tary policy cause movements in the exchange rate in the direction predicted by the sticky-price monetary model. (Dornbuseh overshooting theory).^ In general, it ...
Applied Financial Economics, 2000, 10, 401-412
Asymmetries in the conditional mean and conditional variance in the exchange rate: evidence from within and across economic blocks MARIA SOPHIA AGUIRRE and REZA SAIDI Department of Bminess and Economics, The Catholic University of America, Washington. DC 20064. USA E-mail: aguirreio^cua.edu and saidi(aj,cua.edu
The paper tests the hypothesis that both the conditional mean and the conditional variance of exchange rates are asymmetric functions of past information. This hypothesis is tested by estimating an Asymmetric Threshold GARCH model for fifteen currencies. The empirical evidence suggests that both the conditional mean and the conditional variance respond asymmetrically to past information, with an AR(1) structure within blocks and an ARMA(1,1) structure for the EU currencies against the dollar. It is found that the conditional mean is an asymmetric function of past innovations, rising proportionately more during appreciation periods within and across blocks. This implies that, on average, the market incorporates positive news (depreciations) more quickly than negative news (appreciations). The conditional variance is an asymmetric function of past innovations as well, rising proportionately more during depreciations within blocks and appreciation periods across blocks. Furthermore, asymmetries in the conditional mean are linked to asymmetries in the conditional variance because the more rapid adjustment of the market to depreciations causes greater volatility during these periods. This, in turn, causes within blocks a slower speed of adjustment in the variance to devaluations than to appreciations. Finally, greater efficiency in currency markets is found within blocks than across blocks.
I. INTRODUCTION Recently, a great deal of research has focused on the large increases in the volume of trading in foreign exchange markets. One issue addressed by this research is the behaviour of exchange rate volatility. This behaviour is important to economists and investors as well as to policy makers because of the effects that volatility can have on the balance of trade, currency substitutabiUty, portfolio risk, and foreign exchange policy. The economic relevance of this issue becomes more important when it is analysed within a framework of economic integration. One expects central banks and traders in countries within integrated markets to be more sensitive to changes in the exchange
rate trend or mean and changes in the volatility of member countries' currencies than to other currencies' volatility. Exchange rate misalignments can seriously affect the trade balance, especially within free trade areas, while a risk difference will not only affect allocation of investment and capital movement but also the current exchange rate policies. Furthermore, while the sign of the asymmetric coefficients in the mean during appreciations and depreciations provides evidence of central banks" leaning-against(into)the-wind-behaviour, the speed of adjustment of past values indicates how soon it takes for the central banks' interventions to be absorbed by the market or overshooting and undershooting to disappear. At the same time sensitivity of
Applied Fimmcial Economics ISSN 0960-3107 print/ISSN 1466-4305 online © 2000 Taylor & Francis Lid http://www.tandr.co.uk/journals
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402 market operators to positive and negative innovations (expectational errors that cause overshooting and undershooting) indicate if credibility affects the variance and, thus, the exchange rate risk exposure^ over time. This in turn, can be an indicator of how they view the sustainability of the foreign currency in a given country. Eurthermore, asymmetries in the conditional mean are linked to asymmetries in the conditional variance because the more rapid adjustment of the market to past depreciations (appreciations) could cause greater (smaller) volatility during these periods. Finally, the persistence of this "news" or the speed of adjustment at which this news is absorbed is an indicator of the efficiency in a given market. The purpose oi' this paper is to shed light on these issues by investigating asymmetries in the mean and variance of fifteen foreign exchange currencies. Studies on the dynamics of exchange rate behaviour and its volatility have found four characteristics: (1) in the short run, the exchange rate behaves randomly," However, recent studies such as those of Huizinga (1987) and Mark (1995) have challenged the hypothesis of exchange rate predictability. They find that the exchange rate predictability improves with time. (2) The exchange rate, on average, moves in the opposite direction and fails to move in line with predictions of the forward discount or interest differential." (3) Research has supported the concept that some fundamentals are able to affect the direction of the spot exchange rate, especially for the long horizon,'' They find that a nonlinear combination of the monetary model and the lagged spot rate performs better than the simple lagged spot rate model. (4) Unexpected changes in monetary policy cause movements in the exchange rate in the direction predicted by the sticky-price monetary model (Dornbuseh overshooting theory).^ In general, it is found that the short-term reaction is less than the medium-term reaction. In this way, Eichenbaum and Evans (1993) and Grilli and Roubini (1993) find that it takes two years for the exchange rate to adjust to an unexpected change in monetary policy. Clarida and Galli (1994) also find a lag before the "peak effect". These findings suggest that currency appreciates gradually in the aftermath of an increase in the interest rate differentials, rather than contemporaneously as the rational-expectations form of the over-
M. S. Aguirre and R. Saidi shooting model predicts. During this time, the currency is most likely to appreciate, thus explaining why interest rate differentials or the forward discount., on average, point in the opposite direction. Even if this is the case, two new questions arise concerning why a currency appreciates gradually rather than suddenly, and whether the market reacts with the same speed to positive and negative news within and across blocks. Research on the microstructure of the foreign exchange markets has shed some light on the first question. One of the characteristics found is that exchange rate volatility changes over time.'' Another characteristic observed is a systematic pattern in the intra-day volatility. This is lower when the volume traded is low, such as over the weekend and during the lunch hour, and it is very high during the first hour of Monday trading for each currency in its own market. Goodhart and Giugale (1993) explain this finding by suggesting that residents have a comparative advantage at processing news regarding their own currencies, or that trading is largely unrelated to news, or that, perhaps, trading activity per se generates volatility. Engle et al. (1990, 1992) examine intra-day volatility in four markets and find that upswings in volatility in one market are passed on as higher volatility in the next market. Finally, Ito et al. (1997) present evidence on the microstructure and volatility patterns of the Japanese market. They conclude that traders trade based on additional infonnation they possess and they resolve differences in the perception of information content in times of higher volatility. ARCH models, which allow the conditional variance to be a function of past square innovations, have been used to address time-varying variances. Bollerslev et al. (1992) offer a general survey. Yet, in the presence of positive and negative innovations, both in the mean and in the variance, negative coefficients, or innovation persistence, these models fall short. In this paper, we intend to clarify and unite some of the macroeconomic and microeconomic findings in the literature on the short-term behaviour of the exchange rate. With this aim, we estimate a partial adjustment model for spot exchange rates. The method used is an Asymmetric Threshold GARCH model (T-GARCH) which is applied to daily foreign exchange rates of fifteen
' Here foreign currency risk exposure is understood as the gain or loss to which foreign currency holders are exposed because currency values fluctuate over time. •'Meese and Rogoif (1983), Camphell and Clarida (1987), and Flood and Rose (1993) support these findings. It is interesting to note, however, that the fact that the expectational error component is significantly ditTereni from zero could also entail that, if" the exchange rate follows approximately a random walk, the market participants do not expect future exchange rates to follow the same process (Takagi, 1988 and 1991). ' Froot and Frankel (1989) and Hodrick (1988) survey the empirical work on risk premium, while Takagi (1991) and Lewis (1989) survey the departure from rational expectations. ""These studies include Woo (1987). Somanath (1986), Mark (1990), Dominguez and Franke! (1993), and Aguirre and Saidi (1998). "^Dornbuseh (1976) first presents this model. Some of the studies that support this mode! are Woo (1985), Somanath (1986), and Mark (1995). '^See Takagi (1991), Lewis (1989), and Kahya et al. (1994).
Asymmetries in exchange rates coutitries.^ This allows us to study the effect that positive and negative news as well as past errors have on the mean value and on the volatility of the daily exchange rate over time, both within and across blocks. The period covered is January 1987 to April 1997. The findings show that an Asymmetric T-GARCH model successfully captures the behaviour of exchange rates over time within and across blocks, thus providing information about the exchange rate trend and its relation to central bank interventions, as well as about the behaviour of the market towards exchange rate risk.*^ When considering the exchange rates' asymmetries, it is important to keep in mind that we are dealing with a bilateral concept. Thus, an appreciation of a country's currency corresponds to a depreciation of another country's currency and vice versa. The determinants of the exchange rate vary between the ASEAN block and NAFTA, and the European Union (EU). While the last one, with the exception of France, shows an ARMA(1,1) structure, the first two blocks show only an AR(I) structure. This indicates the influence of past exchange rate changes on the present spot exchange rate for ASEAN and NAFTA countries but it does not indicate the influence of past deviations. We also find that, while the T-GARCH model adequately captures the exchange rate dynamics of currencies of the same block in terms of the leader country within that block, i.e. United States for NAFTA., Germany for the EU. and Japan for ASEAN, the same does not hold for currencies across blocks. The model was not found significant for ASEAN countries' currencies against the dollar or against the deutschmark (DM). As expected, Japan is an exception, and the yen is significant against the dollar. Neither Canadian nor Mexican currencies show significance against the DM or the yen. Finally, the model is significant in explaining EU member currencies vis-d-vis the dollar. Italy is an exception on all accounts. Overall, past appreciations are more significant than past depreciations in the determination of the mean and the sign for both coefficients is negative. Past positive expectational errors or positive innovations (which capture currency overshooting), on the other hand, are more significant in the determination of the volatihty of the exchange rate. The presence, for currencies within blocks, of this type of asymmetry in the mean suggests a leaningagainst-the-wind policy on the part of central banks, while the type of asymmetry in the variance suggests a loss of credibility and, therefore, a perception of an increase of risk in the face of currency overshooting. The opposite to the pattern described for currencies within blocks tends to be the behaviour of currencies across blocks. Finally, in all
403 cases, the speed of adjustment is faster for the variance than for the mean. Such behaviour is compatible with overshooting models, but the underlying consequences are different. We flnd differences in the persistence of European currencies' overshooting and undershooting when comparing them across blocks rather than within blocks. The speed of adjustment is significantly smaller against the deutschmark than against the dollar. Japan's persistence, on the other hand, is significantly smaller than the European countries' vis-d'vis the dollar. The adjustment takes place in a matter of hours within blocks, while it can take a day or more across blocks. This indicates lower possibilities of arbitrage within blocks than across blocks. We find that the probability distribution of the error term in the foreign exchage model to be non-normal for all countries except Canada. We also find a shift in the volatility after 30 August 1993 for some EU countries. This is not the case for ASEAN. Finally, for Mexico and Belgium against the dollar, as well as for all EU countries against the DM, the model fails to capture the dynamics of the exchange rate after August 1993. At this point, the eflect of the speculative attack clearly eliminates any tractability. The rest of the paper is organized as follows. In the second part of the paper we present the review of the method and the data employed. The empirical results are presented in the third part. For the sake of brevity, we concentrate on differences across blocks. We conclude with a discussion of the results and policy implications.
11. M E T H O D A N D D A T A Nelson (1991) developed exponential GARCH (EGARCH) models where the conditional variance is allowed to depend on the magnitude as well as the sign of the innovations (i.e. error term). This means that the variance of the exchange rate changes over time and is an asymmetric function of past error terms. That is, negative and positive innovations can have diflerent impacts on the exchange rate behaviour. This specification, however, does not require the imposition of non-negativity constraints on the coefficients in order to ensure positive variance. The Threshold GARCH model (T-GARCH) introduced by Zakoian (1994) allows the conditional variance to respond asymmetrically to positive and negative innovations as well, but it has some important differences. In the former, the conditional variance is a function of standardized innovations. In the T-GARCH, on the other hand, a linear
The approach employed in this paper follows the work of Koutmos (1998) who applied it to the stock market indices. "^ For policy purposes and for the determination of credibility, it is interesting to look at the behaviour of real exchange rates as well. Lack of available data for daily price indexes, however, prevented us from using them.
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representation of some function of e, is allowed. Thus, a vector (e^,£,"") is introduced and can be shown to follow an ARMA or MA process. As in the EGARCH model, the {£•?} sequence follows an ARMA or MA process.'* The T-GARCH can be written in the presence of an ARMA form as follows:
market efficiency. This measure is a good indicator of the efficiency in a given currency market because the more efficient the market is, the faster it should either return to its current equilibrium level or reach a new equilibrium level. A more intuitive measure of this speed of adjustment is the half-life of an expectational error (an undershooting/ overshooting), or an innovation which can be measured as A,= (1) [ln(0.5)/ln(coefficient)] In order to assure that the T-GARCH model's conwhere A/ is the first difference of the logarithm of the ditional variance is positive, it is necessary to impose exchange rate or approximate rate of return, E,_I is the non-negativity constraints on the variance (Equation 3), past error or deviation which captures an MA(1) process, i.e. QQ > 0, Oj^ > 0, ai > 0, and 0 > 0. The degree of A,^_, = Max(O, A,_|) and A ; 1 , = Min (A,_|,0) are two volatility persistence depends on the size of the coefficient filters, one for the positive and one for the negative AR process. When O"" — 0~ — 6, the process is symmetric, and 4>. For 0 = 1 , the unconditional variance does not exist. Therefore, it is a random walk, implying that the conthe {A;} sequence will follow the ARMA( 1,1) process. ditional variance is an integrated process of the first degree. Thus, the optimal one-step ahead forecast can be obtained If 0 < 1, the unconditional variance exists and is a meanby E(A;|I,_,) -/^o + /:^i£,_i +^A,_|. And when 6*+ ^ ^ , reverting process. the {Af} sequence will be more complex. In this case, the optima] one-step ahead forecast will depend not only on the Equations 1 and 3 are well suited for the test of a MA process but on the sign of A,, i.e. presence of asymmetries in the conditional mean and variance of the exchange rate distribution. The vector ,_, >0 (2a) 0 = {l3.6'^.0~,aQ,a'^,a~,(p,i/) can be estimated by maximizing the log-likelihood over the sample period given {2b)
Asymmetry is significant in the variance of all countries and the speed of adjustment is similar for i^ and e~. As for the case of Europe and for currencies across blocks, overshooting (positive expectational errors) increases the present volatility of the currency while undershooting (negative expectational errors) decrease present volatility. As previously mentioned and replicating the leading ro!e of Germany within Europe, we also find that Japan's leading role in the region is reflected in the increase of volatility in the yen during appreciations. (j> is positive and less than one and its corresponding half-life values reported in Table 2B vary between 9 days and 18 days, revealing a market that is much less efficient than that of the EU. The results are not surprising since many of these markets are developing economies with emerging foreign currency markets.
NA FTA currencies against the dollar Table 3 presents the results of the T-GARCH model for currencies of NAFTA, EU atid Japan against the dollar. We find that while an MA(1) process (/5|) is not present in the mean for NAFTA member countries and Japan, in the EU it exists and it is significant in all countries except for France. The asymmetric AR(1) process {B) is present and significant for most of the countries. Finally, we find a significant downward shift (negative 72) in volatility for Germany and the UK and a significant upward shift (positive 7]) in the mean for Canada. As was the case in ASEAN, we find no MA(I) process for NAFTA members. We find, however, that the AR(1) process is asymmetric and significant in both Canada and Mexico. Another characteristic that is similar to the two other blocks is that we find negative signs for both positive (0+) and negative [9') autoregressive processes. Once again, distributed-lag expectations are present and they suggest central bank interventions in the market. This is consistent with the central bank's support of the peso after Mexico pegged its peso against the dollar in December 1987.''' In addition, the significant difference in the speed of adjustment for past depreciations, supports the clear signs of weakening credibility in the markets before the summer of 1993 when the differential of the interest rates between the tesohonos and the US Treasury bill of similar maturity increased significantly. At this time, pressure started to develop which culminated in the currency crisis of December 1994.'^ This is also captured by the higher values in the variance's speed of adjustment reported in Table 3B for Mexico. Our model is unable to capture the volatility and the mean of the exchange rate after August,
In addition, in January 1989. the government fixed the peso against the dollar, and this became a crawling peg that allowed for some depreciation each year, but at a quite modest rate, given Mexico's past history of inflation. At the end of 1991, a band was introduced that allowed the currency to depreciate potentially while the floor of the band remained constant. '^ For a more detailed study of Mexico's financial collapse of 1994, see Obstfeld and Rogoff (1995).
410 1993. This is not the case for Canada, where results indicate credibility in the currency. It should particularly be noted that the Canadian dollar has been closely aligned to the US dollar over the period under study, a period which coincided first with the signing of the CanadianUS Free Trade Agreement (CUSFTA) in 1987. followed by its expansion through NAFTA in December 1992. Asymmetry in the variance (a) is present and significant in both countries. As was the case for the other two blocks, the impact of past expectational errors on volatility is higher for depreciations than for appreciations (a"*" > a~). Once again, the roie of the dollar as an anchor, a position that results from the US"s leadership role in the region, is also supported in NAFTA. We also find that NAFTA, like the other regions, has a mean reverting process (# < 1) and. its persistence reported in Table 3B indicates, at least, a weak-form of market efficiency where arbitrage can take place. Across blocks comparisons The results reported in Table 3 show that an autoregressive process {0) is found across blocks, but with the exception of France, only European countries show an MA( 1) process {0i) vis-d-vis the dollar as well. It is significant and negative, indicating that market participants expect corrections of deviations in these currencies across blocks when they are caused by expectational errors. The signs and significance of the asymmetry In the mean {9^,6~), however, change significantly when currencies are considered across blocks. In the case of Japan, past appreciations {9^) are not significant against the dollar, while past depreciations (9~) cause an appreciation of the yen in the following period. This is not surprising, given the currency policies followed by both the US and Japan since the Louvre Accord. The European countries, on the other hand, show the opposite behaviour. In the UK and Belgium, past depreciations and appreciations followed distributed-lag expectations' behaviour vi.s-a-vis the dollar. France is an exception, showing bandwagon expectations for depreciations but distributed-lag expectations for appreciations. The half-life of overshooting and undershooting for Japan is significantly smaller than that for the EU countries. Asymmetry is also present in the variance {e'^,e~) for Japan and EU currencies vis-a-vis the dollar and they are significantly different. In both cases, negative innovations dominate, and the speed of adjustment in the variance is higher for depreciations than for appreciations. The rest of the European countries follow Germany with the exception of the UK. whose asymmetry is positively dominated. Germany's pattern of behaviour against the dollar, clearly spills over into the other EU countries under study. Japan's persistence is smaller than the European, indicating higher efficiency in the former market than in the latter market.
M. S. Aguirre and R. Saidi Overall, however, the differences in the persistence of overshooting and undershooting when comparing them across blocks versus within blocks indicate a lower possibility of arbitrage within blocks than across blocks.
Summary After examining Tables 1 to 3, several patterns emerge from the exchange rate behaviour over time: (1) within blocks the asymmetry in the mean tends to appreciate the currency and rises proportionally more during appreciation periods. In addition, a distributed-lag form of expectations is found. (2) Across blocks, currencies' asymmetry in the mean does not necessarily follow distributed-lags nor is it present across blocks except for some countries vis-d-vis the dollar. (3) Asymmetry is found in the variance within blocks, but in this case, positive expectational errors increase volatility more than negative expectational errors decrease it. The opposite holds for the leading country in each region. (4) The speed of adjustment of these coefficients (a and 0) suggests that markets adjust more quickly to depreciations than to appreciations. Observations (I) to (4) indicate that, within blocks, on average, economic agents expect their currencies to be aligned vis-d-vis the leader country and are more sensitive to depreciations. This expectation is independent of the degree of integration within the block. An explanation for such behaviour can be found in the high level of intra-trade and capital mobility that exists in the three blocks considered in this study. On the other hand, this is not necessarily the case, for $. across blocks vi.s-d-vi.s the dollar. While some currency integration (captured in the asymmetry) is present for the yen and some EU currencies i75-c/-v/,v the dollar, no evidence was found for the EU and NAFTA members against the yen or the mark. The prevalence of the dollar in foreign currency markets as well as in trade could explain this finding. (5) The half-fife of an innovation varies across blocks. We find indications of greater efficiency in the EU than in the other two blocks. Once again, this is to be expected since these countries have more developed financial markets. (6) Asymmetries in the conditional mean are linked to asymmetries in the conditional variance because the faster adjustment of the market to depreciations causes higher volatility during these periods. This, in turn, causes a faster speed of adjustment in the variance to depreciations than to appreciations. (7) Different behaviour is found in countries across blocks than with their partners. With few exceptions across blocks, past depreciations follow bandwagon expectations while distributive-lag expectations accompany past appreciations. That is. across blocks, agents expect short-term deviations to take place when past depreciations are involved but not appreciations. In this last case, they will follow the currency's long-term path. However, contrary to what was found to be the case within
Asymmetries in exchange rates blocks, the variance's asymmetry tends to be dorninated by appreciations rather than depreciations. These patterns shed some light upon previous macro and micro foreign currency market studies. When the blocks are compared against the dollar, we find that the EU currency markets are less efficient against the dollar than against the DM. This supports previous findings such as those of Goodhart and Giugale (1993) who suggest a higher efficiency in the domestic markets due to comparative advantage in processing information. Our study suggests that this cornparative advantage extends to blocks as a whole. A significant shift was found in the mean for Canada and in the variance for Gennany and the UK after August 1993 (the speculative attack in Europe.) This reflects that the countries with more developed capital markets were affected independently of their block. This, in turn, suggests spillover effects across blocks, especially across developed financial markets such as those found by Engle et al. (1990. 1992). The fact that past depreciations increase the present volatility of the currency and the opposite holds for past appreciations, is consistent with the findings of Ito ct al. (1997). This suggests that traders perceive depreciations as destabilizing and therefore they seek to resolve differences in the perception of information content. Such activity increases the exchange rate volatility even further. In addition, findings in micro- and macro-economic literature regarding a mean-reverting process are also supported in our study within and across blocks. The link found between the mean and the variance process, point (6), provides new insights to the findings of Eichenbaum and Evans (1993), Grilli and Roubini (1993), and Clarida and Galli (1994). The slower adjustment of the market to past appreciations causes lower volatility during these periods, which, in turn, causes a slower speed of adjusttnent in the variance to appreciations than to depreciations. This set of information seems to suggest that, in the very short run, the Dornbusch overshooting theory is affected by the degree of credibility and efficiency that exists in a given country.
IV. C O N C L U S I O N In this paper, we have tested the hypothesis that the conditional mean and the variance of foreign exchange rates are asymmetric functions of past information. We tested this hypothesis by using a T-GARCH model for fifteen countries by blocks and across blocks. Such a technique allows us to observe market expectations of central bank interventions together with the effects that asymmetry has on the determination of foreign currency risk exposure.
411 The empirical evidence suggests that both conditional mean and conditional variance respond asymmetrically to past information. The conditional mean is found to be an asymmetric function of past depreciations and appreciations, rising proportionally more when there is currency undershooting both within blocks and across blocks. In addition the exchange rate undershooting is more persistent than overshooting. Yet, this persistence is greater across blocks than within blocks. The large values of the speed of adjustment suggest more credibility. The conditional variance is also found to be an asymmetric function of past expectational errors rising proportionally more during depreciation periods within blocks, and declining proportionally more during appreciation periods across blocks. Both behaviours emphasize the importance of credibility in currencies within integrated markets and also suggest a leading role for the large country within each region. Both behaviours are consistent with the Dornbusch overshooting model, but suggests that credibility is the cause of such behaviour in the short-run exchange rate. In agreement with other studies, we find that the behaviour of asymtnetry within blocks responds to central bank interventions or, at least, expectations on the part of the agents that such corrections in the market take place. This confirms what several other studies have found regarding intra-marginal central bank interventions within the EU, Japan and the US.'^ Finally, the present study finds that the European foreign currency markets against the dollar are the most efficient and ASEAN countries the least efficient ones. Erom a policy point of view, these results are of interest to policy makers and central banks, because it furnishes them with information regarding the impact that their interventions can have not only on their own country and on other member countries of the block, but also on other blocks. The presence and degree of asymmetry in their exchange rate, indicates both the credibility in the market and how long it will take for the market to process new information. Our findings suggest that the speed of adjustment is less than a day within blocks and that 'news' affects the currency volatility for less than two hours. On the other hand, across blocks, only movements against the dollar are relevant, and the market takes at least one day to stabihze. Sensitivity of market operators to positive and negative innovations (expectational errors that cause overshooting and undershooting) indicate if credibility affects the variance and, thus, the risk exposure of the exchange rate over time. Volatility tends to increase more during depreciations within blocks, indicating to central banks that traders are more sensitive to their own member countries' currencies
'' A good summary of these studies is presented in Dominguez and Frankel (1993).
412 than to the volatility of currencies outside the block. Therefore, central banks need to avoid such movements actively if they wish to avoid an increase in the volatility of their currencies. This is of special importance for EU countries because it is an indicator of how the market views the sustainability of their currencies. Finally., speed of adjustment at which 'news' or innovations are absorbed is an indicator of the efficiency in a given market. The fundamentals upon which the market defines the trend of the exchange rate as well as its credibility is an important issue that must be examined. This issue was not addressed in this paper because of the lack of availabihty of daily data. However, the study of this matter at the monthly level is the subject of another paper.
ACKNOWLEDGEMENTS We are grateful for the helpful discussion workshop presentations given by the department of economics at both the University of Chicago and the University of Notre Dame, and for the useful comments received from anonymous referees. The usual disclaimer applies.
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