003
14.04.2003
10.07.2003
07.10.2003
02.01.2004
31.03.2004
28.06.2004
23.09.2004
21.12.2004
08.05.2001
08.02.2001
13.11.2000
16.08.2000
19.05.2000
22.02.2000
25.11.1999
30.08.1999
02.06.1999
05.03.1999
08.12.1998
10.09.1998
15.06.1998
18.03.1998
19.12.1997
23.09.1997
26.06.1997
31.03.1997
01.01.1997
18.10.2002
0.8025.04.2002
0.90
23.07.2002
1.0028.01.2002
1.10
25.04.2002
1.2031.10.2001
1.30
28.01.2002
1.4003.08.2001
Fig. 8.2 EUR/JPY nominal exchange rate
31.10.2001
EUR/JPY
03.08.2001
08.05.2001
08.02.2001
13.11.2000
16.08.2000
19.05.2000
22.02.2000
25.11.1999
30.08.1999
02.06.1999
05.03.1999
08.12.1998
10.09.1998
15.06.1998
18.03.1998
19.12.1997
23.09.1997
26.06.1997
31.03.1997
01.01.1997
8.2 Empirical Analysis49
170
160
150
140
130
120
110
100
90
80
EUR/USD
Fig. 8.3 EUR/USD nominal exchange rate
For means of comparison, all nominal exchange rates were scaled to basis 100on 1 January 1997. The paths were then calculated on a continuously compoundedbasis by sequentially adding log first differences.
50
8 Statistical Analysis of Daily Exchange Rate Data
120
110
100
90
80
70
60
EUR/GBP
EUR/JPY
21.12.2004
23.09.2004
28.06.2004
31.03.2004
02.01.2004
07.10.2003
10.07.2003
14.04.2003
15.01.2003
18.10.2002
23.07.2002
25.04.2002
28.01.2002
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03.08.2001
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08.02.2001
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30.08.1999
02.06.1999
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10.09.1998
15.06.1998
18.03.1998
19.12.1997
23.09.1997
26.06.1997
31.03.1997
01.01.1997
50
EUR/USD
Fig. 8.4 Base currency EUR in terms of foreign currencies GBP, JPY, and USD
Every time series was examined in a hypothesis testing framework according tothe following criteria:1. Stationarity2. Normality3. Linearity4. Heteroskedasticity5. NonlinearityIn a hypothesis testing framework, there are always two hypotheses that go together,known as the null hypothesis (H0 ) and the alternative hypothesis (H1 ). The null hypothesis is the statement or the statistical hypothesis that is actually being tested.The alternative hypothesis represents the remaining outcomes of interest. Hypothesis tests were conducted via a test of significance approach which centers on astatistical comparison of the estimated value of the coefficient and its value underthe null hypothesis. Generally, if the estimated value is a long way away from the hypothesized value, the null hypothesis is likely to be rejected. Otherwise, if the valueunder the null hypothesis and the estimated value are close to one another, the nullhypothesis is less likely to be rejected. Note that it is incorrect to state that if thenull hypothesis is not rejected it is accepted since it is impossible to know whetherthe null is actually true or not.
8.2.1 StationarityAs a first step we ensured that the time series data we work with are stationary,which is an important property to apply for statistical inference procedures. A series
8.2 Empirical Analysis
51
is said to be strictly stationary, if the mean, the variance, and the covariance of theunderlying series remain constant over time ([180], p. 45):E(yt ) = μ = const.Var(yt ) = E (yt − μ )2 = σ 2 = const.Cov(yt , yt+τ ) = E ((yt − μ )(yt+τ − μ )) = λ = const.
(8.2)
However, strict stationarity as represented in (8.2) is generally not fulfilled in financial and economic time series. For this reason, econometric theory shows a greaterinterest in whether a time series is (weakly or covariance) stationary, which is thecase if the mean and variance are independent of time t, and if the series’ covarianceonly depends on the time difference τ , but not on t itself:Cov(yt , yt+τ ) = E ((yt − μ )(yt+τ − μ )) = λτ
(8.3)
Statistical tests of the null hypothesis that a time series is nonstationary against thealternative that it is stationary are called unit root tests. The name derives from thefact that a stochastic process is nonstationary if the characteristic polynomial hasa root that does not lie inside the unit circle. The unit root tests applied are theaugmented Dickey–Fuller (ADF) test [98], and the Philipps–Perron (PP) test [324].The Kwiatkowski–Philipps–Schmidt–Shin (KPSS) test [236] which is additionallyapplied differs from the above two tests in that the underlying time series is assumedto be stationary under the null hypothesis.Table 8.1 shows that the results of the three tests are consistent for the series ofexchange rate levels yt . In the case of the unit root tests, we cannot reject the nullhypothesis that daily price data are generated by a nonstationary stochastic process.Otherwise, in the case of the KPSS test, we reject the null hypothesis that dailyprice data are generated by a stationary stochastic process at the 1% significancelevel which matches the result of the unit root tests.A time series that is nonstationary, but can be stationarized by d-fold differentiation is called integrated of order d, i.e., I(d) [58]. Accordingly, log first differencesΔ yt = ln(yt ) − ln(yt−1 ) of the price data for the period from 2 January 1997 to31 December 2003 were taken, as visualized by Figs. 8.5–8.7, the same tests wereTable 8.1 Testing for (non-) stationarityNull hypothesis
NonstationarityStationarity
Test
ADFPPKPSS
Input
yt , Δ ytyt , Δ ytyt , Δ yt
OutputEUR/GBP
EUR/JPY
−2.28, −43.98∗∗∗
−1.31, −40.48∗∗∗
−2.10, −44.14∗∗∗0.79∗∗∗ , 0.03
EUR/USD
−1.34, −44.70∗∗∗−1.33, −40.47∗∗∗ −1.41, −44.64∗∗∗0.90∗∗∗ , 0.070.81∗∗∗ , 0.07
The columns from left to right denote the null hypothesis, name of the test applied, time seriesinput, and the test statistic output per currency pair. ∗ , ∗∗ , and ∗∗∗ indicate rejection of the nullhypothesis at the 10%, 5%, and 1% significance level
23.09.200421.12.2004
21.12.2004
14.04.2003
15.01.2003
18.10.2002
23.07.2002
25.04.2002
28.01.2002
31.10.2001
03.08.2001
08.05.2001
08.02.2001
13.11.2000
16.08.2000
19.05.2000
22.02.2000
25.11.1999
30.08.1999
02.06.1999
05.03.1999
08.12.1998
10.09.1998
15.06.1998
18.03.1998
19.12.1997
23.09.1997
26.06.1997
31.03.1997
01.01.1997
28.06.2004
−0.08
23.09.2004
−0.0631.03.2004
−0.04
28.06.2004
−0.0202.01.2004
0.00
31.03.2004
0.0207.10.2003
0.04
02.01.2004
0.0610.07.2003
Fig. 8.5 Log first differences of daily EUR/GBP nominal exchange rate data
07.10.2003
d(ln(EUR/GBP),1)
10.07.2003
14.04.2003
15.01.2003
18.10.2002
23.07.2002
25.04.2002
28.01.2002
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10.09.1998
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18.03.1998
19.12.1997
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31.03.1997
01.01.1997
528 Statistical Analysis of Daily Exchange Rate Data
0.03
0.02
0.02
0.01
0.01
−0.010.00
−0.01
−0.02
−0.02
−0.03
d(ln(EUR/JPY),1)
Fig. 8.6 Log first differences of daily EUR/JPY nominal exchange rate data
conducted subsequently. Again, the vertical bar drawn at 29 August 2003 dividesthe series into in-sample period (left of the bar) and out-of-sample period (right ofthe bar).Test statistics suggest that now all three exchange rate series are stronglydifference-stationary, i.e., integrated of order one (I(1)).
8.2 Empirical Analysis
53
0.03
0.02
0.01
0.00
−0.01
−0.02
21.12.2004
23.09.2004
28.06.2004
31.03.2004
02.01.2004
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10.09.1998
15.06.1998
18.03.1998
19.12.1997
23.09.1997
26.06.1997
31.03.1997
01.01.1997
−0.03
d(ln(EUR/USD),1)
Fig. 8.7 Log first differences of daily EUR/USD nominal exchange rate data
8.2.2 Normal DistributionIn a second step, the empirical probability distributions of Δ yt were tested for departures from the normal distribution. The normality assumption Δ yt ∼ N(0, σ 2 ) isrequired in order to conduct hypothesis tests about the model parameters. One of themost commonly applied tests for normality is the Jarque–Bera (JB) test [206]. JBuses the property of a normally distributed random variable which characterizes theentire distribution by its first two moments, i.e., the mean and the variance. The standardized third and fourth moments of a distribution are known as its skewness (SK)and kurtosis (KU). The coefficient of SK measures the extent to which a distributionis not symmetric about its mean value and is expressed asSK =
E[y3 ](σ 2 )3/2
(8.4)
A normal distribution is defined to have SK = 0. KU measures how fat the tails ofthe distribution are and is expressed asKU =
E[y4 ](σ 2 )
2
(8.5)
A normal distribution is defined to have a coefficient of kurtosis of KU = 3. Instead of KU, it is also common to use the coefficient of excess kurtosis (EK),given by EK = KU − 3, which must be zero for a normal distribution. JB tests
54
8 Statistical Analysis of Daily Exchange Rate Data
Table 8.2 Testing for normalityNull hypothesis
––Normal Distribution
Test
SKKUJB
Input
Δ ytΔ ytΔ yt
OutputEUR/GBP
EUR/JPY
EUR/USD
0.093.7442.11∗∗∗
−0.346.851106.89∗∗∗
0.063.6229.18∗∗∗
The columns from left to right denote the null hypothesis, name of the test applied, time series input, and the test statistic output per currency pair. ∗ , ∗∗ , and∗∗∗ indicate rejection of the null hypothesis at the 10%, 5%, and 1% significancelevel
whether the coefficient of SK and the coefficient of EK are jointly zero by the teststatistic 2SK(KU − 3)2+JB = T(8.6)624where T is the sample size. The test statistic asymptotically follows a χ 2 (2) underthe null hypothesis that the series is symmetric and mesokurtic.Table 8.2 summarizes the results of our tests for normality. We observe thatEUR/GBP and EUR/USD are slightly skewed to the right (positive signs) whereasEUR/JPY is skewed to the left (negative sign). We further observe that all three series have positive excess kurtosis indicating that there is more weight in both tailsof the distribution than in the normal distribution. Probability distributions exhibiting this phenomenon are said to be leptokurtic or fat-tailed. Among the three series,EUR/JPY exhibits the most leptokurtic behavior whereas EUR/GBP and EUR/USDshow weaker signs of fat tails. This is confirmed by the JB test statistics which reject the null hypothesis of normal data at high levels. The empirical observation ofleptokurtic exchange rate returns has been mentioned in countless studies (see, forinstance, [375], p. 148).
8.2.3 LinearityA major objective when analyzing stationary time series is to detect linear dependencies among the data through identifying an appropriate linear model. Univariatetime series models can only be explained by their own lagged values, i.e., by autoregressive (AR) terms as explanatory variables in their representation. The autoregressive model of order 1, the AR(1) model isyt = c + α yt−1 + εt
(8.7)
where εt ∼ i.i.d.(0, σ 2 ). The constant term c models a trend in the series either upwards (c > 0) or downwards (c < 0). The lag coefficient α determines the stabilityof the process. If |α | > 1 the time series will explode, that is, yt → ±∞ as t → ∞.
8.2 Empirical Analysis
55
The special case |α | = 1 gives the random walk model and it is only when |α | < 1that the process defined by (8.7) will be stationary.Furthermore, if the underlying process is stochastic and stationary, the errors canbe linear combinations of white noise at different lags, so the moving average (MA)part of the model refers to the structure of the error term εt . The first-order movingaverage model MA(1) isyt = c + εt + β εt−1(8.8)where εt ∼ i.i.d.(0, σ 2 ). This model is a stationary representation for any valuesof c or b, since E(yt ) = c. The most general model for a stationary process is anintegrated autoregressive moving average model ARIMA(p, q, r)yt = c + α1yt−1 + α2 yt−2 + . . . + α p yt−p + εt + β1 εt−1 + βqεt−q
(8.9)
with p autoregressive terms, q moving average terms, integration order r, with r = 1in our case, and εt ∼ i.i.d.(0, σ 2 ). ARIMA(p, q, 1) models are also simply denoted asARMA(p, q) models. The parameters p and q are commonly estimated by visuallyinspecting the autocorrelation function (ACF) and partial autocorrelation function(PACF) for MA models and low-order AR models [180]. ACF and PACF functionscharacterize the pattern of temporal, linear dependence that is existent in the series.Since independent variables are always uncorrelated, testing for linear independencyis equivalent to testing for zero autocorrelation. We first calculated Durbin–Watson(DW) test statistics [108] for Δ yt . DW is a test for first order autocorrelation, i.e.,it only tests for a relationship between an error and its immediately previous value.The test statistic is(8.10)DW ≈ 2(1 − ρ̂ )with ρ̂ ∈ [−1, 1] denoting the correlation between Δ yt and Δ yt−1 . Thus if DW = 2ρ̂ = 0 there is no autocorrelation in the residuals and the null hypothesis would notbe rejected. Otherwise if DW = 0 (ρ̂ = 1) there is perfect positive autocorrelationin the residuals and, if DW = 4 (ρ̂ = −1) there is perfect negative autocorrelationin residuals. The DW test does not follow a standard statistical distribution but isdivided up into regions including an upper and lower critical value.Second, Ljung–Box (LB) Q-statistics [266]
τ̂k2∼ χ2T−kk=1m
Q∗ = T (T + 2) ∑
(8.11)
were calculated to test the joint hypothesis that all m of the τk correlation coefficients are simultaneously equal to zero.It is interesting to see from Table 8.3 that for the EUR/JPY series no significant linear relationships among the data can be identified and thus, one would notnecessarily need to estimate a linear model.However, we decided at this point to obtain clearer statistical results by alsoremoving some insignificant linearities through
Δ yt = 0.0287Δ yt−1 + εt(1.20)
(8.12)
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8 Statistical Analysis of Daily Exchange Rate Data
Table 8.3 Testing for linear dependenciesNullTesthypothesis
Input
OutputEUR/GBP
No AC
k = 1: 4.95∗∗ , 0.00
EUR/JPY
EUR/USD
LB
Δ yt ,ARMA- k = 2: 4.98∗ , 0.13Residuals k = 3: 10.43∗∗ , 0.14k = 4: 10.60∗∗ , 0.15of Δ ytk = 5: 10.61∗∗ , 0.17k = 6: 10.65∗ , 0.18k = 7: 10.70, 0.23k = 8: 12.38, 2.23k = 9: 13.39, 3.20k = 10: 14.19, 3.74k = 15: 16.44, 5.58k = 20: 22.49, 11.46k = 24: 32.93, 19.77
k = 1: 1.42, 0.00k = 1: 8.36∗∗∗ , 0.03k = 2: 1.44, 0.02k = 2: 10.49∗∗∗ , 0.07k = 3: 2.00, 0.56k = 3: 11.85∗∗∗ , 0.07k = 4: 2.41, 0.97k = 4: 13.07∗∗ , 0.34k = 5: 3.02, 1.67k = 5: 13.18∗∗ , 0.54k = 6: 4.38, 3.13k = 6: 13.21∗∗ , 0.55k = 7: 4.48, 3.24k = 7: 14.10∗∗ , 1.66k = 8: 5.24, 4.19k = 8: 15.74∗∗ , 2.99k = 9: 5.52, 4.29k = 9: 16.83∗ , 3.06k = 10: 6.59, 5.27k = 10 : 16.79∗ , 4.44k = 15: 9.67, 8.27k = 15: 21.03, 7.55k = 20: 15.70, 14.71 k = 20: 23.88, 11.78k = 24: 15.84, 14.85 k = 24: 26.82, 14.51
BG
ARMA- 0.8617Residualsof Δ yt
0.6836
0.8975
DW
Δ yt ,1.9939, 1.9988ARMAResidualsof Δ yt
1.9995, 1.9995
1.9928, 2.0021
The columns from left to right denote the null hypothesis, name of the test applied, time seriesinput, and the test statistic output per currency pair. ∗ , ∗∗ , and ∗∗∗ indicate rejection of the nullhypothesis at the 10%, 5%, and 1% significance level. k denotes the number of lags included in thetest regression
For the other two series, linear dependencies are obviously existent. To removethem, it was sufficient to specify the following simple linear models
Δ yt = −0.0526Δ yt−1 −0.0562 Δ yt−3 + εt(−2.20)
(−2.35)
(8.13)
for the EUR/GBP series and
Δ yt = −0.5959Δ yt−1 + 0.5323εt−1 + εt(−3.01)
(2.55)
(8.14)
for the EUR/USD series. T-statistics are given in brackets below the parameter estimates. Each model’s residuals series {εt } was retested according to LB and theBreusch–Godfrey Lagrange Multiplier (LM) test which, like LB, is a more general test for autocorrelation up to the rth order. The model for the errors under thistest isεt = β1 εt−1 + β2 εt−2 + . . . + βr εt−r + νt(8.15)
8.2 Empirical Analysis
57
and the null hypothesis is that the current error is not related to any of its r previousvalues, i.e., β1 = β2 = . . . = βr = 0. This kind of test is known as a test on omittedvariables. The test statistic of the Breusch–Godfrey LM test is given by(T − r)R2 ∼ χr2
(8.16)
The test statistics indicate that serial dependencies have now disappeared at anylag. Hence, the models are statistically adequate. However, although linear independency can be inferred for all three series, nonlinear dependencies might still exist.
8.2.4 HeteroskedasticityIn a next step, we investigated the origin of nonnormal behavior by focusing on thephenomenon of heteroskedastic processes [115]. The classical regression model assumes that the error process εt in the model is homoskedastic. In other words, εt isassumed to have a constant variance Var(εt ) = σ 2 . Heteroskedasticity is motivatedby the observation that in many financial time series the variance of et appears tobe related to the variance of recent residuals, i.e., Vart (εt ) = σt2 . This phenomenonwhich is referred to as volatility clustering [45, 273], indicates that returns Δ yt arenot in fact independent but rather, they follow some sort of nonlinear dynamic process. In order to detect these second-moment dependencies, we tested according to[286]. We first calculated the autocorrelations of the squared residuals and computed the LB Q-statistics for the corresponding lags. If there is no autoregressiveconditional heteroskedasticity (ARCH) in the residuals, autocorrelations and partialautocorrelations should be zero at all lags and the Q-statistics should not be significant. In order to find out whether these results can be confirmed by a different test,we apply the ARCH LM test which tests the null hypothesis that there is no ARCHup to order q in the residuals. The test statistic is computed from the auxiliary regression222εt2 = β0 + β1 εt−1+ β2 εt−2+ . . . + βqεt−q+ νt(8.17)where ε is the residual. The F-statistic is used as an omitted variable test for the jointsignificance of all lagged squared residuals.According to Table 8.4, LB outputs reject the null hypothesis of “No ARCH”at very high significance levels for all three series of squared ARMA residuals.The ARCH-LM F-statistics confirms the results for the squared ARMA residuals ofEUR/GBP and EUR/JPY: the null hypothesis of zero heteroskedasticity is clearlyrejected for all selected lags at the 1% level.The result for EUR/USD is less clear: according to both, Q-statistics and ARCHLM testing results, the hypothesis of a constant variance can only be rejected athigher lags and with slightly lower confidence. This brings us to an important result, which has also been reported in literature ([99], p. 10): ARCH processes areleptokurtic, or fat-tailed, relative to the normal. The slightly weaker test statisticsfor EUR/USD may perhaps be explained by a skewness and kurtosis that are closer
58
8 Statistical Analysis of Daily Exchange Rate Data
Table 8.4 Testing for HeteroskedasticityNull
Test
NoARCH LB
Input
Output
(ARMA-,Residuals)2of Δ yt ,(GARCHResiduals)2of Δ yt
ARCH ARMALMResidualsof Δ yt ,(GARCHResiduals)2of Δ yt
EUR/GBP
EUR/JPY
EUR/USD
k = 1: 20.40, 0.06k = 2: 27.86, 0.55k = 3: 52.93∗∗∗ , 0.67k = 4: 56.64∗∗∗ , 0.69k