Chapter 4
Extremes in Exchange Rates and Macroeconomic Fundamentals: Some Evidence from Asia-Pacific Countries

Phornchanok Cumperayot

Roy Kouwenberg

INTRODUCTION
EXTREME VALUE THEORY
THE FELLER TAIL ADDITIVITY THEOREM
ESTIMATION
DATA
EMPIRICAL RESULTS
The External Relationship
CONCLUSION
References

INTRODUCTION

Extreme behavior of exchange rate returns, such as exchange rate realignments or currency crises, show up in the tail areas of the distribution. Although extreme events are rare, they characterize long-term risk that is highly relevant to national welfare, as proven by the financial crises that occurred in the last decade.¹ Understanding the likelihood and the source of extreme behavior is therefore essential for long-term risk management.

This chapter will investigate the distributional characteristics of exchange rate returns and changes in macroeconomic fundamentals and their association in the tail area for eighteen Asia-Pacific Economic Cooperation (APEC) nations. The contribution to the literature is the application of extreme value theory (EVT) in the field of macroeconomics which allows us to study the likelihood of extreme events and their correlations without relying on any specific class of distributions.

In economics, numerous papers have tried to explain the movement of exchange rates.² Many theoretical attempts, however, have failed to determine exchange rates in practice, and the empirical support for the theories has been weak.³ In consensus, it has

¹ Krugman (2001) states that unlike the older-generation crisis, the new-generation crisis tends to be more harmful as it becomes indeterministic and leads to bad consequences for the real economy.

² For the monetary-approach partial equilibrium models, Frenkel (1976), Mussa (1976), and Bilson (1978) discuss the flexible-price model, while Dornbusch (1976), Frankel (1979), Mussa (1979), and Buiter and Miller (1982) consider the sticky-price model. The general equilibrium asset pricing models are studied by Stockman (1980), Lucas (1982), Svensson (1985a, and 1985b), and Hodrick (1989), and extended into the continuous-time stochastic framework by Bakshi and Chen (1997), and Basak and Gallmeyer (1998).

³ For example, Frenkel (1976), Bilson (1978), Hodrick (1978, 1989), Meese and Rogoff (1983, 1988), Backus (1984), Meese (1990), MacDonald and Taylor (1994), Chinn and Meese (1995), Mark (1995), and Flood and Rose (1995, 1999).

generally been acknowledged that economic fundamentals fail to explain the volatility of exchange rates, particularly in the short and medium terms⁴, but there is some evidence of predictability in a long-run relationship.⁵

On the other hand, in the currency crisis literature there are a large number of articles that claim the success of economic fundamentals in determining the time and origin of crises.⁶ These provide evidence on the relationship between exchange rates and macroeconomic fundamentals for extreme events. Thus, this chapter tries to understand the association between irregular movements of the exchange rate and economic fundamentals, while focusing on extreme events in the tail area.

The empirical regularity of heavy-tailed distributions of financial asset returns, including exchange rates, is well known.⁷ Hols and de Vries (1991) show the usefulness of using extreme value theory in the study of the distribution of extreme exchange rate returns. Pozo and Amuedo-Dorantes (2003) point out the usefulness of applying extreme value theory to identify periods of currency crisis.

Because of the low frequency nature of macroeconomic variables, however, there are few studies examining the tail behavior of macroeconomic fundamentals, let alone the extremal relationship between the exchange rate and its fundamentals.⁸ This chapter aims to investigate whether the apparently non-normal exchange rate returns are caused by macroconomic fundamentals or by market noise. Using a similar methodology to Cumperayot (2002) and Cumperayot and de Vries (2006), limit laws are applied to study the distributions of the extreme realizations and non-parametrically estimate the tail index. The tail index helps to measure the mass in the tails and provides information about the underlying distribution of the variables.⁹

The Hill (1975) estimator and the threshold selection method for small-sized samples, developed by Huisman et al. (2001), are used to study the probability of large swings in currency prices and economic fundamentals. Feller's (1971) Convolution Theorem then provides the tail index relationship between exchange rates and its fundamentals, based on the well-known log-linear exchange rate models.

Here, the movements of the exchange rates of seventeen APEC countries, measured as the domestic price of a U.S. dollar, and their available economic fundamentals are studied. The sample includes countries which are different in various aspects, such as

⁴ See, for example, Meese and Rogoff (1983, 1988), Flood and Rose (1995, 1999), and Flood et al. (1991).

⁵ For empirical results, see MacDonald and Taylor (1994), Groen (1999), and Cumperayot (2005).

⁶ For a literature review, see Kaminsky et al. (1998) and Kumar et al. (2003).

⁷ See, for example, Fama (1965), Rothschild and Stiglitz (1970), Dooley and Shafer (1976), Westerfield (1977), Parkinson (1980), Jansen and de Vries (1991), Koedijk et al. (1992), Koedijk and Kool (1994), Loretan and Phillips (1994), Login (1996), Kearns and Pagan (1997), and Bekaert et al. (1998).

⁸ These exclude Cumperayot (2002) and Cumperayot and de Vries (2006), who study the distributional characteristics of exchange rates and economic fundamentals and their extremal associations for eight OECD countries.

⁹ This study differs from the literature in which a currency crisis is generally defined when large exchange rate returns exceed two or three standard deviations above the sample mean. The same method is used in the signalling approach to declare when economic variables are deemed to issue a warning signal. However, the probability of a currency crisis is often predicted using some pre-specified models, such as probit or logit models.

geography, exchange rate regime, and monetary policy. They are divided into a group of developed countries, a group of Latin American countries, and a group of Asian countries.

The results provide strong evidence that the assumption of normality is not appropriate when studying the relationship between exchange rates and economic fundamentals since the variables appear to be heavy-tailed and distributed asymmetrically. According to Feller's (1971) Tail Additivity Theorem, the exchange rate models postulate asymmetric extremal associations between the exchange rate and its fundamentals. It was found that a fixed (or dirty floating) exchange rate regime is not a guarantee of exchange rate stability, as its likelihood of a currency crisis tends to be higher. Economic fundamentals tend to do well in explaining the extreme depreciation of the developed countries' currencies, and the extreme currency appreciation of less developed countries. Moreover, as in Krugman (2001), when comparing the Latin American crises with the Asian crisis of 1997–1998, the latter seems to be more extreme and less likely to be explained by economic fundamentals.

EXTREME VALUE THEORY

According to the theory of heavy tails¹⁰, a distribution function F is said to have heavy tails if its tails vary regularly (slowly) at infinity. Consider a stationary sequence {X₁,X₂,…, X_n} of independent and identically distributed (iid) random values of a variable X with a distribution function F. Suppose one is interested in the upper tail.¹¹ The tail varies regularly at infinity with tail index α flim

It implies that the distribution F approaches infinity at a power rate and the number of existing unconditional moments is finite and equals the integer value of. α¹² A lower α implies a slower rate of approaching infinity and thus, higher accumulating probability (that is, thicker) in the tail area. The tail contains more probability mass, that is, there is a higher chance of extreme events than in the benchmark case of a normal distribution.

Analogous to the Central Limit Theorem, extreme value theory provides, under some conditions, a precise form of heavy-tailed distribution, which is independent of the

¹⁰ For theoretical details on extremal analysis, the reader is referred to Login (1996), and Embrechts et al. (1999).

¹¹ For the lower tail,

¹² For instance, Pareto distributions satisfy the Power law and have a number of bounded moments less than or equal to α. The normal distribution approaches infinity at an exponential rate and its number of existing moments is infinite.

unknown data generating the process of distribution F(x). From the first n random values, the maximum order statistic¹³ is defined as

M_n≡max{X₁, X₂,…, X_n}

The probability that the maximum is below a certain level X is given by the distribution of the maximum conditions on the distribution F(x):

P{M _n≤ x} = Fⁿ (x)

According to extreme value theory, regardless of the precise form of F(x), the limiting distribution is precisely known. The proper linearly scaled order statistic M_n converges to a limiting distribution as n → ∞, such that:

P{a_n (M_n−b_n) ≤ x} → G(x)

where a_n and b_n are appropriate normalizing constants. By reparameterizing one obtains a continuous, unified model for the general extreme value distribution,

G(x) = exp(− (1 +γx)^−1/γ)

If F(x) is heavy-tailed, which means α ¾ 0 and thus γ ¾ 0, given appropriate location and scale parameters, a_n and b_n, G(x) is a Frechet distribution, that is, G(x) exp(−x^−1/γ)for x¾ 0.¹⁴

An important application of extreme value theory is the estimation of extreme probability-quantile() combinations. According to De Haan et al. (1994), given the ascending order statistics of univariate excess, probabilities can be estimated by using the semi-parametric probability estimator:

where the tail cutoff point X_(n−m) is the m−th largest observation from a sample of size n. The extreme probability-quantile combination () measures , where q is a large quantile given, such that q > X_(n−m). The tail probability estimator () depends on the properties of α. A lower α implies higher accumulating probability in the tail area.

¹³ The analysis of the minimum order statistic becomes analogous to the maximum by premultiplying the observations by 1.

¹⁴ Note that in the case of the light-tailed distributions which approach infinity at exponential rates, such as a normal distribution, G(x) follows a Gumbel distribution.

THE FELLER TAIL ADDITIVITY THEOREM

To investigate whether the tails of the economic fundamentals associate with the heavy-tailed exchange rate returns, a theoretical tail-index relationship is derived from the log-linear exchange rate models, such as the flexible-price (Frenkel–Bilson) monetary model, the sticky-price (Dornbusch–Frankel) monetary model, and the (Krugman–Flood–Garber) balance-of-payments currency crisis model.

According to Feller's Tail Additivity Theorem, if X and Y are iid with regularly varying tails and if X has a tail index of and Y has a lighter tail, as s → β

P{X + Y > s} ≈ As^{– α}

The convolution of the functions f_x(.) and f_y(.) is dominated by the heavier tail, while the scalar A may change if the coefficients are not equal to one. Note that if X and Y are linearly dependent, the Feller Convolution Theorem still holds as the tail shape, that is, is α unaffected by linear dependence.

According to the log-linear exchange rate models (Meese and Rogoff, 1983), the exchange rate is an additive function of the economic fundamentals, such as the money supply, real income, interest rates, price levels, domestic credit, international reserves, fiscal imbalances, and external imbalances. Hence, the tail shape of the exchange rate return distribution is determined by the tail properties of those fundamental changes.

The tail shape of exchange rate returns is governed by the heaviest tail of the fundamentals in the set of exchange rate determinants. Nevertheless, there is no consensus on the perfect set of exchange rate determinants. At this point, from the economic variables suggested in the models, economic fundamentals are selected which have similar tail fatness to those of the exchange-rate in order to help explain the extremal exchange rate.

Given the asymptotic normal distribution of the tail estimator, H₀:γ_e ≤ γ_f is tested H₁:γ_e ≤ γ_f where the subscripts e and f denote that the inverse tail index γ belongs to the exchange rate return and the fundamental changes, respectively. If the null hypothesis is rejected, the exchange rate return exhibits a thicker tail than the fundamental changes. According to the theorem, the fundamental variable is not a potential contributor to the extreme movement of the exchange rate.

The variables for which one cannot reject the null hypothesis are subjected to a second one-sided test. This is done to eliminate the fundamentals that have fatter tails than the exchange rate. If the fundamentals have a fatter tail than the exchange rate, one would reject the null hypothesis of H₀:γ_e ≤ γ_f in favor of the alternative H₁: γ_e ≤ γ_f. From the two different confidence bands, if both null hypotheses cannot be rejected, it can be concluded that the tail behavior of the exchange rate returns accords well with the tail behavior of the fundamentals. This would be in line with the monetary model, at least for the larger movements in the variables.

ESTIMATION

To estimate the tail index, several estimators can be used, such as that of Hill (1975), that of Dekkers–Einmahl–de Haan (1989) (DEdH), and that of Smith (1987). However, this study will consistently report the Hill estimates, because firstly, it is conventional and easy to implement, and secondly, it is asymptotically unbiased. It is also demonstrably superior to the other estimators. Koedijk et al. (1992) show that the non-parametric technique appears to be more efficient than the Smith estimator in the case of an unknown underlying distribution, and it is asymptotically unbiased.

Define the ascending order statistics from a sample of size n as X₍₁₎ ≤ X_(2).≤ X_(n). The non-parametric estimator proposed by Hill (1975) is

Where denotes the inverse tail index, that is, and X_(n-m) is a selected threshold. Thus, there are m observations above the threshold. Quoted from Huisman et al. (2001), the tail index is a measure of the amount of tail fatness of the distribution under investigation and fits within extreme value theory. Although the Hill estimator is only valid for the Frechet distribution, in the case of heavy-tailed distributions it has been shown to be more efficient, and is asymptotically normal with mean zero and variance γ². See Mason (1982), Goldie and Smith (1987), and Jansen and de Vries (1991).

One of the most essential steps in computing a tail index is to select the threshold, X_(n-m), or the number of observations m included in the tail area. Too few observations can enlarge the variance of the estimate, while too many observations reduce the variance at the expense of bias because observations in the central range are included. Several methods have been developed to deal with this trade-off problem. Danielsson et al. (2001) recommend a subsample bootstrap technique to find an asymptotic mean-squared-error optimal threshold.

Yet, to yield a reliable estimate a large sample size is necessary. In general, most of the estimators work well for large sample sizes but all suffer seriously from a small-sample bias, including the Hill estimator. To cope with small-sample bias, a modified version of the Hill estimator was used, proposed by Huisman et al. (2001). The new estimator exploits the fact that the asymptotic bias increases in m and always exists for any m exceeding zero. Through a linear approximation of the asymptotic expected value for the Hill estimator, in which the bias term is linear in m, one gets

where k is chosen such that is approximately linear in m. In practice, the Hill estimates are plotted against a number of extremes to select the right endpoint k. An unbiased estimate of γ is thus equal to ρ₀, as m approaches zero. To yield reliable econometric results, Huisman et al. (2001) propose a weighted-least squares (WLS) technique with a (k x k) weighting matrix, that has as diagonal elements and zero

elsewhere, to correct for heteroskedasticity. Therefore, the modified version of the Hill estimator is the weighted average of a set of conventional Hill estimates

where the weights are a function of m. Because of the problem of overlapping data, one needs to take into account the autocorrelation in the error term (m). Here, Newey-West statistics are applied to correct for serial correlation. According to their study, Huisman et al. (2001) claim that the estimator can reduce the bias in the Hill-based tail index estimates drastically for samples as small as one hundred observations.

DATA

The International Monetary Fund's (IMF) International Financial Statistics (IFS) is the main data source for monthly observations from eighteen APEC member states, generally starting from January 1974 to December 2002.¹⁵ While the United States is used as the benchmark foreign country, the nations are grouped into developed countries, countries in South America, and countries in Asia. The first group consists of Australia, Canada, Japan, and New Zealand. The second consists of Chile, Mexico, and Peru, while the third group comprises ten countries in Asia: China, Hong Kong, Indonesia, South Korea, Malaysia, Papua New Guinea, Philippines, Singapore, Thailand, and Vietnam.

Exchange rate fundamentals are selected according to the flexible-price monetary model, the sticky-price monetary model, the portfolio-balance model, and the balance-of-payments currency crisis model. The variables include the exchange rate, money supply (M2), production index, interest rates, price index, international reserves, domestic credit, trade balance, and government deficit.

The exchange rate is quoted as the price of a U.S. dollar in terms of the domestic currency. The economic fundamentals are measured in terms of the domestic variable relative to the U.S. variable. Stationary economic time series are created by taking the first difference of the natural logarithms of the level series, except in the case of interest rates, the trade balance, and government deficit. Interest rates are measured in monthly percentage changes, while the trade balance and government deficit are measured according to their monthly changes relative to the level of national real income.

An advantage of the extreme value theory is exploited by considering left and right tails separately. The left tail represents a negative rate of change, in which the domestic rate of change is below the U.S. rate of change. For the exchange rate, observations in the

¹⁵ Because of the large number of countries and variables involved, there are some variables missing or are only available for different periods of time.

left tail represent a negative rate of return from holding the U.S. dollar, or an appreciation of the domestic currency. For the right tail, the interpretation is just the opposite.

These tail comparisons are implemented based on the relationships given in the exchange rate theories. For example, a depreciation of the domestic currency may occur because of an increase in the money supply or domestic credit, or a decline in real income that causes an overwhelming supply of domestic currency relative to the U.S. dollar, or an increase in prices or a decrease in interest rates relative to the U.S. levels, or a negative change in international reserves, the trade balance, or the fiscal budget.

Therefore, a depreciation of the domestic currency (the right tail of the exchange rate return distribution) is associated with the right tails of the distribution of changes in the money supply, domestic credit, and prices, and the left tails of the distribution of changes in real income, interest rates, international reserves, the trade balance, and the fiscal budget (the latter two variables are relative to the level of real income).

EMPIRICAL RESULTS

This section is devoted to the empirical investigation of the extreme behavior of exchange rates and macroeconomic variables, and their associations. First the tails of each variable are examined. Left and right tail indices are estimated separately. Then the relationship of these variables is studied, based on the Feller Additivity Theorem.

Extreme Behaviors

The results for the estimates of the inverse tail index_, based on the modified Hill estimator, of the exchange rate return and the rate of change of economic fundamentals are provided in the appendix. A higher γ implies more probability mass (thicker) in the tail area and thus a higher chance of extreme events. Given the asymptotic normal distribution of the tail estimator, the significance of tail fatness of random variables is identified by testing H_0: γ = 0 against H_1: γ =0. The t-statistic computed for the tail of series x is

where the asymptotic standard error is estimated by bootstrapping. The number of bootstrap replications is set equal to 1,000. The p–value is presented beneath each estimated inverse tail index.

For each country, the row denoted _dep shows the Hill estimate for the right tail of the exchange rate returns (e) and the Hill estimates for the tails of the economic fundamentals that are associated with the domestic currency depreciation. The row contains the Hill estimates for the right tails of changes in the money supply (m), the price level (p), and domestic credit (DoC), and the estimates for the left tails of changes in real

income (y), short-term (r), and long-term (i) interest rates, international reserves (InR), the trade balance (TB), and the fiscal balance (FB). Analogously, the row γ_app represents a domestic currency appreciation.

For the advanced countries, except for the left tails of the Canadian dollar and the Japanese yen return distributions, all the tails significantly reject the null hypothesis of normality at the 1 percent level, with the maximum number of existing moments equal to 5. For all four countries, the probability of a domestic currency depreciation is higher than the probability of a currency appreciation. In the case of Australia and New Zealand, only the second moment, or unconditional variance, exists.

For the exchange rate fundamentals, at the 5 percent significance level the series that exhibit thin tails are the negative changes in the Australian and Canadian money supply, the Australian and Japanese domestic credit, and the Japanese international reserves, as well as the changes in the fiscal balance. The rest of the series have heavy tails. In every row, the market interest rates tend to have the heaviest tails, except for the Japanese γ_app row, in which the positive growth rate of international reserves has the fattest tail.

For all three Latin American countries, the probability of domestic currency depreciation is also higher than the probability of a currency appreciation. While the tail of the domestic currency appreciation cannot reject the null hypothesis of a thin tail at the 1 percent significance level, the domestic currency depreciation significantly rejects the null. The first moment or unconditional mean exists for the Chilean and Mexican peso, but not in the case of Peru.

The economic fundamentals of the Latin American countries have fatter tails when compared with the previous group. Two-fifths of the fundamental series have the tail index less than 2, which implies the unbounded variance. In Chile, the short-term interest rate exhibits a thin-tailed distribution, whereas the long-term interest rate, the price level, and international reserves are heavy-tailed, with the tail index less than 1.5. The Hill estimates also show asymmetric distributions for all the series, except for the price level, international reserves, and the trade balance, since estimates on the γ_dep row have significantly fatter tails than those of the γ_app row.

For Mexico, the negative growth rates of real income, international reserves, and domestic credit cannot reject the null hypothesis of normality. Unlike Chile, the distribution of the price index shows thin tails, whereas the rest of the series can significantly reject the null. In particular, long-term and short-term interest rates exhibit asymmetric thick-tailed distributions, with heavier tails on the decreasing interest rate. For Peru, while the exchange rate and the short-term interest rate are very heavy-tailed, the economic fundamentals on the γ_dep row cannot reject the null hypothesis of a thin tail at the 5 percent level. The asymmetric distributions appear with heavier tails on the γ_app row.

As far as the Asian countries are concerned, except for Hong Kong with its currency board regime, the probability of large depreciations is obviousely higher than the probability of large appreciations for every country. For China, Indonesia, the Philippines and Vietnam, the right tails of the exchange rate returns have the tail index less than 1.5, while the left tails cannot reject the null hypothesis of normality,

that α equals ∞, at the 10 percent significance level. For the rest of the currencies, the distributions are asymmetric and heavy-tailed, with the number of bounded moments less than 4.

For the Asian countries, although the Hill estimates are quite modest compared with the Latin American countries, more than 50 percent of the fundamental series have heavy tails, with unbounded fourth moments, while the variance does not exist for about 15 percent of the series. The few series that exhibit thin tails are the Malaysian trade surplus and the Philippine deflation rate at the 10 percent level, as well as the left tails of Malaysia's interest rate and the Philippine's international reserves, and the right tails of Hong Kong's trade balance and Singapore's interest rate at the 5 percent level.

Among the three groups, the developed countries, including Singapore, which experience flexible exchange rates tend to have the thinnest tails for the exchange rate return distributions. The exchange rates of the rest of the countries are extremely volatile, especially on the downside. This resembles the probability of boom and bust in asset prices, as the chance of huge asset price increases is lower than the chance of huge asset price decreases. Moreover, in most cases, the macroeconomic fundamentals appear to be asymmetric and heavy-tailed, with only a few existing moments.

The External Relationship

Next, an assessment is made on whether the extreme movements of the exchange rate are influenced by tail behaviors of the economic fundamentals. Based on the Feller Tail Additivity Theorem, the fundamentals that may influence extreme exchange rate movements must have the heaviest tail among other fundamentals in the set of exchange rate determinants, and the same tail fatness as that of the exchange rate.

The results of the distributional relationships presented in the appendix, in which represents the difference between the Hill estimates of the exchange rate return (e) and the changes in economic fundamentals (f). The abbreviations subscripted for the fundamental series are similar to those explained earlier.

Rows denoted Dep. and App. represent the depreciation side and the appreciation side of the domestic currency, respectively. For example, the Dep. row shows the Hill estimate differences between the right tail of the exchange rate returns and the tails of the economic fundamentals that are associated with domestic currency depreciation.

p–values are computed from testing H_0: γ_e≤ γ_f against H_1: γ_e≤ γ_f The t-statistic is calculated from

where the asymptotic standard error is estimated by bootstrapping. The number of bootstrap replications is set at 1,000. To reject the null hypothesis implies that the tail of the fundamentals is thinner than the tail of the exchange rate.

Then, to distinguish the variables that have fatter tails than the exchange rate return from the variables that have similar tail fatness, H_0: γ_e≥ γ_f is tested against H₁: γ_e> γ_f. The p–values for this t-test are one minus p–values, derived from the previous test. Variables that do not reject both null hypotheses have similar tail estimates as the exchange rate return. Thus, they are considered as potential explanations for the extreme movements of the exchange rate.

For the advanced countries, most fundamentals appear to have similar degrees of tail fatness as the tails of the exchange rate returns. The fundamental tails that are thinner than the tails of the exchange rate include both tails of the series for Australia's domestic credit and fiscal balance, the left tails of Australia's international reserves and trade balance, and right tails of New Zealand's money supply and domestic credit. These series thus fail to explain the extreme movements of the exchange rate.

The fundamentals with thicker tails than those of the exchange rate returns are often on the appreciation side. These include long-term interest rates, Canada's and Japan's short-term interest rate and international reserves, Canada's real income and domestic credit, and Japan's money supply and price level. On the depreciation side, the series also include Japan's short-term interest rate and price level. The rest of the fundamental series are potentially related to the extreme exchange rate.

In the Latin American case, the fundamentals that have thinner tails compared with the exchange rate are Peru's money supply and international reserves, and Mexico's real income and price level. Furthermore, there are the negative tails for Chile's real income and trade balance, Mexico's international reserves, domestic credit, and trade balance, and the positive tail for Peru's domestic credit. The fundamentals with thicker tails compared with the exchange rate include both tails of Chile's long-term interest rate and price level, and the right tails of Chile's international reserves and Peru's short-term interest rate.

In the Asian sample, the probability of a Chinese depreciation is related to its international reserves, where the Hong Kong dollar depends upon the trade balance. In the cases of Indonesia, South Korea, and Malaysia, all fundamentals exhibit thinner tails than those of the domestic currency depeciations, while some fundamentals have similar tail fatness as those of the domestic currency appeciation. For Thailand, apart from the long-term interest rate, economic fundamentals fail to explain the extreme movements of the Thai baht in both directions. Because of exchange rate targeting, none of Singapore's fundamentals have a thinner tail than the exchange rate returns.

CONCLUSION

It is well known that exchange rate returns exhibit heavy tails. However, it has been found that economic fundamentals are also heavy-tailed and that the tails of these variables can be larger than those of exchange rate returns. Thus, the assumption of normality is not appropriate when studying the relationship between heavy-tailed exchange rates and economic fundamentals.

From the tail comparison, it is also found that, because of the asymmetric distribution of the variables, the association between extreme exchange rate returns and extreme changes in economic fundamentals is asymmetric. When comparing the APEC countries, their economic fundamentals tend to do well in explaining the extreme depreciation of these developed countries' currencies, and the extreme appreciation in the case of less developed countries.

Additionally, the evidence supports the finding of Koedijk et al. (1992) that currencies under a floating exchange rate regime appear to have thinner tails than those under a fixed or managed-float regime. Moreover, as in Krugman (2001), when comparing the Latin American crisis with the Asian currency crisis, the latter was more extreme and less likely to be explained by economic fundamentals.

References

Backus, D. 1984. Empirical models of the exchange rate: Separating the wheat from the chaff. Canadian Journal of Economics 17: 824–46.

Bakshi, G., and Z. Chen. 1997. Equilibrium valuation of foreign exchange claims. Journal of Finance 52: 799–826.

Basak, S., and M. Gallmeyer. 1998. Currency prices, the nominal exchange rates and security prices in a two-country dynamic monetary economy. Mathematical Finance 9: 1–30.

Bekaert, G., C. Erb, C. Harvey, and T. Viskanta. 1998. Distributional characteristics of emerging market returns and asset allocation. Journal of Portfolio Management: 102–16.

Bilson, J. 1978. The monetary approach to the exchange rate: Some empirical evidence. Working paper, International Monetary Fund, Washington, DC.

Buiter, W., and M. Miller. 1982. Real exchange rate overshooting and the output cost of bringing down in ation. European Economic Review 18: 85–123.

Chinn, M., and R. Meese. 1995. Banking on currency forecasts: How predictable is change in money? Journal of International Economics 38: 161–78.

Cumperayot, P. 2002. International financial markets: Risk and extremes. PhD diss., Erasmus University, Rotterdam.

Cumperayot, P. 2005. Dusting off the perception of risk and returns in FOREX markets. In Exchange rate economics: Where do we stand?, ed. P. de Grauwe. Cambridge: MIT Press.

Cumperayot, P., and C. de Vries. 2006. Large swings in currencies driven by fundamentals. Unpublished working paper.

Danielsson, J., L. de Haan, L. Peng, and C. de Vries. 2001. Using a bootstrap method to choose the sample fraction in tail index estimation. Journal of Multivariate Analysis 26: 226–48.

Dekkers, A., J. Einmahl, and L. de Haan. 1989. A moment estimator for the index of an extreme-value distribution. Annals of Statistics 17: 1833–55.

Dooley, M., and J. Shafer. 1976. Analysis of short-run exchange rate behavior: March 1973 to September 1975. International Finance Discussion Paper 123, Board of Governors of the Federal Reserve.

Dornbusch, R. 1976. Expectations and exchange rate dynamics. Journal of Political Economy 84: 1161–76.

Embrechts, P., C. Kluppelberg, and T. Mikosch. 1999. Modelling extremal events for insurance and finance. Berlin: Springer-Verlag.

Fama, E. 1965. The behavior of stock market prices. Journal of Business 38: 34–65.

Feller, W. 1971. An introduction to probability theory and its applications. New York: Wiley.

Flood, R., and A. Rose. 1995. Fixing exchange rates: A virtual quest for fundamentals. Journal of Monetary Economics 36: 3–37.

Flood, R., and A. Rose. 1999. Understanding exchange rate volatility without the contrivance of macroeconomics. Economic Journal 109: 660–72.

Flood, R., A. Rose, and D. Mathieson. 1991. An empirical exploration of exchange-rate target zones. Carnegie-Rochester Series on Public Policy 35: 7–66.

Frankel, J. 1979. On the mark: A theory of floating exchange rates based on real interest differentials. American Economic Review 69: 601–22.

Frenkel, J. 1976. A monetary approach to the exchange rate: Doctrinal aspects and empirical evidence. Scandinavian Journal of Economics 78: 200–24.

Goldie, C., and R. Smith. 1987. Slow variation with remainder: Theory and applications. Quarterly Journal of Mathematics 38: 45–71.

Groen, J. 1999. Long horizon predictability of exchange rates: Is it for real? Empirical Economics 24: 451–69.

De Haan, L., D. Jansen, K. Koedijk, and C. de Vries. 1994. Safety first portfolio selection, extreme value theory and long run asset risks. In Proceedings from a conference on extreme value theory and applications, 471–87, ed. J. Galambos.London: Kluwer Press.

Hill, B. 1975. A simple general approach to inference about the tail of a distribution. Annals of Mathematical Statistics 3: 1163–74.

Hodrick, R. 1978. An empirical analysis of the monetary approach to the determination of the exchange rate. Reading: Addison-Wesley.

Hodrick, R. 1989. Risk, uncertainty, and exchange rates. Journal of Monetary Economics 23: 433–59.

Hols, M., and C. de Vries. 1991. The limiting distribution of extremal exchange rate returns. Journal of Applied Econometrics 6: 287–302.

Huisman, R., G. Koedijk, C. Kool, and F. Palm. 2001. Tail-index estimates in small samples. Journal of Business and Economic Statistics 19: 208–16.

Jansen, D., and C. de Vries. 1991. On the frequency of large stock returns: Putting boom and busts into perspective. Review of Economics and Statistics 73: 18–24.

Kaminsky, G., S. Lizondo, and C. Reinhart. 1998. Leading indicators of currency crises. Staff Paper 45, International Monetary Fund.

Kearns, P., and A. Pagan. 1997. Estimating the density tail index for financial time series. Review of Economics and Statistics 79: 171–75.

Koedijk, K., and C. Kool. 1994. Tail estimates and the EMS target zone. Review of International Economics 2: 153–65.

Koedijk, K., P. Stork, and C. de Vries. 1992. Differences between foreign exchange rate regimes: The view from the tails.Journal of International Money and Finance 11: 462–73.

Krugman, P. 2001. Crises: The next generation? Paper presented at the Razin Conference, Tel Aviv.

Kumar, M., U. Moorthy, and W. Perraudin. 2003. Predicting emerging market currency crashes. Journal of Empirical Finance 10: 427–54.

Login, F. 1996. The asymptotic distribution of extreme stock market returns. Journal of Business 69: 383–408.

Loretan, M., and P. Phillips. 1994. Testing the covariance structure of heavy-tailed time series. Journal of Empirical Finance 1: 211–48.

Lucas, R. 1982. Interest rates and currency prices in a two-country world. Journal of Monetary Economics 10: 335–60.

MacDonald, R., and M. Taylor. 1994. The monetary model of the exchange rate: Long-run relationship, short-run dynamics, and how to beat a random walk. Journal of International Money and Finance 13: 276–90.

Mark, N. 1995. Exchange rates and fundamentals: Evidence on long-horizon predictability. American Economic Review 85: 201–18.

Mason, D. 1982. Laws of large numbers for sums of extreme values. Annals of Probability 10: 754–64.

Meese, R. 1990. Currency fluctuations in the post-Bretton Woods era. Journal of Economic Perspectives 4: 117–31.

Meese, R., and K. Rogoff. 1983. Empirical exchange rate models of the seventies: Do they t out of sample? Journal of International Economics 14: 3–24.

Meese, R., and K. Rogoff. 1988. Was it real? The exchange rate–interest differential relationship over the modern floating rate period. Journal of Finance 43 (4): 933–48. Mussa, M. 1976. The exchange rate, the balance of payments, and monetary and scal policy under a regime of controlled floating. Scandinavian Journal of Economics 78: 229–48.

Mussa, M. 1979. Empirical regularities in the behavior of exchange rates and theories of the foreign exchange market, policies for employment, prices and exchange rates, 9–57. New York: North-Holland.

Parkinson, M. 1980. The extreme value method for estimating the variance of the rate of returns. Journal of Business 53: 61–65.

Pozo, S., and C. Amuedo-Dorantes. 2003. Statistical distributions and the identification of currency crises. Journal of International Money and Finance 22: 591–609.

Rothschild, M., and J. Stiglitz. 1970. Increasing risk: A definition. Journal of Economic Theory 2: 225–43.

Smith, R. 1987. Estimating tails of probability distributions. Annals of Statistics 15: 1174–1207.

Stockman, A. 1980. A theory of exchange rate determination. Journal of Political Economy 88: 673–98.

Svensson, L. 1985a. Currency prices, terms of trade, and interest rates: A general equilibrium asset pricing cash-in-advance approach. Journal of International Economics 18: 17–42.

Svensson, L. 1985b. Money and asset prices in a cash-in-advance economy. Journal of Political Economy 93: 919–44.

Westerfield, J. 1977. Empirical properties of foreign exchange rates under fixed and floating rate regimes. Journal of International Economics: 181–200.

Appendix: Empirical results