Empirical Analyses of Industry Stock Index Return
Distributions for the Taiwan Stock Exchange
Svetlozar T. Rachev∗, Stoyan V. Stoyanov, Chufang Wu, Frank J. Fabozzi
–––––––––––––––––––––––––––––––
∗Svetlozar T. Rachev (contact person)
E-mails: rachev@pstat.ucsb.edu; zari.rachev@statistik.uni-karlsruhe.de;
zari.rachev@finanalytica.com
Econometrics, Statistics and Mathematical Finance, School of
Economics and Business Engineering, University of Karlsruhe,
D-76128 Karlsruhe, Germany and Department of Statistics and
Applied Probability, University of California Santa Barbara,
CA 93106, USA.
Stoyan V. Stoyanov
Chief Financial Researcher in FinAnalytica, Inc., Seattle, USA.
Chufang Wu
Department of Applied Mathematics, National Donghua University,
Hualien, Taiwan, ROC.
Frank J. Fabozzi
Frederick Frank Adjunct Professor of Finance, School of Management,
Yale University, USA.
–––––––––––––––––––––––––––––––
Acknowledgement. Prof Rachev gratefully acknowledges research support
by grants from Division of Mathematical, Life and Physical Sciences, College of
Letters and Science, University of California, Santa Barbara and the Deutschen
Forschungsgemeinschaft.
1

---

Empirical Analyses of Industry Stock Index Return
Distributions for the Taiwan Stock Exchange
Abstract
We study the daily return distributions for 22 industry stock indexes on the Tai-
wan Stock Exchange under the unconditional homoskedastic independent, identically
distributed and the conditional heteroskedastic GARCH models. Two distribution
hypotheses are tested: the Gaussian and the stable Paretian distributions. The per-
formance of the stable Paretian distribution is better than that of the Gaussian dis-
tribution. A back-testing example is provided to give evidence on the superiority of
the stable ARMA-GARCH to the normal ARMA-GARCH.
Keywords: Stable distributions, ARMA-GARCH, heavy tails, volatility cluster-
ing, Value at Risk.
1

---

Empirical Analyses of Industry Stock Index Return
Distributions for the Taiwan Stock Exchange
1
Introduction
It is well known that financial returns are non-normal and tend to have fat-tailed dis-
tributions. Mandelbrot (1963) strongly rejected normality as a distributional model
for asset returns, conjecturing that financial return processes behave like non-Gaussian
stable processes (commonly referred to as "stable Paretian" distributions). The au-
toregressive conditional heteroskedastic (ARCH) models proposed by Engle (1982)
and the generalized GARCH proposed by Bollerslev (1986) capture the extra proba-
bility mass in the tails. The appealing feature of incorporating conditional volatility
is that it allows for a changing distribution over time. However, the distribution of
conditional residuals is still not normal (Bollerslev, 1987). The implication for the
commonly used risk measure Value-at Risk (VaR) is that risk is still underestimated
at high quantiles for fat-tailed results. Moreover, GARCH models also fail to model
the asymmetric effect of volatility, where negative return shocks generated by bad
news have a larger thrust in increasing future volatility than positive return shocks
caused by good news.
One innovation has focused on the power term by which the data are to be trans-
formed. Ding, Granger and Engle (1993) introduced a generalized asymmetric ver-
sion of the power ARCH (APARCH) model to capture the potentially asymmetric
effects of return shocks on future volatility. To further enhance the robustness of
the estimation results with respect to non-normality, the errors are considered to
2

---

follow a t-distribution, called Student-APARCH. Huang and Lin (2004) analyzed
the VaR for Taiwan stock data. They assumed the asset returns have fat tails and
volatility clustering. At lower VaR confidence levels, the Normal-APARCH model
is preferred. However, at high confidence levels, the VaR forecast obtained by the
Student-APARCH model is more accurate.
Chiang and Doong (1999) used a generalized M-GARCH(1,1) process and found
evidence to reject the hypothesis that the stock excess returns are independent of
the real and financial volatilities. The stock excess returns are explained by the
predicted volatility of macrofactors and the conditional standard deviation. The
volatility of macrofactors consists of the volatilities arising from real (internal) and
financial (external) shocks, whereas the time-series volatility is due to previous shocks.
The stock excess return is associated with the volatility of macrofactors. The finance
industry is more sensitive to a change in economic conditions and has been the leading
industry on the Taiwan Stock Exchange (TSE) in the past decade.
In a study of the TSE, Ammermann (1999) found that the stocks trading ex-
hibit nonlinearity and nonstationarity. To capture the full-sample nonlinear serial
dependencies found within a number of financial time series, the Normal-GARCH,
t-GARCH, and STAR (student t autoregressive) models were fitted and compared
with the dynamic linear models. The inferences obtained varied from model to model,
suggesting the importance of adequately accounting for nonlinear serial dependencies
3

---

(and of ensuring data stationarity) when studying financial time series.
Rachev and Mittnik (2000) give a very detailed description on the stable Paretian
models in finance. The stability property is highly desirable for asset returns. In the
context of portfolio analysis and risk management, the linear combinations of different
return series follow again a stable distribution. In fact, the Gaussian law shares this
feature, but it is only one particular member of a huge class of distributions, which
also allows for skewness and heavy tails.
In this paper, we study industry stock index return data with respect to: (1)
non-Gaussian, heavy-tailed and skewed distributions, (2) volatility clustering (ARCH-
effects), (3) temporal dependence of the tail behavior, and (4) short- and long-range
dependence. Stable models allow us to generalize Gaussian-based financial theories
to build a more general framework for financial modelling. Since asset returns exhibit
temporal dependence, the conditional distributions become of interest. We study the
daily return distributions for 22 industry stock indexes on the TSE under the uncon-
ditional homoskedastic independent, identically distributed (iid) and the conditional
heteroskedastic GARCH (varying-conditional-volatility) cases. Two distribution hy-
potheses were tested: the Gaussian and the stable Paretian distribution. The stable
Paretian distribution performed better than that of the Gaussian distribution.
In section 2, we state the probability models and measures applied in this paper. In
section 3, the numerical analyses results are demonstrated, followed by a back-testing
4

---

example in section 4. The conclusion is provided in section 5.
2
Probability models
The class of autoregressive moving average (ARMA) models is a natural candidate
for conditioning on the past of a return series. These models have the property
that the conditional distribution is homoskedastic. Moreover, since financial mar-
kets frequently exhibit volatility clustering, the homoskedasticity assumption may
be inadequate. On the contrary, the conditional heteroskedastic models, such as
ARCH and the GARCH models, combining with an ARMA model, referred to as an
ARMA-GARCH model, are common in empirical finance. It turns out that ARCH-
type models driven by normally distributed innovations imply unconditional distri-
butions which themselves possess heavier tails. However, many studies have shown
that GARCH-filtered residuals are themselves heavy-tailed, so that stable Paretian
distributed innovations ("building blocks") would be a reasonable distributional as-
sumption.
A random variable X is said to have a stable distribution if there are parameters:
α ∈ (0,2], β ∈ [−1,1], σ ∈ [0,∞), μ ∈ R such that its characteristic function has the
following form:
5

---

⎧ exp{−σα|θ|α(1−iβ(signθ)tan(απ))+iμθ
2
}
if α 6= 1,
ϕ (θ) = ⎪
X
⎨
⎪
⎩ exp{−σ|θ|(1+iβ2(signθ)ln
π
|θ|) + iμθ}
if α = 1.
In the general case, no closed-form expressions are known for the probability den-
sity and distribution functions of stable distributions. The parameter α is called
the index of stability, which determines the tail weight or density’s kurtosis. The
parameters β, σ, and μ are called the skewness parameter, scale parameter, and loca-
tion parameter, respectively. Stable distributions allow for skewed distributions when
β 6= 0; when β is zero, the distribution is symmetric around μ. Stable Paretian laws
have fat tails, meaning that extreme events have high probability relative to the nor-
mal distribution, when α < 2. The Gaussian distribution is a stable distribution, with
α = 2. (For more details on the properties of stable distributions see Samorodnitsky,
Taqqu (1994).)
The general form of the ARMA(p,q)-GARCH(r,s) model is:
p
q
Rr = C + XaiRt−i+Xbjεt−j +εt
i=1
j=1
εt = σtδt
r
s
σ2 = K +
ω
+
υ
t
X kε2t−k X lσ2t−l
k=1
l=1
where ai, bj, ωk, νl, C, K are the model parameters, for i = 1, . . . , p, j = 1, . . . , q,
k = 1, . . . , r, l = 1, . . . , s. δ0 s are called the innovations process and are assumed to
t
6

---

be iid random variables which we additionally assume to be either Gaussian or stable
Paretian. An attractive property of the ARMA-GARCH process is that it allows a
time-varying volatility via the last equation in the above model.
We test the hypotheses in two cases. In the first case, we assume that daily
return observations are iid. In the second case, the daily return observations are
assumed to follow a GARCH(1,1) model. The first case concerns the unconditional
homoskedastic distribution model while the second case belongs to the class of con-
ditional heteroskedastic models.
For both cases, we verify whether the Gaussian hypothesis holds based on the
Kolmogorov distance (KD):
KD = sup |Fe(x) − F(x)|,
x∈R
where Fe(x) is the empirical sample distribution and F (x) is the cumulative distri-
bution function of the estimated parametric density and emphasizes the deviations
around the median of the distribution.
For both the iid and the GARCH cases, we compare the goodness-of-fit for the
Gaussian and the more general stable Paretian hypotheses. We use two goodness-
of-fit measures for this purpose, the KD-statistic and the Anderson-Darling (AD)
statistic. The AD-statistic accentuates the discrepancies in the tails and is computed
as follows:
7

---

AD = sup
|Fe(x) − F(x)| .
x∈R pF(x)(1−F(x))
The data used in this study consist of the daily returns for 22 industry stock
indexes (sectors) from the entire TSE. The industries are listed in the first column of
Table 1. The sample period of this research spans January 1999 through December
2002. Industry stock returns are defined as the first difference in the log of daily
indexes, R(t) = log(S(t)/S(t − 1)), where S(t) is the value at t (the returns are
adjusted for dividends).
3
Main results
There are several methods that can be employed for estimating the parameters of sta-
ble distributions. The most popular methods are Maximum Likelihood (ML), Fourier
Transform (FT), and Fast Fourier Transform (FTT), see Rachev and Mittnik (2000),
Rachev (2003). Only ML easily allows for estimation of the skewness parameter β; it
is also the most accurate method. However it is not the fastest.
3.1
Unconditional homoskedastic iid model
In the simple setting of the iid model, we have estimated the values for the four param-
eters of the stable Paretian distribution using the ML method. Summary statistics of
the various statistical tests and parameter estimates for the entire sample are provided
8

---

in Table 1.
For the in-sample analyses, we use the standard Kolmogorov-Smirnov test based
on the KD. We observe that 59.09% and 27.27% of the industry sectors for which
normality is rejected at confidence levels 95% and 99%. On the contrary, 22.73% and
4.55% of the sectors for which the stable Paretian distribution is rejected at confidence
levels 95% and 99%. Therefore we have evidence that the stable-Paretian hypothesis
is rejected in much fewer cases, hence the stable Paretian distribution fits better than
the normal distribution.
For every industry index in our sample, the KD in the stable Paretian case is
below that in the Gaussian case. The same is true for the AD. The KD implies that
for our sample there is a better fit of the stable Paretian model around the center of
the distribution while the AD implies a better fit in the tails. The substantial differ-
ence between the AD computed for the stable Paretian model relative to the Gaussian
model strongly suggests a much better ability for the stable Paretian model to forecast
extreme events and confirms an already noticed phenomenon: the Gaussian distri-
bution fails to describe observed large downward or upward asset price shifts. That
is, in reality extreme events have larger probability than predicted by the Gaussian
distribution
9

---

Document Outline

• cover.pdf
• twn8.pdf
• tables.pdf

---