Про збіжність GARCH(p,q)
The paper deals with symmetric GARCH(p,q) model. Assuming that there exists defined by this model stationary time series, we have proposed the necessary and sufficient condition for exponential mean square convergence of any stochastic recurrent procedure satisfying this model to the above stationar...
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2018
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| author | Carkovs, J. Gutmanis, N. |
| author_facet | Carkovs, J. Gutmanis, N. |
| author_institution_txt_mv | [
{
"author": "J. Carkovs",
"institution": null
},
{
"author": "N. Gutmanis",
"institution": null
}
] |
| author_sort | Carkovs, J. |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2018-04-11T11:13:01Z |
| description | The paper deals with symmetric GARCH(p,q) model. Assuming that there exists defined by this model stationary time series, we have proposed the necessary and sufficient condition for exponential mean square convergence of any stochastic recurrent procedure satisfying this model to the above stationary time series. A mathematical background of the proposal approach is based on the derived covariance method for mean square exponential stability analysis of linear stochastic difference equations, which permits one to state a mean square convergence criterion for GARCH(p,q) models with any integer positive p and q in the convenient for application form of an integral inequality involving the model parameters. |
| first_indexed | 2025-07-17T10:23:55Z |
| format | Article |
| fulltext |
© J. Carkovs, N. Gutmanis, 2007
120 ISSN 1681–6048 System Research & Information Technologies, 2007, № 1
TIДC
НОВІ МЕТОДИ В СИСТЕМНОМУ АНАЛІЗІ,
ІНФОРМАТИЦІ ТА ТЕОРІЇ ПРИЙНЯТТЯ РІШЕНЬ
УДК 519.21
ON GARCH(p,q) CONVERGENCE
J. CARKOVS, N. GUTMANIS
The paper deals with symmetric GARCH(p,q) model. Assuming that there exists
defined by this model stationary time series, we have proposed the necessary and
sufficient condition for exponential mean square convergence of any stochastic
recurrent procedure satisfying this model to the above stationary time series.
A mathematical background of the proposal approach is based on the derived co-
variance method for mean square exponential stability analysis of linear stochastic
difference equations, which permits one to state a mean square convergence crite-
rion for GARCH(p,q) models with any integer positive p and q in the convenient
for application form of an integral inequality involving the model parameters.
1. INTRODUCTION: STATIONARY GARCH(p,q) MODELS
Over the last decade, there has been a tendency to employ to analysis the financial
time-series data model the regression equation for exogenous variables )(s
tX ,
Nk ,...,,= 21 endogenous variables tY , and residuals tU defined by formulae
,+++= −
==
−
=
∑∑∑ t
s
mtms
N
s
n
m
mtm
n
m
t UXbYabY )(
111
0
2
1
2
1
1
}{0}{ ttttttltl
n
l
t EEUcU σξξξ =Φ/,≡Φ/,+= −−−
=
∑ , (1)
where }{ Ztt ∈,ε is white-noise type time series (that is, i.i.d. random variables
with mean zero and variance one), 1−Φ t is sigma-algebra of information up to
time 1−t , defined by random variables }1{ −≤, tssε , and },{ Ztt ∈ξ is time se-
ries of errors (shocks) with variance, that is given as GARCH ),( qp process
(Generalized Auto Regressive Conditional Heteroskedasticity), that takes the fol-
lowing form [4]:
.++= −−
=
−
=
∑∑ 22
1
2
1
0
2
ktktk
q
k
ktk
p
k
t εσθσϕθσ (2)
The above processes are defined for time moments Zt∈ by 1+q coeffi-
cients }10,0{ 0 qkk ,...,=,≥> θθ , p coefficients }210{ pkk ,...,,=,≥ϕ , mean 0b ,
1+nN linear regression coefficients, conditional variance 2
tσ and distribution of
On GARCH(p,q) convergence
Системні дослідження та інформаційні технології, 2007, № 1 121
random variable 0ε . As it has been shown by [1], under assumption
1
11
<+∑∑
==
k
q
k
k
p
k
θϕ there exists defined by (2) stationary time-series σ̂{ 2
t , }Zt∈
and expectation of deviations σσ ˆ 22
tttu −=: of any other satisfying (2) time series
}{ 2 Ztt ∈,σ converge to zero in the mean with ∞→t , that is
0ˆlim 22 |=−|
∞→
σσ ttt
E . This paper supposes the above inequality +∑
=
k
p
k
ϕ
1
1
1
<+∑
=
k
q
k
θ to be fulfilled. It should be mentioned that parameters of regression
model (2) are mainly defined by the least square method and therefore it is prefer-
able [5] to analyze a behavior of the second moments of iterations (2) with
∞→t . We will say that the stationary GARCH model (2) is exponential mean
square stable if the above second moments exponentially tend to zero as ∞→t ,
that is, there exist such positive numbers λ,M that
}ˆ{}ˆ{ 222)(222 |−|≤|−| −− σσσσ λ
ss
st
tt Me EE (3)
for any Zsst ∈,≥ . The problem arises: to determine a largest set of parameters
of model (2), which guarantees the stability property (3). For GARCH(p,1) mod-
els this problem has been discussed in the paper [2]. Applying some of well
known mathematical results for positive defined matrices, the mentioned paper
derives the necessary and sufficient condition for exponential mean square stabil-
ity in a form of inequality involving forth moment of tε and parameters
11 θϕϕ ,,..., p . In spite of the convenience for application of the proposal there
approach for 1=q , that has been written as an inequality for two specially con-
structed determinants, it becomes very complicated for GARCH ),( qp -models
with 2≥q . To eliminate this lack we will apply another method, developed in
paper [3] for asymptotical stability analysis of linear stochastic difference equa-
tions. It permits us to derive necessary and sufficient exponential mean square
stability condition for any p and q in convenient for application form.
2. INTEGRAL CRITERIA FOR GARCH(p,q) EXPONENTIAL MEAN SQUARE
STABILITY
It is easy to write for the deviations 22ˆ tttu σσ −=: the homogeneous difference
equation
,+= −−
=
−
=
∑∑ ktktk
q
k
ktk
m
k
t yuuau θ
11
(4)
where ,≤<,≤<,≤,+= qkppkqqpka kkkkk if andif},{min if ϕθθϕ
},{max qpm = and 12 −= tty ε . The latter random variables }{ Ztyt ∈, are i.i.d.
J. Carkovs, N. Gutmanis
ISSN 1681–6048 System Research & Information Technologies, 2007, № 1 122
with mean zero and variance 224 1 |−|=: tEs ε defined by distribution of tε .
Formula (4) defines a linear difference equation with random coefficients and the
problem is: to find necessary and sufficient conditions for exponential mean
square decreasing of its solutions. Let sequence }{ Ztut ∈, be a solution of (4).
According to proposal in [3] method first of all we have to define two se-
quences: }{ Ztht ∈, , satisfying for 0>t homogeneous difference equation
)(2211 hha…hahah mtmttt ,+++= −−− under conditions 010 =,= thh for 1−≤t ,
and }0~{ >,tx t satisfying the same homogeneous difference equation
,+++= −−− xa…xaxax mtmttt
~~~~
2211 but for 0≤t is the same as tu , that is,
0~ ≤,= tux tt . Now we should rewrite equation (4) in a following form
,+= −−
==
∑∑ kkjjkt
t
k
q
j
tt uyhgu θ
11
where jsjsjst
j
s
q
j
tt uyhxg −−−
==
∑∑+= θ
11
~ is 0Φ -adop-
ted random sequence for any 0≥t . Squaring the both parts of the above equity
and taking a conditional expectation under condition 0Φ we can reach for condi-
tional second moment }{ 0
2 Φ/||=: tt uEm an equation += 2
tt gm
kkt
t
k
mbs 2
1
4
−
=
∑+ , where jjt
q
j
t hb θ−
=
∑=
1
. Because 2
tg and 2
tb are exponentially
decreasing to zero nonnegative sequences, any satisfying (4) positive sequence
}0{ ≥, tmt may be majorized by sequence }0{ ≥, tct for sufficiently large c .
Therefore to analyze an asymptotic of this sequence we may apply discrete
Laplace transformation multiplying the both parts of equation for tm by tz with
some constant )0( 1−,∈ cz and summarizing by t from 0 to ∞ . This approach
permits to find function ∑∞
=
:= 0)( t t
t mzzM in a form of fraction =)(zM
))(1/()( 4 zBszG −= , where ∑∞
=
:= 0)( t t
t gzzG , ∑∞
=
= 0
2:)( t t
tbzzB . It is obvi-
ously that tm exponentially decreases with ∞→t if and only if the series t
t
m∑
∞
=0
converges. Therefore one can make sure of equivalence the latter assertion to ine-
quality 122 ))1((}1{E −<|−|| Btε involving fourth moment of white noise and
parameters of GARCH ),( qp . Let )(1 zB be a discrete Laplace transformation of
sequence }{ tb , that is, t
t
t
zbzB ∑
∞
=
=:
0
1 )( . Applying the well known Cauchy theo-
rem one can find
∫
∑∑
∑∑
=
=
−
=
= =
−−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
1
11
1
1
11
2
1)1(
z
m
k
k
k
m
k
k
k
q
kj
q
k
kj
dz
zaza
zk
i
B
j
θθ
π
.
On GARCH(p,q) convergence
Системні дослідження та інформаційні технології, 2007, № 1 123
The function )(1 zB is a Z -transformation of series jjt
q
j
t hb θ−
=
∑=
1
. There-
fore applying Z -transformation one can find expression )1(B in an integral form
∫
∑∑
∑∑
=
=
−
=
−
= =−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
−
1
11
1
11
4
11
2
1
z
m
k
km
k
m
k
k
k
m
q
kj
q
k
dz
zaza
zzk
i
S
kj
j
θθ
π
, (5)
where }max{ qpm ,= and ka defined above in formula )( ka . Therefore the nec-
essary and sufficient condition for stationary GARCH ),( qp mean square stability
has a form an inequality 4
4 1E St +<ε . The integral in (5) can be calculated apply-
ing residual theory. For example necessary and sufficient exponential mean
square stability condition of GARCH )2,2( has following complete form:
.
++−−+
+++−−−
+<
211122
2
2
2
1
22
2
11
2
224
)(4)1)((
)1]()()1[(
1E
θθθϕθϕθθ
θϕθϕθϕ
ε t
REFERENCES
1. Bollerslev T. 1986. Generalized autoregressive conditional hederoskedasticity. In
Journal of Econometrics, 307–327.
2. Carkova V., Gutmanis N. 2002. On Convergence of GARCH(p,q). In Statistical
Modelling in Society. Proceedings of the 17th Inernational Workshop on Statisti-
cal Modelling (Chania, Greece, 8–12 July 2002). National and Kapodistrian Uni-
versity of Athens & University of North London, 149–152.
3. Carkova V., Carkovs J. 1969. On Stability of Solutions of Difference Equations
with Random Coefficients. In Latvijskij Matematicheskij Ezhyegodnik, 5,
153–173. Riga: LU
4. Hamilton J. 1994. Time Series Analysis. Princeton: Princeton University Press.
5. He C., Terasvirta T. 1999. Fourth moment structure of the GARCH(p,q) process //
Econometric Theory, 824–846.
6. Swerdan M., Carkov J. 1994. Stability of Stochastic Impulse Systems. RTU, Riga.
Received 11.07.2006
From the Editorial Board: the article corresponds completely to submitted manu-
script.
|
| id | journaliasakpiua-article-127965 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:23:55Z |
| publishDate | 2018 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/13/41879c5854960e69077b8daa0da3a613.pdf |
| spelling | journaliasakpiua-article-1279652018-04-11T11:13:01Z On GARCH(p,q) convergence O сходимости GARCH(p,q) Про збіжність GARCH(p,q) Carkovs, J. Gutmanis, N. The paper deals with symmetric GARCH(p,q) model. Assuming that there exists defined by this model stationary time series, we have proposed the necessary and sufficient condition for exponential mean square convergence of any stochastic recurrent procedure satisfying this model to the above stationary time series. A mathematical background of the proposal approach is based on the derived covariance method for mean square exponential stability analysis of linear stochastic difference equations, which permits one to state a mean square convergence criterion for GARCH(p,q) models with any integer positive p and q in the convenient for application form of an integral inequality involving the model parameters. Рассматривается симметричная модель GARCH(p,q). В предположении, что существует задаваемый этой моделью стационарный временной ряд, предлагается необходимое и достаточное условие сходимости в среднем квадратичном любой итерационной процедуры, удовлетворяющей уравнению GARCH(p,q), к этому стационарному процессу. Предложен ковариационный метод анализа линейных разностных уравнений со случайными коэффициентами, который позволил для произвольных целых неотрицательных p и q сформулировать критерий сходимости в среднем квадратичном в удобной для использования форме в виде интегрального неравенства, содержащего параметры модели. Розглядається симетрична модель GARCH(p,q). Припускаючи існування стаціонарного часового ряду, який задається цією моделлю, пропонується необхідна і достатня умова збіжності у середньому квадратичному будь-якої ітераційної процедури, що задовольняє рівнянню GARCH(p,q), до такого стаціонарного процесу. Запропоновано коваріаційний метод аналізу лінійних різницевих рівнянь із випадковими коефіцієнтами, що уможливило для довільних цілих від’ємних p и q сформулювати критерій збіжності у середньому квадратичному в зручній для використання формі у вигляді інтегральної нерівності із параметрами моделі. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2018-04-04 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/127965 System research and information technologies; No. 1 (2007); 120-123 Системные исследования и информационные технологии; № 1 (2007); 120-123 Системні дослідження та інформаційні технології; № 1 (2007); 120-123 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/127965/122798 Copyright (c) 2021 System research and information technologies |
| spellingShingle | Carkovs, J. Gutmanis, N. Про збіжність GARCH(p,q) |
| title | Про збіжність GARCH(p,q) |
| title_alt | On GARCH(p,q) convergence O сходимости GARCH(p,q) |
| title_full | Про збіжність GARCH(p,q) |
| title_fullStr | Про збіжність GARCH(p,q) |
| title_full_unstemmed | Про збіжність GARCH(p,q) |
| title_short | Про збіжність GARCH(p,q) |
| title_sort | про збіжність garch(p,q) |
| url | https://journal.iasa.kpi.ua/article/view/127965 |
| work_keys_str_mv | AT carkovsj ongarchpqconvergence AT gutmanisn ongarchpqconvergence AT carkovsj oshodimostigarchpq AT gutmanisn oshodimostigarchpq AT carkovsj prozbížnístʹgarchpq AT gutmanisn prozbížnístʹgarchpq |