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The statistical properties of daily foreign exchange rates: 1974–1983
ku_n!al of hternational Ef#xlomk324 (lM8) x29-145. North-HokId
THE S’MK!!.!!.SCAL PROPER’IIES OF DAILY FOREIGN EXCHANGE RATE& 1974498ii’
David A. HSIEH* uniQers&y qfchicugo, C&ago, IL 4M37, USA ReceivedMay 1986,revisedversionreceivedJune 1987 nispaperexaJnilm thcstatisticalpropertksofdaily~ofdrangeoffiveforcigncumndes froJn1974 to 1983.The main purposeir to disrimiMte bctw@Jltwo competingcxplanati0Il~for theobaervadBiea~taitsotthe~~~~thor~datasre~~y~~lromah~~ tail distribaltionwhich tanainsfinsdo~time,andthatthedata~ftomdistributionswhich vary over time. Evidence pint to the r&&n of the Brst hypothcsk Further invcatigatioas showthattbefejcctioncaabeattriitochangingmcaasandvaknce3 illfkdlitf&Whidl canbcdcscribalbyasimpk-model.
1. Intmhth
This paper examines the statistical propertiesof daily rates of change of the prices of five foreign currenciesfrom 1974 to 5983. Knov&dge of this distributionhas importanteccnomic applications:the effectsof exchangerate movementson internationaltrade and capital flows, mean-varianceanalysis of international asset portlolios, and the pricing of options on foreign currencies.In previous studies in the literature,1the concensus is that the distributionof these changesis unim~ *mdhas fatter tails than the normal distribution.It is this second feahulewhich has attractedthe most attention. Two competing explanations have been put forth to munt for the leptokurtosis.One view suggests that the data are independentlydrawn from a fat tail distributionthat remainsfixed over time, while the other proposes come fromdistributionsthat vary over time. There have been few attempts to determinewhich of the two explanations better characterizesthe data. Friedman and Vandersteel(1982) considered WBC author is grateMto Ian Domcwitz, RobertFlood, Jacob Frenkel,Lazs Hansen, Robert Hodrick,Jolm Hukinga, Michael Mussa, Arnold Zellner, the part&pants of the University of Chicago Inuernational Economics Workshop, the Northwestern University 111temational EconomicsWorkshop,smd the NBER Mini-C!onference on InternationalAsset Pricing(1986)for hclpfid comments on aulier draf4s.This re#arch was supported by the Universityof Chicago GraduateSchool of Business The data were provided by the Universityof Chicago Centerfor RescarchonScurity Prices. ‘Sac Burt,Kaen and Booth (1977),WesterGeld(1977),Rogalskiand Vinso (1978).
130
D.A.iisieh, Daily fkign
excha?&erate: 39744983
the following three hypotheses: (a) the data are independent and identically distributed (iid), drawn from a stable symmetric paretian distribution; (b) the data are iid drawn from a mixture of two normal distributions; and (c) the data are independently drawn from a normal distribution whose mean and variance change over time. They found evidence in favor of the third hypothesis. Calderon-Rossell and Ben-Horim (1982) tested directly for the iid property without requiring distributional assumptions. They found that they can reject iid for most currencies. They attributed the rejection to shifts in the mean in 9 out of 13 currencies, and to changes in the skewness in the remaining cases. This paper extends the previous work in several ways. We use ten full years of daily data to investigate in detail the cause of rejection of the iid hypothesis. We find that not only do the distributions of exchange rates differ across days of the week, but that their means and variances also shift over time. Furthermore, we are able to construct a simple statistical model to describe these time-varying means and variances.
ZTbedata The data consist of daily closing bid prices of foreign currencies (in terms of the U.S. dollar) from the interbank market provided by the University of Chicago Center for Research on Security Prices. Five major currencies are selected in this Mudy:British Pound (RP), Canadian Dollar (CD), Deutsche Mark (DIM), Japanese Yen (JY), and Swiss Franc (SF). There are a total of 2510 daily observations, from 2 January 1974 to 30 December 1983.’ We calculate the rate of change by taking the logarithmic difference between the close of two successive trading days. For the first year of our sample, tb.e JY moved very little, if at all, which leads us to believe that the Japanese government might have pegged the exchange rate. We therefore calculate two sets of statistics for the JY, one using the ent%esample of 2510 observations, the other using only the last 9 years of data (2262 observations). Table 1 provides some summary statistics of the data. The means are quite small. The range of daily changes, however, is relatively high, the largest (in absolute value) being -7.0967 percent for the DM on 11November 1978. The data are unimodal and approximately symmetriG with higher peaks and fatter tails than the normal distribution. (Frequency plots are avai!ab!e corn the author upon request.) Noa-normality is confirmed by the ccefficients of kurtosis being larger than three for all five currencies. %ie data actuallybeginon 1 July 1973.We start our analysison 2 January 1974becausewe need some pre-sampledata in our statisticalmodel in section 7. A sample of 1 o’clock (New York)bid prices from 1977to 1983 were also available.Analysis shows that the two data sets behavedin very similarmanners.
D.A. Hsieh,
Daily foreign exchnge rates:1974-1983
131
Table 1 summary statisticsof log pricechan@s: r,= log(SJS,_ *)*100. Sampleperk& 2 January 1974to 30 December1983. BP
CD
-0.aB9 -0.0098 0.2234 -0.3149
-0.0184 0.0000 0.5921 -0.4136
Mean
Median Std. der. SkewnH$
DM O.tMOS O.tMOO
0.6372 -0.4249
JY
SF
0.0077 0.0171 O.oooO OSMIO 0.6260 0.7889 -0.2044 -0.2835
JY2’
0.0116 O.OWO 0.6307 0.1670
zz’
KUKtOSiS
Maximum Minimum studentized range NOBS
C$@)91 -7’0967 14.2052 2510
15.2892 2510
16.8906 2510
15*6!348
2510
14.5132 2510
14.0053 2262
Standarderrorsin squarebrackets. Nore:The standarderrorsare computed as follows: J6m
for the coefficientofskewness,
,/-a-
for the cu&cieat ofkurtosis. 1983.
l2 hlluary 1975 to 30 hxmber
Two explanations have been put forth to account for the fat tail distribution of exchange rate changes. One view is that the data are independent and identically distributed (iid), drawn from a non-normal distribution. A stable symmetric Paretian distribution was used in Westerlleld (1977), and the Student-t distribution in Rogalski and Vinso (1978). In addition to these distributions, Roothe and Glassman (1985) estimated a mixture of two normals with equal means but dill&rentvariances. A second view is that the data are not iid, but drawn from distributions which change over time. Friedman and Vandersteel(l982) suggested a normal distribution with time-varying means and variances. In order to discriminate between these two explanations, we divide the data into two equal subsamples. If the data w?re iid, the two subsamples should have the same distribution. This can be tested by comparing their empirical densities. Partition the event space into k mutually exclusive and collectively exhaustive regions, A 1,. . . , AL.Deline pi&) to be the probability of an observation in the first (second) subsample falling into region Ai. The null hypothesis of equal distribution is expressed as: Ho:pil=pi2,
for i=l,...,k.
132
D.A. Hsieh,Daily fopcisn
exchange rates
19744983
is a test of equality of two multinomial distributions, and can be accomplishedby using a &i-square stati~tic.~[See IIogg and Craig (1970, p. 312).] There is no clear-cut rule on chokng the number of regions (k) in a partition.In order to apply the results from asymptotic distributincsn theory, krease with the sample size. the numberof observations per region should 0n the other hand, the power of the test can be improvedonly by increasing the number of regions with the sample sixe. A possible candidate is k=& where n is the number of observations.This leads to a choice of k= 50 for our sample sixe. Kendall and Stuart (1967, vol. 2, p* 438) suggests k= n21s, which would be k=23. We tried values of 21,31,41 and 51, and found that the results are not sensitive to the choice of k. For a given value of k, we select the partition so as to equal& the number of observations(from the entiresample)in each region. [See Hogg and Craig (1970,pp. 365-366)) The chi-squarestatistics for k=21 and 51 are presentedin the first two rows of table 2. They show very clearly that the d&ibutions zre not equal in the two subsamplesfor all five fzurrenckr. It is possible that the rejection of the tid hypothesis results from halving the sample at 1979, which coincides with the change of the operating procedureof the U.S. Federal Reserve System. We thereforeconduct the iid test using two year subperiods. Table 2 shows that out of the nine SU~~~IX@~S, the iid hypothesisis rejeckd at the 1 percentsignificancelevel 4 times for BP, once for CD, 4 times for DM, 5 times for JY, and 6 times for SF. These results indicate that the rejection of the iid hypothesis for the entire sample period is not related to splittingthe sampleat 1979. This
The rejectionof the iid hypothesissuggeststhat the data are either serially dependent or not identically distributed.In this section we test for serial independence.Table 3 lists selective autocorrelationcoefficients,p(r), for the five currenks. No coefikient exceeds 0.1, implying that there is little serial correlation,which is in agreementwith the literature! Under the null hypothesis that the data are white noise with a constant variance, the standard error for each sample autocorrelationcoefllcient is 3Calderon-Rosselland Ben-Horim(1982) used the ‘Mann-Whitney-Wilcoxontest. Let F(x) be the distributionof X, and qy) tbc Qktribwic~~of I! The ati bypothcsis is F(z) = G(z) for d z, aad the akrnative hypothesk is F(Z)> G(Z) for alI Z, or F(z)
133
D.A. Hsieh, Daily fore&a exchange rates: 19744983
Table 2 Tests for idepmdenccandidentical distibutions, Sample period
BP
CD
DM
N
SF
JY2’
1974-83
308.79
i83.19
110.80
456.01
103.35
324.58
Two-yearsubplea 1974-75 1975-76 197677 1977-78 1978-79 19?9a 19m-81 1981-82 1982-83
(k=21):
3634
(0.0140) 4217 (O.ooza) 33.52 (0.0295) 59.88 (O.oooo) 49.65 (0.5309) (0.4186) (0.ooo3) 31.28 17.39 25m t&1794) 48.42 (OAm4) 4454 *I.* 34J6 (aoolq g&16) :0.2318) a37 16.,9 (0.1339) @ixloO) (0.6601)
49.17 53.21 (OmO3) (0.Ooo1) XI.76 49.47 (0.4114) (O.ooo3) 3267 90.64 (O*Oow W.0367) 64.14 108.05 (O.oooo) (O.ooOO) 21.75 59.46 (0.3542) (0.~) 24.01 18.19 (0.5749) w=O) 4267 40.67 (0.0041) (Om23) 2587 l!.E (09449) (61709) 37.92 40.69 (OsKt41) (0.0091)
Mar#inal~levelsh~~
.2 January1975to 30Deamber 1983.
1 n or 0.01996 for this sample. This would indicate that several coelkients n may he statistically dikent from zero. Furthermore,the null hypothesis implies that the sample autocorrelation coefficients are not correlated asymptotically. Thus, the joint test that the first K autocorrelation cxtefkients are zero can be conducted using the Box-Pierce Q statistic, Tz$=,p(.r)2, which is asymptotically a &square distribution with K degrees of freedom, In table 3 we report the test for the first SOautocorrelation coefkients. (The results are not sensitive to this particular choice of K.) The null hypothesis is rejected at the 1 percent signikance level for CD and JY (and JY2), at the 5 percent level for SF, and not rejected for BP and DM at conventional signigcance levels. These tests seem to indicate the presence of serial correlation in the data. +‘-a +L& conclusion. Heteroscedasticity ma] We caution the reader in aaxp,L, cause the standard error of each sample autocorrelation coefficient to be underestimated by fi. Diebold (1986) gives a heterogeilasticity-consistent estimate of the standard error for the zth sampk autocorre ation coefficient:
D.A. Usi&, Daily fmeign exchfwe rates 1974-BW3
134
Table 3 Autocoz&ation co&kknts ~et~ti~ty~asistent standarderrors]. CD
BP
. .i._ ma
+E!
-ao638b-0.0569
SF -0.0410
1
-0.0216
cQJ=W -0.0025
CW&l
2
caO2931 -0.0075
$=I
3
CamI -0.oM4 [;=I
4
5
Cda2603 -0.0229 cao=l
-a0244
-0.0074
[O&9]
W&l
6
-0.0021
7
-0.0124 [O.w23] 40065 -0.0155 v0.q
[O&261] -0.0074 caoucr? c;Fz
@J&I
co*o2W
8 9 10 20 30 40 50 BOX-pierce
QWJ
Adjusted B
Ctg:;]
%z ttz
W-0-T
[O.O22l]
-a0290 rmwbi L---- -a 0.0385
CQQI
%zl
!ggl -0.0238 caO216l [O&3] caom -0.033 -a0203 -a0405 [a02171 [:.;I %E1 -a0083 W&l %z cKz? -0.0267 ph225] [Ob212] [Ob219] [0.0216] 51.62 77.52 60.67 98.66 (0.4103) (0.0076) (0.1434) (O.Ow 39.26 (0.8631)
54.33 (0.3130)
ph254]
[OD237j
-a0004
-%Z k&l
-0.0259
[0.0253] lP.oz293 70.21 102.85 (ao312) (O.Oaw
46.59
69.70
(0.6110)
(a03421 (a6238)
4627
71.29 (0.0256)
@itlana levels in parentheses. 2 January 1975 to 30 Jkwnk 1983. bsitly dihcnt from zero at the 1 percentlevel (0nHaibd test).
Mm
S(7) = JC MN
1+ yx2(7)/0*),
(2)
where ~~47) is the rth sample autocovariance of the squared data, and u is the sample standard deviation of the data. These adjusted standard errors are reported in table 3, and they show that few of the autocorrelation coekients are statistically different from zero. Diebold (1986) shows that the adjusted Box-Pierce Q statistic, ptoticaliy a &i-square distribution with K degrqs
D.A. ifsieh, Daily fmeign exchangerates:1974-1983
135
of freedom. This is also reported in table 3. The statistics do not reject the null hypothesis of serial independence for the BP, CD, DM and SF data. There appears to be some autocorrelation in JY and JY2, at the 5 percent but not the 1percent signil’lcancelevel. These results indicate that the rejections of serial independence using the standard &sting procedure were caused by heteroscedasticity in the data. Furthermore, even if there is serial correlation in the data, it is of such a small magnitude that it cannot account for the strong rejections of the iid hypothesis. 5. Test& for the ‘day of the week’ e&t The results of the serial correlation test indicates that the rej&ion of the iid hypothesis cannot be caused by serial dependence. We therefore turn our attention to whether the rejection can be explained by changing distributions. One possibility is that the distribution of daily spot rate changes may be differentbetween weekdays and weekends. In the stock market literature, the return to holding stocks between Priday’s close and Monday’s close is shown to be diffbrentfrom the returns on other days of the week.s To check whether the distribution of exchange rates actually changes across days of the week, we divide the data into 5 subsamples according to the day of the week. Within each subsample, we assume that the data are iid, and test for equality of distribution across subsamples using the &i-square test. These pairwise tests are reported in the upper half of table 4. Seven out of the 25 statistics are significant at the 1 percent level, with 8 more sign&ant at the 5 percent level. We therefore reject the null hypothesis that the distribution is iid for each day of the week and iid across different days of the week. To check whether this ‘day of the week’ effect is responsible for the rejection of the iid hypothesis, we split each subsample into two halves and perform the &i-square test. The results are in the lower half of table 4. Fourteen out of the 25 statistics are significant at the 1 percent level, with 5 more sign&ant at the 5 percent level. Thus, even though the distribution of exchange rates is different across the days of the week, it cannot account for the rejection of the iid hypothesis. 6. +lbting for time-varying merursurb vatiuees Another possibility of unequal distributions in the data is that means and variances change over time. This was offered by Friedman and Vandersteel (1982) as an explanation of the fat tail distribution of spot rate changes. They %ecFrench (19&o),Gibbons
and Hess (19$1), and Keim and L3i.wihugh (P984).
DA Hsieh,Daily&w&n exchqe ratesa. 19744983
136
Table 4 Testofthe‘dayoft.heweek’e&ct. BP
CD
DM
~testforiidacromdayaoftheweck(k=21) Monday WL TRY Tuesdayvsa W-Y wedm!sdayvs. TkdY Thunldayva 3Om Friday Fridayvs. Monday Teatof iid within each subsampleii = 2n) 23.a 68.18 17.12 M&Y T-day Wt!&lMShY m&Y
Friday
JYr
JY
SF
33.02 (O-0336) aI.14 (a44921 24.43 (O=l) m.13 (0.4498) 27.28 (0.0385)
23.91 W-2463) 23.67 (OJm 33.15 (QO3rn
75.08 (-1 86.32 WlW S4.11 (aoooo) 8852 (m 8934 taoooo)
39.57 76.49 (aoooo) (aoooo) 13.58 65.61 (0.1930) (aoooo) 54.71 13.17 (02143) (aoooo) 83.20 28.38 (aoozr) (asooo) 71.96 2OA8 WW (aoooo)
22J3 (0.3335) 2uO (Ql499) 35.50 (aOr76) xm 19.04 (0.0218) (Omx2) 72% mm
Mar8iMl~kvelsinparul~ ‘2 January1975to 30 Jhcember 1983.
never directly tested this hypothesis. Instead, they used some moving statistics to summarize these changes in the data Calderon-RosseMand Ren-Ho&n (1982)checked whether changing means and variances could account for the rejection of the iid hypothesis. They divided their data into three subperiods of about 365 daily observations each. They assumed that the means and variances are constant within each subperiod, and tested for differencesacross subperiods. They faund that shifts in mean accounted for 9 rejections out of 13 currencies, but shifts in variances accounted for none of the remaining 4 cases. They, too, never directly tested for changing means and variances. Follotig the procedures in Calderon-Rossell and Ben-Horim (1982), we first checked whether changing means and variances can account for the rejection of the iid hypothesis. Rather than assuming constant means and variances over time intervals as lengthy as 18 months, we assume that means and variances are constant within each month. (An examination of the moving statistics in Friedman and Vandersteei reveals that means and variances change substantially over time.) As a maintained hypothesis,
D.A.Hsieh,Daily@reignexchangerata 1974-1983
assuming that
we arc
data are serially independent
the
137
for the remain& of
this !3ectioIL To check whetherchangesin monthly means can account for the rejection of iid, we center the data within a month by subtractingthe montMymean. We then recompute the &i-square test of iid using the centered data. As shown in the first row (test 1) of table 5, the statistics are stilI highly sign&ant. Clearly, changes in monthly means alone cannot be responsibk for the @a%@ of iid To check whether changes in monthly varianm can account for the r#tion of iid, we M&C the data within a month by dividing by the monthly standard deviation. The results of the &i-square tests are in the second row (test 2) of table 5. These statistics arc much smaller than their counterpar@in the prc&ing row, although all but the CD are still Table5
Testdequdityofmonthlymeansandvuirnoer BP
CD
Equdity of4!istribo~ (b21) Ttil 266s
c2 c!?! x3(20, Jiql8lityofWddtert zvl91
taoooo)
DM
~~1~7237 to-1 38.17
48.50 (0.1533) [O-I lOA 4433 (0-@)131 e958-41 acxa68months 199.75 lN86 (Odwl) (_
(0-1 25.17 (0.1950) 159.09 (-3)
IY
SF
37428 (O.oooo) 78.94 (0*88w 6821 (O=w 181.46
to-1
EaulitY of varhnes-months 8X33 1074All 127732 z (O_ (OAOOO) (0.08w 4.47 4.16 435 E&O] wJ8w (0*88w (O_ 4.72 (Onmoo) rthrgid~_~inpafcn~
02hmlary 1975to 30 Daembef 1983. .
&~tyofdisteuti~ Test 1: Data axecenteredh c&radng monthly meaw deviations. Test2 DataarescakdbydhidingbymonthIy Test% Dataarcstddidbymonthlymumsand~~tio~ Eqdity of vacross months: LcvcneI: M~Levffletest,using47.5procnttrimmcdmeaas. LCVCIICIT: ModiciedLevenctcst,~glOpercent~meaerrs.
mm
Jy2
43.10 (OJ=v 20.69 (0.4lw
263.45 (-1 5264 (O.aKu) 49.74 (0-88021
13629 (0.1327)
165.35 (OiIO32)
to-am
1137.85 to=) X01 (O.oooo)
138
D.A. I&i&, Daily
/me&n
exchange
rates:
19744983
sign&ant at the 1 percent level. It appears that changes in monthly variances can partly account for the rejection of iid. I’o check for the combined effects of changing means and variances, we standardixe the data within a month by subtracting the monthly mean and &idhg by the monthly standard deviation. The results of the &i-square tests are in the third row (test 3) of table 5. These statistics are dramatically lower than the two preceding rows in some cases. The CD, DM and SF are no longer sign&ant, even at the 10 percent level. The BP and JY are still significant at the 1 percent level. It appears that changing means and variances across months can jointly account for the rejection of iid, at least for the CD, DM and SF. To verify that the means and variances actually change over time, we perform some direct statistical tests. Again, we assume that means and variances are constant within each month, and test whether they are equal across the 120 months of data. The test of equality of means is best explained in a regression context. The daily rates of change are w on a constant term and 119 dummy variables, one for each of the 119 months after the first month. If the means arc constant across months, then the coeliicients for all 119 dummy variables should be xero. We test this with the Wald statistic, employing a covariance estimator which is consistent against a large class of heterosce&sticity.6 The statistics are in the fourth row of table 5. The equality of monthly means is rejected for the CD, DM and JY at the 1 percent level, and for the BP at the 5 percent level. It is not rejected for the SF7even at the 10 percent level. This evidence show that means are indeed not constant over time. The test for the equality of variances is more problematic, due to the heavy tails of the distribution. The Bartlett test, which is the standard test of homoscedasticity, is a likelihood ratio test for normally distributed data. It is given in the fifth row of table 5. Since there is strong evidence against normality, we prefer to use tests that are robust against fat tail distributions, such as the mod&d Levene test described in Brown and Forsythe (1974). These are labelled ‘Levene I” and ‘Levene II’ in the last two rows of table 5. The first procedure centers the data within each month by the 47.5 percent trimmed mean, and the second procedure uses the 10 percent trimmed mean. Brown and Forsythe (1974) show that these two procedures perform very well in Monte Carlo experiments under the normal distribution as well as fat tail distributions. Their results also indicate that the Bartlett test rejects homoscedasticity too frequently, whenever the distribution has fatter tails than the normal. As is evident in table 5, the equality of monthly variances is rejected at any conventional significance level for all currencies regardless of the test procedure. ‘See White (1980), Hansen (1982) and Hsieh j1983) for more details.
D.A. H&h, Doily fbrei8n excke
rties: 19744983
139
7. A statistid model of dally exchangerates
It would be of interest to try to describethe data with a statisticalmodel, In particular,the change in variancesappearsto be the strongestcharacteristic in the data. A succeti model of how varianceschange over time could have importantapplicationsin the pricingof foreigncurrencyoptions, which d=qendscriticallyon expectationsof futurevariances. To achieve this purpose,we adopt a version of the ARCH model in Engle (1982).This was used by Domowitx and Hakkio (1985) to model risk premia in the forwardexchange market. Although Domowitx and Hakkio justified their model with theoreticalconsideratiom it is not the case here. We choose this model for its computationaltractability. Our specificationis as follows. Let rt= log(S& I) be the rate of &ange of spot rates between two tradingdays. Conditioningon all informa?ionat time t- 1, we assume that r, is normallydistributedwith mean # and variance& where
(3)
+b
(4)
where &, BTn Dw, and Drptare dummy variables for Monday, Tuesday, Wednesday and Thursday. HOL, is the number of holidays (excluding weekends)between the (t - 1)st and tih tradingday. The log-likelihoodof the datais: . II
log+#log27c-~ r=1 +
P [
r,-_lcr
Q
* l
r
11 /
The maximumlikelihoodestimatesof the coel&zientsare obtainedby using a numerical optimixation procedure described in Rerndt, Hall, Hall and Hausman(1974).f The two lag lengths (P and Q) are chosen to maximize(5). Table 6 contains the estimated parametersand the optimal choice of Q ‘Note that r, will not be independent of past obwvations, since its mean and variance depends on them_ b&wm liieliiood estimation of the (unconditional) distti?x?kn s! rg is the&ore not appropriate. OIIthe other hand, the transformed series (r,-&kg are iadepdent of each other, and hence maximum likelihood estimation is appropriate.
140
D.A.H&h, Dailyfoteignexchange rates 19M-1983 Table6 Jzstimation results.
BP
CD
DM
JY2
SF
[Om8233]b -OEWO1
co.=9841
-0.097387 [O.O867U] -0.048588 c:zE? ~0.022873] 0.033939
‘ZEZ?
[0.018994] -!MM9o79
likclillood
- 1920.23
517.38
-2029.79
- 1814.29
standarderrorsin squarebrackets. *2 January1975to 30tleccmh 1983. %gni!kmtlydiliixcntfromzeroat the 1 percentlevel(one-tailed test).
-2570.37
D.A. Hsieh, Daily jimign exchange mtec lY74-fY83
141
and P. Few parameters in the mean equations have any statistical signifkance. Hence the model is very poor in predicting the &e&on of
exchange rate changes, evca within sample. On the other hand, quite a mmber of parametersin the variance equation are statistically sign&ant. The most important variablein the variance equation is the autoregressive term, 6. In all five currencies,6 is significantlymerent from xero at the 1 percent level. In the case of CD, DM and SF, 6 is less than unity. This means that the effa of a large variance on future varianceswill eventually dampen out. In the case of BP and JY, 6 is greaterthan unity. This means that the elfects of a large VG&%Z-on future variances will be amplified, which is clearly undesirable.As we shall see below, this model fits the data for CD, DM and SF, but not for BP and JY. In addition,the coe@cientsfor the Monday dummy (Vd and the holiday dummy (I$,) in the variance equation are large and sign&ant. The coefllcients for the other daily dummies (I$, Vw and Vn)are typically smaller, although some are also significant This means that the variancesof daily exchange rates are larger whenever the trading days spau a weekend or a holiday.Notice, however,that it is importantto distinguishbetween the two effects.Theincmase in variancedue to a weekend (which has two calendar days)issmallerthantheincreaseinvariance due to a holiday on a weekday (which typicallyhas one calendarday), in the case of the CD, DM and SF. After estimation, we perform some diagnostics to check how well the model fits the data We use the standardixcdresidual (6) e,=(r,-Pt)/6, where P, and a# arc the estimated means and standard deviations. If the model is correctly specSed, then e, should be approximately iid, normal, with 2e~omean and unit variance. Table 7 presents some statistical properties of these standard&ed residuals. The means are close to zero, and the standard
deviations are close to one. There is little evidence of serial correlation.But there is evidenceof non-normality,as indicatedby the coefllcientsof kurtosis being largerthan three. To test for iid of the residuals, we again employ the &i-square test describedin section 3. The results in the first row of table 8 show that the BP and JY reject iid at very high significancelevels. The SF rejects iid at 274 percent,while the CD and DM do not reject at 10 percent.This test shows that the model does not fit the data for BP and JY.* If we test iid across days of the week ancjwithin each day of the week, we also find much lower re@tion rates than the raw data? However, the rejectionof iid does ~wasnotsurprising,;clviewofohereSultsinthethirdrowoftable5. pWe compute the W@@~~ucc levels of the tests as if we knew the true parameter VOWS,
This
is &djf
a0t
tw
since we are using
estimated
parameter
values.
-The a@stnm~,
however, must be very small, because of the urge number of liegfees of freedom for our .
142
Statistical propertim of stmdardhd DM
CD
BP
15.2587 Test of serial cor&tion Box-Pierce 35.25 (0.9433) QUO)
SF
JY2
-0.018 -0.0149 l.oooO 0.1057
-0.0107 -0.0321 0.9999 0.1982
-0.#98 0.0078 0.9999 -0.3142
Meau Median Std. dev. Skewness
residuals..
10.1688
11.6742
129955
125421
39.42 (0.8589)
37.27 (0.9086)
47.06 (0.5921)
39.33 (0.8613)
StfUhddenorSillsqUarebraclcetg Marginal signi6eaneelevels in parentheses. ‘2 January 1975 to 30 Dazmbcr 1983. Table 8 Test for equality of distribution ofstamhhd BP
CD
Equality of distribution (&=21) 98.55 24.43 maays waKI) (0.2241) x3cz0, Test of iid across we&days (k=21) Monday vs. 14.77 23.82 (tL7894) g.tJ3) T-Y Tuesday vs. 26.04 WedlEIday Wednesday vs. n&Y Thursday vs. Friday Friday vs. Monday Test of iid within we&days (k=21) 19.88 54.30 Mondays (O.OOoO) (0.0304) Tuesdays 14.75 27.08 (0.0025) (!8414) WedIleXhIyS 13.17 (0.2143) (O&14) Thursdays 14.78 29.74 Fridays __
_ _.
.-
residuals.
DM
1T2
28.18 (0.1052)
91.84 33.81 (0.OQoO) (0.0274)
20.47 (0.4289) 11.92 @Z8”)
16.38 (0.6928) 17.35 (0.6301) 24.67 (0.2144) 16.47 (0.6871) 20.32 (Q4381)
18.38 (0.5624) 40.91 (O.NI38) 1487 (0.6614) 19.89 w46w 23.46 (0.2668)
35.59
31.16 (g%)
G948) 18.36 (0.5637) 13.51 (0.8544)
22.56 (0.0125) 12.47 (0.2548) 12.49 (0.2536) 14.26 @161§) 13.94 (0.1757)
to.ooaz) 22.54 (oat 26) (E42) 32.04 (3.m) 11.50 (0.3199)
SF
($:85) (Oh57) 19.48 (0.0346) 10.96 (0.3606)
D.A. Iisieh, Daily j’oreign exchange rates: 197c1983
143
not come from shifts in means and variances. Test for the equality of monthly means and variancesof the residualsare in table 9. The equality of monthly means is rejectedonly for the CD at 3.26 percent.The equality of monthly variancesusing the modifiedLevene tests is rejectedonly for the BP at 4.34 percent.
In our investigation,we find that: (1) Exchangerate changesare not independentand identicallydistributed (2) Each day of the week may have a differentdistribution,but this is not suflicientto explain the rejectionof iid (3) Thereis little serial correlationin the data. (4) Means and variances change over time. Jointly they may be able to explain the rejectionof iid for at least rlrreecurrencies- CD, DM and SF. Furthermore,we constructedan ad hoc statistical model to capture these proper& In certain aspects this attempt was suu~&~L For all five currenciesthe model was able to capture the change in means and variances over time. Based on the small numberof statisticallysignificantcoefficientsin the mean equation, we believe that the model would not be able to predict directionsof exchange rate changes. However, based on the large numberof statistically significantcoe&ients in the variance equation, we believe the model should be able to predict volatility. In other aspects, the model was less s~ful. It was able to account for the rejection of iid for CD, DM and SF, but failed to do so for BP and JY. Furthermore,the standardized Table 9 Test of equality of monthly means and varianas for standard&d resi&&. BP
CD
Test of equality of monthly means 148.99 WaldtCSt 133.20 (0.0326) (0.1764) x2(119) Test of equality of monthly variance 422.11 197.34 &utktt (O-000@ (0.Qooo) x2(119) 1.12 0.95 Levenei (0.1820) (0.6342) F(119,239O) Levene II fl119,2390)
I.24 (0.0434)
1.10 (0.2213)
Marginal sigaifiwcc level in parentheses. “2 January 1975 to 30 December 1983.
DM
Jw
SF
139.09 (O.looa)
117.06 (0.5332)
122.33 (0.3986)
216.45
212.19 (0-m) 0.79 (0.9528)
236.08 (0-m)
wQw 1.01 (0.4542) (z6)
0.87 (0.8380)
(E47) 1.15 (0.1325)
144
D.A. Hsieh, Daily jkign exchangerates 19741983
residualsfor all five currenciesare non-normal,even though they have less kurtosis than the raw data.lo Time-varyingmeans and variances therefore are not sufficient to Mly account for the leptokurtosis in exchange rate changes. The search for an appropriatedistribution will be left for future research. ‘@I-heissue of wtion arises wllen the distribution of (r#-rcl)!o f uo#nMLaL genaal, a n&qe&catioll of the distributionfunction may lead to kWnsu&n parametersof interest.In the present context, however,such &on&en& may not a&e. The reaPonisthat~~liltetibood~ganarmal~~~~~t~~of means and warii~ even if the true distributionis not normal, providedcertain conditions are met. [See White (1982) and Gourieroux,Monfortand Tro8non (1984).]
Refm Berndt,ErnstK,B~~H.Hall,RobertE.HallandJenykHo~1974,~end iniluence in nonlinear structuralmadels, Annah of Economic and Social Measurement4, 653-665. Boaz&e,R$ ~~ Glaspman, 1985, The statist&d distribution of exduuQe ratep: economic impllcations#unpublished manlMuipt @epWaWnt of Econont& Universityof Alberta) BroJyc”thtB. and. Alan & Fomythe, 1974, Robust tests for the equality of variwcs, Amencan statrsticel Associah
69,36&367.
~John,FndRKaenaadG.~Bootb,1917,Fotei;gnmarlteteagciencyurrderfladMe exchangerate&JournalofFinance 3291325-1330. Calderon~Rossell, Jorge R and Foshe Ben~&rim, 1982,The behaviorof foreignexdtange rates, Journalof InternationalBusmessStudies 13,99-W. Cornell, Bradford,1977, Spot rates, forward rates, and market elRcieney,JournaI of Finan&l EconomicsS,55-65. Diebold, Francis X., 1986 Testing for serial correlationin the presenceof ARLCZunpublished mantWript(ul&ersity of Pennsylvania). Domowitq Ian and Craig S. Hakkio, 1985, Conditional variamx andtheriskpremiuminthe foreignexchange market,JournalofInternational Economics 19.47-66. Engle, Rohert F, 1982, Automgressiveconditional hctenwcedasticiy with estimates of the variamz of United ICiqIom inflation, u SO,987-107. French,KennethR., 1980,Stock returnsand the weekendeffect,Journalof Financial Econti 8,569. Friedman,Daniel and Stoddard Vans 1982, Short-run fluctuations in foreign exchange rates,Journal ofInternational Economics 13.171-186. Gibbons, Michael R. and PatrickHess, 1981,Days of the week effectsand asset returns,Journal of Business54,579-596. Giddy, Ian H. and Gunter Dufey, 1975, The random behz&r of flexible exskange rates: Implicationsfor forecasting,Journal of Inteeational BusinessStudies 61-32. C+%u.uieroux, Christian, Alain Monfort and Alain Trognon, 1984, Pseudo maximum likelih~ methods:Theory, Econometrica52,681-?!D Hansen, Lars E., 1982, Large sample propertiesof 8enemlized method of moments estimators, Econometrica54 1029-1054. Hog& Robert V. and Alien T. Craig, 1970, Introductionto MathematicalStatistics (Mad#an, New York). liaieh, David .4., 1983, A heteroseedasticity-consistent covariaucematrixestimatorfor time series rqrcssions, Journal of Econometrics22281-290. Kceir;l, Donald B. and Robert F. Stambaqh, 1984,A furtherinvestigationof the weekend effect in stock retsms, Journalof Finance 39,819-837.
D.A.Hsieh,Dailyfirm
exchangerace=1974-1983
145
Kendall, Maurice G. and Alan S. Stuart, 1967, Tble advanced theory of statistics, Vol. 2, Infcrene and relationship(Hafncr,New York), Loguq Dumia E and RichardJ. Sweeney,1977,whitenow in impcrfbt markets:The CBSC of the ti!&!lar exchangeF&F+Jowl! of Fiaane 32,761-768. Logue, Dennis E., RichardJ. Sweeneyand Thonw D. Willctt,1978,Speculativebehavior of foreignexdumge caterduringthe currentfloat,Journald BusinessResearch6,159-174. Rogabki,RichardJ. and JosephD. Vinw, 1978,Empiricalipropertiesof foreignexchangerates, JoumalofIntemationalBwim1Studiu9,69-79. W~JsnictM,19n,Aoeraminotionobfordgrcnchangeriskunderfixcdarndfloa~g rate e JournalofInternational Economics 7.181-200. . . . t wvariance estimator and a direct test for Whitc,Habrt,1%O,Ah&nw&&a +~USIMO ty: &xnwmex r% 8r7-838. white, HaIl%Z Maximumlike&hoode&nation of mbpe&ial models,EconometricaSO, l-26.
THE S’MK!!.!!.SCAL PROPER’IIES OF DAILY FOREIGN EXCHANGE RATE& 1974498ii’
David A. HSIEH* uniQers&y qfchicugo, C&ago, IL 4M37, USA ReceivedMay 1986,revisedversionreceivedJune 1987 nispaperexaJnilm thcstatisticalpropertksofdaily~ofdrangeoffiveforcigncumndes froJn1974 to 1983.The main purposeir to disrimiMte bctw@Jltwo competingcxplanati0Il~for theobaervadBiea~taitsotthe~~~~thor~datasre~~y~~lromah~~ tail distribaltionwhich tanainsfinsdo~time,andthatthedata~ftomdistributionswhich vary over time. Evidence pint to the r&&n of the Brst hypothcsk Further invcatigatioas showthattbefejcctioncaabeattriitochangingmcaasandvaknce3 illfkdlitf&Whidl canbcdcscribalbyasimpk-model.
1. Intmhth
This paper examines the statistical propertiesof daily rates of change of the prices of five foreign currenciesfrom 1974 to 5983. Knov&dge of this distributionhas importanteccnomic applications:the effectsof exchangerate movementson internationaltrade and capital flows, mean-varianceanalysis of international asset portlolios, and the pricing of options on foreign currencies.In previous studies in the literature,1the concensus is that the distributionof these changesis unim~ *mdhas fatter tails than the normal distribution.It is this second feahulewhich has attractedthe most attention. Two competing explanations have been put forth to munt for the leptokurtosis.One view suggests that the data are independentlydrawn from a fat tail distributionthat remainsfixed over time, while the other proposes come fromdistributionsthat vary over time. There have been few attempts to determinewhich of the two explanations better characterizesthe data. Friedman and Vandersteel(1982) considered WBC author is grateMto Ian Domcwitz, RobertFlood, Jacob Frenkel,Lazs Hansen, Robert Hodrick,Jolm Hukinga, Michael Mussa, Arnold Zellner, the part&pants of the University of Chicago Inuernational Economics Workshop, the Northwestern University 111temational EconomicsWorkshop,smd the NBER Mini-C!onference on InternationalAsset Pricing(1986)for hclpfid comments on aulier draf4s.This re#arch was supported by the Universityof Chicago GraduateSchool of Business The data were provided by the Universityof Chicago Centerfor RescarchonScurity Prices. ‘Sac Burt,Kaen and Booth (1977),WesterGeld(1977),Rogalskiand Vinso (1978).
130
D.A.iisieh, Daily fkign
excha?&erate: 39744983
the following three hypotheses: (a) the data are independent and identically distributed (iid), drawn from a stable symmetric paretian distribution; (b) the data are iid drawn from a mixture of two normal distributions; and (c) the data are independently drawn from a normal distribution whose mean and variance change over time. They found evidence in favor of the third hypothesis. Calderon-Rossell and Ben-Horim (1982) tested directly for the iid property without requiring distributional assumptions. They found that they can reject iid for most currencies. They attributed the rejection to shifts in the mean in 9 out of 13 currencies, and to changes in the skewness in the remaining cases. This paper extends the previous work in several ways. We use ten full years of daily data to investigate in detail the cause of rejection of the iid hypothesis. We find that not only do the distributions of exchange rates differ across days of the week, but that their means and variances also shift over time. Furthermore, we are able to construct a simple statistical model to describe these time-varying means and variances.
ZTbedata The data consist of daily closing bid prices of foreign currencies (in terms of the U.S. dollar) from the interbank market provided by the University of Chicago Center for Research on Security Prices. Five major currencies are selected in this Mudy:British Pound (RP), Canadian Dollar (CD), Deutsche Mark (DIM), Japanese Yen (JY), and Swiss Franc (SF). There are a total of 2510 daily observations, from 2 January 1974 to 30 December 1983.’ We calculate the rate of change by taking the logarithmic difference between the close of two successive trading days. For the first year of our sample, tb.e JY moved very little, if at all, which leads us to believe that the Japanese government might have pegged the exchange rate. We therefore calculate two sets of statistics for the JY, one using the ent%esample of 2510 observations, the other using only the last 9 years of data (2262 observations). Table 1 provides some summary statistics of the data. The means are quite small. The range of daily changes, however, is relatively high, the largest (in absolute value) being -7.0967 percent for the DM on 11November 1978. The data are unimodal and approximately symmetriG with higher peaks and fatter tails than the normal distribution. (Frequency plots are avai!ab!e corn the author upon request.) Noa-normality is confirmed by the ccefficients of kurtosis being larger than three for all five currencies. %ie data actuallybeginon 1 July 1973.We start our analysison 2 January 1974becausewe need some pre-sampledata in our statisticalmodel in section 7. A sample of 1 o’clock (New York)bid prices from 1977to 1983 were also available.Analysis shows that the two data sets behavedin very similarmanners.
D.A. Hsieh,
Daily foreign exchnge rates:1974-1983
131
Table 1 summary statisticsof log pricechan@s: r,= log(SJS,_ *)*100. Sampleperk& 2 January 1974to 30 December1983. BP
CD
-0.aB9 -0.0098 0.2234 -0.3149
-0.0184 0.0000 0.5921 -0.4136
Mean
Median Std. der. SkewnH$
DM O.tMOS O.tMOO
0.6372 -0.4249
JY
SF
0.0077 0.0171 O.oooO OSMIO 0.6260 0.7889 -0.2044 -0.2835
JY2’
0.0116 O.OWO 0.6307 0.1670
zz’
KUKtOSiS
Maximum Minimum studentized range NOBS
C$@)91 -7’0967 14.2052 2510
15.2892 2510
16.8906 2510
15*6!348
2510
14.5132 2510
14.0053 2262
Standarderrorsin squarebrackets. Nore:The standarderrorsare computed as follows: J6m
for the coefficientofskewness,
,/-a-
for the cu&cieat ofkurtosis. 1983.
l2 hlluary 1975 to 30 hxmber
Two explanations have been put forth to account for the fat tail distribution of exchange rate changes. One view is that the data are independent and identically distributed (iid), drawn from a non-normal distribution. A stable symmetric Paretian distribution was used in Westerlleld (1977), and the Student-t distribution in Rogalski and Vinso (1978). In addition to these distributions, Roothe and Glassman (1985) estimated a mixture of two normals with equal means but dill&rentvariances. A second view is that the data are not iid, but drawn from distributions which change over time. Friedman and Vandersteel(l982) suggested a normal distribution with time-varying means and variances. In order to discriminate between these two explanations, we divide the data into two equal subsamples. If the data w?re iid, the two subsamples should have the same distribution. This can be tested by comparing their empirical densities. Partition the event space into k mutually exclusive and collectively exhaustive regions, A 1,. . . , AL.Deline pi&) to be the probability of an observation in the first (second) subsample falling into region Ai. The null hypothesis of equal distribution is expressed as: Ho:pil=pi2,
for i=l,...,k.
132
D.A. Hsieh,Daily fopcisn
exchange rates
19744983
is a test of equality of two multinomial distributions, and can be accomplishedby using a &i-square stati~tic.~[See IIogg and Craig (1970, p. 312).] There is no clear-cut rule on chokng the number of regions (k) in a partition.In order to apply the results from asymptotic distributincsn theory, krease with the sample size. the numberof observations per region should 0n the other hand, the power of the test can be improvedonly by increasing the number of regions with the sample sixe. A possible candidate is k=& where n is the number of observations.This leads to a choice of k= 50 for our sample sixe. Kendall and Stuart (1967, vol. 2, p* 438) suggests k= n21s, which would be k=23. We tried values of 21,31,41 and 51, and found that the results are not sensitive to the choice of k. For a given value of k, we select the partition so as to equal& the number of observations(from the entiresample)in each region. [See Hogg and Craig (1970,pp. 365-366)) The chi-squarestatistics for k=21 and 51 are presentedin the first two rows of table 2. They show very clearly that the d&ibutions zre not equal in the two subsamplesfor all five fzurrenckr. It is possible that the rejection of the tid hypothesis results from halving the sample at 1979, which coincides with the change of the operating procedureof the U.S. Federal Reserve System. We thereforeconduct the iid test using two year subperiods. Table 2 shows that out of the nine SU~~~IX@~S, the iid hypothesisis rejeckd at the 1 percentsignificancelevel 4 times for BP, once for CD, 4 times for DM, 5 times for JY, and 6 times for SF. These results indicate that the rejection of the iid hypothesis for the entire sample period is not related to splittingthe sampleat 1979. This
The rejectionof the iid hypothesissuggeststhat the data are either serially dependent or not identically distributed.In this section we test for serial independence.Table 3 lists selective autocorrelationcoefficients,p(r), for the five currenks. No coefikient exceeds 0.1, implying that there is little serial correlation,which is in agreementwith the literature! Under the null hypothesis that the data are white noise with a constant variance, the standard error for each sample autocorrelationcoefllcient is 3Calderon-Rosselland Ben-Horim(1982) used the ‘Mann-Whitney-Wilcoxontest. Let F(x) be the distributionof X, and qy) tbc Qktribwic~~of I! The ati bypothcsis is F(z) = G(z) for d z, aad the akrnative hypothesk is F(Z)> G(Z) for alI Z, or F(z)
133
D.A. Hsieh, Daily fore&a exchange rates: 19744983
Table 2 Tests for idepmdenccandidentical distibutions, Sample period
BP
CD
DM
N
SF
JY2’
1974-83
308.79
i83.19
110.80
456.01
103.35
324.58
Two-yearsubplea 1974-75 1975-76 197677 1977-78 1978-79 19?9a 19m-81 1981-82 1982-83
(k=21):
3634
(0.0140) 4217 (O.ooza) 33.52 (0.0295) 59.88 (O.oooo) 49.65 (0.5309) (0.4186) (0.ooo3) 31.28 17.39 25m t&1794) 48.42 (OAm4) 4454 *I.* 34J6 (aoolq g&16) :0.2318) a37 16.,9 (0.1339) @ixloO) (0.6601)
49.17 53.21 (OmO3) (0.Ooo1) XI.76 49.47 (0.4114) (O.ooo3) 3267 90.64 (O*Oow W.0367) 64.14 108.05 (O.oooo) (O.ooOO) 21.75 59.46 (0.3542) (0.~) 24.01 18.19 (0.5749) w=O) 4267 40.67 (0.0041) (Om23) 2587 l!.E (09449) (61709) 37.92 40.69 (OsKt41) (0.0091)
Mar#inal~levelsh~~
.2 January1975to 30Deamber 1983.
1 n or 0.01996 for this sample. This would indicate that several coelkients n may he statistically dikent from zero. Furthermore,the null hypothesis implies that the sample autocorrelation coefficients are not correlated asymptotically. Thus, the joint test that the first K autocorrelation cxtefkients are zero can be conducted using the Box-Pierce Q statistic, Tz$=,p(.r)2, which is asymptotically a &square distribution with K degrees of freedom, In table 3 we report the test for the first SOautocorrelation coefkients. (The results are not sensitive to this particular choice of K.) The null hypothesis is rejected at the 1 percent signikance level for CD and JY (and JY2), at the 5 percent level for SF, and not rejected for BP and DM at conventional signigcance levels. These tests seem to indicate the presence of serial correlation in the data. +‘-a +L& conclusion. Heteroscedasticity ma] We caution the reader in aaxp,L, cause the standard error of each sample autocorrelation coefficient to be underestimated by fi. Diebold (1986) gives a heterogeilasticity-consistent estimate of the standard error for the zth sampk autocorre ation coefficient:
D.A. Usi&, Daily fmeign exchfwe rates 1974-BW3
134
Table 3 Autocoz&ation co&kknts ~et~ti~ty~asistent standarderrors]. CD
BP
. .i._ ma
+E!
-ao638b-0.0569
SF -0.0410
1
-0.0216
cQJ=W -0.0025
CW&l
2
caO2931 -0.0075
$=I
3
CamI -0.oM4 [;=I
4
5
Cda2603 -0.0229 cao=l
-a0244
-0.0074
[O&9]
W&l
6
-0.0021
7
-0.0124 [O.w23] 40065 -0.0155 v0.q
[O&261] -0.0074 caoucr? c;Fz
@J&I
co*o2W
8 9 10 20 30 40 50 BOX-pierce
QWJ
Adjusted B
Ctg:;]
%z ttz
W-0-T
[O.O22l]
-a0290 rmwbi L---- -a 0.0385
CQQI
%zl
!ggl -0.0238 caO216l [O&3] caom -0.033 -a0203 -a0405 [a02171 [:.;I %E1 -a0083 W&l %z cKz? -0.0267 ph225] [Ob212] [Ob219] [0.0216] 51.62 77.52 60.67 98.66 (0.4103) (0.0076) (0.1434) (O.Ow 39.26 (0.8631)
54.33 (0.3130)
ph254]
[OD237j
-a0004
-%Z k&l
-0.0259
[0.0253] lP.oz293 70.21 102.85 (ao312) (O.Oaw
46.59
69.70
(0.6110)
(a03421 (a6238)
4627
71.29 (0.0256)
@itlana levels in parentheses. 2 January 1975 to 30 Jkwnk 1983. bsitly dihcnt from zero at the 1 percentlevel (0nHaibd test).
Mm
S(7) = JC MN
1+ yx2(7)/0*),
(2)
where ~~47) is the rth sample autocovariance of the squared data, and u is the sample standard deviation of the data. These adjusted standard errors are reported in table 3, and they show that few of the autocorrelation coekients are statistically different from zero. Diebold (1986) shows that the adjusted Box-Pierce Q statistic, ptoticaliy a &i-square distribution with K degrqs
D.A. ifsieh, Daily fmeign exchangerates:1974-1983
135
of freedom. This is also reported in table 3. The statistics do not reject the null hypothesis of serial independence for the BP, CD, DM and SF data. There appears to be some autocorrelation in JY and JY2, at the 5 percent but not the 1percent signil’lcancelevel. These results indicate that the rejections of serial independence using the standard &sting procedure were caused by heteroscedasticity in the data. Furthermore, even if there is serial correlation in the data, it is of such a small magnitude that it cannot account for the strong rejections of the iid hypothesis. 5. Test& for the ‘day of the week’ e&t The results of the serial correlation test indicates that the rej&ion of the iid hypothesis cannot be caused by serial dependence. We therefore turn our attention to whether the rejection can be explained by changing distributions. One possibility is that the distribution of daily spot rate changes may be differentbetween weekdays and weekends. In the stock market literature, the return to holding stocks between Priday’s close and Monday’s close is shown to be diffbrentfrom the returns on other days of the week.s To check whether the distribution of exchange rates actually changes across days of the week, we divide the data into 5 subsamples according to the day of the week. Within each subsample, we assume that the data are iid, and test for equality of distribution across subsamples using the &i-square test. These pairwise tests are reported in the upper half of table 4. Seven out of the 25 statistics are significant at the 1 percent level, with 8 more sign&ant at the 5 percent level. We therefore reject the null hypothesis that the distribution is iid for each day of the week and iid across different days of the week. To check whether this ‘day of the week’ effect is responsible for the rejection of the iid hypothesis, we split each subsample into two halves and perform the &i-square test. The results are in the lower half of table 4. Fourteen out of the 25 statistics are significant at the 1 percent level, with 5 more sign&ant at the 5 percent level. Thus, even though the distribution of exchange rates is different across the days of the week, it cannot account for the rejection of the iid hypothesis. 6. +lbting for time-varying merursurb vatiuees Another possibility of unequal distributions in the data is that means and variances change over time. This was offered by Friedman and Vandersteel (1982) as an explanation of the fat tail distribution of spot rate changes. They %ecFrench (19&o),Gibbons
and Hess (19$1), and Keim and L3i.wihugh (P984).
DA Hsieh,Daily&w&n exchqe ratesa. 19744983
136
Table 4 Testofthe‘dayoft.heweek’e&ct. BP
CD
DM
~testforiidacromdayaoftheweck(k=21) Monday WL TRY Tuesdayvsa W-Y wedm!sdayvs. TkdY Thunldayva 3Om Friday Fridayvs. Monday Teatof iid within each subsampleii = 2n) 23.a 68.18 17.12 M&Y T-day Wt!&lMShY m&Y
Friday
JYr
JY
SF
33.02 (O-0336) aI.14 (a44921 24.43 (O=l) m.13 (0.4498) 27.28 (0.0385)
23.91 W-2463) 23.67 (OJm 33.15 (QO3rn
75.08 (-1 86.32 WlW S4.11 (aoooo) 8852 (m 8934 taoooo)
39.57 76.49 (aoooo) (aoooo) 13.58 65.61 (0.1930) (aoooo) 54.71 13.17 (02143) (aoooo) 83.20 28.38 (aoozr) (asooo) 71.96 2OA8 WW (aoooo)
22J3 (0.3335) 2uO (Ql499) 35.50 (aOr76) xm 19.04 (0.0218) (Omx2) 72% mm
Mar8iMl~kvelsinparul~ ‘2 January1975to 30 Jhcember 1983.
never directly tested this hypothesis. Instead, they used some moving statistics to summarize these changes in the data Calderon-RosseMand Ren-Ho&n (1982)checked whether changing means and variances could account for the rejection of the iid hypothesis. They divided their data into three subperiods of about 365 daily observations each. They assumed that the means and variances are constant within each subperiod, and tested for differencesacross subperiods. They faund that shifts in mean accounted for 9 rejections out of 13 currencies, but shifts in variances accounted for none of the remaining 4 cases. They, too, never directly tested for changing means and variances. Follotig the procedures in Calderon-Rossell and Ben-Horim (1982), we first checked whether changing means and variances can account for the rejection of the iid hypothesis. Rather than assuming constant means and variances over time intervals as lengthy as 18 months, we assume that means and variances are constant within each month. (An examination of the moving statistics in Friedman and Vandersteei reveals that means and variances change substantially over time.) As a maintained hypothesis,
D.A.Hsieh,Daily@reignexchangerata 1974-1983
assuming that
we arc
data are serially independent
the
137
for the remain& of
this !3ectioIL To check whetherchangesin monthly means can account for the rejection of iid, we center the data within a month by subtractingthe montMymean. We then recompute the &i-square test of iid using the centered data. As shown in the first row (test 1) of table 5, the statistics are stilI highly sign&ant. Clearly, changes in monthly means alone cannot be responsibk for the @a%@ of iid To check whether changes in monthly varianm can account for the r#tion of iid, we M&C the data within a month by dividing by the monthly standard deviation. The results of the &i-square tests are in the second row (test 2) of table 5. These statistics arc much smaller than their counterpar@in the prc&ing row, although all but the CD are still Table5
Testdequdityofmonthlymeansandvuirnoer BP
CD
Equdity of4!istribo~ (b21) Ttil 266s
c2 c!?! x3(20, Jiql8lityofWddtert zvl91
taoooo)
DM
~~1~7237 to-1 38.17
48.50 (0.1533) [O-I lOA 4433 (0-@)131 e958-41 acxa68months 199.75 lN86 (Odwl) (_
(0-1 25.17 (0.1950) 159.09 (-3)
IY
SF
37428 (O.oooo) 78.94 (0*88w 6821 (O=w 181.46
to-1
EaulitY of varhnes-months 8X33 1074All 127732 z (O_ (OAOOO) (0.08w 4.47 4.16 435 E&O] wJ8w (0*88w (O_ 4.72 (Onmoo) rthrgid~_~inpafcn~
02hmlary 1975to 30 Daembef 1983. .
&~tyofdisteuti~ Test 1: Data axecenteredh c&radng monthly meaw deviations. Test2 DataarescakdbydhidingbymonthIy Test% Dataarcstddidbymonthlymumsand~~tio~ Eqdity of vacross months: LcvcneI: M~Levffletest,using47.5procnttrimmcdmeaas. LCVCIICIT: ModiciedLevenctcst,~glOpercent~meaerrs.
mm
Jy2
43.10 (OJ=v 20.69 (0.4lw
263.45 (-1 5264 (O.aKu) 49.74 (0-88021
13629 (0.1327)
165.35 (OiIO32)
to-am
1137.85 to=) X01 (O.oooo)
138
D.A. I&i&, Daily
/me&n
exchange
rates:
19744983
sign&ant at the 1 percent level. It appears that changes in monthly variances can partly account for the rejection of iid. I’o check for the combined effects of changing means and variances, we standardixe the data within a month by subtracting the monthly mean and &idhg by the monthly standard deviation. The results of the &i-square tests are in the third row (test 3) of table 5. These statistics are dramatically lower than the two preceding rows in some cases. The CD, DM and SF are no longer sign&ant, even at the 10 percent level. The BP and JY are still significant at the 1 percent level. It appears that changing means and variances across months can jointly account for the rejection of iid, at least for the CD, DM and SF. To verify that the means and variances actually change over time, we perform some direct statistical tests. Again, we assume that means and variances are constant within each month, and test whether they are equal across the 120 months of data. The test of equality of means is best explained in a regression context. The daily rates of change are w on a constant term and 119 dummy variables, one for each of the 119 months after the first month. If the means arc constant across months, then the coeliicients for all 119 dummy variables should be xero. We test this with the Wald statistic, employing a covariance estimator which is consistent against a large class of heterosce&sticity.6 The statistics are in the fourth row of table 5. The equality of monthly means is rejected for the CD, DM and JY at the 1 percent level, and for the BP at the 5 percent level. It is not rejected for the SF7even at the 10 percent level. This evidence show that means are indeed not constant over time. The test for the equality of variances is more problematic, due to the heavy tails of the distribution. The Bartlett test, which is the standard test of homoscedasticity, is a likelihood ratio test for normally distributed data. It is given in the fifth row of table 5. Since there is strong evidence against normality, we prefer to use tests that are robust against fat tail distributions, such as the mod&d Levene test described in Brown and Forsythe (1974). These are labelled ‘Levene I” and ‘Levene II’ in the last two rows of table 5. The first procedure centers the data within each month by the 47.5 percent trimmed mean, and the second procedure uses the 10 percent trimmed mean. Brown and Forsythe (1974) show that these two procedures perform very well in Monte Carlo experiments under the normal distribution as well as fat tail distributions. Their results also indicate that the Bartlett test rejects homoscedasticity too frequently, whenever the distribution has fatter tails than the normal. As is evident in table 5, the equality of monthly variances is rejected at any conventional significance level for all currencies regardless of the test procedure. ‘See White (1980), Hansen (1982) and Hsieh j1983) for more details.
D.A. H&h, Doily fbrei8n excke
rties: 19744983
139
7. A statistid model of dally exchangerates
It would be of interest to try to describethe data with a statisticalmodel, In particular,the change in variancesappearsto be the strongestcharacteristic in the data. A succeti model of how varianceschange over time could have importantapplicationsin the pricingof foreigncurrencyoptions, which d=qendscriticallyon expectationsof futurevariances. To achieve this purpose,we adopt a version of the ARCH model in Engle (1982).This was used by Domowitx and Hakkio (1985) to model risk premia in the forwardexchange market. Although Domowitx and Hakkio justified their model with theoreticalconsideratiom it is not the case here. We choose this model for its computationaltractability. Our specificationis as follows. Let rt= log(S& I) be the rate of &ange of spot rates between two tradingdays. Conditioningon all informa?ionat time t- 1, we assume that r, is normallydistributedwith mean # and variance& where
(3)
+b
(4)
where &, BTn Dw, and Drptare dummy variables for Monday, Tuesday, Wednesday and Thursday. HOL, is the number of holidays (excluding weekends)between the (t - 1)st and tih tradingday. The log-likelihoodof the datais: . II
log+#log27c-~ r=1 +
P [
r,-_lcr
Q
* l
r
11 /
The maximumlikelihoodestimatesof the coel&zientsare obtainedby using a numerical optimixation procedure described in Rerndt, Hall, Hall and Hausman(1974).f The two lag lengths (P and Q) are chosen to maximize(5). Table 6 contains the estimated parametersand the optimal choice of Q ‘Note that r, will not be independent of past obwvations, since its mean and variance depends on them_ b&wm liieliiood estimation of the (unconditional) distti?x?kn s! rg is the&ore not appropriate. OIIthe other hand, the transformed series (r,-&kg are iadepdent of each other, and hence maximum likelihood estimation is appropriate.
140
D.A.H&h, Dailyfoteignexchange rates 19M-1983 Table6 Jzstimation results.
BP
CD
DM
JY2
SF
[Om8233]b -OEWO1
co.=9841
-0.097387 [O.O867U] -0.048588 c:zE? ~0.022873] 0.033939
‘ZEZ?
[0.018994] -!MM9o79
likclillood
- 1920.23
517.38
-2029.79
- 1814.29
standarderrorsin squarebrackets. *2 January1975to 30tleccmh 1983. %gni!kmtlydiliixcntfromzeroat the 1 percentlevel(one-tailed test).
-2570.37
D.A. Hsieh, Daily jimign exchange mtec lY74-fY83
141
and P. Few parameters in the mean equations have any statistical signifkance. Hence the model is very poor in predicting the &e&on of
exchange rate changes, evca within sample. On the other hand, quite a mmber of parametersin the variance equation are statistically sign&ant. The most important variablein the variance equation is the autoregressive term, 6. In all five currencies,6 is significantlymerent from xero at the 1 percent level. In the case of CD, DM and SF, 6 is less than unity. This means that the effa of a large variance on future varianceswill eventually dampen out. In the case of BP and JY, 6 is greaterthan unity. This means that the elfects of a large VG&%Z-on future variances will be amplified, which is clearly undesirable.As we shall see below, this model fits the data for CD, DM and SF, but not for BP and JY. In addition,the coe@cientsfor the Monday dummy (Vd and the holiday dummy (I$,) in the variance equation are large and sign&ant. The coefllcients for the other daily dummies (I$, Vw and Vn)are typically smaller, although some are also significant This means that the variancesof daily exchange rates are larger whenever the trading days spau a weekend or a holiday.Notice, however,that it is importantto distinguishbetween the two effects.Theincmase in variancedue to a weekend (which has two calendar days)issmallerthantheincreaseinvariance due to a holiday on a weekday (which typicallyhas one calendarday), in the case of the CD, DM and SF. After estimation, we perform some diagnostics to check how well the model fits the data We use the standardixcdresidual (6) e,=(r,-Pt)/6, where P, and a# arc the estimated means and standard deviations. If the model is correctly specSed, then e, should be approximately iid, normal, with 2e~omean and unit variance. Table 7 presents some statistical properties of these standard&ed residuals. The means are close to zero, and the standard
deviations are close to one. There is little evidence of serial correlation.But there is evidenceof non-normality,as indicatedby the coefllcientsof kurtosis being largerthan three. To test for iid of the residuals, we again employ the &i-square test describedin section 3. The results in the first row of table 8 show that the BP and JY reject iid at very high significancelevels. The SF rejects iid at 274 percent,while the CD and DM do not reject at 10 percent.This test shows that the model does not fit the data for BP and JY.* If we test iid across days of the week ancjwithin each day of the week, we also find much lower re@tion rates than the raw data? However, the rejectionof iid does ~wasnotsurprising,;clviewofohereSultsinthethirdrowoftable5. pWe compute the W@@~~ucc levels of the tests as if we knew the true parameter VOWS,
This
is &djf
a0t
tw
since we are using
estimated
parameter
values.
-The a@stnm~,
however, must be very small, because of the urge number of liegfees of freedom for our .
142
Statistical propertim of stmdardhd DM
CD
BP
15.2587 Test of serial cor&tion Box-Pierce 35.25 (0.9433) QUO)
SF
JY2
-0.018 -0.0149 l.oooO 0.1057
-0.0107 -0.0321 0.9999 0.1982
-0.#98 0.0078 0.9999 -0.3142
Meau Median Std. dev. Skewness
residuals..
10.1688
11.6742
129955
125421
39.42 (0.8589)
37.27 (0.9086)
47.06 (0.5921)
39.33 (0.8613)
StfUhddenorSillsqUarebraclcetg Marginal signi6eaneelevels in parentheses. ‘2 January 1975 to 30 Dazmbcr 1983. Table 8 Test for equality of distribution ofstamhhd BP
CD
Equality of distribution (&=21) 98.55 24.43 maays waKI) (0.2241) x3cz0, Test of iid across we&days (k=21) Monday vs. 14.77 23.82 (tL7894) g.tJ3) T-Y Tuesday vs. 26.04 WedlEIday Wednesday vs. n&Y Thursday vs. Friday Friday vs. Monday Test of iid within we&days (k=21) 19.88 54.30 Mondays (O.OOoO) (0.0304) Tuesdays 14.75 27.08 (0.0025) (!8414) WedIleXhIyS 13.17 (0.2143) (O&14) Thursdays 14.78 29.74 Fridays __
_ _.
.-
residuals.
DM
1T2
28.18 (0.1052)
91.84 33.81 (0.OQoO) (0.0274)
20.47 (0.4289) 11.92 @Z8”)
16.38 (0.6928) 17.35 (0.6301) 24.67 (0.2144) 16.47 (0.6871) 20.32 (Q4381)
18.38 (0.5624) 40.91 (O.NI38) 1487 (0.6614) 19.89 w46w 23.46 (0.2668)
35.59
31.16 (g%)
G948) 18.36 (0.5637) 13.51 (0.8544)
22.56 (0.0125) 12.47 (0.2548) 12.49 (0.2536) 14.26 @161§) 13.94 (0.1757)
to.ooaz) 22.54 (oat 26) (E42) 32.04 (3.m) 11.50 (0.3199)
SF
($:85) (Oh57) 19.48 (0.0346) 10.96 (0.3606)
D.A. Iisieh, Daily j’oreign exchange rates: 197c1983
143
not come from shifts in means and variances. Test for the equality of monthly means and variancesof the residualsare in table 9. The equality of monthly means is rejectedonly for the CD at 3.26 percent.The equality of monthly variancesusing the modifiedLevene tests is rejectedonly for the BP at 4.34 percent.
In our investigation,we find that: (1) Exchangerate changesare not independentand identicallydistributed (2) Each day of the week may have a differentdistribution,but this is not suflicientto explain the rejectionof iid (3) Thereis little serial correlationin the data. (4) Means and variances change over time. Jointly they may be able to explain the rejectionof iid for at least rlrreecurrencies- CD, DM and SF. Furthermore,we constructedan ad hoc statistical model to capture these proper& In certain aspects this attempt was suu~&~L For all five currenciesthe model was able to capture the change in means and variances over time. Based on the small numberof statisticallysignificantcoefficientsin the mean equation, we believe that the model would not be able to predict directionsof exchange rate changes. However, based on the large numberof statistically significantcoe&ients in the variance equation, we believe the model should be able to predict volatility. In other aspects, the model was less s~ful. It was able to account for the rejection of iid for CD, DM and SF, but failed to do so for BP and JY. Furthermore,the standardized Table 9 Test of equality of monthly means and varianas for standard&d resi&&. BP
CD
Test of equality of monthly means 148.99 WaldtCSt 133.20 (0.0326) (0.1764) x2(119) Test of equality of monthly variance 422.11 197.34 &utktt (O-000@ (0.Qooo) x2(119) 1.12 0.95 Levenei (0.1820) (0.6342) F(119,239O) Levene II fl119,2390)
I.24 (0.0434)
1.10 (0.2213)
Marginal sigaifiwcc level in parentheses. “2 January 1975 to 30 December 1983.
DM
Jw
SF
139.09 (O.looa)
117.06 (0.5332)
122.33 (0.3986)
216.45
212.19 (0-m) 0.79 (0.9528)
236.08 (0-m)
wQw 1.01 (0.4542) (z6)
0.87 (0.8380)
(E47) 1.15 (0.1325)
144
D.A. Hsieh, Daily jkign exchangerates 19741983
residualsfor all five currenciesare non-normal,even though they have less kurtosis than the raw data.lo Time-varyingmeans and variances therefore are not sufficient to Mly account for the leptokurtosis in exchange rate changes. The search for an appropriatedistribution will be left for future research. ‘@I-heissue of wtion arises wllen the distribution of (r#-rcl)!o f uo#nMLaL genaal, a n&qe&catioll of the distributionfunction may lead to kWnsu&n parametersof interest.In the present context, however,such &on&en& may not a&e. The reaPonisthat~~liltetibood~ganarmal~~~~~t~~of means and warii~ even if the true distributionis not normal, providedcertain conditions are met. [See White (1982) and Gourieroux,Monfortand Tro8non (1984).]
Refm Berndt,ErnstK,B~~H.Hall,RobertE.HallandJenykHo~1974,~end iniluence in nonlinear structuralmadels, Annah of Economic and Social Measurement4, 653-665. Boaz&e,R$ ~~ Glaspman, 1985, The statist&d distribution of exduuQe ratep: economic impllcations#unpublished manlMuipt @epWaWnt of Econont& Universityof Alberta) BroJyc”thtB. and. Alan & Fomythe, 1974, Robust tests for the equality of variwcs, Amencan statrsticel Associah
69,36&367.
~John,FndRKaenaadG.~Bootb,1917,Fotei;gnmarlteteagciencyurrderfladMe exchangerate&JournalofFinance 3291325-1330. Calderon~Rossell, Jorge R and Foshe Ben~&rim, 1982,The behaviorof foreignexdtange rates, Journalof InternationalBusmessStudies 13,99-W. Cornell, Bradford,1977, Spot rates, forward rates, and market elRcieney,JournaI of Finan&l EconomicsS,55-65. Diebold, Francis X., 1986 Testing for serial correlationin the presenceof ARLCZunpublished mantWript(ul&ersity of Pennsylvania). Domowitq Ian and Craig S. Hakkio, 1985, Conditional variamx andtheriskpremiuminthe foreignexchange market,JournalofInternational Economics 19.47-66. Engle, Rohert F, 1982, Automgressiveconditional hctenwcedasticiy with estimates of the variamz of United ICiqIom inflation, u SO,987-107. French,KennethR., 1980,Stock returnsand the weekendeffect,Journalof Financial Econti 8,569. Friedman,Daniel and Stoddard Vans 1982, Short-run fluctuations in foreign exchange rates,Journal ofInternational Economics 13.171-186. Gibbons, Michael R. and PatrickHess, 1981,Days of the week effectsand asset returns,Journal of Business54,579-596. Giddy, Ian H. and Gunter Dufey, 1975, The random behz&r of flexible exskange rates: Implicationsfor forecasting,Journal of Inteeational BusinessStudies 61-32. C+%u.uieroux, Christian, Alain Monfort and Alain Trognon, 1984, Pseudo maximum likelih~ methods:Theory, Econometrica52,681-?!D Hansen, Lars E., 1982, Large sample propertiesof 8enemlized method of moments estimators, Econometrica54 1029-1054. Hog& Robert V. and Alien T. Craig, 1970, Introductionto MathematicalStatistics (Mad#an, New York). liaieh, David .4., 1983, A heteroseedasticity-consistent covariaucematrixestimatorfor time series rqrcssions, Journal of Econometrics22281-290. Kceir;l, Donald B. and Robert F. Stambaqh, 1984,A furtherinvestigationof the weekend effect in stock retsms, Journalof Finance 39,819-837.
D.A.Hsieh,Dailyfirm
exchangerace=1974-1983
145
Kendall, Maurice G. and Alan S. Stuart, 1967, Tble advanced theory of statistics, Vol. 2, Infcrene and relationship(Hafncr,New York), Loguq Dumia E and RichardJ. Sweeney,1977,whitenow in impcrfbt markets:The CBSC of the ti!&!lar exchangeF&F+Jowl! of Fiaane 32,761-768. Logue, Dennis E., RichardJ. Sweeneyand Thonw D. Willctt,1978,Speculativebehavior of foreignexdumge caterduringthe currentfloat,Journald BusinessResearch6,159-174. Rogabki,RichardJ. and JosephD. Vinw, 1978,Empiricalipropertiesof foreignexchangerates, JoumalofIntemationalBwim1Studiu9,69-79. W~JsnictM,19n,Aoeraminotionobfordgrcnchangeriskunderfixcdarndfloa~g rate e JournalofInternational Economics 7.181-200. . . . t wvariance estimator and a direct test for Whitc,Habrt,1%O,Ah&nw&&a +~USIMO ty: &xnwmex r% 8r7-838. white, HaIl%Z Maximumlike&hoode&nation of mbpe&ial models,EconometricaSO, l-26.