Comparing U.S. GNP volatility across exchange rate regimes: An application of spahe cracking

STEVEN R. CUNNINGHAM JON VILASUSO L’nicersity of Connecticut Storrs,

Connecticut

Comparing U.S. GNP Volatility across Exchange Rate Regimes: An Application of Saphe Cracking Providing empirical evidence regarding the increased or decreased volatility of domestic real gross national product (GNP) measures following the change from a hxed to a flexible exchange rate regime has proved difficult. Comparing the volatility of GNP measures across the two regimes in conventional ways begs the question. The question is not whether the volatility of the measure has changed, but rather whether the volatility is different than it otherwise would have been. This paper introduces the cosine-squared cepstrum to provide evidence that U.S. real GNP has been less volatile since 1973 than it would have been had the fixed exchange rate regime continued.

1.

Introduction The Bretton Woods Agreement began to come apart with Richard Nixon’s suspension of gold convertibility in August of 1971, and continued to weaken with the Smithsonian Agreement later that year. Following a sequence of abortive efforts, Deputies of the Committee of Twenty met in Washington, D.C., in late November 1972 to reform the international monetary system, laying the groundwork for what would be the closing of the Bretton Woods era in March of 1973. While the resulting “non-system” that emerged has been blamed for a variety of ills, proof that the abandonment of fixed exchange rates actually caused any changes in the behavior of real U.S. output has been elusive. Using a technique new to economics and econometricians, the cosine-squared cepstrum, we find empirical evidence that U.S. gross national product (GNP) has been less volatile since 1973 than it would have otherwise been. Economic theory suggests at least two specific consequences of a change to flexible exchange rates, a trade-jbw efict and an inter-county insulation effect. The flexibility of nominal exchange rates implies their increased volatility. If this results in increased volatility in real exchange rates or increased trading risks, then we would expect trade flows to vary.1 There‘Whether international leave trade flows unaffected,

]ol4?nd of iuacroec-,

firms can eliminate exchange rate risk through hedging, and thereby is an open question. Short term risk can be easily hedged, although

Summer 1994, Vol. 16, No. 3, pp. 445459 Copyright 0 1994 by Louisiana State University Press 0164-0794/94/$1.59

445

Steven R. Cunningham

and Jon Vilasuso

fore, if the trade-flow effect dominates, then a trading nation should experience an increased volatility in domestic product measures. Friedman (1953), among others, has focused on the efficiency gains derived from the increased ability of domestic economies to adapt to and absorb shocks under flexible exchange rates. That is, the flexibility of exchange rates insulates trading partners from the internal events of the other.2 Therefore, if the insulation effect dominates, then a trading nation should experience decreased volatility under flexible exchange rates. Determining which of these two effects has dominated in the US. is a vexing empirical problem. Comparing U.S. GNP volatility across the two regimes does not address the real question. The real question is not whether U.S. GNP volatility has increased or decreased since the inception of the flexible exchange rate regime, but rather is the U.S. GNP series more volatile than it would have been had the fixed exchange rate regime continued. If the abandonment of the fixed exchange rate regime affected output volatility, then observed GNP is represented as the sum of two series: (1) the series that would have eventuated had the fixed exchange rate continued, and (2) a second series associated with the direct impact of the change in regime that arrived in late I972 or early 1973. If the regime had any coherent effect, then the two series must be well correlated. If the second series is positively correlated with the series that would have eventuated, then it increases volatility, in which case the secondary series is said to be in-phase. Volatility declines if the second series is out of phase (negatively correlated), because in this case fluctuations in the series that would have eventuated are dampened. More generally, if a time-varying, stationary series is composed of two such series that are (linear) well correlated, then the autocovariance function contains a global maximum at the zero lag and a local extremum at a lag corresponding to the arrival date of the second series. The problem with the

Note cont. from page 445 not without cost, in forward exchange markets. However it is more difficult to hedge risks beyond a one-year horizon since the required forward markets are virtually nonexistent (Peree and Steinherr 1989). Willet (1986) notes that even in the short run, it is unlikely that firms can eliminate exchange rate risk entirely since most firms can not approximate a full set of contingent contracts. ‘It is now widely recognized that the ability to absorb shocks depends on the origin (foreign or domestic) and the nature (monetary or real) of the shock. See Marston (1965) for a thorough discussion. Dombusch (1963) argues against the insulation effect, claiming that policy measures “cannot fail to spill from one country into another, whatever the exchange rate regime.” Although Dombusch does argue in favor of a flexible rate system as a means of accommodating and permitting differences in trend inflation and productivity, he does not believe that flexible exchange rates are a means of insulating a country from external disturbances.

446

Comparing

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Volatility

autocovariance function is that a local extremum appears as a broad cycle, and it is difficult to detect a second series and determine to what degree the second series is phase-shifted. To overcome these difficulties, we introduce the cepstrum to the economics literature. Cepstra have been used with great success in detecting secondary influences in engineering, particularly communication theory. Intuitively, the cosine-squared cepstrum behaves like an autocovariance function with sharpened resolution to reveal the arrival and phase relationship of a second series with greater precision. Instead of a broad cycle, a local extremum appears as an impulse, the direction of which determines whether the second series supports or dampens fluctuations. Because of this, the cosine-squared cepstrum is well equipped to determine whether GNP volatility is greater than it would otherwise have been had the fixed exchange rate regime continued. The organization of the paper is as follows. In the next section, we discuss the economic problem of the relationship of exchange rate regime to real economic activity, and examine the related literature. In Section 3, we develop the theory and intuition of the cosine-squared cepstrum, and its relationship to conventional autocovariance and spectral methods. In Section 4, the cepstrum is applied to the annualized growth rates of U.S. real GNP. A cepstral peak, 180 degrees out of phase, at the lag corresponding to the first quarter of 1973, confirms the decreased volatility in the series that resulted from the adoption of the new exchange rate regime. Conclusions and extensions are discussed in the final section.

2.

Exchange Rates and Real Activity Theory provides a variety of arguments supporting an inverse relationship between nominal exchange rate volatility and the volume of international trade, and therefore domestic output measures. One possible linkage is through a rational response to risk. The increased variability of exchange rates increases the risk of exporters. If exporters are risk averse and hedging is costly or impossible, then the increase in profit risk should lead to a reduction in trade. In other words, exchange rate volatility increases the risk and uncertainty associated with international transactions, and in this way, reduces trade.3 Another possible linkage is through the real exchange rate. If fluctuations in nominal exchange rates result in fluctuations in real exchange rates, then the variations in the relative prices of goods across trading countries “Franke (1991) offers exchange rate volatility.

an opposing

view

where

exporting

firms

benefit

from

increases

in

447

Steven R. Cunningham and Jon Vila.suso might manifest as GNP volatility. Mussa (1986), Stockman (1983), and Baxter and Stockman (1989) have all found that the volatility of real exchange rates is not neutral with respect to the nominal exchange rate system, and since the collapse of the Bretton Woods system, the volatility of the real exchange rate is markedly higher under the system of flexible rates. Because the real exchange rate measures the relative price of national output, together with the move towards more integrated international capital markets and the growing importance of the foreign sector, the increase in real exchange rate volatility provides suggestive empirical support to the notion that the behavior of real aggregates is not invariant to the exchange rate regime. This trade-flows hypothesis has been subject to extensive empirical testing. While Hooper and Kohlhagen (1978) find no significant effects of nominal exchange rate volatility on trade, subsequent studies have documented numerous negative effects of real exchange rate volatility on bilateral trade flows.4 Cushman (1983, 1988), Kenen and Rodrik (1986), and De Grauwe (1988) find that increased volatility depresses the volume of trade. However, in a survey of this literature, Willet (1986) concludes that while there is strong evidence of an inverse relationship, the effects of exchange rate variability on trade “have not been enormous” (p. S106). Cross-country studies investigating the correlation between measures of foreign output with that of the U.S. suggest greater interdependence under flexible exchange rates, providing evidence inconsistent with the improved insulation properties of flexible exchange rates. Swoboda (1983), using correlation and principle component analysis, finds increased correlation of real output movements following the abandonment of fixed exchange rates. Gerlach (1988) applies cross-spectral methods to industrial production indices of the U.S. and many European countries and concludes that coherences are typically higher following the move to flexible exchange rates. As noted by both Swoboda and Gerlach, it is uncertain whether the increased interdependence observed under flexible exchange rates is attributable to the change in exchange rate regime or the changing nature of underlying disturbances. Hutchison and Walsh (1992) examine the insulation property of flexible exchange rates in a dynamic multivariate model of the U.S. and Japan, so as to distinguish between foreign and domestic shocks. Hutchison and Walsh conclude that in the case of Japan, a flexible exchange rate regime is more effective in insulating the country from foreign distur4Willet (1986) notes econometric specification. heroically and statistical

448

that much of the empirical work is sensitive to the sample period and Furthermore, equations predicting trade activity have not performed proxies for uncertainty are difficult to come by.

Comparing

U.S. GNP Volatility

bances; however, observed cross-regime differences in real output volatility is largely traced to the changing nature of disturbances. The oil shock experienced in the 1970s was certainly international in character. The effects of oil prices on output, however, appear to be concentrated among low frequencies. Perron (1989) pursues this line of reasoning to argue that the impact of the 1973 oil shock on U.S. GNP has resulted in a reduction in the linear trend (slope) following the oil shock, but has not resulted in cyclical or high-frequency changes.5 Similarly, Gerlach (1988) fi n d s sign i fi cant coherences in the low frequency bands in crosscounty industrial production indices. 6 Baron-Adesi and Yeung (1990) also note a decrease in output growth dispersion across industrialized countries following the move to a flexible rate system. While the negative effects of a flexible exchange rate system on trade are widely acknowledged, empirical investigations that examine cross-regime differences in the behavior of U.S. real output have not produced consistent results. That is, researchers have been unable to provide convincing evidence as to whether the trade-flow or inter-country insulation effect dominates. Baxter and Stockman (1989) find little evidence of a systematic difference in the behavior of the industrial production index over different exchange rate regimes.’ In a complementary study, Baxter (1991) notes that GNP volatility is unchanged after first differencing the data, but argues that volatility comparisons across samples are not robust to the detrending method employed.K In fact, when linear detrending was used, the results were consistent with decreased GNP volatility over the post-1973 sample period. Although the technique employed in this paper is not invariant to the detrending procedure, we use first-differences to ensure stationarity and to focus on high frequency components in determining whether GNP volatility is different than it would otherwise have been had the fixed exchange rate regime continued. “Perron’s (1989) paper is directed at the issue of whether GNP contains a unit root. Perron rejects the unit root hypothesis in favor of a trend stationary specification with a decline in slope following the 1973 oil shock. Banerjee, Lumsdaine, and Stock (1992) and Zivot and Andrews (1992), however, demonstrate that if the break point is treated as unknown a priori, then the unit root hypothesis is not rejected. ‘Gerlach (1988) also finds increased coherences at business cycle frequencies, thus supporting the notion of world business cycles. 70utput volatility is measured by the standard deviation of the first difference of the industrial production index for the sample periods 1969-1970 and 1973-1986. ‘Baxter (1991) considers three popular detrending procedures-the linear filter, first differencing, and the Hodrick-Prescott filter. Of these filters, forming first differences places the most emphasis on higher frequency components, while the linear filter the least. Put differently, the first difference procedure filters out the influence of low frequencies to a larger extent than the others, the linear filter the least, with the Hodrick-Prescott in between,

449

Steven R. Cunningham and Jon Vilasuso 3.

The Cepstrum The cepstrum is an integral transform with a long history, if not in economics. Poisson (1823) and Schwarz (1872) applied cepstra to problems involving potential functions with real parts fixed on the unit circle. Szego (1915) and Kolmogorov (1939) used cepstra in the extraction of stable causal systems by factoring power spectra of random processes. The engineering application by Bogert, Healy, and Tukey (1963) most coincides with our interest here. Bogert, Healy, and Tukey (1963) consider a time varying functionf(t), composed of another functionf,(t) and its additive “echo,” f,(t-T), lagged by z periods. That is, f(t) The power spectrum off(t) IF(

=fi(t)

+ afi(t-Tt)

.

(1)

is

= IFl(o)12(1 + a2 + 2ucos(uys)) ,

(2)

where F,(w) is the complex Fourier transform off,(t). The “echo” is manifested as a cosine function riding on the envelope of the power spectrum. The period of the cosine function is the reciprocal of the lag 2. In studies involving autocorrelation analysis in hydroacoustic problems, Griffin, Jones, and Cunfer (1980) have shown that iff(t) and its “echo” are not perfectly correlated (their spectra differ), then the parameter a in Equation (2) varies with frequency. More specifically,f(t) is better represented by fW=“m)

+.L& - 2) >

so that the power spectrum off(t) IF(

fib)

+$2(t)

(3)

is given by

= lF,(o)12 + IF2(o)12 + 21F,(o)lIF2(o)lcos{~,(w)

- Cp2(o) + 02) , (4)

where q,(o) and $2(o) are the phase spectra off’(t) andf,(t), respectively. The modulating cosine is phase-shifted by an amount equal to the differences in the phases of functions composingf(t). Thus, if the functionsf,(t) andf2(t) are not close copies of one another, the argument of the modulating cosine is not constant, with important consequences as we shall soon see. We proceed under the assumption thatf,(t) =f2(t) and return to the more general case later. The functionf(t) can be represented as the linear convolution offi with a train of impulses. Bogert, Healy, and Tukey argue that if the envelope of IF( could be made optimally white, this would be equivalent to making IF,(o)12 into a “boxcar” function, whose Fourier transform is a sine function at the origin .a In the limit, as the envelope of the power spectrum becomes everywhere uniform (at all frequencies), the sine function QRecall

450

that sine x = (sin x)/x.

Comparing

U.S. GNP Volatility

approaches a Dirac delta function. The transform of the modulating cosine would be an impulse whose delay is related to the frequency of the modulating cosine which is equal to the lag betweenfi(t) and its “echo.” Therefore, under ideal conditions and when properly scaled, this resulting series is a time domain function with a global maximum, or “peak” at the origin, and a local maximum or “peak” indicating the arrival time of the “echo.” In practice, the whitening of the power spectrum is performed by application of the natural logarithm, and the inverse Fourier transform is applied. Because it ignores the phase spectrum, and is calculated directly from the log power spectrum, the resultant function is referred to as the power

cepstrum.

Recall now the case presented in Equations (3) and (4). If the component functions are not close copies of one another, then the argument of the modulating cosine is not invariant, and cepstral peaks rapidly degenerate. Therefore, the impulse appears at the appropriate lag of the cepstrum only if the component

functions

are well correlated.

In the discrete case, cepstra are a class of integral transform whose kernel is a function of the z-transform of a real sequence. The discrete power cepstrum of a data sequence r(nT) with z-transform X(z) is given bylo X(nT) = --&

$ log IX(z)W’dz

‘,

where n= O,1,2, . . . , N, enumerates the samples, T is the sampling interval, and C is a closed contour inside the region of convergence of the power series and enclosing the origin. As discussed by Cunningham (1980), this can be extended to the cosine-squared cepstrum by addition of the Signum function according to 2(nT) = f x sgn

PC loglX(z)lz”-‘dz

.

The addition of the Signum function allows the cepstrum to determine not only the arrival time of the secondary series, but also its polarity relative to the original series, making its interpretation entirely analogous to the conventional autocovariance function. Because in applications the real sequences x(nT) are of finite length, the annulus of convergence of X(z) always includes the unit circle, the transforms may be computed by means of the fast Fourier transforms to the advantage of a great reduction in processing time. “See the appendix power cepstnnn.

for a formal

description

linking

the log power

spectrum

and the discrete

451

Steven R. Cunningham and Jon Vilasuso The close relationship between the cosine-squared cepstrum and the autocovariance function becomes obvious when we write the autocovariance function as a function of the z-transform of the same series: xmf (T) = -&- f$c IX(z) lzn-’ dz 2lci

(7)

Equivalently, the autocovariance function is the inverse Fourier transform of the power spectrum of a series. The process of taking logarithms prior to transforming seems to increase the resolution of the autocovariance function, Because the cepstrum is essentially the spectrum of a spectrum, the cepstral domain is a time domain. The terminology easily becomes a bit confused. Bogert, Healy, and Tukey (1963) suggest the following conventions: the term “cepstrum” is an anagram of the word “spectrum.” Likewise, periodicities in the cepstrum are discussed in terms of “quefrencies,” “gamnitudes,” and “repiods,” analogous to “frequencies,” “gain/amplitudes,” and “periods” in the usual time domain. “Filtering” in the cepstral domain is “liftering” and so on. Bogert, Healy, and Tukey like to refer to analysis of data in the cepstral domain as “alanysis,” although in practice the term never caught on. The term “saphe,” pronounced “safe” and related to “phase,” is used to refer to the displacement between the secondary, or lagged series, and the original. Thus, the detection and analysis of cepstral peaks and the “phase” shifts of the lagged series that they imply, is referred to “saphe cracking.”

4.

Application to the Volatility Issue If the collapse of the fixed exchange rate system affected the volatility of the U.S. real GNP series, then the series is representable as the sum of the pre-1973 series and a secondary series arriving in early 1973. If that secondary series is perfectly in phase with the pre-1973 series, then volatility increased relative to what it would have been under the fixed exchange rate regime; if it is perfectly out-of-phase, then volatility decreased. Determining whether or not a secondary series arrived, and whether or not it was in-phase, or out of phase to some degree, is a straightforward application of the cosine-squared cepstrum We begin by forming growth rates of the quarterly real U.S. GNP series, dating from the first quarter of 1970 to the fourth quarter of 1991.u Subtracting the mean avoids the dominance of the zero frequency component in the power spectrum, and applying five-point Hanmng-type cosine “The data for this paper are from All of the computations are performed

452

the Citibase by software

macroeconomic data base called Citibase. developed by one of the authors.

Comparing

0.0

0.1

The Log Power

Spectrum

U.S. GNP Volatility

0.4

0.2A

Figure of the Growth

1. Rates of Real U.S. GNP,

0.5

1970-1991

tapers suppresses possible sidelobes. Next we apply a Cooley-Tukey fast Fourier transform (FFT) and compute the natural logarithm of the sum of the squares of the real and imaginary parts to form the log power spectrum. Following the suggestion of Bogert, Healy, and Tukey, the spectrum is smoothed; we apply a five-point centered moving average for this purpose. The result is shown in Figure 1. The moving average suppresses high quefrency variations that would correspond to relatively long lags in secondaries. The underlying “boxcar” shape of the envelope of the log power spectrum is evident. A deep null in the spectrum at approximately 0.25 cycles per quarter, indicating little evidence of cycles of a year in length, reflects the absence of seasonality. The modulation pattern on the envelope of the log power spectrum anticipates successful results from the cepstrum computation The average “distance” between troughs is approximately 0.083 cycles per quarter, so that the lag associated with the secondary influence on this series is at approximately a 12 quarters lag (2), arriving in the first quarter of 1973. Even if one is patient enough to make and average these measurements, the phase information, that is, whether the secondary had an additive or a dampening effect, is nearly impossible to discern from here. Forming the inverse FFT, forming the sum of squares again, and establishing the sign according to the sign of the real part, the cosine-squared 453

Steven R. Cunningham

and ]on Vikz.suso

-0.16

The Cosine-Squared

Cepstrum

Figure 2. of the Growth Rates

of Real U.S. GNP,

197&1981.

cepstrum is formed. In Figure 2, the first 44 lags of the cosine-squared cepstrum are shown with the time scale replacing the lag numbers for convenience. The first nine points of the cepstrum are resealed so as to enhance the detail’s A delta function spikes downward prominently at the lag corresponding to the first quarter of 1973. This “arrival time” for the secondary is consistent with the final breakdown of the Bretton Woods agreement, and the establishment of flexible exchange rates. The unambiguous nature of the delta function and its negative sign provide evidence of a secondary influence on the real U.S. GNP series that matched it fluctuation by fluctuation, but exactly 180 degrees out of phase with the GNP series.13 That is, the secondary influence that arrived in early 1973 dampened the fluctuations of the GNP series. In this sense, the volatility of real U.S. GNP was reduced. Because the year 1973 was also marked by world-wide oil shocks, it is natural to wonder if it is the influence of the oil shocks that is being seen in this analysis. If the influence of the 1973 oil shock is concentrated among low ‘?he first nine cepstral estimates are set equal to zero to enhance the detail. This operation essentially removes the delta function at the origin, which corresponds to the global maximum. ‘3The relationship of the polarity of the delta functions to the phase shifts is discussed more fully in Cunningham (1980).

454

Comparing U.S. GNP Volatility

Squared

Coherences

of the Growth

Figure 3. Rates of Real U.S. GNP

and Oil Prices,

19741991

frequency components, then there is little possibility of confusion between the effects of the changing exchange rate regime and oil prices. Moreover, movements in GNP and oil prices must exhibit near perfect correlation across all frequencies, otherwise the influence of oil prices degenerates in the cepstrum. Figure 3 plots the squared coherences between the growth rates of real GNP and oil prices over 1974-1991. Squared coherences measure the variation in real GNP that can be explained by changes in oil prices at corresponding frequency components. l4 Clearly oil price movements and output fluctuations are not closely correlated across all frequencies, and it appears that the two series are most closely related over a low frequency range.15 This argument is made stronger by the recognition that while real U.S. GNP may be described as cyclical, it is clearly not perfectly periodic. That is, the cepstrum has provided evidence of an influence that matches the irregular fluctuations of the series. Thus, we have evidence of a systematic and continuing influence that has dampened GNP fluctuations in the U.S. since the first quarter of 1973. 14S 4 uared coherences are analogous to the square of the correlation coefficient. “Fuller (1976) suggests the statistic K = 4&/[2( 1 - c)] for testing whether the coherence, c, at a given frequency is zero. K is approximately distributed as pd where d is the width of the spectral window. The critical value of c is 0.248 for a 5% level, as indicated in Figure 3.

455

Steven R. Cunningham

and Jon Vilasuso

5.

Conclusions The negative delta function in the cosine-squared cepstrum has provided evidence that the flexible exchange rate regime began to impact real economic activity in the U.S. in the first quarter of 1973, making real U.S. GNP volatility less than it would have been had the fixed exchange rate regime survived. Apparently, adjustments in exchange rates made possible by the new regime insulated the real U.S. economy from shocks generated by other economies with the result that the inter-country insulation effect was dominant. The cosine-squared cepstrum introduced here shows promise in application wherein the effects of economic events and policy changes on time series can be modeled as additive time series well correlated with the series under study. There are important limitations to this approach. For example, the dampening effect of the new exchange rate regime may not be permanent. It may be that the (dampening) response in the GNP series may be pulse-like, with adjustments to capital mobility causing increases in volatility that magnify the trade-flows effect, eventually overtaking the inter-country insulation effect. Given that these adjustments likely take time, it is probably reasonable to interpret our result as supporting reduced volatility for at least most of the 1970s. Without recovering the secondary series, the method requires outside information for the identification of the cepstral peaks. While a prominent cesptral peak, like the one in this study, is relatively unambiguous in terms of its timing and phase characteristics, it takes additional economic insight to isolate the possible economic causes of the event. Therefore, without the recovery of the secondary, the cepstrum is limited to studies in which the response to a possible event can be isolated in time. Receiwd: Febma y 1993 Final version: November 1993

References Banerjee, Anindya, Robin L. Lumsdaine, and James H. Stock. “Recursive and Sequential Tests of the Unit-Root and Trend-Break Hypotheses: Theory and International Evidence.” Journal of Business G Economic Statistics 10 (July 1992): 271-87. Barone-Adesi, Giovanni, and Bernard Yeung. “Price Flexibility and Output Volatility: The Case for Flexible Exchange Rates.“]oumal of Znternational Money and Finance 9 (September 1990): 276-98. 456

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Baxter, Marianne. “Business Cycles, Stylized Facts, and the Exchange Rate Regime: Evidence from the United States.” JoumaE of International Money and Finance 10 (March 1991): 71-88. Baxter, Marianne, and Alan C. Stockman. “Business Cycles and the Exchange-Rate Regime: Some International Evidence.“JournaZ ofMonetary Economics 23 (May 1989): 377400. Bogert, Bruce P., M. J. R. Healy, and John W. Tukey. “The Quefrency Alanysis of Time Series for Echoes: Cepstrum, Pseudo-Autocovariance, Cross-Cepstrum, and Saphe Cracking.” In Proceedings of a Symposium on Time Series Analysis, edited by Murray Rosenblatt. New York: John Wiley & Sons, Inc., 1963. Cunningham, Steven R. “Application of Cepstral Techniques to Rayleigh Wave Multipathing.” U.S. National Technical Znformution Service (NTZS), ENSCO DCS-STR-80-45, 1980. Cushman, David 0. “The Effects of Real Exchange Rate Risk on Intemational Trade.” Journal of International Economics 15 (August 1983): 4563. ~ “U.S. Bilateral Trade Flows and Exchange Rate Risk During the Floating Period.” Journal of International Economics 24 (May 1988): 31730. De Grauwe, Paul. “Exchange Rate Variability and the Slowdown in Growth of International Trade.” ZMF StaJf Papers 35 (March 1988): 63-84. Dombusch, Rudiger. “Flexible Exchange Rates and Interdependence.” ZMF Staff Papers 30 (March 1983): 330. Franke, Giinter. “Exchange Rate Volatility and International Trading Strategy.” Journal of International Money and Finance 10 (June 1991): 29% 307. Friedman, Milton. “The Case for Flexible Exchange Rates.” In Essays in Positive Economics, edited by Milton Friedman. Chicago: University of Chicago Press, 1953. Fuller, Wayne A. Introduction to Statistical Time Series. New York: John Wiley & Sons, Inc., 1976. Gerlach, H. M. Stefan. ‘World Business Cycles Under Fixed and Flexible Exchange Rates.“Joumal of Money, Credit, and Banking 20 (November 1988): 62132. Griffin, James N., Robert L. Jones, and Ronald S. Cunfer. “Experimental Procedures for Estimating Depths and Yields of Underwater Explosions Using Recordings of Limited Dynamic Range.” Final Report, Project T/8152/, ENSCO, 1980. Hooper, Peter, and Steven W. Kohlhagen. “The Effect of Exchange Rate Uncertainty on the Prices and Volume of International Trade.“JournaZ of International Economics 8 (November 1978): 483511. 457

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and ]on Vilasuso

Hut&son, Michael, and Carl E. Walsh. “Empirical Evidence on the Insulation Properties of Fixed and Flexible Exchange Rates: The Japanese Experience.“]ournal of International Economics 32 (May 1992): 241-63. Kenen, Peter B., and Dani Rodrik. “Measuring and Analyzing the Effects of Short-Term Volatility in Real Exchange Rates.” Review of Economics and Statistics 68 (May 1986): 311-15. Kolmogorov, A. N. “Sur l’interpolation et extrapolation des suites stationnaires.” Comptes Rendus Hebdomadaires des Siances de I’Acadimie o!es Sciences (1939): 204345. Marston, Richard C. “Stabilization Policies in Open Economies.” In Handbook of Znternational Economics, Vol. IZ, edited by Ronald W. Jones and Peter B. Kenen. New York: North-Holland, 1985. Mussa, Michael. “Nominal Exchange Rate Regimes and the Behavior of Real Exchange Rates: Evidence and Implications.” In Real Business Cycles, Real Exchange Rates and Actual Policies, edited by Karl Brunner and Allan H. Meltzer. Carnegie-Rochester Conference Series 25. Amsterdam: North-Holland, 1986 Peree, Eric, and Alfred Steinherr. “Exchange Rate Uncertainty and Foreign Trade.” European Economic Review 33 (July 1989): 1241-64. Perron, Pierre. “The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis.” Econometrica 57 (November 1989): 1361401. Poisson, S. D. “Sur la distribution de la chaleur dansles corps solides.“JournaZ de I’Ecole R. Polytechnique 19 (1823): 1-162. Schwarz, H. A. “Zur Integration de partiellen DifferentiaIgleichung.“JoumuZ fur die Reine und Angewandte Mathematik (1872): 21854. Stockman, Alan C. “Real Exchange Rates Under Alternative Nominal Exchange-Rate Systems.” Journal of International Money and Finance 2 (August 1983): 147-66. Swoboda, Alexander K. “Exchange Rate Regimes and European-U.S. Policy Interdependence.” ZMF Staff Papers 30 (March 1983): 75-102. Szego, G. “Ein Grenzwertsatz uber die Toeplitzschen Determinanten einer reellen positiven Fur&ion.” Mathematische Annalen 76 (1915): 490503. Willet, Thomas D. “Exchange-Rate Volatility, International Trade, and Resource Allocation: A Perspective on Recent Research.“]ournal of Znternational Money and Finance 5 (March 1986): SlOl-S112. Zivot, Eric, and Donald W. K. Andrews. “Further Evidence on the Great Crash, the Oil-Price Shock, and the Unit-Root Hypothesis.” ]ountaZ of Business G Economic Statistics 10 (July 1992): 251-70. Appendix The log power spectrum is computed by taking the natural logarithm of Equation (2); that is, 458

Comparing U.S. GNP Volatility loglF(o)12 = loglF,(o)12 + log(l+ a2 + 2ucos(6x)) ,

(Al)

where log{l+ a2+ ~UCOS(CIX)} is approximated by 2ucos(m) - a2cos(2ww) ,

(A.2)

to quadratic accuracy. The term a%os( 2xa) represents the second harmonic and contributes a4/2 to the variance of the “echo” (Bogert, Healy and Tukey, 218). The fundamental, 2ucos(e.x), contributes 2u2 to the variance. Although harmonics are to be expected, the contributions of the harmonic terms depends on the magnitude of a. In practice, it is usually difficult to determine the importance of harmonics in the log spectrum. With the cepstrum, the fundamental manifests as an impulse at lag 2, while the second harmonic manifests as a local maximum at lag 22. The second harmonic is discernible in Figure 2, but is clearly dominated by the fundamental. Given these results, there is not likely a problem here. This suggests that the log power spectrum can be written as loglF(o)12 = loglF,(0)12 + Sacos(cx)

(A3)

In the discrete case, the power spectrum is then given by Equation (5).

459