equation (7) makes clear the forward-looking nature of the solution for the exchange rate. However, it does not suggest that e must follow any particular stochastic process to satisfy market efficiency. For example, e can be functionally related to its past values if they help predict its current and future values. For example, suppose that there is no risk premium and m follows a first-order autoregressive process, i.e., AR(1), with a coefficient k. Then, equation (7) can be simplified to,

et={β/[1+φ(1k)]}mt+zt.(8)

Clearly, if m is AR(1) and zt is white noise, e is also AR(1). As to the behavior of the forward rate, we see from equation (3) that the forward rate is simply the appropriately updated spot exchange rate plus a risk premium. Consequently, if we assume that the risk premium is zero and m follows AR(1), we have:

tft+n={βkn/[1+φ(1k)]}mt.(9)

Thus, if m is AR(1), the forward rate is also AR(1). Yet, in either case, the presence of serial correlation in no way implies market inefficiency because the behavior of the exchange rate satisfies the condition of market efficiency given by equation (3).

III. Time-Series Properties of Exchange Rates

1. Random walk tests

Despite the lack of any theoretical necessity for e or f to follow any particular stochastic process, recent empirical investigations have almost unanimously found that the exchange rate follows a process that is closely approximated by a random walk. One straightforward way to see this “random walk” nature of empirical exchange rate behavior is to calculate the “F” and “t” statistics proposed by Dickey and Fuller (1979, 1984) that are extensions of the conventional F and t tests to non-stationary time-series. Like the conventional tests, Dickey and Fuller’s F test involves a test of the joint restriction that the intercept is zero and the coefficient is unity when e or f is regressed on its lagged value; the statistic in excess of a critical value would result in a rejection of the random walk hypothesis. Dickey and Fuller’s t test is also analogous to the conventional one, involving a test of the hypothesis that the slope coefficient is unity in the above regression.

Table 1 and Table 2 report Dickey-Fuller test statistics based on daily and monthly exchange rates over four subperiods during 1975-86 for six bilateral exchange rates. 8/ Tests were performed for spot exchange rates as well as 3-month and 12-month forward exchange rates. Two observations emerge from the tables. First, daily and monthly data, as well as the spot and forward rates, show broadly similar patterns. Second, except in a few cases, one cannot reject the hypothesis that the spot and forward exchange rates followed random walks during different subperiods of 1975-86. 9/ It is interesting to note that two of the cases where the null hypothesis of a random walk was rejected involved the Japanese yen during the first subperiod, although it is not clear to what extent the presence of capital controls might have been a contributing factor. 10/

Table 1.

Dickey-Fuller Statistics on Daily Exchange Rates

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Note: a) **(*) indicates that the statistic is significant at 5(10) percent.b) For each exchange rate, the first entry is the “F” statistic and the second entry in the parenthesis is the “t” statistic (with an intercept).
Table 2.

Dickey-Fuller Statistics on Monthly Exchange Rates

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Note: a) **(*) indicates that the statistic is significant at 5(10) percent.b) For each exchange rate, the first entry is the “F” statistic and the second entry in the parenthesis is the “t” statistic (with an intercept).

Similar unit root tests on other frequency data or on currency futures prices reached essentially the same conclusion. On the basis of weekly data for the Swiss franc, deutsche mark, and Canadian dollar during 1976-81, Meese and Singleton (1982) found similar random walk patterns in the spot and forward exchange rates of these currencies against the U.S. dollar. More recently, Diebold and Nerlove (1986) reached the same conclusion for the weekly spot rates of the deutsche mark, Japanese yen and Canadian dollar against the U.S. dollar during 1973-85. As to the behavior of foreign currency futures prices, Doukas and Rahman (1987) found similar random walk behavior on the basis of the daily U.S. dollar exchange rates of the deutsche mark, Canadian dollar, Swiss franc and Japanese yen during 1977-83.

Caution should be exercised, however, before concluding that the exchange rate should of necessity follow or did in fact follow a random walk. For one thing, we have already established that a random walk is not a necessary implication of market efficiency. In fact, a random walk requires quite stringent conditions about the time-series process of underlying economic variables that are unlikely to be met in practice. 11/ For another, it has been argued that the Dickey-Fuller types of unit root tests have low power against borderline stationary alternatives. Hakkio (1986) have shown, on the basis of a Monte Carlo study, that four popular types of random walk tests, including the Dickey-Fuller test, have an extremely low rejection rate when the true model follows a stationary process that is close to a random walk. Frankel and Meese (1987) have suggested a rough rule of thumb that a sample size of over 700 is required for Dickey-Fuller’s t test to be able to distinguish between a random walk and an autoregressive process with a coefficient of .99.

2. Autocorrelations of exchange rate changes

To examine the possibility that the exchange rate displays a small deviation from a random walk, it may be informative to obtain basic descriptive statistics of innovations in exchange rate series. Following standard practice, we define innovations to be the first differences of the logarithms of nominal exchange rates, which can be roughly interpreted as percentage changes. For the spot rate, the innovation can be expressed as:

Δet=(β/1+φ)Σj=0(φ/1+φ)j(Etmt+jEt1mt+j1)+Σj=0(φ/1+φ)j+1(Etrt+jEt1rt+j1)+Δzt,(10)

where Δ is a first difference operator. For the forward rate, we have:

Δft=(β/1+φ)Σj=0(φ/1+φ)j(Etmt+jEt1mt+j)+Σj=0(φ/1+φ)j(Etrt+jEt1rt+j1).(11)

Comparison of (11) with (10) reveals that Δf is simply an updated version of Δe except that Δf has the additional term of (rt-rt-1) and does not have Δzt. Both expressions become white noise if r and m follow random walks; otherwise, there is no reason for them to be white noise. Note that the celebrated martingale property of speculative prices refer to the property of (Etmt+j-Et-lmt+j) and not of (Etmt+j-Etmt+j-1). In general, therefore, exchange rate changes reflect changes in expectations about the future path of relevant economic variables as well as changes in risk premia, whatever their stochastic processes might be.

Autocorrelations of Δe and Δf based on daily and end-of-month data were calculated up to five lags each (Table 3 and Table 4). We find that the magnitude of autocorrelations is generally small and statistically insignificant. There are, however, exceptions to this observation, although some of the statistically significant coefficients may admittedly reflect type II errors or heteroskedasticity. 12/ For daily time-series, we find a few sample autocorrelations at different lags that exceed 0.2, implying that at least 4 percent of the variance of daily changes can be explained by the linear relationship between present and lagged changes. For monthly time-series, some sample autocorrelations are as large as 0.3-0.5, implying that between 10 and 25 percent of the variance of a current exchange rate change is linearly related to one of its lagged values.

Table 3.

Autocorrelations of Daily Exchange Rate Changes

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Note: **(*) Indicates that the statistic is significant at 5(10) percent.
Table 4.

Autocorrelations of Monthly Exchange Rate Changes

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Note: **(*) Indicates that the statistic is significant at 5(10) percent.

Three conclusions can be drawn from the tables. First, the sample autocorrelations of both daily and monthly changes in exchange rates are virtually identical for spot, 3-month forward, and 12-month forward rates of the same period. Second, the values of sample autocorrelations are generally small and imply that much less than 1 percent of the variance can be explained by the linear relationship between present and lagged changes, although each exchange rate series has at least one statistically significant sample autocorrelation. Third, when sample autocorrelations are statistically significant, their values tend to be much larger for monthly data (amounting to as much as 0.5) than for daily data (amounting to no more than 0.26). This probably indicates the presence of low frequency movements which are not detected by daily data. 13/

The presence of some time dependency in high frequency data is also apparent from the comparison of the standard deviations of daily and monthly changes (Tables 9 and 10). If we think of a month as consisting of about 22 trading days, strict independence implies that the monthly variance would be 22 times the daily variance; that is, the monthly standard deviation should be roughly 5 times the daily standard deviation. However, some of the monthly standard deviations were in practice somewhat smaller than what independence would suggest, implying that the sample autocorrelations were on average negative. This conclusion would be strengthened if we think of a month as consisting of 30 calendar days.

3. Time-series representations of exchange rate changes

Deviations from random walks may be better detected if one simply fits an nth order AR process to the exchange rate series as,

dt=Σj=1nλjdtj+ɛt,(12)

where dt can be either Δet or Δft, λj ’s are coefficients, and εt is a white-noise shock. Table 5 and Table 6 report the t-values of auto-regressive coefficients with order 5. We find not surprisingly that the estimated AR coefficients provide similar information as that provided by the estimated autocorrelations. As we found from autocorrelation analysis, the presence of statistically significant coefficients in monthly data indicates the presence of systematic low frequency movements in the exchange rates which could not be detected by daily data. For both daily and monthly exchange rate changes, one can almost always find an AR representation that fits the data better than pure white noise. However, the deviations of the first-differenced time-series from white noise are generally very small.

Table 5.

Autoregressive Representations of Daily Exchange Rate Changes

(t-values)

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Note: **(*) Indicates that the t-value Is significant at 5(10) percent.
Table 6.

Autoregressive Representations of Monthly Exchange Rate Changes

(t-values)

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Note: **(*) Indicates that the t-value Is significant at 5(10) percent.