one, the lower the estimated tail index is. In addition, the variability of the estimated tail index increases with increasing V s0.99 as well as increasing c .
g 1
The effect of decreasing is demonstrated by comparing the left and right panels; all other factors being equal, the estimated tail index undergoes a downward shift,
which parallels the finding of GK using EGARCH models with non-zero asymme- try parameter. Note that this shift is not linear in a , with the drop being far more
Ž .
pronounced for large a-values near two , and minimal for smaller values of a.
4. An application to East Asian currencies
We now examine whether returns on East Asian currencies can be modeled by the aforementioned stable-GARCH modeled and whether the estimated innova-
tions are adequately described by the stable law. During the Asian financial crises in the late 1990s, the exchange rates of many Asian countries experienced violent
fluctuations, rendering the resulting daily return series particularly recalcitrant. The empirical results for one currency, the Thailand bhat, will be discussed in
some detail. Results for other series, namely the Singapore dollar, the South
Ž .
Korean won and the Sri Lanka rupee all against the US dollar are briefly summarized toward the end of this section.
The Thai bhat is particularly interesting because its massive devaluation in July 1997 is considered to have triggered the financial crisis that led to the collapse of
foreign exchange and equity prices in many East Asian countries. During the 1980s, Thailand’s capital markets became increasingly liberalized and opened up
for foreign investors.
9
As in other countries in the region, the Thailand economy experienced high economic growth and large inflows of private capital throughout
the 1990s, while maintaining an effectively pegged exchange nominal rate. The capital inflow led to an appreciation of the real exchange rate, which, in turn,
negatively affected firms’ exports and profit margins and, ultimately, led to cash flow shortages. From 1990 to 1997, Thai companies increased the US-de-
nominated foreign debt outstanding from international bond market activities by a
Ž .
factor of about 10, to US12.9 billion cf. Harvey and Roper, 1999 . In February 1997, Somprasong became the first Thai company which failed to serve foreign
debt. In May of that year, the Thai bhat came under heavy attacks from speculators acting on the economic slow down as well as political instability. The
largest finance company, Finance One, failed shortly thereafter. In June, Thailand’s finance minister, who had strongly resisted a devaluation of the bhat, resigned. On
July 2, the Bank of Thailand announced a managed float of the bhat and requested Atechnical assistanceB from the International Monetary Fund. The resulting deval-
9
Ž .
For a more detailed account of the Asian financial arises, see, e.g., Corsetti et al. 1999a,b and Ž
. Harvey and Roper 1999 .
uation from 24.52 to 30.18 bhat per US dollar is considered to have triggered the East Asian crisis. Subsequent to that negative news from the Thai economy but
also from other countries in the region led to violent swings for the bhat, reaching a low point on January 12, 1998 with 56.10 bhat per US dollar.
The 2205 daily returns of the Thai bhat–US dollar exchange rate from the period January 1, 1990 to December 31, 1998 are illustrated in the top panel of
Fig. 8. Visibly striking is the massive increase in volatility occurring at the onset of the crisis. Although at first sight, it might appear that separate models for the
Ž two distinct periods would be necessary or the use of more complicated models,
. which incorporate regime switching, structural breaks, etc. , we find that the
proposed class of stable-GARCH models are adequate. In particular, we fit an
d
Ž .
d
Ž .
Ž . S
A-PARCH 1, 1 and an S Q-GARCH 2, 1 model, each with an MA 1
a , b a , b
Ž .
d
Ž .
Ž .
Fig. 8. The returns top and fitted S Q-GARCH 2, 1 residuals middle of the daily Thai bhat–US
a , b
dollar exchange rate. The bottom graph shows the estimated a index for the residuals as a function of the summing interval s and the approximate 2-standard deviations confidence intervals.
component for the mean. Summary statistics are provided in Table 1 and include the estimated tail index and skewness parameter of the innovations process, a , and
ˆ ˆ
ˆ ˆ
b, respectively, along with approximate standard errors, SE. Similarly, V stems
ˆ
Ž . Ž
. from either Eqs. 7 or 11 , with calculation of SE as discussed in Section 2.3.
Ž .
The corrected AIC AICC criterion: 2T k q1
Ž .
AICC sy2 L L q
, 13
Ž .
T yky2 Žwhere k denotes the number of estimated parameters and T the number of
. Ž
Ž ..
observations and the AD statistic Eq. 12 are reported as comparative good-
ness-of-fit measures between the two fitted models. Given the considerably lower value of the AICC statistic for the Q-GARCH
specification, we restrict discussion just to the results of this model. This yielded a
ˆ
stable tail index of a s1.65 and persistence parameter Vs0.996. The residuals
ˆ
from this model are plotted in the middle panel of Fig. 8; they show no sign of a structural break at the crisis onset and appear to be consistent with iid stable
Ž .
realizations with a s1.65. Correlogram graphs not shown for the residuals,
ˆ
absolute residuals and squared residuals do not provide any evidence against the independence hypothesis.
We are now interested in testing whether these residuals are adequately described by a stable law. Under the null hypothesis, the a-estimates of sums of
iid stable variates should be invariant with respect to the number of summands, S, while, for non-stable data, the index should increase towards two as S increases. A
Ž .
test based on this property has been proposed in Paolella 2000 and shown to have not only appropriate nominal size, but also reasonable power against both
Student’s t and mixed normal alternatives. It is based on the linear increase of a
ˆ
Ž vs. S, where a is calculated using the Hill-intercept tail estimator see Appendix
ˆ
. A for details . The lower panel of Fig. 8 plots a vs. S along with 2 times the
ˆ
estimated standard error. The small increase in a turns out to be well within the
ˆ
range expected from simulated iid stable variates with a s1.65. In particular, the Ž .
stability test delivers both the test statistic, t a , and the appropriate cutoff
T
Ž .
values as a function of the estimated stable tail index a and sample size T at the Ž .
90, 95 and 99 percent levels. The bhat Q-GARCH residuals yield t a s3.23,
T
Ž .
for which the null hypothesis of stability cannot be rejected at the 90 or higher percent level. Thus, to the extent to which the chosen model captures the
Ž .
dominating signal in the data and recovers approximately iid innovations, the stable hypothesis seems quite plausible.
A similar analysis is conducted for the returns on three other East Asian currency series, namely the Singapore dollar, South Korean won and Sri Lanka
rupee, with estimation results also shown in Table 1. As with the Thai bhat, the AICC clearly favors the Q-GARCH over the A-PARCH model for these three
currencies as well. The estimates for the tail index obtained with the Q-GARCH model range from 1.58 to 1.62 and are marginally lower than for the Thai bhat.
S. Mittnik
et al.
r Journal
of Empirical
Finance 7
2000 389
– 416
410
Table 1 Selected parameter estimates, goodness-of-fit and summability inference measures
a d
d
Ž .
Ž .
Return S
A-PARCH 1, 1 S
Q-GARCH 2, 1
a , b a , b
series
ˆ ˆ
ˆ ˆ
Ž .
Ž .
Ž .
Ž . Ž
. Ž
. Ž
. Ž .
a SE b SE
V SE AICC
AD t
a a SE
b SE V SE
AICC AD
t a
ˆ ˆ
T T
Thailand 1.54
0.0303 0.982
y877.35 0.0862
3.11 1.65
y0.0094 0.996
y931.5 0.0670
1.65 Ž
. Ž
. Ž
. Ž
. Ž
. Ž
. 0.0034
0.044 0.0037
0.011 0.055
0.00067 Singapore
1.51 y0.149
0.992 240.83
0.0668 5.35
1.62 0.214
1.011 198.12
0.0542 4.44
Ž .
Ž .
Ž .
Ž .
Ž .
Ž .
0.011 0.068
0.0039 0.030
0.057 0.0033
Korea 1.51
0.083 0.981
y1137.1 0.0644
5.70 1.58
0.0600 0.995
y1150.9 0.0944
3.30 Ž
. Ž
. Ž
. Ž
. Ž
. Ž
. 0.0026
0.16 0.0053
0.0071 0.053
0.00080 Sri Lanka
1.50 0.043
1.053 82.289
0.1244 3.82
1.58 0.0805
1.002 59.724
0.1540 1.71
Ž .
Ž .
Ž .
Ž .
Ž .
Ž .
0.009 0.044
0.011 0.021
0.045 0.0028
a
Ž Thailand, Singapore, Korea and Sri Lanka denote the Thai bhat–US dollar, Singapore dollar–US dollar, South Korean won–US dollar January 1, 1990 to
. Ž
. Ž .
December 31, 1998 and Sri Lanka rupee–US dollar January 1, 1992 to December 31, 1998 , respectively. Column t a is the summability test statistic; no
T
Ž .
stars means we cannot reject the null of stability at the 90 and then also the 95 and 99 . The null of stability at 95 level cannot be rejected here.
The null of stability at 99 level cannot be rejected here.
ˆ
Only for the Singapore dollar is the estimate of the skewness parameter, b s y0.214, significantly different from zero, implying left skewness. For the persis-
ˆ
Ž .
tence measure V, we obtain values around unity—slightly above below one for Ž
. Ž .
the Singapore dollar Sri Lanka rupee . The values for the t a statistic are
T
insignificant; an exception is the Singapore dollar, for which we reject the SuS hypothesis at the 95 level. With respect to sum stability, the A-PARCH model
Ž comes to somewhat different results. For two of the four currencies South Korean
. won and Singapore dollar , we would reject the SuS hypothesis at the 95 level
and for the two other currencies at the 90 level. This discrepancy for the two models indicates the importance of the chosen specification for the GARCH
equation. It appears that the higher degree of non-linearity and the higher lag
Ž .
length of the Q-GARCH 2, 1 specification is more capable of producing iid-like Ž
. residuals than the A-PARCH 1, 1 model. This coincides to some extent with
Ž .
other findings. Lane et al. 1996 demonstrated the superiority of the Q-GARCH model over that of bilinear models in terms of both model fit and forecasting
ability for data sets exhibiting significant non-linearities; while Franses and van Ž
. Dijk 1996 showed that the Q-GARCH model markedly outperformed several
rival GARCH models in an extensive forecasting comparison. For either model, however, we strongly reject the normality assumption for the innovation process.
5. Conclusions
