The Parameter Variation Of ARCH-Type Models

73 Pages·2009·3.32 MB·English

by  Alex Günsberg

The Parameter Variation Of ARCH-Type Models

Alex Günsberg

A thesis submitted in partial fulfilment of the degree of Master of Science

Hanken School of Economics
Department of Finance and Economical Statistics
Vaasa
2009
Abstract
The purpose of this paper is to test the parameter variation in the ARCH-type models
and to determine which models are statistically on the firmest ground. Moreover this
paper introduces the key features and restrictions that the ARCH-type models possess.
A selection of previous literature is reviewed, focusing on papers studying and
comparing the various volatility models. The findings of this paper show that the
parameters are relatively stable in all ARCH-type models. The PARCH model has the
most stable parameters, however, the GARCH model that is more intuitive and
parsimonious exhibits nearly as stable parameters as the PARCH model.

Keywords: GARCH; TARCH; EGARCH; PARCH; NASDAQ; Paired t-test;
NASDAQ Composite index; Dow Jones Industrial Average
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Table of Contents
Abstract
1 Introduction............................................................................ 3
! !
2 Theory and Previous Research............................................. 6
! !
2.1! Stylized Facts......................................................................................................7!
2.2! General Features of ARCH-Type Models......................................................11!
2.2.1! Nesting ARCH-Type Models....................................................................13!
2.2.2! Restrictions of the ARCH-Type Models.................................................14!
2.3! Previous Research............................................................................................15!
3 Methods................................................................................. 26
! !
3.1! Paired t-test.......................................................................................................27!
4 Data ....................................................................................... 29
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4.1! The NASDAQ Composite and Dow Jones Industrial Average Indices.......30!
4.2! Descriptive Statistics........................................................................................33!
5 Results................................................................................... 37
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6 Conclusion............................................................................. 41
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7 Swedish Summary (Svensk sammanfattning)................... 43
! !
7.1! Motivering och syfte.........................................................................................44!
7.2! Tidigare forskning, material och metoder.....................................................44!
7.2.1! Metoder......................................................................................................47!
7.2.2! Datamaterial..............................................................................................47!
7.3! Redogörelse för genomförandet av undersökningen....................................48!
7.4! Resultat..............................................................................................................48!
7.5! Sammanfattning...............................................................................................50!
References.................................................................................. 52
!
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List of Tables
Table 1: Stylized Facts..................................................................................................................................................7!
Table 2: Nesting of the ARCH Models......................................................................................................................14!
Table 3: Previous Literature......................................................................................................................................25!
Table 4: Descriptive Statistics for Daily Return Series............................................................................................35!
Table 5: Descriptive Statistics for Weekly Return Series........................................................................................36!
Table 6: Paired t-tests.................................................................................................................................................39!
Table 7: Components of Dow Jones Industrial Average.........................................................................................56!
Table 8: Daily Parameter Estimates For NASDAQ.................................................................................................57!
Table 9: Daily Parameter Estimates For Dow Jones................................................................................................59!
Table 10: Weekly Parameter Estimates For NASDAQ...........................................................................................61!
Table 11: Weekly Parameter Estimates For Dow Jones..........................................................................................63!
Table 12: Mean Equations For Daily NASDAQ Series...........................................................................................65!
Table 13: Mean Equations For Daily Dow Jones Series..........................................................................................66!
Table 14: Mean Equations For Weekly NASDAQ Series........................................................................................67!
Table 15: Mean Equations For Weekly Dow Jones Series......................................................................................68!
Table 16: Sign Bias Tests............................................................................................................................................69!

List of Figures
Figure 1: Sectors within NASDAQ OMX Group Decomposed...............................................................................30!
Figure 2: Daily Trading Volume & Adjusted Close Prise of the NASDAQ Composite.......................................31!
Figure 3: NASDAQ Daily Return Series...................................................................................................................31!
Figure 4: Daily Trading Volume & Adjusted Close Prise of the Dow Jones Industrial Average........................32!
Figure 5: Dow Jones Daily Return Series..................................................................................................................33!

Appendix I: Components of NASDAQ-100
Appendix II: Parameter Estimates and Mean Equations
Appendix III: Sign Bias Tests
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1 Introduction
The field of volatility modelling has been one of the most studied arias in finance
during the preceding two decades. The endless pool of studies comparing the different
models has not been able to conduct a clear-cut winner among the models. However,
the family of autoregressive heteroskedasticity models have probably caught the most
attention.
The previous papers studying the ARCH-type models are many. To name a few,
Kuen and Hoong (1992), Walsh and Tsou (1998), Maukonen (2002) and Kumar (2006)
conclude that the integrated generalized autoregressive conditional heteroskedasticity
(IGARCH1) model is superior to the other models whereas for instance Akgiray (1989),
McMillan, Speight and Apgwilym (2000) and Zhang and Pan (2006) find that the
generalized autoregressive conditional heteroskedasticity (GARCH) is the best model.
The previous literature generally consists of four types of papers. First, older papers that
focus mainly on symmetrical ARCH-type models, naive models and exponential
smoothening models (see e.g. Akgiray (1989), Kuen and Hoong (1992) and Figlewski
(1994)). Second, newer papers that compare asymmetrical ARCH-type models with
symmetrical ARCH-type models, different exponential smoothening models and
sometimes even the naive models (see e.g. McMillan, Speight and Apgwilym (2000),
Hansen and Lunde (2001) and Balaban, Bayar and Faff (2006)). Third, papers that test
different option based models (see e.g. Yu (1999), and Bluhm and Yu (2001)) and
compares them with the models mentioned above and finally papers that suggest and
test new models or new specifications of the existing models (see e.g. Ederington and
Guan (2004), and Chou and Wang (2007). This last type of research reflects the nature
of volatility modelling. The quest for the universally best model is still ongoing. It has
been repeatedly shown that the ARCH-type models perform well within various
models, however, the previous literature has not been able to agree with one universal
model that would outperform or even perform well in all studies. One reason for this is
that the different error statistics favour different models resulting in inconclusive
results. However, it is most likely that the existing error statistics will never agree on

1 Sometime referred to equally weighed moving average (EWMA), although commonly in the EWMA,
the smoothening parameters are not refined during the forecast horizon. For instance RiskMetrics uses
0.94 as a smoothening parameter for all equities and time horizons.
4
one superior model, hence the literature on the field should focus on finding new
methods to compare the models rather than trying to find new ones to model the second
moment.
The purpose of this study is to compare the IGARCH, GARCH, threshold
GARCH (TARCH), exponential GARCH (EGARCH) and asymmetric power GARCH
(PARCH) by using the simple yet powerful method of paired t-tests2. Thus the focus of
this paper is laid on the supposedly constant parameters. By using this method the
distinction of the models that are statistically on a firm basis can be made and finally
they offer more reliable comparison and conclusion of the models.
This paper adds to the existing literature in two ways. First, a new method is used
to see if the models fulfil the statistical assumption of constant parameters whereafter
they are placed in order according to their stability. Second, the stability of the
parameters are compared in bear, bull and normal markets. Thus the study is
particularly interesting for those who are concerned with the statistical background of
the ARCH-type models and for those interested in previous literature within the field.
This study uses the quasi maximum likelihood (QML) estimation method to
conduct the volatility forecasts where QML function is optimized using the Marquardt
optimization algorithm. The data set for this study is retrieved from Thomson Data
Stream and includes 4393 daily and 891 weekly observations from both the NASDAQ
Composite and Dow Jones Industrial Average indices. The data are dated from January
1992 to February 2009.
The results of this paper show that the NASDAQ Composite index is more
unstable than the Dow Jones Industrial Average index. This mirrors the fact that the
irregular price variation is larger on the NASDAQ. For the purpose of 1-day ahead
forecasts the PARCH model is considered superior to the other models. However, the
GARCH model is intuitive and parsimonious and performs well in all samples, with the
exception of one subsample in the weekly NASDAQ series, where it is the worst model.
The plan of this term paper is as follows. Section 2, Theory and Previous
Research starts by introducing the statistical assumptions, stylized facts and the general

2 In this study the TARCH and PARCH are defined as in the computer software EViews. Thus the
TARCH estimates variances instead of the volatility (sometimes referred to as the GJR-GARCH). The
PARCH on the other hand includes the asymmetry parameter hence older literature refer to it as the
asymmetric power GARCH. For further details see Sections 2.2.1 and 3.
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characters of the ARCH-type models. This is followed by a review of the most essential
literature focusing on the comparison of the volatility models. Section 3 presents the
methods, namely the volatility models and the paired t-test. The data together with
descriptive statistics are discussed in Section 4. Finally, Section 5 presents the results
and Section 6 discusses and concludes the results.
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2 Theory and Previous Research
There are two statistical assumptions that are crucial in order to model financial time
series and consequently the volatility. The first assumption is Ergodicity, which states
that the sample moments of a realization {x }T approach the process {x }"
t 1 t !"
counterparts when the realization approaches infinity. The second assumption is
Stationarity, which implies that a shift in the time period will not affect the sample
moments of the distribution. The assumption of stationarity allows us to pick data from
different time periods (realizations) and regardless of the time point provide good
estimates of the sample moments of the returns.
Formally, a strictly stationary process x can be written as
t
( ) ( )
f x ,x ,...,x = f x ,x ,...,x , (1)
t1 t2 tm t1+k t2+k tm+k
where f denotes the multivariate density function. Equation (1) implies that the time
series is independently and identically distributed (i.i.d.) in all conditional moments of
the distribution.
The i.i.d. assumption is, however, too restrictive if the objective is to model the
higher moments of the distribution. Thus, the restrictions are commonly loosened from
being independent and identical to only being independent, consequently implying a
martingale process. Martingale processes allow non-linear modelling and dependent
structures in the higher moments of the distribution. The sub-martingale process is
widely accepted among scholars as the statistical definition of the process generating
the distributions of the financial asset returns. Consequently, this paper is taking the
sub-martingale process as given and will not discuss the consequences of loosening the
i.i.d. assumption to being independent.
Mathematically the martingale properties for a process {x }T can be written as
t 0
E! x #<% &t (2)
" t $
and
E#x ! %= x , (3)
$ t t"1& t"1
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where ! is the information set at time point t-1. Consequently the sub-martingale
t"1
process can be written as
E#x ! x " %'0 (4)
$ t t!1 t!1&
where it can be observed that the expected future value is higher than the present value.
In practice it can be agreed that this is true for instance for the risk-free investment as it
yields the risk-free interest rate.
2.1 Stylized Facts
A variety of studies report findings of some common non-normal patterns for daily
stock and index returns. These characteristics (commonly referred to as stylized facts)
are discussed in greater detail for instance in Taylor (2005) and in Cont (2001). An
overview of the stylized facts for financial time series will be presented in Table 1.
Table 1: Stylized Facts
# Stylized Fact Description
i Absence of Autocorrelation The returns of financial time series rarely exhibit any
significant autocorrelations
ii Heavy Tails The tails of the unconditional distribution are heavier
compared to what the Gaussianity assumes
iii Gain/Loss Asymmetry The upward movements of stocks and stock indices are not
as substantial as the downward movements
iv Aggregational Gaussianity The shape of the distribution will get closer to a normal
Gaussian distribution as the sample size increases
v Intermittency The returns vary substantially at any point in time
vi Volatility Clustering The volatility is significantly autocorrelated for several
days
vii Conditional Heavy Tails The tails of the conditional distributions are heavier than
the Gaussianity assumes
viii Slow Decay of Autocorrelation in The autocorrelation of the absolute returns decays slowly,
Absolute Returns which indicates a long memory
ix Leverage Effect The asset returns are negatively correlated with the
volatility of the asset
x Volume/Volatility Correlation The volatility is correlated with the trading volume
xi Asymmetry in Time Scales The rough (with respect to time) volatility estimates make
better predictions of the fine-scale volatility than the fine-
scale predictions of the rough-scale volatility
Source: Cont (2001)
The autocorrelation of returns between time point t and t+k can be estimated by
the sample autocorrelation !ˆ at time point k,
k,r
8
n"k n
!ˆ = #(r "r)(r "r)/#(r "r)2 k >0 , (5)
k,r t t+k t
t=1 t=1
where n is the number of observations. The return series, however rarely exhibit
significant autocorrelations between lags, thus implying the absence of autocorrelation
between returns for different days. Due to the statistical assumptions, it can be proven
that the autocorrelation should be zero for all lags k, following the Taylor (2005)
derivation,
cov(r,r )=cov(!u ,! u )
t t+k t t t+k t+k
= E"!u! u $& E"!u $E"! u $
# t t t+k t+k% # t t% # t+k t+k%
= E"!! $E"u $E"u $& E"! $E"! $E"u $E"u $ (6)
# t t+k% # t% # t+k% # t% # t+k% # t% # t+k%
=0&0
=0.
In contrast to the returns that are generally not autocorrelated, the absolute returns
d
r are empirically known to exhibit significant autocorrelations for several days.
t
( )2
Mathematically the significant autocorrelation can be proven. Let s= r !µ , then
t t
cov(s,s )=cov(!2u2,!2 u2 )
t t+k t t t+k t+k
= E"!2u2,!2 u2 $& E"!2u2$E"!2 u2 $
# t t t+k t+k% # t t % # t+k t+k%
= E"!2!2 $E"u2$E"u2 $& E"!2$E"!2 $E"u2$E"u2 $ (7)
# t t+k% # t % # t+k% # t % # t+k% # t % # t+k%
=cov(!2,!2 )
t t+k
>0,

as shown by Taylor (2005).
Usually the autocorrelations in absolute returns are significant when d=2,
however, the autocorrelations are commonly most severe when d!1 (Ding and Granger,
1996). This is thus an argument against the ARCH-models that estimate the conditional
heteroskedasticity from the second moment instead of the first moment. Due to the
uncertainty of the best fitting value for d the PARCH specification is especially
attractive compared to the other conditional heteroskedasticity models. PARCH
estimates the power function ! from the data, thus adding more dynamics to the model
(Ding, Granger and Engle, 1993).