Back Matter

APPENDIX: Sources of Data

The daily time-series of the U.S. dollar exchange rates of the pound sterling, deutsche mark, and Japanese yen between January 1, 1975 and June 30, 1986 were all obtained from Data Resources, Inc. The rates are noon closing rates in New York and London as quoted by the Bank of America.

The rates actually used in the paper are averages of the bid and ask rates, and the cross rates were calculated from these average U.S. dollar rates. The monthly rates are end-of-month rates, taken from the rates reported on the last trading day of each month.

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*

The author is on a two-year leave of absence from the International Monetary Fund and is currently at the Institute for Monetary and Economic Studies of the Bank of Japan. He would like to thank, without implicating, Paul De Grauwe, Michael P. Dooley, Mitsuhiro Fukao, Michio Hatanaka, Hideo Hayakawa, Kazuhiro Igawa, Gyoichi Iwata, Kazuo Ogawa, Kunio Okina, Robert F. Owen, Yoshiyuki Takeuchi and Adrian Tschoegl for useful comments on an earlier draft. The views expressed in the paper do not necessarily reflect those of the International Monetary Fund or the Bank of Japan.

1/

The paper does not discuss strong form efficiency. At least in the context of the foreign exchange market, very little work has been done in this area.

2/

An analogous expression holds for the relationship between the n-period forward rate and the n-period interest rate differential for any arbitrary n.

3/

Both Japan and the United Kingdom lifted capital controls in the late 1970s. However, covered interest parity has always held as a matter of course in the Eurocurrency market; it is said that Euro interest rates and forward rates are set according to the covered interest parity formula.

4/

One conceptual difficulty encountered in testing the hypothesis of forward market efficiency is the so-called “peso problem.” It is the small sample problem of a large change with a small probability not occuring during the sample period, causing the forward rate to show an apparent bias.

5/

In addition to market inefficiency, capital controls can produce serial correlation in the left-hand-side variable by introducing serial correlation in the residuals.

6/

One direct way of testing for market efficiency is to use a filter rule suggested by Alexander (1961). Dooley and Shafer (1983), for example, found that 1, 3, and 5 percent filter rules would have yielded large profits for several major currencies during 1973-81. This type of efficiency tests have at least two conceptual problems: (1) one must know the extent of transactions costs in currency trading; and (2) one must know the riskiness of speculation in the foreign exchange market in order to judge what constitutes excessive profits.

7/

Mussa (1984) shows that the same solution form can be obtained by concentrating on balance of payments flows rather than money market conditions.

8/

Of the six bilateral exchange rates reported in the paper, only three are independent. Data on the exchange rates are obtained from DRI’s Financial and Credit Statistics Information Service. Its source is Bank of America, San Francisco, based on 10 a.m. opening New York markets.

9/

We also note that the frequency of rejection is greater for the “F” test, suggesting that the “F” test has more power than the “t” test.

10/

Japan completely liberalized foreign exchange transactions in December 1980.

11/

Takagi (1986) has shown in a formal model that a random walk implies the absence of correlation between the exchange rate and the interest rate differential.

12/

Heteroskedasticity can understate the true variance (Hsieh, 1987). Tschoegl (1987) has also shown that, when the observed price consists of the equilibrium price and noise, observed prices can show first-order negative serial correlation. Thus, some of the statistically significant negative first-order daily serial correlations may simply reflect the presence of noise in the observed time-series. If this is the case, the absence of statistically significant autocorrelations may in some cases mean that negative correlations due to noise are largely offset by positive correlations. Another possible source of bias in daily data is leptokurtosis, which may create more “significant” t-statistics when autocorrelation coefficients are estimated (see Friedman and Vandersteel, 1982 and the next section).

13/

Some of the larger absolute values of monthly coefficients relative to those of daily coefficients when the coefficients are negative can be a statistical artifact. It can be shown that, if the true process is AR(1) in daily data with the AR coefficient close to unity, first-order autocorrelation coefficients are negative and the monthly coefficients are larger in absolute value than the daily coefficients.

14/

Westerfield (1977) also analyzed the distribution of weekly exchange rate changes for the fixed exchange rate period. Her analysis shows that the degree of leptokurtosis was greater for fixed exchange rates; the estimated characteristic exponents were lower than the estimates for the flexible period, ranging between 1.00 and 1.44, when a stable Paretian distribution was fitted (see the next subsection).

15/

So (1987) found similar leptokurtic distributions for the daily future prices of the U.S. dollar against the pound sterling, deutsche mark and Japanese yen during 1974-82.

16/

In a general distribution of Bookstaber and McDonald (1987), all the three hypotheses can be considered as special cases of the mixed distribution hypothesis.

17/

Pareto’s law is a generalization of the central limit theorem to the case where there is no finite variance.

18/

Both the stable Paretian and Student distributions can be thought of as special cases of the class of mixed normal distributions: if the variance follows an inverted gamma-2 distribution, the resulting distribution becomes a Student distribution; if the variance follows a strictly positive stable distribution, the resulting distribution becomes a symmetric stable Paretian distribution (Islam, 1982). For a more comprehensive discussion on mixed distributions, see Bookstaber and McDonald (1987) and McDonald and Butler

19/

However, McFarland, Petit, and Sung (1982) consistently found the estimated characteristic exponent to be considerably less than 2 for different daily distributions and concluded that the mixed distribution was not a normal but a stable Paretian with changing parameters.

20/

It is noteworthy that some of the values of kurtosis in daily data were close to 3 when the sample period was three years, whereas the values of kurtosis in longer sample periods in other studies were hardly ever smaller than 10. This suggests the possibility that even the daily distribution can be normal during a short enough period and that the leptokurtosis is generated by mixing several normals with different parameters.

21/

White (1984, Chapter 5) reviews several versions of the central limit theorem that specify the conditions needed for finite variance distributions to converge to normality.

22/

Diebold and Nerlove (1986) suggested the estimated conditional variance as a measure of volatility in the foreign exchange market. Domowitz and Hakkio (1985) applied the ARCH process to explain the risk premium. They hypothesized that the risk premium is a positive function of the conditional variance of the forecasting error. Based on monthly data for 1973-82, however, they concluded that the conditional variance did not entirely explain the risk premium.

23/

The asymptotic distribution of the Dickey-Fuller statistics is invariant to ARCH.

On the Statistical Properties of Floating Exchange Rates: A Reassessment of Recent Experience and Literature
Author: International Monetary Fund