Rational Expectations, Risk Premia, and the Market for Spot and Forward Exchange

International Finance Discussion Papers Number 165

July 1980

RATIONAL EXPECTATIONS, RISK PREMIA, AND THE MARKET FOR SPOT AND FORWARD EXCHANGE

by

Richard A. Meese and Kenneth J. Singleton*

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

Rational Expectations, Risk Premia, and the Market For Spot and Forward Exchange

by

Richard A, Meese and Kenneth J. Singleton

1. Introduction

Implicit in models of exchange rate determination that assume markets for spot and forward rates are "efficient" are the assumptions of rational expectations, costless transactions, risk neutral market participants, and the absence of legal restrictions on transactions. Since these assumptions do not literally describe the foreign exchange market, the efficiency hypothesis also cannot be literally true. While this observation alone does not imply that the efficient markets model is a poor approximation to the true model, the recent empirical studies of Geweke and Feige (1979), Hakkio (1979), and Hansen and Hodrick (1980a, b) suggest that there have been significant departures from efficiency during the 1970's.

The primary purpose of this paper is to investigate the extent to which risk aversion, as manifested in a time varying risk premium, is capable of explaining these departures from efficiency. In an efficient market, the expected n-period holding period return equals the proportional n-period forward premium. We begin our empirical analysis by testing this version of the efficiency hypothesis and find that it is rejected by the data for the countries considered. To investigate the possibility that the failure of the efficiency hypothesis is a consequence of a time varying, ex ante risk premium, a testable lower bound on the variance of the ex ante risk premium is derived. For the countries considered, the test results support the hypothesis of a time varying, ex ante risk premium.

In the process of testing these hypothesis, we also discuss two practical issues that heretofore have received little formal attention in the literature. First, we attempt to overcome the potentially important problem

of data alignment that is unique to foreign exchange markets. Three month

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forward contracts, for example, are not usually contracts for the delivery of foreign exchange ninety days: hence. As a means of avoiding biases due to misalignment of forward rates with their respective future spot rates, we have constructed a new data set from daily data in which the rates are properly aligned.

Second, we formally address the problem of non-stationarity of spot and forward exchange rates. Our econometric tests of exchange market efficiency require the assumption of covariance stationarity! of the variables in the model. Before studying the behavior of holding period returns, we test the null hypothesis that the natural logarithm of the exchange rate is well approximated by a random walk using the recently developed testing procedures of Fuller (1976), Dickey and Fuller (1979), and Hasza and Fuller (1979). The results of this analysis suggest that the n-period holding period return, which is approximately the n-th difference in the logarithm of the exchange rate, is covariance stationary and, hence, that the asymptotic distribution theory we employ to study risk premia is appropriate.

The paper is organized as follows. In section 2 we derive a relationship between the holding period return, the proportional forward premium, and the ex ante risk premium under the null hpothesis of efficiency. Then in section 3 we discuss the proper alignment of present forward and future spot rates, review the estimation problems inherent in exchange rate models where the sampling interval differs from the forward contract length, and present evidence that the aligned holding period return is covariance stationary. In section 4 we test and reject the holding period version of the efficiency hypothesis and present evidence of a time varying risk premium. Concluding

remarks comprise section 5.

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2. Rational Expectations and the Behavior of Risk Premia.

If the rational expectations hypothesis applies to the market for forward exchange, if this market is perfectly competitive, and if market

participants are risk neutral, then f"(t) = E(s(t +n) [4(t)), (1)

where f"(t) is the n-period forward price of foreign exchange observed in period t, s(t +n) is the spot price at period (t +n), and E(s(t + n) |¢(t)) denotes the conditional expectation of the spot price in period (t +n) based on the current information o(t). The set $(t) includes at least

current and past s(t) and £"(t). Equation (1) implies s(t +n) = f(t) +n™(t +n), (2)

where n™(t + n) is the error from forecasting s(t +n) using information dated t. and earlier, and is therefore uncorrelated with £"(t). Since equation (2) can be estimated using standard regression techniques, it is frequently employed to conduct statistical tests of (1); see for example Frenkel (1976, 1977, 1979, 1980).

In this paper we focus on a different implication of (1), and utilize an approach to test the null hypothesis that has not been employed previously. To start with, the spot rate s(t) is contained in the information set o(t),

so equation (1) implies,

n itp) sto) SG tas) | 9ty). (3)

Equation (3) says that in a risk neutral world with rational expectations, the proportional n-period forward premium, H(t) = (f(t) - s(t))/s(t), is an unbiased predictor of the holding period return, H(t +n) = (s(t +n) s(t))/s(t) 2! This version of the efficiency hypothesis implies that the ex

ante risk premium, which can be defined (Frankel (1979), p.6) as

E(H(t + n){6(t)) - H(t) = (E(s(t + n)]o(t)) - £(t))/s(t), (4)

is identically zero.

Statistical tests of the null hypothesis that the ex ante risk premium is identically zero can be formulated in several ways. First observe

that equation (3) implies H(t +n) = H(t) + e“(t +n), (5)

where et +n) is the error from forecasting H(t + n) using information in o(t), and is thus uncorrelated with H(t). From (5) it is clear that a regression of the forecast error or ex post risk premiun, et +n) =

(A(t +n) - H(t)), on H(t) and a constant should yield zero coefficients on the regressors. Formulated in this way, the regression test of (5)

corresponds to a test of the variance inequality,

var(H(t +n)) > var(H"(t)), (6)

3/

which is also implied by (5) when both series have finite variance.— Both

tests are based on the orthogonality of H(t) and the forecast error e"(t +n). Since (6) would still hold if cov (H(t), a(t +n)) were positive, the

variance inequality (6) is a necessary but not sufficient condition for the null

4/

hypothesis to obtain.—

5=

Second, consider the variable H(t) defined as the linear least squares projection of H(t +n) on P(t), where P(t) contains present and a finite

number of past H(t). Again, we can represent H(t +n) as the sum of a predictor

and forecast error,

. 7 H(t +n) = H(t) + v(t +n), (7)

where cov(H"(t), v(t +n)) = 0 by construction. Equating the right hand sides

of (5) and (7) we can infer H(t) = H(t) +v"(t $n) - M(t +n), (8)

n,. “ where cov(v (t + n)-e (t +n), H(t)) = 0 because P(t) is contained in the

information set $¢(t). In order to couch the regression test of (7) in terms of

directly observable variables, note that

H(t +n) - H(t) = (H(t +n) - H'(t)) + (a(t) - #(t)). 9)

The forecast error (H(t +n) - H"(t)) is uncorrelated with the elements of

. n “n n n . P(t). Since (H (t) - H(t)) = (v (t +n) -e€ (t +n)) from (8), it too is uncorrelated with the constitutents of P(t). This implies that a(t +n) = (H(t +n) - H"(t)) is uncorrelated with P(t), so a regression of the observable (H(t +n) - H(t)) on any of the constituents of P(t) can be used to test (7). Again, we can associate this regression test with a test of a variance inequality,

since under the null hypothesis equation (8) implies

var(H'(t)) > var(E"(t)). (10)

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All of the implications of the zero ex ante risk premium hypothesis that we have discussed thus far can be tested jointly by performing a test of the null hypothesis a” = (a, 5 Ory 5 a5) = 07 in the regression equation,

(H(t +n) - H(t)) =a

i+ aH (t) + a,H(t) + o'(t +n). (i1)

In section four we report the results of the regression test (11) based on weekly data for three currencies vis a vis the U.S. dollar.

If the foreign exchange market is not efficient, i.e.. if the ex ante risk premium is not identically zero, then the ex ante risk premjnum has non-zero variance which is a testable implication. To be more precise, let E(e (t + n)|o(t)) denote the ex ante risk premium (4), and define Q(t) as the set of past and present ex post risk premia or forecast errors, Q(t) = [e"(t), e(t -1),...]. Consider the linear least squares projection of a(t +n) on Q(t), denoted

e"(t). As before we can represent a(t +n) as the sum of two components, n “n e(t+#n) =e (t) +O (t +n), (12)

n ras where the forecast error © (t +n) is uncorrelated with the predictor e"(t).

Clearly, (12) implies var(e"(t +n)) > var(e*(t)), and we can also deduce var(E(e"(t + n)|[o(t))) > var(e*(t)) > 0, (13)

as the projection set Q(t) is contained in ¢(t). Equation (13) provides

a least lower bound on the variance of the ex ante risk premium, since the inclusion of more variables in the projection set Q(t) cannot decrease the variance of the predictor (tr). A test of the hypothesis that the ex ante risk premium is time varying, a sufficient but not a necessary condition for (1) not to obtain, can be conducted by performing a test that var(e"(t)) > 0

using the asymptotic distribution theory derived by Singleton 198). The

results of this test are reported in section four.

a] -

Both the regression equation (11) and the variance inequality (13) can be interpreted as preliminary tests of the "portfolio-balance" approach to exchange rate determination, an approach that relies on the assumption of

imperfect substitutability of domestic and foreign assets. In a recent paper Frankel (1979) does not reject the hypothesis of perfect asset substitutability and a zero ex ante risk premium. To conduct his analysis, Frankel specifies asset demand functions of the portfolio-balance type, and then inverts these functions to obtain an expression for the ex ante risk premium as a function

of asset stocks. Under the null hypothesis of a zero risk premium and rational expectations, a regression of the ex post risk premium E(t +n) on the determinants of the ex ante risk premium should yield zero coefficients on

the regressors. Since the coefficients on the various regressors in Frankel's estimating equations are insignificantly different from zero, he accepts the hypothesis of a zero risk premium and perfect substitutability.

One of the problems associated with empirical tests of the portfoliobalance models of exchange rate determination is that regression equations involve relative asset stocks and wealth variables. Since data for these variables is not readily available, it is advantageous to work with an estimating equation unencumbered by possible measurement error. To circumvent this

problem note that past values of the forecast error are contained in the information set, so we can test Frankel's (1979) hypothesis using a regression equation of the form

n K n n 3/

€ (t +n) = Yo + yt +, 20 V4 (t -i+2) +o (t +n).~ (14) Under the null hypothesis of efficiency, a zero ex ante risk premia, and perfect substitutability, all regression coefficients are zero. In section

four we also report the results of this test.

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3. The Problems of Data Alignment, Serial Correlation and Covariance Stationarity.

Existing empirical studies of the relationship between forward rates and future spot rates have largely ignored the institutional features of the exchange market which induce irregularities in the alignment of the data.—! For example, when two parties agree to a spot transaction of Swiss francs for dollars, the transaction is for a value date two business days following the day the transaction was closed -/ To be an eligible spot value date, it must be a business day in both the United States and Switzerland as both currencies are involved in the transaction. Similiarly, today's U.S. dollar-Swiss franc three month forward transaction involves settlement of a contract three months from the corresponding spot value date, provided it is an eligible day. The future spot rate that corresponds to this forward value date was contracted two business days prior to the forward value date. The use of spot and forward rate quotations to test the hypothesis that the forward rate is an unbiased predictor of the future spot rate requires knowledge of all banking holidays in all countries whose currencies were transacted over the entire sample period.

All results reported in section four are for Wednesday twelve o'clock Swiss franc-U.S. dollar, German mark-U.S. dollar, and Canadian dollar-U.s. dollar spot bid rates, and the mid-point of the twelve o'clock bid-offer spread on forward rates. These series are collected by the Federal Reserve System as part of their daily monitoring of the New York foreign exchange market. The spot and forward rates were aligned using back issues of Morgan Guarranty Trust Company's World Calendar of Holidays. Our sample period is 1976 through 1979 inclusive. The length of this sample was dictated by our inability to secure

additional back issues of the World Calendar of Holidays.

The institutional features of the foreign exchange market that necessitate separate alignment of each forward rate and its corresponding future spot rate introduces an additional estimation problem. The time period between s(t +n) and f"(t) cannot be considered constant, and some of the econometric procedures we employ assume a constant forward contract length n. When necessary, we use the largest value of n for each currency over the sample period. This value is n = 14 for weekly data and a three month foward contract. In addition, we report empirical results based on weekly data although daily observations are readily available. Weekly data are employed to circumvent the weekend or holiday unobservability problem, ‘as economic and political events that occur when the exchange market is closed can affect the market when it re-opens. Last, the slight incomparability of spot (bid rates) and forward (mid-point of the bid-offer spread) is a potential source of distortion in our empirical results.

Estimation of equations (11) and (14) is further complicated by the fact that the regressors are predetermined, not exogenous, and the disturbances may be serially correlated. Under the null hypothesis of efficiency, the disturbances can be serially correlated whenever the interval of observation t is finer than the contract interval n. In equation (11) for example, the disturbance. 6 (t +n) may be correlated with s(t +i), 0 <i < 2n. Since the regressors in this equation are not strictly exogenous, conventional two-step generalized least squares (GLS) estimation techniques cannot be used to test the zero ex ante risk premium hypothesis. This point is ably made in a recent paper by Hansen (1980), where he develops the asymptotic distribution theory for

the estimation strategy employed in this paper. This strategy involves

ordinary least squares (OLS) estimation of (11) with a correction for the asymptotic bias in the OLS estimator of the covariance matrix of regression coefficients .&/

The asymptotic distribution theory we employ to test the ex ante risk premium hypothesis requires that the regressors in equations (11) and (14) be covariance stationary.’ Since there is considerable debate whether the levels (or logarithmic levels) of exchange rates have finite variance, an analysis of the stationarity assumption is clearly warranted.

Mussa (1978) has conjectured that the natural logarithm of the exchange rate is well approximated by a random walk. To support this conjecture, we report in table 1 the results of a test of the null hypothesis that the univariate autoregressive (AR) representations of 1n(s(t)) and In(£"(t)) have unit roots .22/ In all cases we reject the null hypothesis that these univariate autoregressions have two unit roots. We cannot, however, reject the null hypothesis that these autoregressions have one unit root, except for the German Mark-U.S. dollar spot exchange rate. Since fitting autoregressive processes of different orders had no effect on the hypothesis tests reported in tale 1, these tests suggest that ln(s(t)) and In(£"(t)) do not have stable univariate autoregressive representations, even after mean and linear trend have been removed. Using a consistent estimator of lag length ,2L/ the order selected for the univariate autoregressions of In(s(t)) and In(f"(t)) was always one. This result coupled with our failure to reject the hypothesis that these autoregressions contains a unit root lends

credence to Mussa's conjecture.

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The foregoing analysis suggests that whenever empirical tests of (1) are based on information sets with only past and present s(t) and £"(t), it is important to work with transformations of s(t) and £"(t) for which covariance stationarity is a reasonable assumption. If the resuits of table 1 are taken to imply s(t) and f(t) are nonstationary processes, then we can rule out conventional tests of the efficiency hypothesis like equation (2), as the predetermined regressor In(f"(t)) will not satisfy the requisite assumption of stationarity. If In(s(t)) follows a random walk, the holding period return H(t) is approximately a finite moving average process and hence is convariance stationarity .22/ Thus the asymptotic distribution theory required to test the efficiency hypothesis is appropriate when H(t) is used as a regressor. The three tests of efficiency we report in the next section, equations (11), (14), and the variance inequality (13), require H(t) and H(t) to be jointly covariance stationary 22! Tests for the existence of unit roots in the joint autoregressive representation of s(t) and f(t) are not yet available, so we offer no empirical evidence on the joint

stationarity of H(t) and H(t) 24/

‘The regression tests (11) and (14) can be conducted using only present and past values of H(t) as regressors. This formulation results in test statistics qualitatively the same as those

reported below.

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Table 1

Tests for unit roots All results refer to the equation: y(t) = 8) + Byt + Boy(t - 1) + B,(y(t -2) - y(t - 1)) + error. Switzerland

y(t) = In(s(t))

Bo By By B coefficient ~1093 1016-3 -9699 -.5593-3 OLS standard error -6104-1 5612-4 1706-1 «7335-1 en T= 191 R2 = .9924 (1)

#,(2) = 5343.7**, reject Hy)” © } see table notes below. t= -1.764 , accept Hy

y(t) = In(£"(t)) .

Bo By Bo By coefficient - 1041 - 1013-3 9714 -4130-2 OLS standard error -5906-1 2 5627-4 1649-1 7337-1 Sana T= 191 R2 = .9928

$3(2) = 4787.5**, reject HS? ty = 71.734 , accept a)”

Table 1 continued

Canada

y(t) = In(s(t))

Bo By B. B. coefficient - 2388 -.6263-4 .9484 1789 OLS standard error 9710-1 -2630-4 .2091-1 -7161-1

T = 190

R2 = ,9933 (1) $,(2) = 156953.**, reject Hy

=. (2) t) 2.468 >» accept Hy

y(t) = In(£"(t))

Bo By Bo By coefficient - 2807 -.6950-4 .9393 1790 OLS standard error - 1050 22687-4 (2267-1 7157-1

T,= 190

R2 = ,9921 (1) $5(2) = 207874.#*, reject Hy

=. (2) t 2.678 >» accept Hy

Germany

y(t) = In(s(t))

a a “ “

Bo BiB 8, coefficient 3407 ~ 2049-3 -9062 1525 OLS standard error «1040 26244-4 - 2874-1 7278-1

—_—— T = 191

R2 = .9934 (1) $,(2) = 126907.*#, reject H

th = -3.264* , reject a” at the 10% significance level

Table 1 continued

y(t) = In(£"(t))

a a a a

By By Bo 84 coefficient . 3038 . 1898-3 9164 1336 OLS standard error 9877-1 6161-4 2728-1 0727941 T= 191 R2 = .9936

$,(2) = 96537.**, reject H (1)

=- (2) t. 3.065 , accept Hy

Table 1 notes

H (1) denotes the null hypothesis 8, =8, = 1. The derivation of the test sPatistic and a tabulation of its afstriBution is given in Hasza and

Fuller (1979). In all cases the test statistics exceed 12.31, the critical value for a 1% significance level, T = 100, (see page 1116 of Hasza and Fuller), so ** denotes rejection at well below the 1% significance level.

H (2) denotes the null hypothesis 8, = 1,8, <1. The derivation of the tést statistic and a tablulation of its distribution is given in Dickey and Fuller (1979) and Fuller (1976). WhenT = 100 the critical value of

ty = -3.15 for a 10% significance level, (see Fuller (1976) p.373), so * indicates rejection at the 10% significance level. t_ is the conventional t-test for 8. = 1. T

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4. Some Empirical Evidence of Time Varying Risk Premia.

The results of the regression test (11) for bilateral U.S. exchange rates with Germany, Switzerland and Canada are reported in table 2. In each case the covariance matrix of the estimated parameters was corrected for serial correlation using a frequency domain procedure suggested by

Hansen (1980) .22/ Our results are not supportive of the null hypothesis of

a zero ex ante risk premium. The marginal confidence levels mci )22/ for the tests g = 0 in (11) are for Germany, Switzerland, and Canada, 1.000, .8936 and

-9670, respectively.

In table 3 we report the results of the regression test (14) using the same estimation procedure. The empirical work summarized in table 3 offers no support for the null hypothesis for Germany and Canada, as the marginal confidence levels of the chi-square statistics are both 1.000. The marginal rejection of the null hypothesis for Switzerland, table 2, is not reinforced by the results of table 3 where the MCL of the chi-square Statistic is .5982.

Since the coefficients in out regression equations need not remain constant after a change in policy regime, we examined the subsample from November 1, 1978 to the end of 1979 to see whether the ex ante risk premium has been affected by President Carter's November 1, 1978 announcement of a dollar support package. While the marginal confidence levels of the German and Canadian regression tests (1L) and (14) have not changed for this subsample period, the MCL for the Swiss regression tests are now both

1.000. Since the pcst November 1, 1978 subsample tests are based on

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approximately 33 observations, the asymptotic distribution theory may not provide accurate confidence levels for the tests we employ over this recent sample period.

Last, in table 4 we report the results of the test of the hypothesis

that the lower bound on the variance of the ex ante risk premium (13) is zero. These test statistics, distributed as standard normals under the null hypothesis,

indicate rejection of the null hypothesis at high marginal confidence levels for

all three countries.

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Table 2

Regression tests of (10

Switzerland Xo Oy oo) corrected t-statistic* 2.410 -2.259 -.7169 T= 179 R2 = .1114 x2(3) = 6.219 Marginal confidence level = .8986 Germany io a, a, corrected t-statistic 4.297 -.9657 -2.854 T = 179 R2 = .2040 x2(3) = 28.62 Marginal confidence level = 1.000 Canada oy a, é, corrected t-statistic -1.714 4127-2 1033 T = 178 R2 = 0002 X2(3) = 8.739 Marginal confidence level = .9670

Table 2 notes

*The heading corrected t-statistic refers to the ratio of the OLS estimate of a, to the square root of the ith diagonal element of (X'x)71g 760) (KIX)~ “1 the corrected covariance matrix of Gs see footnote 15.

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Table 3

Regression tests of (13)

Switzerland

Y5 ‘4 Yo ¥3 Yn Y5 a OLS coefficient 2946-1 -, 1028-3 2070 -.9407-1 -.1407 -.7986-1 corrected t-statistic* 1.308 -.5188 -7819 -1.190 -1.835 -.4393 T = 177 R2 = .0276 X2(5) = 6.194 Marginal confidence level = .5982

German Y Y

Yo Yq Yo Y3 4 5 OLS coefficient -4025-1 -.1405-3 - 1092 -.2285 - 8789-1 -.2239 corrected t-statistic* 3.703 -1.288 6273 -2.680 -1.352 -2.531 a Le T=177 R* = .2151 X2(6) = 160.8 Marginal confidence level = 1.000

Canada

Yo vy Yo Y3 Yu Ys OLS coefficient <-.1774-1 -6461-4 8 -.3245 - 1383-1 4455-1 22521 corrected t-statistic -1.776 7878 -2.293 22202 6608 2.170 T = 176 R2 = .0799 X2(6) = 94.61 Marginal confidence level = 1.000

*see table 2 notes.

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Table 4 Variance bound tests of (12), var(e"(t)) = 0.

Switzerland

Point estimate of var(e"(t))* = 3.471 - 7

test statistic = 4.391 Marginal confidence level = 1.000 T = 193 Germany

A.

Point estimate of var(e(t)) = 6.356 - 7

test statistic = 4.667 Marginal confidence level = 1.000 T= 193 Canada

Point estimate of var(e"(t)) = 2.586 - 8

test statistic = 4.879 Marginal Confidence level = 1.000

T = 192

*Var(e"(t)) is the variance of the n = 14 period ahead predictor of e*(t)

based on its own past. This variance was estimated and the null hypothesis

var(e'(t)) = 0 tested using the procedures derived in Singleton (1980). The

spectral density of e"(t) was estimated using an inverted V smoothing window

with 27 periodogram ordinates.

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5. Concluding Remarks.

In this paper we have examined the extent to which departures from exchange market efficiency over the period 1976-1979 can be attributed to a time varying, ex ante risk premium. In Lhe process of testing this hypothesis special attention was paid to the important practical protlems of data alignment unique to the foreign exchange market, and the assumptions underlying the asymptotic distribution theory required to conduct our statistical tests. The empirical results for Germany and Canada presented in tables 2-4 provide no support for the null hypothesis of a zero ex ante risk premium. The results do not, of course, imply a lack of rationality in the market for foreign exchange, since the null is a composite hypothesis of rational expectations, risk neutrality, and competitive markets without transactions costs. The empirical results for Switzerland are not nearly as clear cut. While the results of table 2 are unsupportive of the null hypothesis as the R and individual t-statistics are quite large relative to the overall chi-square test, the Swiss regression results of table 3 do not reinforce the results of table 2. We can however, reject the hypothesis that the lower bound on the variance of the Swiss ex ante risk premium is zero at a very large confidence level using Singleton's (1980) variance bound procedure.

The mixed results for Switzerland and the decisive results for Canada and Germany suggest that exchange rate models need to explicitly account for factors that generate risk premia. This is hardly a new idea, as other authors have suggested the need to model exchange rates in a dynamic competitive equilibrium framework where factors which drive a wedge between forward rates

and expected future spot rates can be explicitly modeled. Cur split

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sample, post November 1, 1978 tests indicate that these factors appear to be growing, not diminishing in importance, as the estimate of the lower bound on

the variance of the ex ante risk premium increased for each currency in the

More recent sample period,

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Footnotes

* International Finance Division and Carnegie-Mellon University respectively.

1/ A time series [x(t)] is said to be covariance stationary if cov(x(t),x(2)) depends only on (t - £). Two time series [x(t)] and [y(t)] are said to be covariance stationary if each is covariance stationary and if cov(x(t),y(2)) depends only on t - g.

2/ Since H(t +n) and H(t) are good approximations for In(s(t + n)/s(t)) and In(f (t)/s(t)) respectively, implication 3) of the null hypothesis avoids convexity problems associated with "Siegel's Paradox" (1972). Rogoff (1979) shows that the empirical relevance of Seigel's paradox cannot be dismissed a priori, without making specific assumptions on the joint distribution of s(t) and £"(t). Most authors assume (1) holds for the natural logarithm of the spot and forward exchange rates, so the convexity problem does not arise.

3/ From (5), var(H(t + n)) = var (H"(t)) + var(e (t + n)), as cov(H (t),e (t +n)) = 0

by construction. Since var(e (t + n)) is non-negative, (6) follows immediately. The variance inequalities (6) and (10) are analogous to those derived by Leroy and Porter (1980), Shiller (1979), and Singleton (1980) for present value relations implied by rational expectations models of stock prices and the

term structure of interest rates.

4/ This point has been noted independently by Geweke (1979), and it suggests

that regression tests are robust to potentially interesting alternative hypotheses that variance bound tests lack power to reject. While regression tests may dominate variance bound tests on power considerations, the latter may be more useful when data alignment problems exist. For example, if a regression relation requires contemporaneous observations on the regressand and regressors, while actual data on these variables is published on different dates, a bias in estimation may ensue. This problem would not affect a variance bound test

as all series are assumed to be covariance stationary. Variance bound tests

are also unaffected by the serial correlation of the regression disturbance

term. ,

5/ The choice of a finite K is arbitrary, and we use three lags of e(t) in table 3 below.

6/ The work of Dooley and Shafer (1976) is a notable exception. 7/ See Reihl and Rodriguez (1977) for a discussion of foreign exchange markets.

8/ When the fraction (n - 1)/n of the total number of observations are omitted by sampling every nth observation, equation (11) can be estimated and the null hypothesis tested using the output from an ordinary least squares regression package. In this case the sequence of every nth error §"(t) is serially uncorrelated so OLS is appropriate. Hansen and Hodrick (1980a) discuss the relative power of regression tests of (1) with and without all observations. Under a reasonable class of alternative hypotheses, they show that the modified

OLS procedure described in the text dominates an OLS procedure using sampled data,

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9/ Sufficient conditions on the regressors in (11) and (14) to obtain Hansen's (1980) results are stationarity and ergodicity.

10/ See Fuller (1976), Dickey and Fuller (1979) or Hasza and Fuller (1979) for a discussion of these tests. These authors supply asymptotic ¢cistribution theory for testing the null hypothesis that autoregressions (with and without trend and constant) have unit roots.

il/ There is a growing literature on the determination of the order of an autoregression. Since the estimators of autoregressive order are derived in the context of a stationary and ergodic time series with linear innovations, their use is less justifiable in our context. Using OLS point estimates of the autoregressive parameters, to calculate the roots of the lag polynominals we fit, it was always the case that these roots had squared moduli greater than one. While point estimates of the AR processes indicate borderline stationarity, we noted above that in virtually all cases we could not reject the hypothesis that one root is unity.

12/ Suppose In(s(t)) follows the process In(s(t)) = In(s(t - 1)) + m(t),

where n(t) is the error from predicting 1n(s(t)) using past values of ln(s(t)). Then

n H(t + n) * In(s(t +n) - In(s(t)) = E(t + i), i=1

a finite, noninvertible moving average process of order (n - 1).

In related work, Bldttbetg and Gonedes (1974) show that stock price holding period returns are better characterized by a Student t distribution than fat tailed symmetric-stable (Paretian) distributions for which second and higher order moments typically fail to exist. Ina recent study of exchange rate changes, Westerfield (1977) concludes that exchange rate changes are better characterized by stable Paretian distributions, without finite second and higher order moments, than by the normal distribution. She does not, however, employ

distributions with fatter tails than the normal whose second and higher order moments are finite.

13/ In addition, Singleton's (1980) variance bound test procedures require that the process ¢ (t) = (H(t +n) - H (t)) have autoregressive representation. The foregoing analysis suggests that H(t +n) itself, does not have an AR representation.

14/ Throughout the analysis, we have restricted attention to an information set $(t) with only current and past exchange rates. Since interest rates via covered interest parity, current account imbalances, relative inflation rates, and other variables are known to influence spot and forward exchange rates, future

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empirical work in this area might best be conducted using vector autoregressive techniques, Our finding of unit roots in the univariate AR representations of s(t) and £ (t) may be an artifact of our choice of the constituents of $(t).

15/ Let x(t) = (1, He (t) ,H(t)) be the tt th now of the observation matrix X. Define Z(t +n) = oo (t +n)x(t), and let T denote sample size. Hansen (1980) shows that the asymptotic covariance matrix of the OLS estimator a appropriate for hypothesis tests of (11) is (X'R) “1g_ (0) (x! x)7t The middle term,

s (0), is the estimated spectral densify matrix of Z(t) evaluated at the zero frequency. We estimate S_(0) using T (see table 2) harmonic periodigram ordinates and a lag window (Daniell) of width 14 ordinates.

16/ The marginal confidence level is the probability of observing a random variable less than or equal to the calculated test statistic, given the random variable has the distribution of the test statistic under the null hypothesis. The marginal significance level is one minus the marginal confidence level.

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