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Fixed versus flexible exchange rates and the measurement of exchange rate instability
Journal
of International
THE
Economics
FIXED VERSUS MEASUREMENT
16 (1984)
295-306.
North
Holland
FLEXIBLE EXCHANGE RATES AND OF EXCHANGE RATE INSTABILITY
David A. BRODSKY* UNCTAD,
Received
May
CH-1211
1982, revised
Geneva
IO, Switzerland
version
received
October
1982
A recent study by Rana that examined exchange rate instability in eight Asian countries concluded that owing to the non-normality of the underlying exchange rate distributions, the standard deviation was an erratic and misleading measure of variability. The present paper argues that the choice of a suitable measure of instability cannot be made on mathematical grounds alone. Rana’s preferred measure is shown to be inappropriate, since it excludes from consideration precisely those observations of greatest significance. Conversely, given the assumption of risk aversion, the standard deviation is an entirely consistent measure. Results are also presented for an expanded study covering more than 150 countries.
1. Introduction
A recent article in this journal [Rana (1981)] explored the question of exchange rate risk under generalized floating in eight Asian countries India, South Korea, Malaysia, Nepal, Philippines, Singapore, Taiwan and Thailand. The author noted that earlier, and more comprehensive studies, had yielded conflicting results, a possible explanation being that they had used ‘inappropriate’ measures of variability. Using what he considered to be more appropriate measures of instability, the author analyzed exchange rate instability in these eight countries in the last years of the Bretton Woods system of fixed (albeit adjustable) exchange rates, and the first years of the system of generalized floating of exchange rates among the major currencies. This paper examines the question of appropriate measures of instability from a somewhat more practical point of view than that employed by Rana; in addition, the results of his research are updated and extended to more than 150 countries. It is shown that mathematical considerations alone cannot dictate the choice of an appropriate measure of instability. By focusing on largely irrelevant issues such as the degree of kurtosis and nonnormality of the underlying exchange rate distributions, Rana’s analysis serves to divert attention from the far more fundamental issues of the nature, and effect, of instability. Indeed, when the economic content of the *The author would like to thank Carlos Diaz., Bill Keeton and Dani Rodrik for helpful comments on an earlier draft of the paper. The views expressed are those of the author and not necessarily those of the United Nations. OC22-1996/84/$3.00
0
1984, Elsevier
Science
Publishers
B.V. (North-Holland)
296
D.A. Brodsky,
Fixed
us. j7exible
exchange
rates
measurement of instability is more fully taken into account, the measure dismissed by Rana as ‘erratic and misleading’ (i.e. the standard deviation) is easily seen to be far superior to the measure whose use he recommends. 2. Background
In his article, Rana calculated import-weighted effective exchange rates, in both nominal and real terms, for the eight Asian countries cited above. In order to investigate the extent to which exchange rate instability had increased in the post-Bretton Woods era of generalized floating, the period July 1967 to May 1977 was divided into two sub-periods: the adjustably pegged period extends from July 1967 to August 1971, while the generalized floating period was taken as beginning in April 1973 and extending to May 1977. Rana’s analysis begins by examining the first four sample moments, as well as the W-statistic, of the quarterly changes in nominal effective exchange rates for each country. From this he observes that: Most distributions are leptokurtic - the kurtosis values are greater than 3, the value for normal populations. This indicates a heavy cluster of observations near the mean value of the distribution. In addition, since sample kurtosis values are the fourth moment about the mean divided by the square of the second moment, high values also indicate that there are relatively more observations at the tails of the distribution than in the case of the normal distribution [Rana (1981, p. 461)]. Because of the underlying non-normality of the distributions of exchange rate changes, Rana concludes that ‘the use of the conventional measures of variability can give misleading results’. This is due to the fact that ‘in nonnormal stable Paretian distributions, the second moment does not exist (is infinite) . . . in such distributions the sample standard deviation is unstable and does not converge as the sample size increases’. Based on these considerations, the author applies two ‘appropriate’ measures of instability to his data: the scale measure of variability’ and the Gini mean difference coefficient, the latter defined as the arithmetic average of the absolute differences between all possible pairs of values. For the eight countries in the study, the values given by these two statistics are compared to those given by the standard deviation, the latter set of values being included only to ‘indicate how misleading the measure can be when used in non-normal cases’. The author observes that ‘calculations . . . confirm the misleading nature of the sample standard deviation measure of variability in non-normal samples’, since according to this measure there was decreased exchange rate instability ‘Which
the reader
is told ‘is 44% of an interfractile
range’.
D.A. Brodsky,
Fixed
vs. flexible
exchange
rates
291
during the generalized floating period for Nepal and the Philippines.(in both nominal and real terms) and for Malaysia (real only); a second deficiency of the standard deviation, he maintains, is that it provides substantially different rankings of countries than those given by either of the other two measures. Both of these findings, according to Rana (p. 465). can be explained by the fact that in non-normal samples the standard deviation measure, by taking into account all observations in the sample, gives a higher estimate of variability than the scale measure, which ignores the extreme observations. This bias of the sample standard deviation measure is more pronounced in the samples of the pegged period than the generalized floating period because the former have higher kurtosis values and hence a larger number of observations at the tails of the distribution [emphasis added]. 3. Discussion
In examining the instability of quarterly percentage changes in the effective exchange rate indices of the eight countries in his study, Rana argues that the standard deviation is an inappropriate measure to use since there are too many observations in the tails of the distributions (which implies that the distributions are non-normal). In practical terms (i.e. other than observing that these distributions are characterized by leptokurtosis) what does this mean? Inspection of the underlying data reveals that in both of the anomalous cases cited by Rana (i.e. Nepal and the Philippines)’ the explanation is provided by the fact that during the first time period there was a single large unilateral change in the ‘fixed’ exchange rate vis-a-vis the U.S. dollar. In the case of the Philippines, in February 1970 there was a 65 percent devaluation of the peso,3 leading to an immediate change in the effective exchange rate index by virtually the same amount; in only one other month did the effective exchange rate index change by more than 2 percent (the exception being a 6 percent change in March 1970). In the case of Nepal, in November 1967 there was a 33 percent devaluation of the rupee in terms of the dollar; in no other month during the first time period did the Nepalese effective exchange rate index change by more than 1 percent. For both of these countries, the situation during the first time period was therefore characterized by a virtually constant effective exchange rate index, with the exception of one single, and very large, change. This is graphically depicted, in somewhat simplified terms, in fig. 1, which shows the values of ‘The reversal in the case of the real effective exchange rate index for Malaysia will not be dealt with since it is clearly marginal (i.e. a decrease from 0.029 to 0.028, as compared to an increase from 0.028 to 0.032 given by the Gini coefficient). 3Measured in terms of the peso price of U.S. dollars.
298
D.A. Brodsky,
Fixed
vs. flexible
exchange
rates
EER
150
100
.
.
.
.
.
.
123..
.
.
50 5152
.
.
.
101
Fig. 1
an effective exchange rate index during a period with 101 observations, with no change in any sub-period except the 51st, at which point the index jumps by 50 percent. In other words, EER( 1) = EER(2) = . . . = EER(50) = 100, while EER( 5 1) = EER( 52) = . . . = EER(lOl)= 150. In this case, the mean change between observations is 0.5 percent, so that the distribution of percentage changes is non-normal in the extreme (100 observations in the left-hand tail, and 1 observation in the right-hand one). For comparison with the situation depicted in fig. 1, fig. 2 portrays a situation essentially representing the exchange rate experience of each of these two countries in the second time period, i.e. a continual succession of moderate exchange rate changes. Thus, it is assumed that the percentage change alternates between -2 percent and +2 percent. In this case the EER
101 99
........ .......
I
t 123.
.
.
50 5152 Fig. 2
.
.
.
101
D.A. Brodsky,
Fixed
mjlexible
exchange
rates
299
distribution can also be seen to be non-normal. The mean percentage change (in absolute value) between observations is 2 percent, i.e. four times that in the situation illustrated in fig. 1. Which of these two exchange rate situations is more unstable? Little light can be shed on this question by reference to considerations of kurtosis or non-normality. Rather, the fundamental question is the relationship - in terms of ‘discomfort’ - between one large change and a number of smaller ones. In other words, what is the relative weight which should be given to ‘extreme’ observations? Considerations of risk aversion, as well as common sense, would suggest that greater weight should be given to extreme observations, for a sudden large devaluation (or revaluation) of a country’s currency can unquestionably affect some people in a deleterious (indeed, devastating) manner. For this reason Cline (1976, p. 19) asserted that as ‘risk averters the LDCs should prefer frequent small changes in the exchange rates of developed countries over infrequent but very large changes’. Similarly, studies of portfolio choice under conditions of uncertainty have generally utilized quadratic cost functions, thereby effectively giving greater weight to extreme observations. [See, for example, Tobin (1958).] The first measure used by Rana not only fails to give added weight to extreme observations, but gives them absolutely no weight at all: for each time series, the scale measure completely excludes from consideration both the lower 28 percent and the upper 28 percent of all observations. By its very nature this measure will invariably reveal an absence of instability in any period of adjustably pegged exchange rates, since such an exchange rate. system is characterized by a relatively small number of (large) exchange rate changes.4 To show the inherent limitations of use of the scale measure in this context, we need only consider the following hypothetical situation: in 28 percent of the observations there is an arbitrarily large increase in the effective exchange rate index, in 28 percent there is an arbitrarily large decrease, and in the remaining 44 percent the effective exchange rate index does not change. According to the scale measure, instability in this case is zero, i.e. identical to the situation in which the effective exchange rate index is absolutely constant. Since there was ‘only’ one significant exchange rate alteration for Nepal and the Philippines during the 1967-1971 period, the scale measure indicates that effective exchange rate instability for both of these countries during this period was essentially non-existent - so that virtually by definition there had to be increased instability in the second period. Use of the scale measure would similarly lead to the conclusion that such momentous events as the 4This conjecture paper.
is quantitatively
confirmed
by the results
presented
in the final
section
ol the
300
D.A.
Brodsky,
Fixed
vs.flexible
exchange
rates
realignment of exchange rates following the Smithsonian Agreement in December 1971, as well as the United Kingdom’s 17 percent devaluation in 1967, were statistically imperceptible. The second measure proposed by Rana, the Gini mean difference, gives equal weight to all observations. In practical terms, the essential difference between this measure and the standard deviation lies not in their mathematical weighting schemes, which are identical, but in the fact that the Gini coefficient is based on absolute differences, whereas the standard deviation is based on second-order (i.e. squared) differences. Letting dEER(j) represent the percentage change in the effective change rate index between the jth and (j+ 1)st observations, dEER the average percentage change in EER, and n the total number of observations, we have: GMD=xxIdEER(j)-dEER(i)I/{(n-l)(n-2)} ij
and SD=
x(dEER(j)-dEER)2/(n-2). J j
We could equally well define a Gini coefficient which uses squared deviations, as well as a standard deviation which uses absolute deviations: T(dEER(j)-dEER(i))‘/{(n-
l)(n-2)}
and SDl=
xIdEER(j)-dEER(/(n-2). j
In fact, it can be shown that GMD2 and SD are equivalent
measures, in that
GMD2 = *SD.’
Table 1 compares the instability according to these four indices, as well as for the scale measure, for each of the two situations depicted above. For both the standard deviation and the Gini mean difference measures, using absolute deviations will have the effect of giving only equal weight to extreme observations and hence is not consistent with the assumption of risk aversion; in this case, the situation depicted in fig. 1 will still be considered a relatively stable one. Using squared deviations will result in effectively giving greater weight to extreme observations and hence is consistent with the assumption of risk aversion; in this case, both measures show a higher degree 5For a proof of this, see Kendall and Stuart (1963, p. 47).
D.A. Brodsky,
Fixed
vs.Jexible
exchange
301
rates
Table 1 Instability of eflective exchange rate index under ‘adjustably pegged (fig. 1) and ‘generalized floating’ (fig. 2) exchange rate systems, according to various measures. Scale
GMD
GMD2
SD1
SD
Adjustably pegged
0.00
1.00
1.07
1.00
5.00
Generalized floating
2.42
2.02
2.84
2.02
2.01
of instability in the first (‘adjustably (‘generalized floating’) case.
pegged’) as compared
to the second
4. Summary and conclusions
Contrary to Rana’s argument, the question of the underlying normality of the distributions of changes in the index of effective exchange rates is irrelevant to the choice of a measure of instability, as long as no statistical inferences are drawn from the results. In particular, whether or not the distribution is normal in the strict mathematical sense has no bearing on the appropriateness of using the calculated standard deviation as a measure of instability; it merely implies that we cannot use the standard deviation to set up confidence intervals in hypothesis testing with regard to the means of the populations. Indeed, a measure of instability is nothing more than a mathematical formula which - when applied to time series data - provides an indication of the degree of dispersion of the underlying sample. Different measures of instability may well provide different rankings of countries, as well as reversals between time periods. This cannot (or should not) be taken as an indication that one measure is ‘right’ and the other ‘wrong’. The choice of an appropriate measure of instability must instead be based on one’s subjective value judgments concerning the nature of instability. In particular, if economic agents are risk averse, as is commonly assumed, then the ‘erratic and misleading’ results given by the standard deviation were seen to be entirely reasonable. Rana’s focus on the purely mathematical properties of the relevant exchange rate distributions obscures the (unstated) assumptions underlying both the scale and Gini mean difference measures. In particular, an examination of the practical implications of the use of his preferred measure - the scale measure - reveals it to be totally unsuited to the measurement of exchange rate instability, since by definition it excludes from consideration precisely those observations of greatest significance. J.I.E.-
D
302
D.A. Brodsky,
Fixed
os.jlexible
exchange
rates
5. Generalization of the results
Rana states that ‘previous studies, using a more comprehensive sample of developing countries by Black (1976) and Cline (1976) yielded conflicting results’, a possible explanation being that they may have used ‘inappropriate’ measures of instability. In his study, Black examined the instability of import-weighted effective exchange rates for developing countries in two time periods: January 1970 to May 1972 and June 1972 to April 1974. His general conclusion was that ‘since mid-1972, floating exchange rates have increased the variance of effective exchange rates and therefore of traded-goods prices’. However, the measure used by Black was in certain respects inappropriate, though not for the reasons cited by Rana. By calculating the variance of monthly observations of effective exchange rate indices, rather than the variances of changes in such indices, no allowance was made for the possibility of steady movement along an upward or downward trend line. This had the effect of providing unrealistically large values of instability for a number of countries and hence distorting the comparisons between time periods.‘j The Cline study cited by Rana noted (pp. 19-20) that: Preliminary evidence suggests that the variation in rates under floating has indeed been smaller than the shocks experienced under fixed rates . . . . Nevertheless, it is too early to reach a reliable conclusion on the relative volatility of floating versus fixed rates. The only quantitative results presented pertain to the bilateral instability of the deutsche mark, yen, pound sterling and French franc vis-8-vis the dollar in the period 1959-1975. Using the standard deviation and coefficient of variation,’ Cline showed that the dollar values of the first three currencies were more stable in the 1973-1975 period than they had been in the 19591973 period. This result would also appear to be at least partly attributable to the use of a measure which does not take trend movements into account.8 Given this background, it would seem to be a useful exercise to test the conclusions reached by Rana with regard to both a larger sample of countries and a wider range of measures of instability. For this purpose, import-weighted effective exchange rate indices have been calculated for the period January 1966 to June 1979 for all countries, both developing and developed, for which the requisite data are available. Real effective exchange ‘In the case of Brazil, for example, Black’s figures show a ten-fold decrease in instability between his first and second time periods; if we adjust for trend, however, we lind a three-fold increase in instability between the two periods. ‘That is, the standard deviation divided by the mean. ‘The present author’s calculations have shown that the standard deviations of percentage changes in bilateral exchange rates among these currencies have increased substantially in the post-Bretton Woods era. See Brodsky, Helleiner and Sampson (1981, table 1).
D.A. Brodsky,
Fixed
vsjlexible
exchange
303
rates
rates have also been calculated, although a lack of data on appropriate price indices limited the coverage to approximately 100 countries. The period January 1966 to June 1979 was divided into three sub-periods: January 1966 to August 1971 (i.e. until the U.S. suspension of dollar/gold convertibility); an interim period from September 1971 to March 1973; and April 1973 to June 1979. The first and last sub-periods will be taken as corresponding to the ‘adjustably pegged’ and ‘generalized floating’ periods. Instability indices were calculated on the basis of quarterly observations.9 When Rana’s scale measure was applied to the data, the results indicated that in every one of the 152 cases in which a comparison was possible, nominal effective exchange rate instability was greater in the generalized floating period than it had been in the adjustably pegged period. Moreover, with one exception, in every case the increase exceeded 100 percent.” In view of the earlier discussion, however, it is clear that these results reflect more the properties of the scale measure itself than those of the exchange rate distributions whose instability it purports to measure. Tables 2-6 report the results with respect to three other measures of instability: the Gini mean difference and the standard deviation - both applied to percentage changes in quarterly observations in order to adjust for trend - and the standard error from the exponential trend regression equation. This last measure has frequently been used in analyses of export instability and in addition satisfies, a highly useful decomposition property not satisfied by either of the other two measures.“i Since it is based on squared deviations, we would expect this measure to give results broadly Table 2 Number of countries experiencing increased (+) and decreased (-) nominal effective exchange rate instability between 1966-1971 and 1973-1979. Gini mean difference
Standard deviation
Standard error measure
+-+-+ Total Developed Developing
141 27 114
11 1 10
132 24 108
20 4 16
130 23 107
22 5 17
‘The results were essentially unchanged when monthly data were used instead of quarterly data. “‘The exceptional case was Morocco, whose nominal instability increased by ‘only’ 35 percent. “That is, for a linear combination aX +bY of two random variables X and Y: ~*(aX+bY)=a21*(X)+b21*(Y)+2(a)(b)cov*(X,
Y),
where the covariance term refers to the trend-corrected covariance [Brodsky (1980)]. For an application of this decomposition to an analysis of the sources of exchange rate instability, see Brodsky and Sampson (1984).
304
D.A.
Number decreased
Brodsky,
of (-)
Fixed
vs. flexible
exchange
rates
Table 3 countries experiencing increased real effective exchange rate instability 1966-1971 and 1973-1979. Gini mean difference
Standard deviation
Standard measure
+-+-+ Total Developed Developing
74 22 52
(+) and between
error
20 3 17
73 20 53
21 5 16
72 18 54
22 7 15
Table 4 Comparison between number of countries in which nominal (N) or real (R) effective exchange rate instability was larger: January 1966August 1971. Gini mean di&rence
Total Developed Developing
Standard deviation
Standard measure
error
N
R
N
R
N
R
2 1 1
95 24 71
16 4 12
81 21 60
27 10 17
70 15 55
Table 5 Comparison between number of countries in which nominal (N) or real (R) effective exchange rate instability was larger: April 1973-June 1979. Gini mean difference
Total Developed Developing
Standard deviation
Standard measure
error
N
R
N
R
N
R
22 14 8
91 11 80
26 15 11
87 10 77
39 16 23
74 9 65
D.A.
Brodsky,
Fixed
us.jlexible
Table
exchange
305
rates
6
Comparison between number of countries in which relative increase in nominal (N) or real (R) effective exchange rate between 19661971 and 1973-1979 instability was greater periods. Gini mean difference
Total Developed Developing
Standard measure
error
N
R
N
R
N
R
77 22 55
17 3 14
64 17 47
30 8 22
60 17 43
34 8 26
Source: Calculations International Financial
various
Standard deviation
based on information Statistics and
available Direction
in IMF, of
Trade,
issues.
similar to those given by the standard deviation of quarterly percentage changes. Based on the information presented in the tables, the following results can be cited. (1) For the large majority of countries, both developed and developing, nominal effective exchange rate instability increased between the two periods (table 2). This result held for 93 percent of countries when the Gini mean difference measure was used; for the other two measures the corresponding figure was somewhat less. In the majority of cases in which instability has declined in the post-Bretton Woods period, as was suggested earlier, the explanation is that there had been a substantial change (usually a devaluation) in the exchange rate during the period 1966-1971. (2) For the large majority of countries, real effective exchange rate instability also increased in the second period, with all three measures providing similar breakdowns (table 3). (3) For most countries, real effective exchange rate instability has exceeded nominal instability in both the adjustably pegged and generalized floating periods (tables 4 and 5). However, according to all three measures, for the majority of developed countries the opposite result has held during the generalized floating period. (4) According to all three measures, in the generalized floating period nominal effective exchange rate instability has increased proportionately more than instability in real terms (table 6). The standard deviation and standard error measures provide nearly twice as many exceptions, however, as does the Gini mean difference. Of these findings, perhaps the most significant is the differing experience between developed and developed countries in the period of generalized floating: for developed countries, relative price changes and nominal
306
D.A.
Brodsky,
Fixed
vs. flexible
exchange
rates
exchange rate movements would appear to have offset each other to a considerably larger extent than has been the case for the developing countries.
References Black, S.W., 1976, Exchange policies for less developed countries in a world of floating rates, Princeton Essays in International Finance 119. Brodsky, D.A., 1980, Decomposable measures of economic instability, Oxford Bulletin of Economics and Statistics 42, 361-374. Brodsky, D.A. and G.P. Sampson, 1984, The sources of exchange rate instability in developing countries: Dollar, French franc and SDR pegging countries, Weltwirtschaftliches Archiv 120, forthcoming. Brodsky, D.A., G.K. Helleiner and G.P. Sampson, 1981, The impact of the current exchange rate system on developing countries, Trade and Development 3, 31-52. Cline, W., 1976, International monetary reform and the developing countries (Brookings Institution, Washington, D.C.). Kendall, Maurice G. and Alan Stuart, 1963, The theory of advanced statistics, vol. 1. (Charles Grifhn and Company, London). Rana, P.B., 1981, Exchange rate risk under generalized floating: Eight Asian countries, Journal of International Economics lL459-466. Tobin, James, 1958, Liquidity preference as behaviour towards risk, Review of Economic Studies 25, 6586.
of International
THE
Economics
FIXED VERSUS MEASUREMENT
16 (1984)
295-306.
North
Holland
FLEXIBLE EXCHANGE RATES AND OF EXCHANGE RATE INSTABILITY
David A. BRODSKY* UNCTAD,
Received
May
CH-1211
1982, revised
Geneva
IO, Switzerland
version
received
October
1982
A recent study by Rana that examined exchange rate instability in eight Asian countries concluded that owing to the non-normality of the underlying exchange rate distributions, the standard deviation was an erratic and misleading measure of variability. The present paper argues that the choice of a suitable measure of instability cannot be made on mathematical grounds alone. Rana’s preferred measure is shown to be inappropriate, since it excludes from consideration precisely those observations of greatest significance. Conversely, given the assumption of risk aversion, the standard deviation is an entirely consistent measure. Results are also presented for an expanded study covering more than 150 countries.
1. Introduction
A recent article in this journal [Rana (1981)] explored the question of exchange rate risk under generalized floating in eight Asian countries India, South Korea, Malaysia, Nepal, Philippines, Singapore, Taiwan and Thailand. The author noted that earlier, and more comprehensive studies, had yielded conflicting results, a possible explanation being that they had used ‘inappropriate’ measures of variability. Using what he considered to be more appropriate measures of instability, the author analyzed exchange rate instability in these eight countries in the last years of the Bretton Woods system of fixed (albeit adjustable) exchange rates, and the first years of the system of generalized floating of exchange rates among the major currencies. This paper examines the question of appropriate measures of instability from a somewhat more practical point of view than that employed by Rana; in addition, the results of his research are updated and extended to more than 150 countries. It is shown that mathematical considerations alone cannot dictate the choice of an appropriate measure of instability. By focusing on largely irrelevant issues such as the degree of kurtosis and nonnormality of the underlying exchange rate distributions, Rana’s analysis serves to divert attention from the far more fundamental issues of the nature, and effect, of instability. Indeed, when the economic content of the *The author would like to thank Carlos Diaz., Bill Keeton and Dani Rodrik for helpful comments on an earlier draft of the paper. The views expressed are those of the author and not necessarily those of the United Nations. OC22-1996/84/$3.00
0
1984, Elsevier
Science
Publishers
B.V. (North-Holland)
296
D.A. Brodsky,
Fixed
us. j7exible
exchange
rates
measurement of instability is more fully taken into account, the measure dismissed by Rana as ‘erratic and misleading’ (i.e. the standard deviation) is easily seen to be far superior to the measure whose use he recommends. 2. Background
In his article, Rana calculated import-weighted effective exchange rates, in both nominal and real terms, for the eight Asian countries cited above. In order to investigate the extent to which exchange rate instability had increased in the post-Bretton Woods era of generalized floating, the period July 1967 to May 1977 was divided into two sub-periods: the adjustably pegged period extends from July 1967 to August 1971, while the generalized floating period was taken as beginning in April 1973 and extending to May 1977. Rana’s analysis begins by examining the first four sample moments, as well as the W-statistic, of the quarterly changes in nominal effective exchange rates for each country. From this he observes that: Most distributions are leptokurtic - the kurtosis values are greater than 3, the value for normal populations. This indicates a heavy cluster of observations near the mean value of the distribution. In addition, since sample kurtosis values are the fourth moment about the mean divided by the square of the second moment, high values also indicate that there are relatively more observations at the tails of the distribution than in the case of the normal distribution [Rana (1981, p. 461)]. Because of the underlying non-normality of the distributions of exchange rate changes, Rana concludes that ‘the use of the conventional measures of variability can give misleading results’. This is due to the fact that ‘in nonnormal stable Paretian distributions, the second moment does not exist (is infinite) . . . in such distributions the sample standard deviation is unstable and does not converge as the sample size increases’. Based on these considerations, the author applies two ‘appropriate’ measures of instability to his data: the scale measure of variability’ and the Gini mean difference coefficient, the latter defined as the arithmetic average of the absolute differences between all possible pairs of values. For the eight countries in the study, the values given by these two statistics are compared to those given by the standard deviation, the latter set of values being included only to ‘indicate how misleading the measure can be when used in non-normal cases’. The author observes that ‘calculations . . . confirm the misleading nature of the sample standard deviation measure of variability in non-normal samples’, since according to this measure there was decreased exchange rate instability ‘Which
the reader
is told ‘is 44% of an interfractile
range’.
D.A. Brodsky,
Fixed
vs. flexible
exchange
rates
291
during the generalized floating period for Nepal and the Philippines.(in both nominal and real terms) and for Malaysia (real only); a second deficiency of the standard deviation, he maintains, is that it provides substantially different rankings of countries than those given by either of the other two measures. Both of these findings, according to Rana (p. 465). can be explained by the fact that in non-normal samples the standard deviation measure, by taking into account all observations in the sample, gives a higher estimate of variability than the scale measure, which ignores the extreme observations. This bias of the sample standard deviation measure is more pronounced in the samples of the pegged period than the generalized floating period because the former have higher kurtosis values and hence a larger number of observations at the tails of the distribution [emphasis added]. 3. Discussion
In examining the instability of quarterly percentage changes in the effective exchange rate indices of the eight countries in his study, Rana argues that the standard deviation is an inappropriate measure to use since there are too many observations in the tails of the distributions (which implies that the distributions are non-normal). In practical terms (i.e. other than observing that these distributions are characterized by leptokurtosis) what does this mean? Inspection of the underlying data reveals that in both of the anomalous cases cited by Rana (i.e. Nepal and the Philippines)’ the explanation is provided by the fact that during the first time period there was a single large unilateral change in the ‘fixed’ exchange rate vis-a-vis the U.S. dollar. In the case of the Philippines, in February 1970 there was a 65 percent devaluation of the peso,3 leading to an immediate change in the effective exchange rate index by virtually the same amount; in only one other month did the effective exchange rate index change by more than 2 percent (the exception being a 6 percent change in March 1970). In the case of Nepal, in November 1967 there was a 33 percent devaluation of the rupee in terms of the dollar; in no other month during the first time period did the Nepalese effective exchange rate index change by more than 1 percent. For both of these countries, the situation during the first time period was therefore characterized by a virtually constant effective exchange rate index, with the exception of one single, and very large, change. This is graphically depicted, in somewhat simplified terms, in fig. 1, which shows the values of ‘The reversal in the case of the real effective exchange rate index for Malaysia will not be dealt with since it is clearly marginal (i.e. a decrease from 0.029 to 0.028, as compared to an increase from 0.028 to 0.032 given by the Gini coefficient). 3Measured in terms of the peso price of U.S. dollars.
298
D.A. Brodsky,
Fixed
vs. flexible
exchange
rates
EER
150
100
.
.
.
.
.
.
123..
.
.
50 5152
.
.
.
101
Fig. 1
an effective exchange rate index during a period with 101 observations, with no change in any sub-period except the 51st, at which point the index jumps by 50 percent. In other words, EER( 1) = EER(2) = . . . = EER(50) = 100, while EER( 5 1) = EER( 52) = . . . = EER(lOl)= 150. In this case, the mean change between observations is 0.5 percent, so that the distribution of percentage changes is non-normal in the extreme (100 observations in the left-hand tail, and 1 observation in the right-hand one). For comparison with the situation depicted in fig. 1, fig. 2 portrays a situation essentially representing the exchange rate experience of each of these two countries in the second time period, i.e. a continual succession of moderate exchange rate changes. Thus, it is assumed that the percentage change alternates between -2 percent and +2 percent. In this case the EER
101 99
........ .......
I
t 123.
.
.
50 5152 Fig. 2
.
.
.
101
D.A. Brodsky,
Fixed
mjlexible
exchange
rates
299
distribution can also be seen to be non-normal. The mean percentage change (in absolute value) between observations is 2 percent, i.e. four times that in the situation illustrated in fig. 1. Which of these two exchange rate situations is more unstable? Little light can be shed on this question by reference to considerations of kurtosis or non-normality. Rather, the fundamental question is the relationship - in terms of ‘discomfort’ - between one large change and a number of smaller ones. In other words, what is the relative weight which should be given to ‘extreme’ observations? Considerations of risk aversion, as well as common sense, would suggest that greater weight should be given to extreme observations, for a sudden large devaluation (or revaluation) of a country’s currency can unquestionably affect some people in a deleterious (indeed, devastating) manner. For this reason Cline (1976, p. 19) asserted that as ‘risk averters the LDCs should prefer frequent small changes in the exchange rates of developed countries over infrequent but very large changes’. Similarly, studies of portfolio choice under conditions of uncertainty have generally utilized quadratic cost functions, thereby effectively giving greater weight to extreme observations. [See, for example, Tobin (1958).] The first measure used by Rana not only fails to give added weight to extreme observations, but gives them absolutely no weight at all: for each time series, the scale measure completely excludes from consideration both the lower 28 percent and the upper 28 percent of all observations. By its very nature this measure will invariably reveal an absence of instability in any period of adjustably pegged exchange rates, since such an exchange rate. system is characterized by a relatively small number of (large) exchange rate changes.4 To show the inherent limitations of use of the scale measure in this context, we need only consider the following hypothetical situation: in 28 percent of the observations there is an arbitrarily large increase in the effective exchange rate index, in 28 percent there is an arbitrarily large decrease, and in the remaining 44 percent the effective exchange rate index does not change. According to the scale measure, instability in this case is zero, i.e. identical to the situation in which the effective exchange rate index is absolutely constant. Since there was ‘only’ one significant exchange rate alteration for Nepal and the Philippines during the 1967-1971 period, the scale measure indicates that effective exchange rate instability for both of these countries during this period was essentially non-existent - so that virtually by definition there had to be increased instability in the second period. Use of the scale measure would similarly lead to the conclusion that such momentous events as the 4This conjecture paper.
is quantitatively
confirmed
by the results
presented
in the final
section
ol the
300
D.A.
Brodsky,
Fixed
vs.flexible
exchange
rates
realignment of exchange rates following the Smithsonian Agreement in December 1971, as well as the United Kingdom’s 17 percent devaluation in 1967, were statistically imperceptible. The second measure proposed by Rana, the Gini mean difference, gives equal weight to all observations. In practical terms, the essential difference between this measure and the standard deviation lies not in their mathematical weighting schemes, which are identical, but in the fact that the Gini coefficient is based on absolute differences, whereas the standard deviation is based on second-order (i.e. squared) differences. Letting dEER(j) represent the percentage change in the effective change rate index between the jth and (j+ 1)st observations, dEER the average percentage change in EER, and n the total number of observations, we have: GMD=xxIdEER(j)-dEER(i)I/{(n-l)(n-2)} ij
and SD=
x(dEER(j)-dEER)2/(n-2). J j
We could equally well define a Gini coefficient which uses squared deviations, as well as a standard deviation which uses absolute deviations: T(dEER(j)-dEER(i))‘/{(n-
l)(n-2)}
and SDl=
xIdEER(j)-dEER(/(n-2). j
In fact, it can be shown that GMD2 and SD are equivalent
measures, in that
GMD2 = *SD.’
Table 1 compares the instability according to these four indices, as well as for the scale measure, for each of the two situations depicted above. For both the standard deviation and the Gini mean difference measures, using absolute deviations will have the effect of giving only equal weight to extreme observations and hence is not consistent with the assumption of risk aversion; in this case, the situation depicted in fig. 1 will still be considered a relatively stable one. Using squared deviations will result in effectively giving greater weight to extreme observations and hence is consistent with the assumption of risk aversion; in this case, both measures show a higher degree 5For a proof of this, see Kendall and Stuart (1963, p. 47).
D.A. Brodsky,
Fixed
vs.Jexible
exchange
301
rates
Table 1 Instability of eflective exchange rate index under ‘adjustably pegged (fig. 1) and ‘generalized floating’ (fig. 2) exchange rate systems, according to various measures. Scale
GMD
GMD2
SD1
SD
Adjustably pegged
0.00
1.00
1.07
1.00
5.00
Generalized floating
2.42
2.02
2.84
2.02
2.01
of instability in the first (‘adjustably (‘generalized floating’) case.
pegged’) as compared
to the second
4. Summary and conclusions
Contrary to Rana’s argument, the question of the underlying normality of the distributions of changes in the index of effective exchange rates is irrelevant to the choice of a measure of instability, as long as no statistical inferences are drawn from the results. In particular, whether or not the distribution is normal in the strict mathematical sense has no bearing on the appropriateness of using the calculated standard deviation as a measure of instability; it merely implies that we cannot use the standard deviation to set up confidence intervals in hypothesis testing with regard to the means of the populations. Indeed, a measure of instability is nothing more than a mathematical formula which - when applied to time series data - provides an indication of the degree of dispersion of the underlying sample. Different measures of instability may well provide different rankings of countries, as well as reversals between time periods. This cannot (or should not) be taken as an indication that one measure is ‘right’ and the other ‘wrong’. The choice of an appropriate measure of instability must instead be based on one’s subjective value judgments concerning the nature of instability. In particular, if economic agents are risk averse, as is commonly assumed, then the ‘erratic and misleading’ results given by the standard deviation were seen to be entirely reasonable. Rana’s focus on the purely mathematical properties of the relevant exchange rate distributions obscures the (unstated) assumptions underlying both the scale and Gini mean difference measures. In particular, an examination of the practical implications of the use of his preferred measure - the scale measure - reveals it to be totally unsuited to the measurement of exchange rate instability, since by definition it excludes from consideration precisely those observations of greatest significance. J.I.E.-
D
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os.jlexible
exchange
rates
5. Generalization of the results
Rana states that ‘previous studies, using a more comprehensive sample of developing countries by Black (1976) and Cline (1976) yielded conflicting results’, a possible explanation being that they may have used ‘inappropriate’ measures of instability. In his study, Black examined the instability of import-weighted effective exchange rates for developing countries in two time periods: January 1970 to May 1972 and June 1972 to April 1974. His general conclusion was that ‘since mid-1972, floating exchange rates have increased the variance of effective exchange rates and therefore of traded-goods prices’. However, the measure used by Black was in certain respects inappropriate, though not for the reasons cited by Rana. By calculating the variance of monthly observations of effective exchange rate indices, rather than the variances of changes in such indices, no allowance was made for the possibility of steady movement along an upward or downward trend line. This had the effect of providing unrealistically large values of instability for a number of countries and hence distorting the comparisons between time periods.‘j The Cline study cited by Rana noted (pp. 19-20) that: Preliminary evidence suggests that the variation in rates under floating has indeed been smaller than the shocks experienced under fixed rates . . . . Nevertheless, it is too early to reach a reliable conclusion on the relative volatility of floating versus fixed rates. The only quantitative results presented pertain to the bilateral instability of the deutsche mark, yen, pound sterling and French franc vis-8-vis the dollar in the period 1959-1975. Using the standard deviation and coefficient of variation,’ Cline showed that the dollar values of the first three currencies were more stable in the 1973-1975 period than they had been in the 19591973 period. This result would also appear to be at least partly attributable to the use of a measure which does not take trend movements into account.8 Given this background, it would seem to be a useful exercise to test the conclusions reached by Rana with regard to both a larger sample of countries and a wider range of measures of instability. For this purpose, import-weighted effective exchange rate indices have been calculated for the period January 1966 to June 1979 for all countries, both developing and developed, for which the requisite data are available. Real effective exchange ‘In the case of Brazil, for example, Black’s figures show a ten-fold decrease in instability between his first and second time periods; if we adjust for trend, however, we lind a three-fold increase in instability between the two periods. ‘That is, the standard deviation divided by the mean. ‘The present author’s calculations have shown that the standard deviations of percentage changes in bilateral exchange rates among these currencies have increased substantially in the post-Bretton Woods era. See Brodsky, Helleiner and Sampson (1981, table 1).
D.A. Brodsky,
Fixed
vsjlexible
exchange
303
rates
rates have also been calculated, although a lack of data on appropriate price indices limited the coverage to approximately 100 countries. The period January 1966 to June 1979 was divided into three sub-periods: January 1966 to August 1971 (i.e. until the U.S. suspension of dollar/gold convertibility); an interim period from September 1971 to March 1973; and April 1973 to June 1979. The first and last sub-periods will be taken as corresponding to the ‘adjustably pegged’ and ‘generalized floating’ periods. Instability indices were calculated on the basis of quarterly observations.9 When Rana’s scale measure was applied to the data, the results indicated that in every one of the 152 cases in which a comparison was possible, nominal effective exchange rate instability was greater in the generalized floating period than it had been in the adjustably pegged period. Moreover, with one exception, in every case the increase exceeded 100 percent.” In view of the earlier discussion, however, it is clear that these results reflect more the properties of the scale measure itself than those of the exchange rate distributions whose instability it purports to measure. Tables 2-6 report the results with respect to three other measures of instability: the Gini mean difference and the standard deviation - both applied to percentage changes in quarterly observations in order to adjust for trend - and the standard error from the exponential trend regression equation. This last measure has frequently been used in analyses of export instability and in addition satisfies, a highly useful decomposition property not satisfied by either of the other two measures.“i Since it is based on squared deviations, we would expect this measure to give results broadly Table 2 Number of countries experiencing increased (+) and decreased (-) nominal effective exchange rate instability between 1966-1971 and 1973-1979. Gini mean difference
Standard deviation
Standard error measure
+-+-+ Total Developed Developing
141 27 114
11 1 10
132 24 108
20 4 16
130 23 107
22 5 17
‘The results were essentially unchanged when monthly data were used instead of quarterly data. “‘The exceptional case was Morocco, whose nominal instability increased by ‘only’ 35 percent. “That is, for a linear combination aX +bY of two random variables X and Y: ~*(aX+bY)=a21*(X)+b21*(Y)+2(a)(b)cov*(X,
Y),
where the covariance term refers to the trend-corrected covariance [Brodsky (1980)]. For an application of this decomposition to an analysis of the sources of exchange rate instability, see Brodsky and Sampson (1984).
304
D.A.
Number decreased
Brodsky,
of (-)
Fixed
vs. flexible
exchange
rates
Table 3 countries experiencing increased real effective exchange rate instability 1966-1971 and 1973-1979. Gini mean difference
Standard deviation
Standard measure
+-+-+ Total Developed Developing
74 22 52
(+) and between
error
20 3 17
73 20 53
21 5 16
72 18 54
22 7 15
Table 4 Comparison between number of countries in which nominal (N) or real (R) effective exchange rate instability was larger: January 1966August 1971. Gini mean di&rence
Total Developed Developing
Standard deviation
Standard measure
error
N
R
N
R
N
R
2 1 1
95 24 71
16 4 12
81 21 60
27 10 17
70 15 55
Table 5 Comparison between number of countries in which nominal (N) or real (R) effective exchange rate instability was larger: April 1973-June 1979. Gini mean difference
Total Developed Developing
Standard deviation
Standard measure
error
N
R
N
R
N
R
22 14 8
91 11 80
26 15 11
87 10 77
39 16 23
74 9 65
D.A.
Brodsky,
Fixed
us.jlexible
Table
exchange
305
rates
6
Comparison between number of countries in which relative increase in nominal (N) or real (R) effective exchange rate between 19661971 and 1973-1979 instability was greater periods. Gini mean difference
Total Developed Developing
Standard measure
error
N
R
N
R
N
R
77 22 55
17 3 14
64 17 47
30 8 22
60 17 43
34 8 26
Source: Calculations International Financial
various
Standard deviation
based on information Statistics and
available Direction
in IMF, of
Trade,
issues.
similar to those given by the standard deviation of quarterly percentage changes. Based on the information presented in the tables, the following results can be cited. (1) For the large majority of countries, both developed and developing, nominal effective exchange rate instability increased between the two periods (table 2). This result held for 93 percent of countries when the Gini mean difference measure was used; for the other two measures the corresponding figure was somewhat less. In the majority of cases in which instability has declined in the post-Bretton Woods period, as was suggested earlier, the explanation is that there had been a substantial change (usually a devaluation) in the exchange rate during the period 1966-1971. (2) For the large majority of countries, real effective exchange rate instability also increased in the second period, with all three measures providing similar breakdowns (table 3). (3) For most countries, real effective exchange rate instability has exceeded nominal instability in both the adjustably pegged and generalized floating periods (tables 4 and 5). However, according to all three measures, for the majority of developed countries the opposite result has held during the generalized floating period. (4) According to all three measures, in the generalized floating period nominal effective exchange rate instability has increased proportionately more than instability in real terms (table 6). The standard deviation and standard error measures provide nearly twice as many exceptions, however, as does the Gini mean difference. Of these findings, perhaps the most significant is the differing experience between developed and developed countries in the period of generalized floating: for developed countries, relative price changes and nominal
306
D.A.
Brodsky,
Fixed
vs. flexible
exchange
rates
exchange rate movements would appear to have offset each other to a considerably larger extent than has been the case for the developing countries.
References Black, S.W., 1976, Exchange policies for less developed countries in a world of floating rates, Princeton Essays in International Finance 119. Brodsky, D.A., 1980, Decomposable measures of economic instability, Oxford Bulletin of Economics and Statistics 42, 361-374. Brodsky, D.A. and G.P. Sampson, 1984, The sources of exchange rate instability in developing countries: Dollar, French franc and SDR pegging countries, Weltwirtschaftliches Archiv 120, forthcoming. Brodsky, D.A., G.K. Helleiner and G.P. Sampson, 1981, The impact of the current exchange rate system on developing countries, Trade and Development 3, 31-52. Cline, W., 1976, International monetary reform and the developing countries (Brookings Institution, Washington, D.C.). Kendall, Maurice G. and Alan Stuart, 1963, The theory of advanced statistics, vol. 1. (Charles Grifhn and Company, London). Rana, P.B., 1981, Exchange rate risk under generalized floating: Eight Asian countries, Journal of International Economics lL459-466. Tobin, James, 1958, Liquidity preference as behaviour towards risk, Review of Economic Studies 25, 6586.