Several Econometric Tests of Exchange Rate Efficiency for a Few European Countries

This paper uses an efficiency specification model of the spot and forward foreign exchange markets and tests the hypotheses for random walk (which cannot be rejected), general efficiency, and unbiasedness by using a regression estimation and various specification and diagnostic tests for the series and the error terms (residuals). Whereas the forward rate is usually viewed as an unbiased predictor of the future spot rate, the unbiased forward rate hypothesis has failed to be rejected for the Canadian dollar, although more research is needed in this particular area so that better statistical inferences can be drawn in the future.


Introduction
Economic theorists posit that the forward exchange rate will be an unbiased predictor of the future spot rate whenever we have the condition of efficient markets coupled with rational expectations (i.e., correct on average). This begs the question, however, about which market is efficient. According to Eugene Fama (1970), a market can be termed "efficient" if its prices always "fully reflect" all information available to its participants. Economists, though, have not even reached agreement yet on major economic issues such as how the general resources and the ownership of the economy's capital stock should be allocated. Up to this point, we have merely depended on whatever our economic system deems to be optimal markets and price mechanisms. For example, all our models today assume that market efficiency exists; but does it actually exist? An understanding of market efficiency and any improvements in it are important to government policymakers, central bankers, managers of multinational corporations, and international investors. Market behavior is of the greatest importance to government policymakers in particular so that they can design appropriate macro-policies to achieve the goals of efficient resource allocation, steady growth, full employment with price stability, and improvement in their fellow citizens' health and standard of living.
Fifteen years after Fama's definition, Samuelson and Nordhaus (1985) further described an efficient market as one in which new information would be quickly absorbed by market participants and also be immediately reflected in market prices. The academic domestic finance literature has subsequently developed this efficient markets hypothesis extensively, with its underlying importance coming from the assumption that, if a market is efficient, the current price of an asset will fully reflect all available information regarding its valuation. The prices of financial assets thus provide signals for portfolio allocation, but is the pertinent "available information" the full information that people absolutely need?
In addition to domestic finance, the efficiency hypothesis has been used in many foreign exchange market studies. This hypothesis itself suggests that there are no unexploited profit opportunities and, particularly in the foreign exchange market, implies that the forward rate summarizes all relevant and available information that could be used in a forecast of the future spot rate. Analyzing this aspect of efficiency requires an equilibrium model of pricing in the foreign exchange market. Consequently, any empirical test of efficiency is a joint test of efficiency (full information) and the equilibrium (harmony) (Note 1) model. The hypothesis of market efficiency in the foreign exchange rates market states that, in general, the expected value of the future spot rate is the current forward rate (Hakkio 1981). Hansen andHodrick (1980, 1983), Fama (1984), and Domowitz and Hakkio (1985) have recently conducted tests showing that the evidence supporting the unbiased forward rate hypothesis is notably scant, finding that an inconstant risk premium exists in several major foreign exchange markets, with the implication being that one cannot directly use the forward rate as an accurate and consistent predictor of the future spot rate. Robichek and Eaker (1978) concluded that the forward rate is a biased predictor of the future spot rate and that speculative positions do not receive a return above that expected in the Capital Asset Pricing Model (CAPM) framework. On the other hand, Chiang's (1988) empirical analysis, based on the full-sample estimation covering January 1974 through August 1983, confirms the unbiased forward rate hypothesis for France, Canada, and the United Kingdom, although his evidence from the Brown-Durbin-Evans test and the Chow test cannot support the constant coefficient hypothesis in the exchange rate regression model and his empirical results from the subsample study using joint-rolling regressions also reject the unbiasedness hypothesis in most cases. Leachman and El Shazly (1992) found empirical evidence supporting the efficiency criterion in four out of five countries, although Chan, Gup, and Pan's (1992) results show that currency futures markets are multi-market inefficient and that currency futures prices appear to be a random walk. Fittingly, Hopper (1994) answered the question about the existence of market efficiency with the response "Maybe." In this paper, we start from an equilibrium state in the foreign exchange markets and then try to study the model's stochastic coefficients' dynamics used in testing the unbiased efficiency hypothesis while performing statistical and time series tests on the model's variables and many diagnostic tests on both the model's underlying assumptions and the adequacy of its specifications.
The paper is organized as follows. In the second section, the model is developed. The one after that provides some basic statistics regarding the model's variables and the fourth one gives the empirical results. The next section deals with the model's different specifications and diagnostic testing, with the final section providing a summary and concluding remarks.

The Derivation of the Basic Model
The notion of market efficiency is usually affiliated with market expectations' rationality. Our method of examining this issue is to decide on the possibility of market participants systematically earning an excess profit. In foreign exchange markets, current prices reflect all available information. Therefore, the efficient market approach paired with rational expectations implies that economic agents' expectations about the future values of exchange rate determinants are fully reflected in the forward rates. It follows that, working under these conditions, an investor cannot earn an outsized profit by exploiting this available information.
The assumptions underlying this conclusion are that the conditions of market equilibrium can be stated in terms of expected returns and that equilibrium expected returns are formed on the basis of the full information set II t such that there exists neither systematic unexploited profits over time nor any irrationality in the market. Following Fama (1970), Mishkin (1983), and Levich (1985), we can write: where R e t+1 is the expectation derived from the forecast from one period ahead of the actual value of asset returns R t+1 and e is the expectations operator conditioned on the information set II t available at the end of period t. (Note 2) The hypotheses that the exchange rate follows a random walk and that the forward rate is an unbiased predictor of the future spot rate can be derived from the use of the following international parity conditions: Purchasing Power Parity International Fisher Parity where notations expressed in lowercase letters are natural logarithms, with the only exception being the interest rates; s t and f t are the spot and forward exchange rates, respectively; p t denotes the price level; (Note 4) and i t and r t are the nominal and real rates of interest, respectively.
Taking the mathematic expectation of equation (7) and substituting equations (3) and (4), assuming also that ∆p t e = ∆p* t e = 0 and that equation (5) holds, we have Substituting equation (8) into equation (1), we obtain Equation (10) suggests that if we have an efficient market then a currency's current price will reflect all available information affecting that currency. The unexpected change in the spot rate, s t+1 -s t , is essentially caused by the random shock t+l which hits the market between time periods t and t+1. Market rationality suggests that a market participant or investor would discern no particular pattern from studying the history of t+l . (Note 5) By taking equation (2) forward for one period and then taking the mathematic expectation, adding and subtracting r t , and substituting the relationship into equations (2), (3), and (5), we receive E(s t+1 ) = p t + ∆p t e -(p t * + ∆p t *e ) = p t + ∆p t e -(p t * + ∆p t *e ) +r t -r t * = p t -p t * + r t + ∆p t e -(r t * + ∆p t *e ) (11) = s t + i t -i t * = f t Substituting equation (11) into equation (1), we obtain In equation (13), the notion of rational expectations without a risk premium is formally expressed and is usually called the "simple efficiency" hypothesis. Some people have argued that the forward rate may also contain a risk premium, RP t+l , if the economic agents are assumed to be risk averse; this mathematical relationship (the "general efficiency" hypothesis) (Note 6) can be stated: We are initially testing equations (10), (13), and (14) as the following: The unbiased efficiency hypothesis is assumed to hold if α 0 = β 0 = ϓ 0 = δ 0 = 0, α 1 = β 1 = δ 1 ,ϓ 1 + ϓ 2 = 1, and δ 2 = 0; the relationship between s t and s t-1, f t-1 and "news" is linear; the s t 's, f t 's, and "news" are nonrandom variables whose values are fixed, and σ 2 st 0, σ 2 ft 0, σ 2 "news" 0 and finite; and E( t ) = 0, E( ) =σ 2 , and E( , ) = 0, meaning that , , , and ~ N (0, σ 2 ).

Simple Testing of the Model and Basic Statistics
The data include monthly figures for the spot and forward rates of the U.S. dollar ($) with respect to the Canadian dollar (C$), the British pound (£), and the French franc (FF) as well as to three-month U.S. Treasury bill rates or other interest rates. All the data come from Main Economic Indicators of the Organization for Economic Cooperation and Development (OECD) and cover the period March 1973 through June 1994 inclusive (256 months).
We started out testing the random walk hypothesis by calculating the mean value, the variance, and the coefficient of variation of the error term ( ), and these results are in Table 1. As is shown, both the E( ) and the variance are small but are not constant over time. Then, the general efficiency hypothesis was tested and its results are presented in Table 2. Table 3 shows the exchange rates' correlation matrix. Some basic statistics are next provided in Table 4, (Note 7) namely, mean values, standard deviations, maximum, minimum, skewness, kurtosis, correlation, normality test statistics, autocorrelation and partial autocorrelation, cross correlation, and roots (stationary) tests.
To predict the , we must use as the best predictor available because is small. In these cases, the forward rate cannot predict the future spot rate very well (i.e., there is no efficiency). A negative RP means that the forward rate contains a risk premium, as is the case in all three countries sampled here. A positive RP means that the forward rate does not contain a risk premium and investors are accepting a lower exchange rate in return for the forward market's safety (meaning that they pay for the certainty of the forward market and prefer the forward market over the spot market, e.g., Canada contains a risk and investors therefore require a risk premium). The smallest risk premium in the forward market appears in France (RP t+1 = -.0004) and the largest in the United Kingdom, where RP t = -.00l. A risk premium in the spot market is required in Canada (RP t+1 = .002).
The foreign exchange market is not very efficient. The most efficient one (RP→0) is in France (1-month forward) and the least efficient one is in the U.K. because of its large risk premium (3-month forward). The most stable market (σ RP →0) is in Canada (current spot market, ) and the U.K. and France equally display the most unstable markets (the largest at σ RPt+2 ).

The Empirical Results
We estimate equations (15), (16), (17), and (18) by using Ordinary Least Squares (OLS) and Instrumental Variable (N) methods. As instruments, we use constant, time, time squared, and lagged values of the spot and forward rates. The expected interest rate differential is computed from a regression of the interest differential on a constant, two lagged values of the interest differential, two lagged spot exchange rates, and time. The results from those four equations' estimations are shown in Tables 5, 6, 7, and 8 respectively. The overall results are robust and we also have good statistics.

Specifications and Diagnostic Tests of the Model
The final equations of the model (Equations (15) through (18)) are subjected to general specification and diagnostic tests so as to determine the statistical specifications' adequacy. We conduct a Wald test to test the hypothesis involving the restriction on the explanatory variables' coefficients and then add an extra variable to the existing equations and ask whether this makes a significant contribution. We next test the residuals of our equations, testing for serial correlation, autocorrelation and partial autocorrelation, autoregressive conditional heteroskedasticity (ARCH), and for white heteroskedasticity. Finally, we did some specification and stability tests, which were: a Ramsey test of specification error; Chow tests by splitting the data into three sets, namely from March 1973 through May 1979, June 1979through February 1985, and March 1985through June 1994; a Chow forecast test by estimating the equation with the observations up to March 1991 and predicting the values of the dependent variables in the remaining data points; and Cusum tests to examine the parameters' stability. The results appear below in Tables 9, 10, 11, and 12.   -instability in the parameters of the equation  t  N(0, 2 I) -instability in the parameters of the equation  t  N(0, 2 I)

Summary and Concluding Remarks
In this efficiency specification model of spot and forward exchange markets, we argued that the forward rate fully reflects the limited available information about exchange rate expectations and the forward rate because of the lack of complete and correct global knowledge or 'wisdom.' Therefore, the forward rate is usually viewed by the market as an unbiased predictor of the future spot rate. The conventional test of the unbiasedness hypothesis that we used was a regression estimation by fitting the current spot on the one-period lagged spot rate, on the one-period lagged forward rate, on the one-period lagged spot and forward rate, and on the one-period lagged forward rate and the "news" (the difference between actual and expected interest differentials). These tests involve the joint hypothesis that the constant terms do not differ from zero, that the coefficients on the one-period lagged spot and forward rates do not significantly differ from one, that the sum of the coefficients of the one-period lagged spot and forward rates do not significantly differ from one, that the coefficient of the "news" is not different than zero, and that the error terms pass some statistical tests (serial correlation, normality, ARCH, etc.).
We cannot reject the unbiased hypothesis for Canada, but we can do so for the U.K. and France. The results imply that we can use the forward rate as a proxy for the prediction of the spot rate next period. There is some instability in the parameters of almost all the equations of the model, but, from a forecasting point of view, this is consistent with the least cost approach to the economic agents, although it may not yield the minimum forecast error due to interventions, incomplete and partial knowledge (incorrect information), and simplicity in modeling. The overall results show that Canada's, foreign exchange market is fairly efficient whereas the market efficiency of the United Kingdom and France is questionable. France's spot rate also follows a random walk but its variances are not constant.