Int. Fin. Markets, Inst. and Money 19 (2009) 597–615
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Journal of International Financial Markets, Institutions & Money j o ur na l ho me pa ge : w w w . e l s e v i e r . c o m / l o c a t e / i n t f i n
Asymmetric volatility in the foreign exchange markets Jianxin Wang ∗, Minxian Yang The Australian School of Business, The University of New South Wales, Sydney, 2052, Australia
a r t i c l e
i n f o
Article history: Received 30 May 2008 Accepted 2 October 2008 Available online 19 October 2008 JEL classiﬁcation: F3 Keywords: Exchange rates Asymmetric volatility Leverage effect Realized variance Continuous and jump components of volatility
a b s t r a c t We examine the presence or absence of asymmetric volatility in the exchange rates of Australian dollar (AUD), Euro (EUR), British pound (GBP) and Japanese yen (JPY), all against US dollar. Our investigation is based on a variant of the heterogeneous autoregressive realized volatility model, using daily realized variance and return series from 1996 to 2004. We ﬁnd that a depreciation against USD leads to signiﬁcantly greater volatility than an appreciation for AUD and GBP, whereas the opposite is true for JPY. Relative to volatility on days following a positive onestandarddeviation return, volatility on days following a negative onestandarddeviation return is higher by 6.6% for AUD, 6.1% for GBP, and 21.2% for JPY. The realized volatility of EUR appears to be symmetric. These results are robust to the removal of jump component from realized volatility and the subsamplings deﬁned by structuralchanges. The asymmetry in AUD, GBP and JPY appears to be embedded in the continuous component of realized volatility rather than the jump component. Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved.
1. Introduction It is well known that volatility in equity markets is asymmetric, i.e. negative returns are associated with higher volatility than positive returns. Robert Engle in his 2003 Nobel Lecture emphasizes the importance of asymmetric volatility. For a portfolio of S&P500 stocks, Engle (2004) shows that ignoring the asymmetry in volatility leads to a signiﬁcant underestimation of the value at risk (VaR). In the foreign exchange markets, however, the consensus seems to be that there is no asymmetric volatility. Bollerslev et al. (1992) suggest that “[W]hereas stock returns have been found to exhibit some degree of asymmetry in their conditional variances, the twosided nature of the foreign exchange market makes
∗ Corresponding author. Email addresses:
[email protected] (J. Wang),
[email protected] (M. Yang). 10424431/$ – see front matter. Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.intﬁn.2008.10.001
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such asymmetries less likely.” All of the studies in their survey adopt symmetric models for exchange rate volatility. Since then the theoretical advances in volatility models, together with the availability of intraday exchange rate data, led to a proliferation of studies of exchange rate volatility. Almost all of them do not consider asymmetric volatility models. Recently Andersen et al. (2001, 2003a) (ABDL hereafter) provide an extensive examination of the statistical properties, modelling and forecasting of realized volatility of foreign exchange rates. Again, the possibility of asymmetric volatility is not investigated in their articles. The “twosided nature of the foreign exchange market” is probably the primary reason for the overwhelming choice of symmetric models for exchange rate volatility. For bilateral exchange rates, because positive returns for one currency are necessarily negative returns for the other, “good news” and “bad news” appear indistinguishable. This implies that exchange rate volatility should have symmetric responses to positive and negative shocks in exchange rate return. Furthermore, it is unclear how the standard explanations for asymmetric volatility in equity markets, i.e. the leverage effect and the volatility feedback effect, apply to the currency markets. The debttoequity ratios in equity markets vary from zero to several hundred percents. With the exception of some small open economies, the debttoGDP ratios for most countries are below 5%, and the debttonational asset ratios are much lower.1 If an investor anticipates higher volatility, say for USD/AUD rate, it is unclear whether she should sell USD or AUD if she holds both currencies. Empirically, the standard asymmetric GARCH models regularly detect asymmetric volatility in daily equity returns. However, these models typically fail to detect asymmetry in daily exchange rate volatility. This is probably another reason for favouring symmetric volatility models for bilateral exchange rates. Despite the bilateral nature of exchange rates, there are at least two reasons for the presence of asymmetry in bilateral exchange rates. First, some currencies have greater economic importance than others. For example, many companies and ﬁnancial institutions use the US dollar (USD) as the base currency for proﬁt and loss calculations but few uses the Australian dollar (AUD). For these institutions, higher expected USD/AUD volatility implies greater risk in AUDdenominated assets but not necessarily in USDdenominated assets. This may lead to the sale of AUDdenominated assets, which lowers USD/AUD exchange rate. This basecurrency effect is likely to be stronger in some currencies than in others, depending on the size and development level of the local economy. For example, the Euro (EUR) area and the United States are of similar sizes and levels of economic development. The basecurrency effect should be weaker for the USD/EUR rate than for USD/AUD, because higher expected USD/EUR volatility may lead Europeans to sell USDdenominated assets and Americans to sell EURdenominated assets. Second, a unique feature of the foreign exchange markets is central bank intervention. Most studies report that interventions lead to higher volatility.2 Since central banks intervene on one side of the market but not the other, interventions may lead to an asymmetric relationship between exchange rate return and volatility. For example, if a central bank were concerned with the depreciation of its currency, it would buy its domestic currency and sell USD. As a result, the higher volatility from market intervention would be associated with past depreciation of the domestic currency. This could be the case when the Reserve Bank of Australia intervened to support AUD in mid2001 when USD/AUD dropped to a historical low of 0.49 (Fig. 1). Conversely, if the central bank were to sell its domestic currency to slow down its appreciation, the resulting higher volatility would be associated with past appreciation. This could be the case for the Japanese yen (JPY) since the Bank of Japan is known to be a heavy seller of JPY over our sample period. Clearly this explanation does not apply to currencies that did not have any central bank intervention during the sample period, e.g. GBP. Given the common perception against asymmetric volatility and the above arguments for its presence, this paper sets out to empirically test for asymmetric volatility in major currencies. The issue
1 When external debt is denominated in the home currency, like the United States, there could be a reverse leverage effect: higher debt may lead to a currency depreciation, which would strengthen net export and enhance the country’s ability to service its debt. This, in turn, may reduce exchange rate uncertainty. We thank the referee for this insight. 2 Examples include BonserNeal and Tanner (1996), Dominguez (1998, 2006), Beine et al. (2002), Beine and Laurent (2003), Galati et al. (2005), Frenkel et al. (2005), and Beine et al. (2007). On the other hand, Beattie and Fillion (1999) report that unexpected interventions by the Bank of Canada reduced intraday volatility. Beine et al. (2003) ﬁnd that when the market is highly volatile concerted interventions decrease volatility.
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Fig. 1. Exchange rates and realized volatility.
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is important for several reasons. First, the foreign exchange markets are several times larger than the equity markets and present a substantial risk to investors. As argued by Engle (2004), the presence of asymmetric volatility, if unaccounted for, will lead to the underestimation of VaR. Second, an empirical examination of asymmetric volatility will enhance our understanding of exchange rate dynamics, particularly in the second moment. This in turn may improve volatility forecasting and derivative pricing. Last but not least, the presence of asymmetric volatility in bilateral exchange rates will provide support for alternative explanations for asymmetric volatility, as listed in the previous paragraph. An early study of the volatilities of bilateral exchange rates by Hsieh (1989) shows that EGARCH models, which accommodate asymmetric effect, produce slightly better ﬁts than GARCH models. However, the asymmetry parameter estimates of the EGARCH models are not reported in his article. McKenzie (2002) ﬁnds some support for the hypothesis that central bank intervention cause asymmetric volatility in exchange rates, using USD/AUD daily return series from 1983 to 1997. Several authors have documented some evidence for asymmetric volatility in exchange rates while studying other issues. For instance, Byers and Peel (1995) document asymmetric volatility in European exchange rates during 1922–1925. Andersen et al. (2003b) show asymmetric responses of major exchange rates to economic announcements in the United States. Asymmetric volatility has also been documented for Malaysian ringgit (Tse and Tsui, 1997), Australian dollar (McKenzie, 2002), and Mexian peso (Adler and Qi, 2003), all against US dollar. To our knowledge, the presence or absence of asymmetric volatility in major exchange rates has not been systematically examined, using realized volatility series, in the literature. The goal of this paper is to formally test for the presence of asymmetric volatility in the four major bilateral exchange rates (AUD, EUR, GBP and JPY against USD), using intraday quotes from January 1996 to March 2004. Speciﬁcally, we examine the relationship between current daily realized volatility and lagged daily return. Realized volatility based on intraday quotes is a consistent and highly efﬁcient estimator of the underlying true volatility. Our long series of realized volatility contain rich information about the underlying true volatilities of the four currencies. Hence, models directly built on realized volatility offer an efﬁcient framework to describe the dynamics of the underlying volatility and to test relevant hypotheses. As a contrast, the dynamics of the underlying volatility cannot be efﬁciently captured by the conventional GARCH type models because they are based solely on daily return series and do not explore intraday observations. Therefore, our realizedvolatilitybased approach is able to uncover volatility features (asymmetric volatility in particular) that the conventional GARCH type models fail to reveal. The model used in this paper is a variant of the heterogeneous autoregressive realized volatility (HARRV) model of Corsi (2004) and Andersen et al. (2005), Andersen et al. (2005) hereafter. In addition to its simplicity, the model adequately accounts for the longmemory autocorrelations observed in realized volatility series. We report the following ﬁndings. First, our testing reveals that equalsized positive and negative returns have signiﬁcantly different effects on the nextday realized volatility for AUD, GBP and JPY. A depreciation against USD leads to more volatility than an appreciation for AUD and GBP, whereas the opposite is true for JPY. These results are robust to alternative speciﬁcations and subsamplings deﬁned by structural changes. They are in sharp contrast to those of the GARCH type models estimated from daily return series, which uniformly imply the absence of asymmetry for all currencies considered. This contrast is consistent with the argument that daily realized volatility is much more informative about underlying true volatility than daily return. It may serve as an explanation for general lack of support for asymmetric volatility in exchange rates in the GARCH literature. Second, the realized volatility of EUR appears symmetric to equalsized negative and positive returns. Other statistical properties of EUR are also different: its return distribution is closer to normal and its realized volatility is less persistent than those of other currencies. Third, using the procedure proposed by Andersen et al. (2005) and BarndorffNielsen and Shephard (2006a,b), we estimate the jump components of realized volatility. We show that the asymmetry in AUD, GBP and JPY is embedded in the continuous component of realized volatility, rather than the jump component. In the analyses above, we ﬁnd that the longterm (i.e. weekly, monthly or quarterly) price trends, measured by absolute returns, have signiﬁcant explanatory power for realized volatility. Section II provides details on the data, the calculation of daily realized volatility, and summary statistics of daily returns and realized volatility. Section III presents GARCH type models with return
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Table 1 Summary statistics for reuters quotes. AUD Total quotes (million) Quotes per weekday Average Median Maximum
GBP
2.49 1238 845 5841
JPY 13.6
6,777 3,920 34,101
EUR 18.1
9064 5,775 32,794
27.6 22,048 20,763 59,798
series and explores why GARCH type models fail to capture asymmetry effects. Tests for asymmetric volatility are carried out in section IV. Section V is sensitivity analysis. Concluding remarks are given in section VI. 2. Data and descriptive statistics Our primary data are intraday Reuters FXFX quotes for Australian dollar (AUD), Euro (EUR), British pound (GBP) and Japanese yen (JPY), all against US dollar (USD), kindly provided by the Securities Industry Research Center of Australia (SIRCA). The samples for AUD, GBP, and JPY are from 1 January 1996 to 31 March 2004 for a period over 8 years. The sample for EUR goes from 1 January 1999 to 31 March 2004 for a period over 5 years. AUD, GBP and EUR are quoted as USD/AUD, USD/GBP, and USD/EUR respectively, while JPY is quoted as JPY/USD. Quotes are ﬁltered for anomalies, e.g., outofrange price or spread. 2.1. Construction of daily return and realized volatility Reuters quotes are used for the construction of daily return and realized volatility. We adopt the same 30min sampling interval as Andersen et al. (2003a), who argue that “the use of equallyspaced thirtyminute returns strikes a satisfactory balance between the accuracy of the continuousrecord asymptotics underlying the construction of our realized volatility measures on the one hand, and the confounding inﬂuences from microstructure frictions on the other.”3 The midpoint of the bid and ask quotes at each 30min interval is computed as the linear interpolation of the quotes immediately before and after the 30min time stamp. Following the convention in Bollerslev and Domowitz (1993) and Andersen et al. (2003a), a trading day starts at 21 GMT, or 4 p.m. New York time, and ends at 21 GMT on the next day. Weekend quotes, from 21 GMT on Friday to 21 GMT on Sunday, are excluded. Halfhourly return is the percentage log return, i.e., 100 times the change in the halfhourly log price (bidask midpoint). Daily returns are the sum of halfhourly returns over the trading day. Daily realized volatility m 2 or realized variance (RV) is the sum of squared halfhourly returns over a trading day, rvD r , t = j=1 t,j where rt,j is the intraday return for interval j and m is the number of intraday sampling intervals. Some trading days may have less than 48 halfhourly observations because of a regional holiday, slow trading or Reuters system stoppage. The days with more than 3.5 h of missing data are excluded from our samples. This process leaves 1960 daily observations for AUD, 1966 for GBP, 1970 for JPY, and 1247 for EUR. 2.2. Data summary Table 1 provides a summary of quote activities. Based on a sample from December 1986 to June 1999, Andersen et al. (2003a) report the average daily number of quotes around 2000 for JPY and 4500
3 Recently several studies have proposed procedures for correcting microstructure noise, e.g. AitSahalia et al. (2005), Bandi and Russell (2006), and Hansen and Lunde (2006). Hansen and Lunde (2006) report that at 20–30 min sampling intervals, microstructure noise is independent of asset prices, and such independence fails at higher frequencies. BarndorffNielsen and Shephard (2006b) show that the difference in realized volatilities from alternative sampling frequencies, e.g. 1 min versus 10 min, is theoretically small. Ghysels and Sinko (2006) ﬁnd that correcting microstructure noise does not improve volatility forecasting.
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Table 2 Daily return summary. AUD Return based on prices at 21 GMT Observations 1960 Median 0.025 Mean 0.007 S.D. 0.677 Skewness −0.012 Kurtosis 5.806 Q(20) 14.363 88.367 Q2 (20)
GBP
JPY
EUR
1966 0.017 0.016 0.487 0.009 4.435 19.798 88.772
1968 0.009 −0.003 0.697 −0.528 6.059 23.702 152.932
1247 0.007 0.018 0.672 0.052 3.499 15.511 38.109
Q(20) and Q2 (20) are LjungBox statistics for testing autocorrelation in return and squared return respectively for the ﬁrst 20 lags. The 5% critical value of 2 (20) distribution is 31.41.
for Deutschemark. The quote intensity has increased substantially since. EUR is clearly the most active: the median number of quotes for EUR is over three times that for JPY, over ﬁve times that for GBP, and approximately 25 times that for AUD. Fig. 1 depicts the exchange rates and the realized volatility over the sample period. The most notable feature from Fig. 1 is the exceptionally high volatility for JPY in early October 1998. On 7 October 1998, JPY jumped from around 130 to 120 in 1 day. The realized volatility is 11.3 for 7 October and 34.6 for 8 October. Since this is regarded as “onceinageneration” volatility,4 these 2 days are treated as outliers and are excluded from the econometric analysis in the rest of this paper (1968 observations for JPY). Table 2 provides some summary statistics for daily returns based on Reuters quotes at 21 GMT. In Table 2, compared to standard deviations, the means of returns are approximately zero. The standard deviations of AUD, JPY and EUR are approximately the same (≈0.68), while that of GBP is less (0.49). All return series are thicktailed with kurtosis greater than 3. JPY returns appear to have pronounced left skewness and greatest kurtosis. The LjungBox statistics indicate no signiﬁcant autocorrelation in return series but strong autocorrelation in squared return series. Interestingly, EUR returns seem to be at odds with the other series in that its skewness is close to 0 and kurtosis is close to 3. The top panel of Table 3 reports summary statistics for daily realized volatility. The average realized volatility for JPY, 0.514 is almost identical to 0.538 reported in Andersen et al. (2001) for the 1986–1996 period. JPY has the highest volatility of volatility, while GBP has the lowest. The LjungBox Q(20) statistics show that realized volatility is highly persistent for all four currencies. The middle panel of Table 3 summarizes the daily log realized volatility, which is the primary variable we study. The skewness and the kurtosis of log realized volatility are much smaller than those of realized volatility. The log realized volatility has higher LjungBox statistics than the realized volatility, which in turn has higher LjungBox statistics than squared returns. These characteristics are consistent with the ﬁndings for DEM/USD and JPY/USD by Andersen et al. (2001, 2003a). Fig. 2 shows the autocorrelation functions of the log realized volatility for lags up to 100 days for the four currencies. While there are minor differences in the autocorrelations, the characteristic of longmemory is apparent for all series with the autocorrelations being signiﬁcantly different from zero even at lags up to 100. 3. Asymmetric GARCH models for daily returns Previous studies using GARCH type models have typically found little volatility asymmetry in the foreign exchange markets. We revisit this issue by estimating asymmetric GARCH models with daily return series. We draw comparison between realized volatility and GARCHestimated daily volatility, which is solely based on daily returns, in terms of statistical properties and shortterm dynamics. Two asymmetric GARCH models for daily returns are used to test for asymmetric volatility. The ﬁrst is the exponential GARCH, EGARCH hereafter, model of Nelson (1991) and the second is that of Glosten et al.
4
See Cai et al. (2001) for events surrounding these days.
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Table 3 Daily realized volatility summary. AUD
GBP
JPY
EUR
Realized volatility Median Mean S.D. Skewness Kurtosis Q(20)
0.394 0.524 0.519 4.557 37.506 3056
0.215 0.263 0.262 13.681 329.342 513
0.354 0.514 0.570 4.366 31.117 3364
0.384 0.464 0.377 6.163 81.159 550
Log realized volatility Median Mean S.D. Skewness Kurtosis Q(20)
−0.930 −0.945 0.757 0.085 3.331 9159
−1.539 −1.543 0.629 −0.084 5.186 3258
−1.039 −0.998 0.768 0.373 3.677 5873
−0.958 −0.972 0.633 −0.101 4.131 1249
0.442 0.471 0.211 0.543 2.673 29738
0.233 0.242 0.076 0.657 3.650 19800
0.440 0.488 0.217 1.490 5.697 32660
0.445 0.459 0.112 0.819 3.432 19535
EGARCH cond. variance Median Mean S.D. Skewness Kurtosis Q(20)
EGARCH conditional variance series are based on the point estimates reported in Table 4. Q(20) is LjungBox statistic for testing autocorrelation in the series for the ﬁrst 20 lags. The 5% critical value of 2 (20) distribution is 31.41.
(1993), GJR hereafter. Given that there is no autocorrelation in return series, the mean speciﬁcation is simply: rt = + εt ,
εt /ht ∼ iid t(),
(1)
where ht is the conditional variance of the daily return series rt and t() is Student’s tdistribution with degree of freedom . The t distribution is suitable for capturing the thicktails in the return series.
Fig. 2. Autocorrelations of log realized volatility. The Bartlett bands (2standarderror) for these series range from ±0.0451 to ±0.0566.
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Table 4 GARCH family models with t, rt = + ε EGARCH ln(ht ) = ω + ˛ GJR ht = ω
2 + ˛rt−1
ε
√t−1

ht−1
2 + S t−1 rt−1
ε
εt /h t ∼ iid t(),
√t−1
+
ht−1
+ ˇ ln(ht−1 ).
+ ˇht−1 .
ω
˛
Coeff. S.E.
0.0155 0.0129
−0.0789*** 0.0166
0.0963*** 0.0197
−0.0067 0.0108
0.992*** 0.0043
6.14*** 0.7762
GBP
Coeff. S.E.
0.0148 0.0100
−0.120*** 0.0275
0.110*** 0.0224
−0.0059 0.0135
0.974*** 0.0106
6.73*** 1.07
JPY
Coeff. S.E.
0.0129 0.0135
−0.0609*** 0.0150
0.0716*** 0.0176
−0.0103 0.0101
0.991*** 0.0045
6.16*** 0.857
EUR
Coeff. S.E.
0.0228 0.0184
−0.0523** 0.0221
0.0557** 0.0220
0.0096 0.0099
0.988*** 0.0096
Coeff. S.E.
0.0171 0.0130
0.0025* 0.0015
0.0412*** 0.0115
0.0019 0.0126
0.955*** 0.0095
5.97*** 0.745
GBP
Coeff. S.E.
0.0154 0.0100
0.0051** 0.0022
0.0375** 0.0129
0.0184 0.0169
0.934*** 0.0156
6.83*** 1.09
JPY
Coeff. S.E.
0.0144 0.0136
0.0041** 0.0020
0.0213** 0.0107
0.0097 0.0120
0.965*** 0.0095
6.05*** 0.813
EUR
Coeff. S.E.
0.0215 0.0184
0.0039 0.0035
0.0318*** 0.0117
0.0139 0.0129
0.967*** 0.0145
EGARCH AUD
GJR AUD
ˇ
11.2** 4.42
11.1** 4.33
The bold numbers are coefﬁcients for volatility asymmetry. The asterisks *, **, and *** represent statistical signiﬁcance at 10%, 5%, and 1%, respectively.
EGARCH speciﬁcation is
ln(ht ) = ω + ˛
εt−1 
ht−1
+
εt−1
ht−1
+ ˇ ln(ht−1 ),
(2)
GJR speciﬁcation is ht = ω + ˛ε2t−1 + S t−1 ε2t−1 + ˇht−1
(3)
where St = 1 if rt < 0; St = 0 otherwise. Engle and Ng (1993) show that EGARCH and GJR are superior relative to other asymmetric volatility models. In both models, the coefﬁcient captures the asymmetric effect of lagged return on volatility. The results of EGARCH and GJR models are reported in Table 4. These results are in line with the stylized characteristics of daily GARCH models. For example, the coefﬁcients ˛ and ˇ are highly signiﬁcant and ˇ are close to one for all currencies. Importantly, the coefﬁcient for asymmetric volatility, , is decisively insigniﬁcant for all currencies. EGARCH and GJR models with εt /ht ∼ iid N(0,1)are also estimated, where the standard errors are computed using the quasiML approach of Bollerslev and Wooldridge (1992). The results (not presented) are qualitatively the same as those in Table 4. In particular, the asymmetric effect, , is insigniﬁcant for all currencies. In summary, the volatility asymmetry in bilateral exchange rates is not present in either daily EGARCH models or GJR models. The GARCH results seem to conﬁrm the “twosided nature of the foreign exchange market”: good news for AUD is bad news for USD and the volatility of the USD/AUD rate is symmetric. However, as discussed in the introduction, there are factors that may break the symmetry of bilateral exchange rate. An alternative explanation for failing to detect asymmetric volatility in GARCH type models is that the volatility measure (conditional variance) in these models is based on squared or absolute return, which is too noisy to reveal the asymmetric relationship between volatility and return. In contrast,
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Fig. 3. Squared return, realized volatility and EGARCH conditional variance for AUD.
realized volatility constructed from intraday returns contains much more information on the return process. In a context of forecasting, Andersen et al. (2003a) point out that “the noise in daily squared returns necessarily renders imprecise measurements of the current volatility innovation” and realized volatility “provides a relatively precise and quickly adapting estimate of volatility because it exploits valuable intraday information.” Therefore, utilizing both daily realized volatility and daily return offers a better alternative for revealing the relationship between current volatility and lagged return than relying solely on daily return. Fig. 3 provides a visual comparison of daily squared return, realized volatility and EGARCH volatility (conditional variance) for AUD5 . EGARCH volatility may be viewed as predicted volatility based solely on the past daily returns, whereas realized volatility may be viewed as an observation on the current volatility. Visually, EGARCH volatility captures the low frequency component adequately but represents high frequency component poorly. EGARCH volatility is a ﬁltered version of past daily returns, where the ﬁltering process is unable to recover useful signals about asymmetric volatility because of the high noise level in daily returns. The noisiness of daily returns is partially represented by the volatile squared returns in Fig. 3. In contrast, realized volatility converges in probability to the underlying volatility when the intraday sampling frequency increases. With a high signalnoise ratio, realized volatility carries much more information about underlying volatility than daily return and GARCH volatility (which is based on daily return) do. The summary statistics of EGARCH volatility are presented at the bottom panel of Table 3. Compared to realized volatility (top panel of Table 3), EGARCH volatility has similar median; much smaller standard deviation, skewness and kurtosis; and much larger LjungBox statistic for all currencies. EGARCH volatility appears much smoother and more persistent (larger LjungBox statistic) than realized volatility. These summary statistics reenforce the characteristics depicted in Fig. 3. 4. Testing for asymmetry in realized volatility As argued in the previous section, daily realized volatility is a much better data source for examining the relationship between daily volatility and lagged daily return than daily return. In this section, we directly model daily realized volatility and test for volatility asymmetry in foreign exchange rates.
5 GJR volatility and GARCH volatilities for other currencies are not presented because they are similar to EGARCH volatility presented in Fig. 3.
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4.1. Longmemory and HARRV model As shown in Fig. 2, the autocorrelations in daily log realized volatility decay extremely slowly. This is known as longmemory dependency, a salient characteristic of daily realized volatility. To account for the longmemory dependency, Andersen et al. (2003a) model the realized volatilities of exchange rates as a fractionally integrated process. Earlier, Müller et al. (1993, 1997) propose a heterogeneous ARCH (HARCH) model, based on their “heterogeneous market hypothesis”, to capture the longmemory dependency in squared return series. Recently, Corsi (2004) and Andersen et al. (2005) adapt the HARCH model to realized volatility, known as heterogeneous autoregressive realized volatility (HARRV) model, and show that HARRV models provide superior forecasting performance. In what follows, we use a variant of the HARRV model to describe realized volatility series and regard the model as data generating processes. The basic HARRV model includes past volatilities aggregated over different time horizons as explanatory variables. Let rvD t be the realized volatility on day t. The average realized volatility in
t
the past h days (including day t) is rvt,h = 1/h
monthly (h = 22), and quarterly (h = 66) volatilities as model of Corsi (2004) may be written as rvD t =ω+
Q
rvD s . We W rvt , rvM t ,
s=t−h+1
ˇk rvkt−1 + t
denote the average weekly (h = 5), and rvQ t , respectively. The HARRV
(4)
k=D
where t is an error term; ω and ˇk are parameters; and k = D (day), W (week), M (month), and Q (quarter). The motivation for (4) is that, because market participants are heterogeneous in that some are shortterm investors and others are middle or longterm investors, the difference in their behavior induces the dependency of current daily volatility on previous daily, weekly, monthly and quarterly volatilities. HARRV models do not feature the “longmemory dependency” deﬁned by a fractionally integrated process. However, the mixing of a small number of realized volatilities with different aggregation frequencies produces an excellent approximation to longmemory dependencies found in data [see Corsi, 2004; Andersen et al., 2005]. 4.2. Test for asymmetry in HARlogRVR model Following Andersen et al. (2003a) and Andersen et al. (2005), we use log realized volatility as our primary variable, so that positivity restrictions on the parameters and the error term in (4) are avoided. Further, as shown by Andersen et al. (2003a), log realized volatility tends to have a distribution close the normal. To investigate the relationship between current daily volatility and previous daily return, we include the lag of daily return and absolute return as additional explanatory variables in the model. Explicitly, our model, labeled as HARlogRVR hereafter, is given by ln(rvD t )=ω+
Q
D D ˇk ln(rvkt−1 ) + ˛D rt−1  + D rt−1 + t .
(5)
k=D
The second term on the righthand side captures the autocorrelation and longmemory dependency in ln(rvD t ). The third and fourth terms capture the size and directional impact of the lagged return rt−1 . In particular D identiﬁes the asymmetric effect of returns on volatility. If D < 0, negative returns lead to greater volatility than positive returns. Since JPY is quoted in JPY/USD, positive returns indicate the depreciation of JPY and may also lead to greater volatility. Therefore, the hypotheses of interest are H0 : D = 0 and H1 : D = / 0. Table 5 reports the estimation results of model (5), where the standard errors are given by the NeweyWest heteroscedasticity and autocorrelation (HAC) robust estimates. The null H0 : D = 0 is convincingly rejected for AUD, GBP and JPY with pvalues being 0.0053, 0.0238 and 0.0000, respectively. The point estimates for D are −0.047, −0.061 and −0.138 for AUD, GBP and JPY, respectively. D D We note that the difference between the effect of a positive rt−1 and that of a negative rt−1 is 2 D .
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Table 5 HARlogRVR models )=ω+ ln(rvD t
Q
D  + D rD + . ˇk ln(rvkt−1 ) + ˛D rt−1 t t−1
k=D
AUD
GBP
Coeff
S.E.
ω ˇD ˇW ˇM ˇQ ˛D D
−0.252*** 0.191*** 0.334*** 0.194*** 0.177*** 0.145*** −0.047***
0.030 0.028 0.050 0.072 0.058 0.028 0.017
Corr(rt , ˆ t ) adj. R2 Q(20)
−0.070*** 0.023 0.502 27.4
JPY
Coeff
EUR
S.E.
Coeff
S.E.
−0.456*** 0.175*** 0.336*** 0.159** 0.125** 0.212*** −0.061***
0.067 0.039 0.060 0.065 0.069 0.044 0.027
−0.306*** 0.188*** 0.320*** 0.080 0.286*** 0.143*** −0.138***
0.037 0.035 0.051 0.061 0.056 0.032 0.021
0.025
0.023
−0.175*** 0.023 0.452 13.9
0.312 27.8
Coeff
S.E.
−0.362*** 0.080* 0.373*** 0.194* 0.153 0.158*** −0.016
0.057 0.041 0.071 0.101 0.094 0.038 0.025
0.027
0.029 0.250 16.0
Q(20) is LjungBox statistic for testing autocorrelation in the residual series for the ﬁrst 20 lags. The 5% critical value of 2 (20) distribution is 31.4. The standard errors are the NeweyWest HAC robust estimates. The bold numbers are coefﬁcients for volatility asymmetry. The asterisks *, **, and *** represent statistical signiﬁcance at 10%, 5%, and 1%, respectively.
Further, we ﬁnd a strong and negative contemporaneous correlation between return and the residual of model (5) for AUD and JPY, indicated by Corr(rt , ˆ t ) in Table 5. EUR appears to be symmetric as H0 : D = 0 cannot be rejected. D < 0, i.e. the quoted price The negative estimates for D imply that volatility increases when rt−1 6 of the denominator currency decreases. Since JPY is quoted as JPY/USD but AUD and GBP are quoted as USD/AUD and USD/GBP, respectively, therefore volatility increases when JPY appreciates and when AUD and GBP depreciate against USD. One plausible explanation for the difference is central bank intervention. Data from Japan’s Ministry of Finance show that for most of the sample period, the Bank of Japan (BOJ) was selling JPY and buying USD. The ad hoc evidence suggests that BOJ intervention was driven by JPY appreciation. If intervention was associated with greater shortrun volatility as documented in most studies, it may partially explain the direction of volatility asymmetry for JPY. Since Bank of England did not intervene over the sample period, intervention cannot explain the asymmetry in GBP, therefore is not the only source for the difference in asymmetric volatility observed here. The estimates of ˇk and ˛D in model (5) are all positive and mostly signiﬁcant. The strong impact from lagged longhorizon volatilities to daily volatility is in line with Müller et al. (1997), Corsi (2004), and Andersen et al. (2005). Although Corsi (2004) and Andersen et al. (2005) do not include the lagged quarterly volatility in their studies, we ﬁnd that the lagged quarterly volatility appears to be important for AUD and JPY (signiﬁcant at 5%). Further, the lagged weekly volatility exhibits the greatest impact on the current daily volatilities, a ﬁnding consistent with that of Andersen et al. (2005). The models in Table 5 ﬁt data adequately in the sense that the strong dependencies (autocorrelations) in the log realized volatility are largely accounted for. The Q(20) statistics for the residuals are strikingly smaller than those in Table 3 and are not signiﬁcant at the 5% 2 (20) critical value. Fig. 4 contains the autocorrelations of the residuals of model (5) for the four currencies. Unlike Fig. 2, the autocorrelations in Fig. 4 ﬂuctuate around zero with few being outside the Bartlett (2standarddeviation) bands. Overall, there is little evidence for the presence of autocorrelations in the residuals. Table 5 establishes the statistical signiﬁcance of asymmetric volatilities for AUD, GBP, and JPY. We now assess the economic or practical signiﬁcance of the estimated parameters. Let be D the standard deviation of the daily return. Equation (5) can be written as ln(rvD t = f (rt−1 ) with
6
We thank the referee for this observation.
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Fig. 4. Residual autocorrelations of the HARlogRVR models.
D on rvD can f(− ) = f(0) + (˛D − D ) and f(+ ) = f(0) + (˛D + D ) . The relative asymmetric effect of rt−1 t be measured by
exp[f (− )] − exp[f (+ )] = exp(−2 D ) − 1 exp[f (+ )] D = − versus r D = + on the next day’s realized volatility. Based This is the percentage impact of rt−1 t−1 on the sample standard deviations of the returns in Table 2 and the point estimates in Table 5, the values of this measure are 6.6%, 6.1% and 21.2% for AUD, GBP and JPY, respectively. At daily frequency, the magnitude of these differences indicates strong economic signiﬁcance for all three currencies.
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5. Sensitivity analysis In this section, we consider an alternative speciﬁcation of the HARlogRVR model and the removal of the jump component from realized volatility. We also consider the subsamplings deﬁned be structural change tests. These serve as robustness checks on the ﬁndings presented in the previous section.
5.1. Modiﬁed HARlogRVR models Theory (e.g. Forsberg and Ghysels, 2004) and empirical evidence (e.g. Ghysels et al., 2006) suggest that absolute returns have predictive powers for future increments in quadratic variation or realized volatility. Longterm (e.g. quarterly) absolute returns may also be viewed as price trends that increase volatility; see Campa et al. (1998) and Johonson (2002). These considerations indicate that lagged weekly, monthly and quarterly absolute returns are relevant in explaining the behavior of daily log realized volatility. We check whether or not the asymmetric effects are sensitive to the inclusion of longterm absolute returns in the HARlogRVR model by estimating the following modiﬁed version:
ln(rvD t )=ω+
Q
k D (ˇk ln(rvkt−1 ) + ˛k rt−1 ) + D rt−1 + t
(6)
k=D
where rtk = (1/h)
t
rD s=t−h+1 s
and h = 1 for k = D, 5 for k = W, 22 for k = M, and 66 for k = Q. Again, the
hypothesis of symmetric volatility is H0 : D = 0. The estimation results of model (6) are presented in Table 6. The pvalues of D are 0.0024, 0.0101 and 0.0000 for AUD, GBP and JPY, respectively. In essence, the results are qualitatively the same as those in Table 5, that is, H0 is rejected for AUD, GBP and JPY but cannot be rejected for EUR. Therefore, the inclusion of absolute longterm returns does not alter the conclusion from model (5). Interestingly, the lagged absolute longterm returns appear to have appreciable explanatory power for the current log realized volatility (e.g. most of their coefﬁcients are signiﬁcant and the adjusted R2 are greater than those in Table 5).
Table 6 HARlogRVR models with long term absolute returns )=ω+ ln(rvD t
Q
k ) + D r D + . (ˇk ln(rvkt−1 ) + ˛k rt−1 t t−1
k=D
AUD Coeff −0.414***
ω ˇD ˇW ˇM ˇQ ˛D ˛W ˛M ˛Q D
0.184*** 0.287*** 0.174** 0.200*** 0.123*** 0.173** 0.301** 0.818*** −0.050***
Corr(rt , ˆ t ) adj. R2 Q(20)
−0.077*** 0.510 26.1
GBP S.E.
JPY
Coeff −0.610***
S.E.
Coeff −0.569***
0.044 0.027 0.049 0.071 0.056 0.028 0.080 0.145 0.241 0.016
0.166*** 0.282*** 0.114* 0.205*** 0.181*** 0.189* 0.743*** 0.717* −0.067***
0.076 0.038 0.059 0.064 0.070 0.044 0.097 0.211 0.377 0.026
0.023
0.022
0.023 0.323 24.3
0.174*** 0.259*** 0.013 0.333*** 0.113*** 0.334*** 0.628*** 0.594** −0.131***
EUR S.E. 0.057 0.034 0.050 0.062 0.056 0.032 0.076 0.161 0.296 0.019
−0.170*** 0.023 0.466 13.2
Coeff
S.E.
−0.571*** 0.069 0.304*** 0.143 0.236** 0.137*** 0.252** 0.517*** 0.643* −0.022
0.076 0.042 0.070 0.106 0.098 0.040 0.101 0.170 0.371 0.025
0.019
0.029 0.263 16.6
Q(20) is LjungBox statistic for testing autocorrelation in the residual series for the ﬁrst 20 lags. The 5% critical value of 2 (20) distribution is 31.41. The standard errors are the NeweyWest HAC robust estimates. The bold numbers are coefﬁcients for volatility asymmetry. The asterisks *, **, and *** represent statistical signiﬁcance at 10%, 5%, and 1%, respectively.
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Table 7 HARlogRVR models with long term returns and absolute returns ln(rvD )=ω+ t
Q
k  + k rk ) + . (ˇk ln(rvkt−1 ) + ˛k rt−1 t t−1
k=D
AUD
GBP
Coeff
S.E.
ω ˇD ˇW ˇM ˇQ ˛D ˛W ˛M ˛Q D W M Q
−0.429*** 0.182*** 0.274*** 0.173** 0.206*** 0.123*** 0.182** 0.310** 0.897*** −0.030 −0.082 −0.039 −0.105
0.046 0.027 0.049 0.072 0.057 0.028 0.080 0.144 0.242 0.019 0.057 0.103 0.170
Corr(rt , ˆ t ) adj. R2 Q(20) pValue
−0.076*** 0.023 0.510 25.5 0.00774
JPY
Coeff
EUR
S.E.
Coeff
S.E.
−0.621*** 0.162*** 0.280*** 0.113* 0.208*** 0.179*** 0.208** 0.753*** 0.730** −0.039 −0.160*** 0.013 0.138
0.078 0.037 0.058 0.063 0.070 0.044 0.098 0.209 0.395 0.028 0.063 0.149 0.276
−0.565*** 0.173*** 0.254*** 0.043 0.323*** 0.112*** 0.328*** 0.672*** 0.654** −0.134*** −0.029 0.145 0.156
0.057 0.034 0.051 0.064 0.060 0.032 0.076 0.162 0.304 0.021 0.052 0.113 0.206
0.020
0.023
−0.170*** 0.023 0.466 12.6 1.85e11
0.324 25.0 0.00642
Coeff
S.E.
−0.602*** 0.065 0.308*** 0.139 0.218** 0.138*** 0.272*** 0.565*** 0.711** −0.003 −0.099 0.037 −0.142
0.078 0.042 0.071 0.108 0.101 0.040 0.100 0.194 0.388 0.029 0.068 0.141 0.211
0.019
0.029 0.263 16.6 0.369
Q(20) is LjungBox statistic for testing autocorrelation in the residual series for the ﬁrst 20 lags. The 5% critical value of 2 (20) distribution is 31.41. The standard errors are the NeweyWest HAC robust estimates. Here, “pValue” is the probability value of the Wald statistic for testing the joint null hypothesis D = W = M = Q = 0. The bold numbers are coefﬁcients for volatility asymmetry. The asterisks *, **, and *** represent statistical signiﬁcance at 10%, 5%, and 1%, respectively.
Models (5) and (6) estimate the asymmetric volatility impact from lagged daily return. We now examine the robustness when lagged weekly, monthly and quarterly returns included: ln(rvD t )=ω+
Q
k k (ˇk ln(rvkt−1 ) + ˛k rt−1  + k rt−1 ) + t
(7)
k=D
where the joint null hypothesis H0 : D = W = M = Q = 0 is tested. The estimation results of Eq. (7) are reported in Table 7. The joint null is convincingly rejected for AUD, GBP and JPY but cannot be rejected for EUR. The pvalues of the Wald tests for joint H0 being 0.0077, 0.0064, 0.0000 and 0.3691 for AUD, GBP, JPY and EUR, respectively. Therefore, the generalization (7) does not change the conclusion about the signiﬁcance of the asymmetric effects in model (5) for all currencies. In Table 7, D for JPY remains signiﬁcant with the same sign as in Table 5. The signiﬁcant coefﬁcient for GBP shifts to W with the same sign as D in Table 5. The coefﬁcients for AUD all have the same sign as D in Table 5 and are jointly signiﬁcantly different from zero. Individually, however, the pvalues of the AUD coefﬁcients are 0.1182, 0.1512, 0.7016 and 0.5391 for daily, weekly, monthly and quarterly lagged returns, respectively. For AUD, there appears to be uncertainty about which lagged returns provide the explanatory power in model (7). Further, in terms of adjusted R2 , little or no improvement over model (6) is achieved in model (7) for all currencies. 5.2. Continuous and jump components of realized volatility Asset price jumps have signiﬁcant impact on asset allocation (Liu et al., 2003) and option pricing (Eraker et al., 2003). Recently BarndorffNielsen and Shephard (2004, 2006a,b) propose a procedure that allows for a direct nonparametric decomposition of the realized volatility into a continuous component and a jump component. As jumps may be viewed as infrequently observed largemagnitude intraday returns, it is of interest to examine if the asymmetric volatility in model (5) is caused by these
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Table 8 Jump component in realized volatility JtD = I{Zt < −za }(rvD − bvD ). t t AUD
GBP
JPY
EUR
0.077 150
0.051 100
0.053 105
0.081 101
Whole jump series Mean S.D.
0.022 0.101
0.012 0.137
0.018 0.115
0.023 0.097
Nozero jumps Min Median Max Mean S.D.
0.024 0.203 1.429 0.284 0.247
0.002 0.134 5.380 0.235 0.566
0.018 0.232 1.967 0.345 0.370
0.028 0.227 1.101 0.282 0.209
a = 0.001 Proportion of days with jumps Number of days with jumps
infrequent “outliers”. In particular, we check whether or not the asymmetric effects in Table 5 are sensitive to the removal of jump component from realized volatility. BarndorffNielsen and Shephard (2004) show that, as the lengths of intraday sampling interval shrink to zero (and m → ∞), the realized volatility rvD t converges in probability to the sum of a component (known as “integrated volatility”) associated with a continuous diffusion process and a component associated with a jump process. The bipower variation, deﬁned as bvD t = m ( /2) j=2 rt,j rt,j 1 , where m is the number of intraday sampling intervals and rt,j is the intraday return for interval j, converges in probability to the “integrated volatility” or the continuous component. BarndorffNielsen and Shephard (2006a,b) suggest the test statistic: Zt =
D m1/2 (bvD t /rvt − 1)
( 2 /4 + − 5) · max{1, (qvD t )
1/2
/bvD t }
(8)
for testing the presence of a jump component in the daily realized volatility, where qvD t = m ( 2 /4)m j=4 rt,j rt,j−1 rt,j−2 rt,j−3  is known as quadpower variation. The Zt statistic converges in distribution to a standard normal random variable when the lengths of intraday sampling intervals shrink to zero and there is no jump. Therefore, the null of no jump is rejected when Zt is too negative. Let za be the standard normal upper tail critical value for a given signiﬁcant level a (e.g. z0.001 = 3.09). As suggested by Andersen et al. (2005), the jump and continuous components of daily realized volatility may be constructed by D JtD = I{Zt < −za }(rvD t − bvt )
D and CtD = rvD t − Jt
(9)
respectively, where I{A} is the indicator function that is 1 if A is true and 0 otherwise. We choose the signiﬁcant level a = 0.001 and construct the continuous and jump components CtD and JtD as described above. Some descriptive statistics of JtD are given in Table 8. The proportions of the days with jumps identiﬁed by the above procedure (out of all days in the sample) range from 5.1% to 8.1%. While these proportions are small in absolute terms, they are higher than the signiﬁcant level a = 0.001 = 0.1% and indicate the presence of jump components in these days. However, the sizes of jump components do not appear to be large. The medians of positive JtD as the percentages of the medians of rvD t range from 51.1% to 65.5% for the four currencies. To check whether the asymmetric effects in Table 5 are driven by infrequent jumps, we estimate the HARlogRVR model (5) with the continuous component CtD replacing rvD t and report the results in Table 9. Since the results in Table 9 are qualitatively the same as those in Table 5, it is clear that the asymmetric effects in model (5) are robust to the removal of jump components from rvD t and are not caused by infrequent jumps. The above results indicate that the asymmetry in realized volatility is essentially embedded in its continuous component. Bollerslev et al. (2005) report similar results for realized volatility of the S&P500 index futures.
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Table 9 HARlogRVR models with continues component ln(CtD ) = ω +
Q
k ) + ˛D r D  + D r D + . ˇk ln(Ct−1 t t−1 t−1
k=D
AUD
GBP
JPY
EUR
Coeff
S.E.
Coeff
S.E.
Coeff
S.E.
Coeff
S.E.
ω ˇD ˇW ˇM ˇQ ˛D D
−0.281*** 0.189*** 0.320*** 0.188*** 0.193*** 0.179*** −0.042**
0.031 0.027 0.048 0.071 0.056 0.028 0.017
−0.451*** 0.167*** 0.334*** 0.190*** 0.118 0.240*** −0.063**
0.070 0.035 0.054 0.065 0.075 0.044 0.027
−0.310*** 0.172*** 0.344*** 0.071 0.289*** 0.148*** −0.135***
0.037 0.037 0.052 0.061 0.055 0.032 0.020
−0.396*** 0.083* 0.346*** 0.210** 0.148 0.180*** −0.012
0.058 0.045 0.071 0.100 0.094 0.038 0.025
Corr(rt , ˆ t ) adj. R2 Q(20)
−0.060*** 0.023 0.499 21.689
0.027
0.023
−0.178*** 0.453 17.842
0.023
0.018
0.029
0.309 27.308
0.247 17.695
Q(20) is LjungBox statistic for testing autocorrelation in the residual series for the ﬁrst 20 lags. The 5% critical value of 2 (20) distribution is 31.41. The standard errors are the NeweyWest HAC robust estimates. The bold numbers are coefﬁcients for volatility asymmetry. The asterisks *, **, and *** represent statistical signiﬁcance at 10%, 5%, and 1%, respectively.
5.3. Structural change and subsamples To check whether or not the HARlogRVR model (5) is stable over the sampling period, we use the supW (maximum Wald) statistic to test for the presence of a structural change. Let be a potential change date, from which onward the parameters in model (5) may change. The parameters of the unrestricted model in the subsample [1, − 1] are different from those in the subsample [, T], where T is the fullsample size. On the other hand, the parameters of the restricted model are the same in the two subsamples (the models in Table 5 are restricted ones). The Wald statistic is computed as W() = T(SSRR − SSRU )/SSRU , where SSRR and SSRU are the sums of squared residuals for the restricted model and the unrestricted model, respectively. The supW statistic is then given by sup W = max {W ()}, where is between 0 T and (1 − 0 )T and 0 ∈ (0,1) is the fraction of trimming.
A popular choice is 0 = 0.15, which is used in this paper. The null hypothesis of no change is rejected if supW is too large (see Hansen (1997) for more details). In the case that the null of no change is rejected, the ˆ that maximizes W() is the estimated date of change. The supW statistics, pvalues and estimated dates of change are reported in Table 10(a). For GBP and EUR, the null of no change cannot be rejected at any conventional level of signiﬁcance. For AUD Table 10 Testing for structural changes in HARlogRVR models.
(a) Testing with whole samples supF statistic pValue Estimated change date
GBP
JPY
EUR
28.181 0.0052 09/Jan/98
12.473 0.5930
32.987 0.0008 30/Oct/02
12.123 0.6287
AUD estimated change date: 09/Jan/98
JPY estimated change date: 30/Oct/02
Prechange
Postchange
Prechange
Postchange
1469 18.011 0.1645
1667 (−66) 14.345 0.4125
301 20.498 0.0785
(b) Testing with subsamples Observations 491 (−66)a supF statistic 18.023 pValue 0.1648 a
AUD
At the start of the sample, 66 observations are needed to compute quarterly realized volatility.
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Table 11 Subsamples of AUD and JPY: HARlogRVR models )=ω+ ln(rvD t
Q
D  + D rD + . ˇk ln(rvkt−1 ) + ˛D rt−1 t t−1
k=D
AUD Estimated change date: 09/Jan/98 Prechange
JPY Estimated change date: 30/Oct/02
Postchange
Prechange
Postchange
Coeff
S.E.
Coeff
S.E.
Coeff
S.E.
ω ˇD ˇW ˇM ˇQ ˛D D
−0.302** 0.132** 0.305** −0.110 0.619*** 0.348*** −0.059
0.123 0.059 0.087 0.155 0.180 0.079 0.046
−0.258*** 0.188*** 0.331*** 0.273*** 0.042 0.114*** −0.043**
0.033 0.031 0.052 0.066 0.056 0.029 0.018
−0.318*** 0.198*** 0.317*** 0.058 0.268*** 0.139*** −0.155***
0.035 0.035 0.054 0.058 0.052 0.035 0.022
Corr (rt , ˆ t ) Observations adj. R2 Q(20)
−0.058 0.049 491 (−66) 0.426 17.809
−0.077***
0.026 1469 0.406 20.105
−0.221*** 0.024 1667 (−66) 0.489 14.641
Coeff −1.680*** 0.085 0.267** 0.118 −0.648** 0.286** −0.042 0.012
S.E. 0.360 0.097 0.115 0.182 0.327 0.113 0.074 0.057 301 0.105 16.588
Q(20) is LjungBox statistic for testing autocorrelation in the residual series for the ﬁrst 20 lags. The 5% critical value of 2 (20) distribution is 31.41. The standard errors are the NeweyWest HAC robust estimates.
and JPY, there is strong evidence (tiny pvalues) for rejecting the null of no change. The estimated dates of change for AUD and JPY are 09 January 1998 and 30 October 2002, respectively, which are close to the fullsample start and end, respectively. To check if there are structural changes in the subsamples divided by the estimated dates of change, we also carry out the supW test for the subsamples of AUD and JPY and report the results in Table 10(b). In all subsamples in Table 10(b), the pvalues are above 5% and the null of no change cannot be rejected at the 5% level of signiﬁcance.7 For AUD and JPY, the HARlogRVR model is reestimated for each subsample and the results are reported in Table 11. For the longer subsamples (post 09 January 1998 for AUD and pre 30 October 2002 for JPY), the asymmetric effects ( D ) are qualitatively the same as those of the fullsample in Table 5. For the shorter subsamples (pre 09 January 1998 for AUD and post 30 October 2002 for JPY), the asymmetric effects are not statistically signiﬁcant but have the same sign as those of the fullsample in Table 5. Given the above observations and the fact that the numbers of data points in the shorter subsamples are substantially fewer than those in the longer subsamples, we conclude that the asymmetric effects are reasonably stable over the fullsamples. 6. Conclusion We present in this paper some new evidence for asymmetric volatility in the exchange rates AUD, GBP and JPY (against USD). The results are robust to a number of variations in model speciﬁcation and data series. However, the economic reasons for asymmetric volatility in exchange rates are not clear at this point. Plausible explanations include basecurrency effect and central bank intervention effect. The results for AUD and GBP appear to be consistent with the basecurrency effect, whereas the results for JPY seem to be related to the effect of central bank intervention. Future research should explore these and other economic factors affecting the asymmetric relationship between volatility and price movements of exchange rates.
7 Hashimoto (2005) uses 10min returns of the JPY/USD rate to carry out monthly estimations of two asymmetric GARCH models for the second half of 1997. He shows that the sign of the coefﬁcients of lagged return shocks changed in November 1997. Our analysis is based on daily realized volatility and lagged daily returns. Studies have shown that the structure of GARCH models is frequencyspeciﬁc and does not survive temporal aggregation [e.g. Drost and Nijman, 1993]. This may explain why at daily frequency asymmetric GARCH models fail to detect volatility asymmetry, and why we identify a different structural break.
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