See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/5141743
Autoregressive Gamma Process Article in Journal of Forecasting · March 2006 DOI: 10.2139/ssrn.757358 · Source: RePEc
CITATIONS
READS
72
99
2 authors: Christian Gourieroux
Joann Jasiak
University of Toronto
York University
287 PUBLICATIONS 7,719 CITATIONS
53 PUBLICATIONS 1,215 CITATIONS
SEE PROFILE
SEE PROFILE
All content following this page was uploaded by Joann Jasiak on 19 September 2014. The user has requested enhancement of the downloaded file.
Les Cahiers du CREF
ISSN: 1707-410X
Autoregressive Gamma Processes Christian Gouriéroux Joann Jasiak CREF 05-03 April 2005
Copyright © 2005. Christian Gouriéroux, Joann Jasiak Tous droits réservés pour tous les pays. Toute traduction et toute reproduction sous quelque forme que ce soit sont interdites. Les textes publiés dans la série «Les Cahiers du CREF» de HEC Montréal n'engagent que la responsabilité de leurs auteurs. La publication de cette série de rapports de recherche bénéficie d'une subvention du programme de l'Initiative de la nouvelle économie (INE) du Conseil de recherches en sciences humaines du Canada (CRSH).
Autoregressive Gamma Processes Christian Gouriéroux Professor CREF, CREST, CEPREMAP and University of Toronto 15, boulevard Gabriel Péri 92245 Malakoff France E-Mail:
[email protected]
Joann Jasiak Professor CREF and York University Department of Economics 4700 Keele Street Toronto, Ontario M3J 1P3 Canada E-Mail:
[email protected]
April 2005
Les Cahiers du CREF CREF 05-03
Les Cahiers du CREF
CREF 05-03
Autoregressive Gamma Processes Abstract We introduce a class of autoregressive gamma processes with conditional distributions from the family of noncentered gamma (up to a scale factor). The paper provides the stationarity and ergodicity conditions for ARG processes of any autoregressive order p, including long memory, and closed-form expressions of conditional moments. The nonlinear state space representation of an ARG process is used to derive the filtering, smoothing and forecasting algorithms. The paper also presents estimation and inference methods, illustrated by an application to interquote durations data on an infrequently traded stock listed on the Toronto Stock Exchange (TSX). Keywords: Frequency
Intertrade Durations, Autoregressive Gamma, CIR, High
JEL Classification : C51
Processus gamma autorégressifs Résumé Nous présentons une classe de processus gamma autorégressifs dont la distribution conditionnelle appartient à la famille de gamma décentrées. L’article fournit les conditions de stationnarité et d’ergodicité de ces processus autorégressifs de tous ordre p, incluant ceux à mémoire longue, et des expressions fermées des moments conditionnels. La représentation non linéaire de l’espace d’état d’un processus gamma autorégressif est utilisée pour dériver des algorithmes de filtrage, de lissage et de prévision. L’article présente également des méthodes d’estimation et d’inférence, illustrées par une application aux données de durée inter-cotations d’une action du Toronto Stock Exchange transigée peu fréquemment. Mots clés : Durées inter-transactions, Gamma autorégressifs, CIR, Haute fréquence
___________________________ Acknowledgments: The authors gratefully acknowledge financial support of the Natural Sciences and Engineering Research Council of Canada.
! "
# $%&! ' ( ' ( )! * $%&
* " $%&
$ $%& Æ $ +,&!
- .//0! +,& $%&
+,&
Æ
1 $ $(! 2 )/
/
3! + + / 4!
Æ 3/! 5 * $ )4! $(! # $%& +,& $( 6 $( $%& +,& * # + . $( !! ! + . + 0 $( + 4 $(
$( %#( + 7 + 8 $( $(!! $( # + 3 $( + + )
+ %
+ / 9 $
3 - 7 3 # 6 # # - 3 ) - # 3 8 # 9 533 - # 3 #
: ! Æ 2 ; Æ ! Æ ! * < Æ 9 < " Æ ! 9 ! Æ = ! ; Æ ! <
Æ
! > !
Æ ?Æ = ! @
Æ
. !
<
> Æ =
> Æ = .
..!
! $( !! ; Æ !
& : $( ! #
Æ
Æ 2
> Æ .
6 $( <
>
=
! ! ! A + /!B $( C $ $( !
< ; Æ
> >
>
!
9 < $
! <
: $( ! > > > > D 9 <
>
> = = = ! >
< > / >
# < 0
! $ $( ! ; Æ ! > ! <
! ; Æ !
! <
;Æ /
!
Æ + 80
$( #
6 9 . <
$( ! <
! !
> Æ =
> Æ = .
<
!
>
!
>
!Æ = ! ! Æ = . !
4
" $( ! $( ! $( ! <
! ! / ÆÆ ! = . Æ
! =
< E > Æ ! # Æ Æ " Æ ! ! " Æ Æ Æ ! 9 4
* $( !
" Æ $( ! " Æ $( #
6 $( ! $( <
! > %
! >
0 !
# $ . <
! > %
! >
:5
= .Æ = ! .Æ = 0
0.!
3 ; 4 8 9 53 ) 3 4 # # #
F Æ ! >
! <
Æ
Æ
?Æ !
!
!
! Æ ?Æ = ! @
Æ
00!
04!
F Æ ! > F Æ ! =
<
!
< F Æ ! >
?Æ!? = !
?Æ = !
Æ ! Æ !
Æ
!
* 2 <
! >
Æ
!
?Æ = ! ?Æ = !? = ! @
< 8
! > / % A ! !B > / >
! >
9 7
<
F Æ !
> F Æ ! =
F Æ ! F Æ !
" < A ! B > !
$( ! $( ! 6 ! ! > AÆ B $( ! Æ Æ
. Æ Æ = ! ! ! ! > Æ = ! = . Æ Æ = ! . $( ! Æ
!
C
! ! <
A B
3
<
AÆ = B
6 # 6 ! : G
> / > > > $(
# ! !
!
! ! ! <
?Æ = !? = !
! >
)
?Æ = !? = !
9 < + $ 4
! " !
! :
<
! >
! = !Æ
½
?Æ = = ! ? = !?Æ = !
<
> Æ = >
=
9 < + $ 7
6
! " $( ! %# ( %#(! 6 # %#( $( * * * " <
A
>
.
Æ=
Æ
> ! B
Æ
>
>
> >
!
> !
! Æ = . ! !
.
$( ! %#( % )7!<
< >
>
/ > Æ
7 ! . $ >
! =
2 2
.!
$( *
%#(
$( %#( $( # /
# !" $( " $(!
#
$%
!
A$( !B > ! ; Æ ! > = =
/ ! >
H $( ! $(! Æ " $(! > ! : 2 <
"
" !
> > >
"
! A" " B " ! " A = " B " " ! = "
Æ
$ ! "!
2
> A
= " B
Æ
/ ./
8 !
<
-
./
/ 0 . /
/ ./
12
#
$(! $! &
.//.!
$( ! "! > / " $ 9
<
< = = = ! + $ 8
#
&
$( ! $( ! Æ
Æ
!
> 2 > " > > 2 A $ 2 B<
A" !
B > A"! = "!B
"! > " = "! "! > Æ = "! 2 * <
B > AÆ "! = "!B ! Æ <
A"
!
Æ "! >
"
= "
" Æ "! > /
& .//.! :
A" ! B
.
$ : $( ! $%& +,& - )!
! Æ
! 2 $( ! * Æ " A % B<
!B A F Æ !B I A F Æ !BI F Æ !B
! > A F Æ
=
> ! # Æ
Æ ! > Æ !
Æ ! >
Æ !
Æ <
#$%
Æ ! >
Æ !
2 A/ B $ & Æ D <
Æ & ! > Æ !
Æ & ! >
Æ !
#$%
! > A#$% Æ !Æ & !B =#$% A Æ ! Æ !Æ & !B 0
> A#$% Æ !Æ & !B > A Æ !Æ & !B > ! Æ !
<
Æ & ! > !
#$''
% & Æ Æ 6 - )/!
$ % # + $ G # $ $(! 2 )/! 2 <
>
=
/ ! " (! * (! H $( $( < ! D $( G
4
# ! $( $( > ! 6
> $( ! * 2 $( $( % % ! # 2 <
A" !
B > A"! = "! A"!BB
2 H < $ : 9 2 )/!< $( - .//0!< $( + /! 4!
"! > " "! > " = "!
& 2 * ) # :: $(
'
$( ! <
7
> Æ =
> Æ = .
% Æ ! > Æ
6
9:2 <
J ÆJ J >
Æ
'
Æ . Æ = . . Æ = .
( )
<
Æ ! >
F Æ !
) !
$( ! H ! <
! Æ ! > = ! = Æ ?Æ = ! @ ).! ! <
Æ ! >
F Æ !=
A =
Æ
F Æ ! F Æ !B
# ! ! * $ 8
! K 9 <
Æ ! >
=
?Æ = !
Æ
Æ
+ ! 9 ! > !
9 9 !
' ( # L !
$( & : + ' 1 ) # 9 = -# 5 4 -# - 3 4 - 4 3 53 #4 A = "!Æ
B < A B
> = "!Æ > = "!Æ
!
= "
!
" = "
M'& 2 2 * > . " <
..
A" !
B
> A A" !
B B
> = "!Æ A > = "!Æ =
" = "
B
" ! ="
> A = = !"BÆ > = "!Æ
Æ
=
"# " " &
.0
" <
" = = !"
" = "
" = " " = "
! > % >
!
!
> A
> A
!
! B
!
! B = A ! B
> Æ = !
!
!
! = A Æ =
> Æ = Æ ! = . = Æ ! = < >
!>
!
Æ Æ = ! Æ = ! !
# <
! Æ Æ = ! ! ! = Æ!ÆÆ = !Æ = .! . !
! > =
=
!
Æ Æ = !Æ = .!Æ = 0!
.4
! Æ Æ =
!
!
B
>
!
Æ Æ = !
!ÆÆ =
! = .
!Æ =
!Æ = .!
= Æ = .!Æ = 0! Æ Æ = ! >
!
Æ Æ = ! AÆ Æ = ! .Æ = !Æ = .! = Æ = .!Æ = 0!B
= .Æ Æ = ! = .Æ = !Æ = .!B > >
! ÆÆ =
!A. = 4Æ = ! B
! . ÆÆ =
> /!
!A = .Æ = !B
= .Æ = ! .Æ = 0
"# '! + $( * %#( 9 ! %#(
( " & ! &
$ H <
>
! =
<
! > !
! = ! .
.7
2 ! 2 < ! = ! > ( ! .
!
( D Æ < ( > > ! N H <
! < !
! = ! = ! > / .
! = ! = ! > / .
2 < .
> < .
>
! >
.
!
! > !
$ !
H < .
!
.
.
<
.! ! = ! = ! > / . !
! =
! = ! > /
$.!
> Æ # H $.!
* 2 A$ * + 3/! ..8 7B <
! >
Æ
!
?Æ = ! ?Æ = !? = ! @ .8
$0!
K Æ !
A$ * + 3/! ... .B <
! Æ
! Æ >
?Æ = ! ?Æ !? = !
?Æ ! " * <
! > Æ
?Æ!? = !
!
$4!
Æ
?Æ = !
6 H
$ ! <
! >
?Æ!? = !
?Æ = !
Æ
!
$7!
> / ( > ! # H < ! > ! < ( > : F Æ ! ! 2 80!! <
F Æ ! > F Æ ! =
! !
F Æ ! ; Æ / !
$8!
"# 2 - ! ! < .3
! >
A- ! !B
" <
!
>
- !
!B A- ! !B
A
- ! ! > - ! !
>
! !
>
! !
< ! ! <
! > !
F
.)
F
- ! !
! ! >
- ! !
! ! ? = ! ?Æ= ! ! ! ?Æ = ! ? = ! ?Æ = !? = !
Æ
Æ
>
Æ
Æ
?Æ = !? = !
"# ' " ! " <
.
>
!
! !- ! ! ? = ! ?Æ = ! A = !B
> > >
>
Æ
Æ
>
½
½
? = !?Æ = ! Æ
Æ
½
= !Æ
! > = !Æ
½
Æ
½
?Æ = = ! ½ ? = !?Æ = !
?Æ = = ! ? = !?Æ = !
½
"# "
" 9 <
.
>
. = . = .
.
.
>
. = .
>
= .
. = .
/ /! . . ! <
0/
.
> .
= .
.
> .
=.
>
. # . ! >
= .
.
: . ! <
. >
=. = . = = . $ <
>
= = = =
> / >
= = !
# = = ! . ! A/ B > / " * # = = ! . !
0
= = !
)*+*)*,* $ * : + # - " < &
3/
D - 2 & 77 $ ' + + 1 9
& # $ N 9 # < / 7 D 2 9 3 2 $%& : < $ $ DP$ M 9 NQ+' + 8/< 3 4 % 5
4 $ D & 6 7< 00 44
% - # - ( + )7 $ + # ( 70< 0)74/) & + % - - .//. + 2 % $ : %('+ &9 & R % 8 : , 9 + ( < $ N $ 30< )7. 7 '
84 9 ' D & ./) . 7
07<
' ( ( - ) $ % & :< $ N : # + & 88< .3 8. & 2 9 )/ 6 1 $ + 9 9 ! .< 3.3347 ' % - - .//0 + , & :