Journal of Forecasting
J. Forecast. 25, 129–152 (2006)
Published online in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/for.978
Autoregressive Gamma Processes
CHRISTIAN GOURIEROUX1 AND JOANN JASIAK2*
1
2
CREST, CEPREMAP and University of Toronto, Canada
York University, Canada
ABSTRACT
We introduce a class of autoregressive gamma processes with conditional distributions from the family of noncentred 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). Copyright © 2006 John Wiley
& Sons, Ltd.
key words intertrade durations; autoregressive gamma; CIR; high
frequency
INTRODUCTION
The liquidity of a financial asset on a given day and market depends on how frequently that asset is
traded, or equivalently on the times between consecutive transactions, called intertrade durations
(Gourieroux et al., 1999). The times to trade recorded sequentially in time form time series which
possess interesting dynamic properties, such as time-varying moments and serial correlation. The
duration analysis of financial assets is a relatively new topic of research, motivated by the interest
in assessing future asset liquidity, in the sense of time it can take to purchase or sell a given quantity of shares. While some stocks are traded instantaneously, others require a considerable amount
of time to trade, which varies randomly throughout a trading day. In the literature, there exist few
models designed for intertrade durations. The most popular among these is the autoregressive conditional duration (ACD) model, introduced by Engle and Russell (1998) [see also Bauwens, Giot
(1997)]. The conditional hazard function of the ACD suggests classifying this model as an accelerated hazard model with lagged durations as explanatory variables. While the ACD is quite easy to
estimate, it relies on a strong and unrealistic assumption of path-independent overdispersion. This
means that the conditional standard deviation exceeds the conditional mean by a fixed amount, unrelated to the history of the process. Another limitation of the ACD is the difficulty in deriving the
stationarity and ergodicity conditions for models with temporal dependence beyond lag one. An alter* Correspondence to: Joan Jasiak, York University. Department of Economics, 4700 Keele Street, Toronto, ON M3J 193.
Canada. E-mail: [email protected]
Copyright © 2006 John Wiley & Sons, Ltd.
130
C. Gourieroux and J. Jasiak
native model, called the stochastic volatility duration (SVD), was developed by Ghysels et al. (2003).
The SVD model improves upon the ACD in that it allows for independent variation of the conditional mean and variance and yields a time-varying over- or underdispersion. The SVD is a nonlinear factor model, with two dynamic heterogeneity factors that determine the mean and dispersion,
respectively. Therefore, the estimation is technically difficult, and requires advanced statistical tools.
The aim of this paper is to introduce a class of models, which accommodate various nonlinearities in inter-durations dynamics and have easily verifiable stationarity conditions and straightforward
estimation procedures. Our approach borrows from the early literature on dynamic processes for
gamma distributed variables developed for applications to ruin processes, such as the gamma autoregressive (GAR) [see, e.g., Gaver and Lewis (1980); the survey by Grundwald et al. (2000);1 the
model of Sim (1990, 1994); the work by Griffiths (1970) and Kotz and Adams (1964)]. The new
model, called the autoregressive gamma model (ARG), is designed specifically for intertrade duration analysis and forecasting, but can also be used in application to asset volatility and traded
volumes, for example. In contrast to the ACD, and similar to the SVD, an ARG can fit any data that
feature time-varying over- or underdispersion and can accommodate complex nonlinear dynamics.
Finally, an ARG can represent processes with long memory more parsimoniously than ACD and
SVD.
The paper is organized as follows. In the next section, we introduce the autoregressive gamma
process of order one (ARG(1)), with a conditional distribution from the family of noncentred gamma
(up to a scale factor), and the noncentrality parameter written as a linear function of lagged durations. This section provides the forecast function and discusses the stationarity. The temporal dependence is examined in the third section by investigating the conditional moments, the first- and
second-order autocorrelograms and nonlinear canonical correlations. The ARG model is viewed as
a nonlinear state space model, with an integer-valued latent state variable in the fourth section. The
ARG-based state space representation allows us to provide the forecasting, filtering and smoothing
algorithms. The relationship between the discrete time ARG process and the continuous time
Cox–Ingersoll–Ross process is covered in the fifth section. The sixth section shows the ARG models
of autoregressive order higher than one (ARG(p)) and long memory ARG models. In the seventh
section, the ARG is compared to other autoregressive processes for gamma distributed variables
existing in the literature. Statistical inference is discussed in the eighth section. The application to
intertrade durations is presented in the ninth section. Contrary to empirical literature on intertrade
durations, which is focused on very frequently traded assets and time deformation, we consider an
infrequently traded stock for which the risk of low liquidity, i.e. of a long waiting time to trade, is
significant. A final section concludes the paper. Proofs are gathered in the appendices.
THE AUTOREGRESSIVE GAMMA PROCESS
The autoregressive gamma processes are Markov processes with conditional distributions from the
family of noncentred gamma (up to a scale factor) and with path-dependent noncentrality coefficients. Let us recall that a variable Y follows the distribution g (d, b, c), if and only if Y/c follows
the noncentred gamma distribution g (d, b ) characterized by three nonnegative parameters: the degree
of freedom d, noncentrality parameter b and scale parameter c. The noncentred gamma distribution
1
The use of GAR processes for the analysis of financial data is limited due to the assumption of conditional homoscedasticity (constant conditional variance), which underlies the model.
Copyright © 2006 John Wiley & Sons, Ltd.
J. Forecast. 25, 129–152 (2006)
DOI: 10.1002/for
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131
arises as a Poisson mixture of gamma distributions: we say that Y follows the distribution g (d, b)
if, and only if, there exists a Poisson variable Z ~ P(b), such that the conditional distribution of Y
given Z is g (d + Z ). The density of the g (d, b, c) distribution is:
y • È yd + k -1
exp- bb k ˘
fd ,b ,c ( y ) = 1y > 0 expÊ - ˆ Â Í d + k
Ë c ¯ k = 0 Î c G (d + k )
k!
˚˙
(1)
and the two first moments of a noncentred gamma distributed variable are:
EY = cd + cb , VY = c 2d + 2c 2 b
(2)
ARG(1)
Definition 1 The process (Yt) is an autoregressive gamma process of order one (ARG(1)) if, and
only if, the conditional distribution of Yt given Yt-1 is g (d, btYt-1, ct), where bt, ct are deterministic
functions of time.
Definition 1 is wide in the sense that it comprises heterogeneous Markov ARG(1) processes and
allows for variation in time of all parameters, except for the degree of freedom. In particular, the
scale ct and the parameter bt appearing in the noncentrality coefficient are time-varying, although
path-independent. The history of the process determines the entire noncentrality coefficient, btYt-1,
which is written as a linear function of the lagged value of the process. Let us recall that all parameters are assumed nonnegative and, for convenience, define a new parameter rt = btct, which will
later be interpreted as an autoregressive coefficient.
From the mixture of gamma interpretations given at the beginning of this section, it follows that
the dynamics of the ARG process can be written as:
Zt
Yt = Â Wi ,t + e t
i =1
where et, Zt, Wi,t are independent conditional on Yt-1 and distributed as g (n, ct), P(btyt-1), g (1, ct),
respectively (see Sim, 1990). This dynamic representation of an ARG is very convenient for simulations of the process trajectories.
A closed-form expression of the conditional distribution of the current value of an ARG(1) process
Yt given Yt-h is available for any h.
Proposition 1 The conditional distribution of Yt given Yt-h is noncentred gamma up to a scale
factor: g (d, bt|t-hYt-h, ct|t-h), where
t
r t t - h = b t t - h ct t - h = ’t =t - h +1 r t
t
ct t - h =
Â
(ct r t +1 . . . rt )
t = t - h +1
Proof. see Appendix A.
Copyright © 2006 John Wiley & Sons, Ltd.
J. Forecast. 25, 129–152 (2006)
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C. Gourieroux and J. Jasiak
Stationarity condition for ARG(1)
To derive the stationarity condition, let us consider a homogeneous Markov ARG(1) process, with
time-invariant parameters bt = b, ct = c and rt = bc = r. By Proposition 1, the parameters of the conditional distribution of Yt given Yt-h become:
rt t -h = r h
ct t - h = c(1 + r + . . . + r h -1 ) = c
1- rh
1- r
with the limiting values:
lim r t t - h = 0, lim ct t - h =
hƕ
hƕ
c
, if r < 1
1- r
It follows that:
Proposition 2
An ARG(1) process is stationary, when the conditional distribution of Yt given Yt-1 is g (d,
btYt-1, c) with r = bc < 1.
(ii) The conditional distribution of Yt given Yt-h is:
(i)
h
1- rh ˘
È r (1 - r )
Yt - h , c
g˜ Íd ,
, "h
h
1 - r ˙˚
Î c(1 - r )
(iii) The marginal invariant distribution is gamma, up to a scale factor:
c ˆ
g˜ Ê d , 0,
Ë
1- r¯
The limiting case of an autoregressive coefficient equal to one will be discussed later.
SERIAL DEPENDENCE
There exist various methods for examining serial dependence in stationary ARG processes. In this
section, we consider the first- and second-order conditional moments, autocorrelograms and nonlinear canonical decomposition.
Conditional moments
From Proposition 2 it follows that:
Proposition 3 The first two conditional moments of a stationary ARG(1) process are:
E(Yt Yt -1 ) = cd + rYt -1
V (Yt Yt -1 ) = c 2d + 2 rcYt -1
Copyright © 2006 John Wiley & Sons, Ltd.
J. Forecast. 25, 129–152 (2006)
DOI: 10.1002/for
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and more generally:
E(Yt Yt -1 ) =
V (Yt Yt - h ) =
c(1 - r h )
d + r h Yt - h
1- r
c 2 (1 - r h )
(1 - r )
2
2
d + 2r h
c(1 - r h )
Yt - h
1- r
We see that the stationary ARG(1) process is a weak AR(1) with conditional heteroscedasticity specified as a linear function of the lagged value.2 The expressions of conditional moments indicate that
an ARG(1) can be under- or overdispersed. The conditional overdispersion exists if and only if:
V (Yt Yt -1 ) > E(Yt Yt -1 )
2
¤ 0 > c 2d (d - 1) + 2 rcYt -1 (d - 1) + r 2 Yt 2-1
The discriminant equation of the polynomial on the right-hand side of the last expression is:
D ¢ = - r 2 c 2 (d - 1)
It is negative for d > 1, and positive for d < 1. When d > 1, we have V(Yt|Yt-1) < E(Yt|Yt-1)2, "Yt-1.
2
)
(
When d < 1, the product of the roots c d d - 1 is negative, while the two roots are real. Proposi2
r
tion 4 below, summarizes the results on dispersion of an ARG(1).
Proposition 4 When d > 1, the stationary ARG(1) process features conditional and marginal underdispersion. When d < 1, the stationary ARG process features marginal overdispersion. In parallel,
the process may feature either conditional under- or overdispersion, depending on the value of Yt-1.
First- and second-order autocorrelograms
From the weak AR(1) interpretation of an ARG process, it follows that:
r (h) = Corr(Yt , Yt - h ) = r h
(3)
The second-order autocorrelogram represents the serial dependence in squared values of the process.
It is proven in Appendix B that:
r ( 2 ) (h) = Corr(Yt 2 , Yt -2 h ) = r h
r h + 2(d + 1)
2d + 3
(4)
Nonlinear canonical decomposition
The nonlinear canonical decomposition (see Barret, Lampard (1955), Lancaster (1958, 1963), Wong
(1962), Eagleson (1964)) allows for the analysis of nonlinear time dependence in Markov processes
2
However, the process doesn’t admit a strong AR(1) representation with i.i.d. innovations as in the general class of models
considered by Grunwald et al. (2000).
Copyright © 2006 John Wiley & Sons, Ltd.
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C. Gourieroux and J. Jasiak
to which belong the ARG. The formula of canonical decomposition is derived in Appendix C using
the relationship between the ARG and the continuous-time Cox–Ingersoll–Ross process [see also
Sim (1990, p. 329) for an approach based on the joint Laplace transform].
Let us denote by:
d
d -1
È (1 - r)yt ˘ È1 - r ˘ yt
f (yt ; d , c, r) = 1yt >0 expÍ ˙˚ ÍÎ c ˙˚ G (d )
Î
c
(5)
the marginal p.d.f. of Yt, and by:
f1 ( yt yt -1 ; d , c, r ) = 1yt > 0 exp( - yt c)
k
•
È ytd + k -1
( ryt -1 c) ˘
X Â Í d +k
exp( - ryt -1 c)
˙
k!
G (d + k )
k =0 Î c
˚
(6)
the conditional p.d.f. of Yt given Yt-1.
Proposition 5 The conditional p.d.f. of Yt given Yt-1 admits the following nonlinear canonical
decomposition:
•
¸
Ï
f1 (yt yt -1 ; d , c, r) = f (yt ; d , c, r)Ì1 + Â r ny n (yt , d , c, r)y n (yt -1 , d , c, r)˝
˛
Ó n =1
where:
G (d )G (n + 1) ˆ
y n ( y; d , c, r ) = Ê
Ë G (d + n) ¯
12
L(nd -1) Ê
Ë
(1 - r)y ˆ
c ¯
and Ln is a generalized Laguerre polynomial:
n
k
L(nd -1) (z) = Â (-1)
k =0
G (d + n)
zk
G (d + k )G (n - k + 1) k!
The polynomials yn, n varying, satisfy the conditions:
Ey n (Yt ) = 0, Cov[y n (Yt ), y m (Yt )] = 0, "n π m
Vy n (Yt ) = 1
Proposition 5 leads directly to the following corollaries.
Corollary 1
The conditional p.d.f. of Yt given Yt-h admits the nonlinear canonical decomposition:
•
¸
Ï
fh (yt yt -h ; d , c, r) = f (yt ; d , c, r)Ì1 + Â r hny n (yt ; d , c, r)y n (yt -h ; d , c, r)˝
˛
Ó n =1
Copyright © 2006 John Wiley & Sons, Ltd.
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DOI: 10.1002/for
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The canonical decomposition allows for easy forecasting of any polynomial transformation of Yt.
Corollary 2
We get E[yn(Yt)|Yt-1] = rnyn(Yt-1), "n.
Thus, there exist several nonlinear transformations of the ARG(1) process that follow a weak
AR(1) process. For instance,
y 1 (Yt ) =
(1 - r )Yt ˘
1 È
d12 Í
˙˚
c
d Î
is AR(1) with autoregressive coefficient r, and
12
2
(1 - r)Yt (1 - r) Yt 2 ˆ
È 2 ˘ Ê d (d + 1) (
)
+
+
1
y 2 (Yt ) = Í
d
Á
˜
Î d (d + 1) ˚˙ Ë 2
c
2c 2 ¯
is AR(1) with autoregressive coefficient r2.
Corollary 3 The joint distribution of (Yt, Yt-1) is symmetric, which means that the process is timereversible.
THE STATE SPACE REPRESENTATION
The dynamics of (Yt) can be written in the form of a nonlinear state space representation, with the
mixing variable (Zt) as the state variable and the transition equation:
Zt Yt -1 ~ P [ bYt -1 ]
The measurement equation is:
Yt Zt ~ g [d + Zt , c]
The latent variable Zt has an interesting interpretation in finance. In some applications, it may be
interesting to rank the stocks traded on a given market with respect to their risk of liquidity, that is,
to distinguish stocks which trade quickly from those with long waiting times to trade. For example,
if the liquidity risk is measured in terms of intertrade durations (which are the waiting times to trade),
the integer-valued Zt can provide a liquidity rating of a stock. Then, by imitating the Moody’s system
of risk rating, we can assign the highest rating to the stock with the lowest liquidity risk, such as Zt
= 0 = AAA, a lower one to the stock with Zt = 1 = AA, etc. The ARG model can yield the current
rating Zt given the lagged observed liquidity Yt-1, and provide a forecast of future liquidity Yt+h given
the current rating Zt.
In the next subsection, we focus on the dynamics of the latent mixing variable (Zt). The first part
concerns the smoothing procedure and the conditioning variable is the observable process (Yt). The
second part shows the forecasts of Zt based on its own past values.
Copyright © 2006 John Wiley & Sons, Ltd.
J. Forecast. 25, 129–152 (2006)
DOI: 10.1002/for
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C. Gourieroux and J. Jasiak
Smoothing
The smoothing algorithm allows us to recover the path of the latent process (Zt).
Proposition 6
(i) The values Z1, . . . , ZT of the state variable are independent conditional on Y0, Y1, . . . , YT.
(ii) The conditional distribution of Zt given Y0, Y1, . . . , YT depends on the observable variables
Yt-1, Yt only.
(iii) This conditional p.d.f. is given by:
zt
1
È byt yt -1 ˘
G (d + zt )G ( zt + 1) ÍÎ c ˙˚
l( zt yt -1 , yt ) = •
z
Ï
1
Ê byt yt -1 ˆ ¸
Ì
˝
Â
) (
)Ë c ¯ ˛
z = 0 Ó G (d + z G z + 1
Proof. See Appendix D.
Marginal distribution of the state variable
Proposition 7 The latent process (Zt) is a Markov process such that the transition density:
z
l( zt zt -1 ) =
(bc) t
d + zt + zt -1
(bc + 1)
G (d + zt + zt -1 )
G ( zt + 1)G (d + zt -1 )
is the density of a negative binomial distribution with parameters:
n = d + zt -1 ,
p=
1
1 + bc
Proof. See Appendix E.
The transition density can be used for examining the probability of an asset migrating from one
liquidity rating to another, by analogy to the migration of bond issuers between various classes of
risk rating in finance.
CONTINUOUS TIME LIMIT OF ARG(1)
The continuous time counterpart of the ARG(1) process is the Cox–Ingersoll–Ross (CIR) model,
commonly used in theoretical finance to represent the volatility of financial assets. In order to prove
that the CIR model is a limit of ARG when the time increments are allowed to shrink to zero, we
need to consider the infinitesimal drift and volatility functions. This means that we need to study the
behaviour of a prediction of Yt at horizon h, when h tends to zero. We get:
Copyright © 2006 John Wiley & Sons, Ltd.
J. Forecast. 25, 129–152 (2006)
DOI: 10.1002/for
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1
lim hÆ 0 [ E(Yt Yt - h = y) - y]
h
1 È c(1 - r h )
˘
d + r h y - y˙
= lim hÆ 0 Í
h Î 1- r
˚
= lim hÆ 0
1 - r h È cd
- y˘˙
Í
h Î1 - r
˚
cd
= - log r Ê
- yˆ
Ë1- r
¯
lim hÆ 0
1
V (Yt Yt - h = y)
h
2
Ï 1 (1 - r h )
r h c(1 - r h ) ¸
= lim hÆ 0 Ì c 2
d +2
y˝
2
h 1- r
(1 - r )
Ó2
˛
2c log r
=y
1- r
This leads us to the conclusion that the limiting continuous time process associated with the ARG(1)
process is the CIR process (Cox et al., 1985):
dYt = a(b - Yt )dt + s Yt dWt
(7)
where:
a = - log r > 0, b =
cd
-2 log r
, s2 =
c
1- r
1- r
A more general result can be established (see, e.g., Lamberton and Lapeyre, 1992) as well.
Proposition 8 The stationary ARG process is a discretized version of the CIR process.
The link between the ARG and CIR processes points to the importance of the ARG model for
applications to intertrade durations. Indeed, it is well-known that asset volatility and liquidity are
strongly related. Therefore, it comes as no surprise that the basic model for asset volatility is related
to the model for intertrade durations.
ARG(p)
The ARG can be extended to an autoregressive order strictly greater than one. We will see that the
forecast function and ergodicity conditions for an ARG(p) can easily be obtained.
Copyright © 2006 John Wiley & Sons, Ltd.
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C. Gourieroux and J. Jasiak
Definition of ARG( p)
Definition 2 The process (Yt) is an autoregressive gamma process of order p (ARG(p)), if, and
only if, the conditional distribution of Yt given Yt-1 = (Yt-1, Yt-2, . . . , Yt-p)¢ is g (d, b¢Yt-1, c), where
b¢Yt-1 = b1Yt-1 + . . . + bpYt-p, and bj ≥ 0, j = 1, . . . , p.
The main difference between the ARG(1) and ARG( p) is the form of the noncentrality coefficient,
which becomes a linear function of past values up to and including lag p. We can rewrite the ARG( p)
easily in the form of a p-dimensional process Yt = (Yt, . . . , Yt-p+1)¢, which is a Markov process of
order one, and has the conditional Laplace transform given by:
E[exp( -u ¢Yt ) Yt -1 ]
= E[exp( -u1Yt . . . - u p Yt - p+1 ) Yt -1 ]
= exp[ -u2 Yt -1 . . . - u p Yt - p+1 ]E[exp( -u1Yt ) Yt -1 ]
- cu1
-d
= [1 + cu1 ] exp( -u2 Yt -1 . . . - u p Yt - p+1 ) expÈÍ
b ¢Yt -1 ˘˙
˚
Î1 + cu1
(from Lemma 1 in Appendix A)
[
= [1 + cu1 ] exp - A(u)¢ Yt -1
-d
(8)
]
where:
cu1
cu1
cu1
È cu1
˘
A(u) = Í
b1 + u2 ,
b 2 + u3 ; . . . ,
b p-1 + u p ,
bp
Î1 + cu1
1 + cu1
1 + cu1
1 + cu1 ˙˚
(9)
Stationarity condition
The stationarity of ARG( p) is determined by the function A(u) (Darolles et al., 2002).
Proposition 9 The ARG(p) process is stationary if limhÆ0 Aoh(u) = 0, for any u with nonnegative
components, where Aoh denotes the function A compounded h times with itself.
As the corollary to Proposition 9 we get:
Corollary 3
The stationarity condition is: c(b1 + b2 + . . . + bp) < 1.
Proof. See Appendix F.
Long memory
The ARG(1) can feature long memory, under conditions similar to the standard AR(1), that is, when
the autoregressive coefficient r is equal to one, and when r is stochastic.
(i)
Autoregressive coefficient r = 1
Let us fix the scale parameter at c = 1. When r = b = 1, the conditional Laplace transform of
the process becomes (see Appendix A, Lemma 1):
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139
E[exp( -uYt ) Yt -1 ] = exp[ - a(u)Yt -1 + b(u)]
where a(u) = u/(1 + u), b(u) = -d log(1 + u).
The conditional Laplace transform at horizon h is:
E[exp( -uYt + h -1 ) Yt -1 ] = exp[ - a oh (u)Yt -1 + bh (u)] (say)
where the compounded function aoh satisfies:
a oh (u) =
u
1 + hu
We see that limh-• aoh(u) = 0, which implies that the process is weakly ergodic (Darolles et al.,
2002). Moreover, the conditional expectations E[exp(-uYt+h-1)|Yt-1] tend to their limiting value
at a hyperbolic rate in h. This suggests that the process has an autocorrelation function with a
hyperbolic rate of decay, that it is features long memory. At the same time, Yt is a stationary
Markov process.
The long memory property is of relevance for applications to intertrade durations of frequently
traded stocks, in which a long range of persistence is generally observed. The long-range persistence can be captured parsimoniously by an ARG(1), while it requires the presence of a fairly
large number of lags in the competing ACD or SVD models (Jasiak, 1998).
(ii) Stochastic autoregressive coefficient
Let us consider an ARG(1) process parameterized by d, r and c. When r is fixed, the bivariate
density of yt, yt-h can be written as (see Corollary 1):
fh ( yt , yt - h ) = f [ yt ; d , c (1 - r )] f [ yt - h ; d , c (1 - r )]
•
¸
Ï
¥ Ì1 + Â r hn Yn [ yt ; d , c (1 - r )]Yn ( yt - h ; d , c (1 - r )]˝
˛
Ó h =1
where g = c/(1 - r). In this expression, parameters d and g appear in the marginal distribution
and canonical variates, while parameter r determines the canonical correlations and temporal
dependence. The first- and second-order marginal moments of the model
E(Yt d , r, g ) = m(d , g ), V (Yt d , r, g ) = s 2 (d , g )
depend on d and g only. The autocovariance function is given by:
Cov(Yt , Yt - h d , r, g ) = r hs 2 (d , g )
and is shown to include r in its expression.
Let us now assume that the autoregressive parameter r is stochastic and takes values from the
interval [0, 1]. Also, let us suppose that r has the probability density p, and that d and g remain
fixed. By integrating the stochastic parameter r out of the first- and second-order moments, we
obtain:
Copyright © 2006 John Wiley & Sons, Ltd.
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DOI: 10.1002/for
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C. Gourieroux and J. Jasiak
E(Yt d , p , g ) = m(d , g ), V (Yt d , p , g ) = s 2 (d , g )
and the autocovariance function
Cov(Yt , Yt - h ) = E[ Cov(Yt , Yt - h d , r, g ) d , p , g )] + Cov[ E(Yt d , r, g ), E(Yt - h d , r, g ) d , p , g )]
= E[ Cov(Yt , Yt - h d , r, g ) d , p , g ]
= E[ r hs 2 (d , g ) d , p , g )]
= Ep ( r h )s 2 (d , g )
The autocorrelation function is no longer a power function of r, but instead an expected value
of r power h:
Corr(Yt , Yt - h d , p , g ) = Ep ( r h )
Consequently, the autocorrelation function features hyperbolic decay when the distribution p
of coefficient r assigns sufficiently large probabilities to values close to one. For example, this
is the case when the distribution of r is beta (Granger and Joyeux, 1980).
ARG AND GAR: COMPARISON
In the time series literature, there exist a variety of dynamic models for gamma processes. [See e.g.
Jacobs, Lewis (1978), Lawrance, Lewis (1981), Smith, Miller (1986)]. So far, none of them has been
used in an application to intertrade durations. As mentioned earlier in the text, the dynamic properties of financial duration data are quite particular, and the available dynamic gamma models don’t
account for features such as conditional heteroscedasticity, for example. In this section, we discuss
the gamma autoregressive process introduced by Gaver and Lewis (1980). Gaver and Lewis considered the following linear autoregression:
Yt = rYt -1 + e t
where the innovations are i.i.d. random variables. They proved that for 0 £ r < 1, there exists a distribution of the error term et such that the marginal distribution of Yt is exponential (or more generally a gamma). When the marginal distribution of Yt is exponential g(1, l), the innovation distribution
is a mixture of a point mass at zero with probability r and an exponential distribution g (1, l) with
probability 1 - r.
There are three main differences between the GAR and the ARG processes:
(i)
By construction, the GAR is conditionally homoscedastic and doesn’t allow for independent
variation of the conditional mean and variance of Yt. In application to intertrade durations, the
assumption of constant duration volatility is very strong and unrealistic.
(ii) The conditional distribution of GAR admits a discrete component. This means that the GAR
process can become deterministic Yt = rYt-1 with probability r.
(iii) The functional dependence between serial correlation and the distribution of innovations is hard
to interpret. For example, in the unit root case r = 1, the GAR process (Yt) takes value zero, at
all t, and remains stationary.
Copyright © 2006 John Wiley & Sons, Ltd.
J. Forecast. 25, 129–152 (2006)
DOI: 10.1002/for
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141
Table I. Autoregressive models for gamma processes
a(u) = ru
Gaver and Lewis (1980): GAR
Gourieroux and Jasiak (2003): ARG
Sim (1990, 1994)
a(u) = ru/(1 + cu)
Let us point out that the ARG and GAR processes are special cases of Car (compound autoregressive) processes. Indeed, an autoregressive process with marginal gamma distribution is easy to
construct by considering a conditional Laplace transform of the type:
E[exp( -uYt ) Yt -1 ] = exp[ - a(u)Yt -1 + c(u) - c[ a(u)]]
where c is the log-Laplace transform of the marginal gamma distribution. The models differ essentially by the form of function a, which determines serial dependence, as shown in Table I.
STATISTICAL INFERENCE
Let us consider a sample of discrete time observations y1, . . . , yT , of size T. It will be shown that
standard estimation methods such as GMM and the maximum likelihood can be used for estimation
of ARG processes.
Moment estimators
The parameters of the ARG( p) process can be estimated from the first- and second-order conditional
moments, since:
E(Yt Yt -1 ) = cd + cb ¢Yt -1
V (Yt Yt -1 ) = c 2d + 2c 2 b ¢Yt -1
Consistent estimators of cd and cbj, j = 1, . . . , p, are obtained by regressing Yt on 1, Yt-1, . . . ,
Yt-p. The efficiency of these estimators can be improved by taking into account conditional
hereroscedasticity. For instance, we can apply a pseudo-maximum likelihood method based on a
Gaussian pseudo-family. The PML estimators are the solutions of:
(cˆ, dˆ , bˆ )¢ = arg max c ,d ,b
2
Ï 1
1 ( yt - cd - cb ¢ yt -1 ) ¸
2
2
Ì- log(c d + 2c b ¢ yt -1 ) ˝
Â
2 c 2d + 2c 2 b ¢ yt -1 ˛
t = p +1 Ó 2
T
Maximum likelihood estimators
The log-likelihood function:
T
LT (d , c, b ) = Â log f1 ( yt yt -1 ; d , c, b )
(10)
t =1
is of a complicated form, and needs to be approximated in practice. The approximations of the loglikelihood function for ARG(1) given below are based on three different methods.
Copyright © 2006 John Wiley & Sons, Ltd.
J. Forecast. 25, 129–152 (2006)
DOI: 10.1002/for
142
(i)
C. Gourieroux and J. Jasiak
Truncating the series expansion of conditional density:
k
T
T
ÈK
(ryt -1 c) ˘
1
ytd + k -1
L(T1) (d , c, r) = - Â (yt + ryt -1 ) + Â log Í Â d + k
˙
k!
t =1 c
t =1
Î k =0 c G (d + k )
˚
(11)
(ii) Truncating the nonlinear canonical decomposition:
T
T
N
È
˘
L(T2 ) (d , c, r ) = Â log f ( yt ; d , c, r ) + Â log Í1 + Â r ny n ( yt ; d , c, r )y n ( yt -1 ; d , c, r )˙
Î n=0
˚
t =1
t =1
m
Intuitively this is a method of moments involving all marginal and cross moments E(Y tmY t-1
),
for m, n £ N. A drawback of this approach is the possibility of obtaining negative values of
the log-likelihood function due to truncation.
(iii) Using the simulated maximum likelihood method based on artificial drawings of the latent
Poisson state variable:
T
L(T3) (d , c, r ) = - Â
t =1
s
T
S
yt
ytd + zt -1
È
˘
+ Â log ÍÂ d + z s
s ˙
t
c t =1
)
(
c
G
d
+
z
Î s =1
t ˚
where S is the number of drawings and (z1t, . . . , zSt) are independently drawn in the Poisson distribution P(ryt-1/c) = P(byt-1).
The advantage of the last approach is that it can easily be extended to any autoregressive order p
higher than one. This can be done by performing the drawings of the Poisson state variable in the
Poisson distribution P(b¢ yt-1).
APPLICATION TO FINANCIAL DURATION SERIES
In empirical literature on high-frequency data, the analysis of intertrade durations typically concerns
very frequently traded stocks, which feature long memory. However, as the liquidity risk is much
higher in infrequently traded (illiquid) stocks, accurate forecasts of future times to trade may become
valuable inputs to any trading strategy. The aim of this section is to examine the dynamics of an
infrequently traded stock with long waiting times to the next upcoming quote or transaction. Therefore, the number of observations available in our sample is considerably lower than in other applications with a sampling period of similar length.
The ARG model is estimated from data on interquote durations of the Dayton Mining stock traded
on the Toronto Stock Exchange in October 1998. The choice of a stock from the mining industry is
not arbitrary. In the past, a false announcement of a gold mine discovery has undermined the credibility of the whole mining sector, and diminished the frequency at which those stocks were traded.
Prior to estimation, durations between quotes with the same time stamp were aggregated.3 The
times between market openings and closures were deleted from the sample. We also removed from
3
In particular, we disregard simultaneous trades at either market opening or during the day, when a large buy (sell) order is
filled by several sell (buy) orders.
Copyright © 2006 John Wiley & Sons, Ltd.
J. Forecast. 25, 129–152 (2006)
DOI: 10.1002/for
143
0.0004
0.0
0.0002
frequency
0.0006
Autoregressive Gamma Processes
0
5000
10000
15000
20000
durations
Figure 1. Dayton Mining, durations
the sample all observations on quotes recorded during market preopenings. Thus, we are mainly concerned with the intraday dynamics of interquote durations. Dayton Mining belongs to infrequently
traded stocks, and is not frequently quoted either. The sample consists of 381 observations, with
mean 1162.5 sec. [approx 20 min]4 and variance 4,762,851 [standard error of about 35 min]. The
kernel smoothed marginal density of the Dayton interquote durations is displayed in Figure 1. It
features a typical form of a gamma density function with a small degree of freedom.
To choose the best fit, several ARG models with different numbers of lags were estimated by OLS
in the first step. The Dayton Mining series displays a rather short range of serial dependence in
interquote durations, typical for infrequently traded stocks. In this, it differs from very frequently
traded assets which feature long memory. Since the coefficients on lags greater than two were found
insignificant, we retained for further analysis the ARG(1) and ARG(2) models. Both models were
then estimated by QMLE. Table II summarizes the results obtained from fitting the ARG(1) and
ARG(2) models to the sample of Dayton Mining using the regression approach and QMLE.
We observe that the estimated coefficients r1 and r2 are positive, less than one, and sum up to
less than one in the ARG(2) model. This means that both models are stationary. Also, both models
fit the data fairly well, and all their coefficients are significant. While the mean log-likelihood of
ARG(2), equal to (-8.154), shows little improvement compared to that of ARG(1), equal to (-8.175),
the R-squared doubles when the second lag is added (0.030 compared to 0.018).5
In the next step, the parameters of the ARG(2) are used in a simulation experiment. It consists in
fixing the vector of initial durations at two consecutive values observed at the end of the sample:
4
5
To compare with an average intertrade duration less than 1 min for a frequently traded stock, such as the IBM.
The value of R2 is small, since it measures linear serial dependence in a nonlinear dynamic framework.
Copyright © 2006 John Wiley & Sons, Ltd.
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C. Gourieroux and J. Jasiak
Table II. Estimation of ARG(1) and ARG(2)
Parameter
ARG(1)
OLS
10.024E2
(1.25E2)
0.132
(0.050)
—
—
cd
r1
r2
QMLE
10.527E2
(0.26E2)
0.0794
(0.057)
—
—
ARG(2)
OLS
8.925E2
(1.35E2)
0.1196
(0.051)
0.1104
(0.051)
QMLE
9.396E2
(0.24E2)
0.0201
(1.262)
0.1612
(0.393)
Table III. One-step predictions from ARG(2)
Mean
Variance
y(T + 1)
2 quotes
3 quotes
4 quotes
1,793.93
8,929,084.7
1,985.58
10,497,642
1,815.43
12,056,015
1,940.20
13,455,554
y(T) = 5226 and y(T - 1) = 7512, and simulating the next four out-of-sample observations y(T + 1),
y(T + 2), y(T + 3), y(T + 4). We replicate this experiment 1000 times. Next we compute the average
predicted waiting times for 1, 2, 3 and 4 quotes, given by y(T + 1), [y(T + 1) + y(T + 2)]/2, [y(T +
1) + y(T + 2) + y(T + 3)]/3 and [y(T + 1) + y(T + 2) + y(T + 3) + y(T + 4)]/4, consecutively. These
forecasts provide a natural measure of future liquidity. More precisely, if we assume that the quotes
result in trades of the same volume V, say, we can claim that our forecasts represent the average
time necessary to trade the basic quantity V of assets when the volumes V, 2V, 3V, 4V , etc. are to
be traded (Gourieroux et al., 1999). By comparing the distributions of forecasted durations, we see
how the liquidity cost measured by time necessary to trade a given volume without causing a change
in the price of asset, depends on the traded quantity. The means and variances of forecasted
durations are reported in Table III.
CONCLUSIONS
This paper introduced the autoregressive gamma processes ARG for high-frequency financial data.
The ARG can represent the dynamics of various positively valued time series such as intertrade
(interquote) durations or series of squared returns, which approximate asset volatility in the same
way intertrade durations approximate asset liquidity. Alternatively, an ARG model can fit a series of
volumes per trade, which is an alternative proxy for liquidity. The ARG is sufficiently flexible to
accommodate short as well as long memory. The stationarity and ergodicity conditions and forecast
functions are available for ARG processes of any autoregressive order. The estimation relies on a
variety of methods, ranging from a simple regression to more sophisticated simulation-based
estimators.
Finally note that the ARG specification for positively valued univariate processes can easily be
extended to a multivariate framework. This approach leads to the Wishart autoregressive (WAR)
processes for sequences of symmetric, positive definite stochastic matrices (Gourieroux, 2004;
Gourieroux et al., 2003).
Copyright © 2006 John Wiley & Sons, Ltd.
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145
APPENDIX A: CONDITIONAL DISTRIBUTION AT ANY HORIZON
Let us prove the result for h = 2 by using the conditional Laplace transforms. The same argument
holds for any horizon H.
Lemma 1
b t ct u ¸
-d
E[exp( -uYt ) Yt -1 ] = (1 + ct u) expÏÌ-Yt -1
˝
1 + ct u ˛
Ó
Proof. We have:
E[exp( -uYt ) Yt -1 ]
= E[[ E exp( -uYt ) Zt ] Yt -1 )
[
= E (1 + ct u)
- (d + Zt )
]
Yt -1 , where Zt Yt -1 ~ P [ b t Yt -1 ]
1 ˆ¸
-d
= (1 + ct u) expÏÌ- b t Yt -1 Ê1 ˝
Ë
1 + ct u ¯ ˛
Ó
b t ct u ¸
-d
= (1 + ct u) expÏÌ-Yt -1
˝
1 + ct u ˛
Ó
QED
Let us now consider the conditional Laplace transform at horizon h = 2. We get:
E[exp(-uYt ) Yt -2 ]
= E[ E[exp(-uYt ) Yt -1 ] Yt -2 ]
b t ct u ¸
-d
È
˘
= (1 + ct u) E ÍexpÏÌ-Yt -1
˝ Yt -2 ˙
Î
˚
1 + ct u ˛
Ó
b t ct u ˆ
= (1 + ct u) Ê1 + ct -1
Ë
1 + ct u ¯
-d
-d
b t ct u ¸
Ï
b t -1ct -1
Ô
1 + ct u Ô
expÌ-Yt -2
˝
b t ct u Ô
Ô
1 + ct -1
Ó
1 + ct u ˛
b t ct b t -1ct -1u ¸
-d
= [1 + (ct + ct -1 b t ct )u] expÏÌ-Yt -2
˝
1 + (ct + ct -1 b t ct )u ˛
Ó
b t t -2 ct t -2 u ¸
-d
= (1 + ct t -2 u) expÏÌ-Yt -2
˝
1 + ct t -2 u ˛
Ó
APPENDIX B: SECOND-ORDER AUTOCORRELOGRAM
We have:
g ( 2 ) (h) = Cov(Yt 2 , Yt 2-h )
2
= E(Yt 2 Yt -2 h ) - (EYt 2 )
2
= E[ E(Yt 2 Yt -h )Yt 2-h ] - (EYt 2 )
Copyright © 2006 John Wiley & Sons, Ltd.
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C. Gourieroux and J. Jasiak
[
]
) + E[(c
2
2
= E[V (Yt Yt -h )Yt 2-h ] + E E(Yt Yt -h ) Yt 2-h - (EYt 2 )
(
2
= E (ct2t -hd + rt t -h ct t -h Yt -h ) Yt 2-h
2
]
2
d + rt t -h Yt -h ) Yt 2-h - (EYt 2 )
t t -h
= ct2t -hd (1 + d )m2 + 2 rt t -h ct t -h (1 + d )m3 + rt2t -h m4 - m22
where:
m j = E(Yt j ) =
cj
(1 - r )
j
d (d + 1) . . . (d + j - 1)
It follows that:
2
g ( 2 ) (h ) = c 2
(1 - r h ) 2
c2
2
d (d + 1)
2
2
(1 - r )
(1 - r )
+ 2r hc
+ r 2h
=
c
c3
1- rh
(1 + d )d (d + 1)(d + 2)
3
1- r
(1 - r )
c4
(1 - r )
4
(1 - r )
4
d (d + 1)(d + 2)(d + 3) 4
[
c4
(1 - r )
4
d 2 (d + 1)
2
2
d (d + 1) (1 - r h ) d (d + 1) + 2 r h (1 - r h )(d + 1)(d + 2)
+ r 2 h (d + 2)(d + 3) - d (d + 1)]
=
c4
(1 - r )
4
d (d + 1)[ r 2 h [d (d + 1) - 2(d + 1)(d + 2) + (d + 2)(d + 3)]
+ r h ( -2d (d + 1) + 2(d + 1)(d + 2)]]
=
=
c4
(1 - r )
4
d (d + 1)[ 2 r 2 h + 4(d + 1)r h ]
4
2 r hd (d + 1)[ r h + 2(d + 1)]
c4
(1 - r )
= g ( 2 ) (0 )r h
r h + 2(d + 1)
2d + 3
APPENDIX C: THE NONLINEAR CANONICAL ANALYSIS
Since an ARG process is a discretized version of the CIR process (see Proposition 9), we can derive
the nonlinear canonical decomposition of the conditional distribution of Yt given Yt-1 from the spectral analysis of the infinitesimal generator associated with the CIR process.
Copyright © 2006 John Wiley & Sons, Ltd.
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(1) Spectral decomposition of the infinitesimal generator
The infinitesimal generator A associated with the diffusion equation:
dYt = a(b - Yt )dt + s Yt dWt
is given by:
y Æ Ay ( y) = a(b - y)
dy
1
d 2y
( y) + s 2 y 2 ( y)
dy
2
dy
Let us verify whether the eigenfunctions of the infinitesimal generator are polynomials.
(i)
Let us first consider a polynomial of degree n, yn say, satisfying the condition:
a( b - y )
1
fd 2y n
dy n
( y) = l ny n ( y)
( y) + s 2 y
2
dy
dy 2
where ln is the corresponding eigenvalue. By identifying the coefficients of yn, we see that: ln
= -na = n log r. This implies the necessary form of the eigenvalue.
(ii) Next we need to solve the differential equations:
a (b - y )
dy n
1
d 2y n
(y) + s 2 y
( y ) + nay n ( y ) = 0
dy
2
dy 2
or: (b - y)
dy n
1 s 2 d 2y n
( y) +
(y) + ny n (y) = 0
y
dy
2 a
dy 2
Let us introduce the change of variable:
z=
2a
2a
y, y n ( y) = y *n Ê 2 yˆ = y *n ( z )
2
Ë
s
s ¯
(C.1)
where: 2a/s 2 = (1 - r)/c.
The differential equation becomes:
2
2a
dy n*
1 s 2 (2 a) d 2y n*
(
)
(
)
(z) + ny n (z) = 0
+
b
y
z
y
dz
2 a (s 2 )2 dz 2
s2
2
Ê 2 ab - zˆ dy n* (z) + z d y n* (z) + ny (z) = 0
n
Ë s2
¯ dz
dz 2
(C.2)
2 ab
= d . It is known that the solution of differential equation (C.2) is proportional to
s2
the generalized Laguerre polynomial (Abramowitz and Stegun, 1970, formula 22.6.15):
where:
Copyright © 2006 John Wiley & Sons, Ltd.
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C. Gourieroux and J. Jasiak
n
L(nd -1) ( z ) = Â ( -1)
k =0
k
G (d + n)
zk
G (d + k )G (n - k + 1) k!
(C.3)
Upon integrating with respect to a g (d) distribution, the polynomials satisfy (Abramowitz and
Stegun, 1970, formula 22.2.12):
Ú
•
0
2
[ L(nd -1) (z)]
1
G (d + n)
exp(- z)z d -1dz =
G (d )
G (d )G (n + 1)
We derive the standardized polynomials:
12
G (d )G (n + 1) ˘ (d -1)
( )
L*n d -1 ( z ) = ÈÍ
Ln ( z )
Î G (d + n) ˙˚
(C.4)
Finally the eigenfunctions of the initial differential equation are derived by applying the change
of variable (C.1):
12
G (d )G (n + 1) ˘ (d -1) Ê (1 - r ) y ˆ
y n ( y) = ÈÍ
Ln
Ë
c ¯
Î G (d + n) ˚˙
(C.5)
Therefore yn, n = 0, 1, . . . forms a basis of eigenfunctions of the infinitesimal generator associated with the eigenvalues ln= log rn.
(iv) It is known that for a univariate diffusion equation the functions yn, n varying, are also eigenfunctions of the conditional expectation operator: y Æ E(y(Yt)|Yt-1 = y), associated with the
eigenvalues: exp ln= rn. Moreover, the process is time-reversible. The conditional distribution
of Yt given Yt-1, f1(yt|yt-1; d, r, c) (say) can be written as (see Lancaster, 1963):
•
¸
Ï
f1 ( yt yt -1 ; d , r, c) = f ( yt ; d , r, c)Ì1 + Â r ny n ( yt )y n ( yt -1 )˝
˛
Ó n =1
(C.6)
c ˆ
Ê
.
where f(yt; d, r, c) is the marginal distribution of Yt, i.e. the distribution g˜ Ë d , 0,
1- r¯
APPENDIX D: PROOF OF PROPOSITION 7
Let us denote by g(yt|zt) the conditional p.d.f. of Yt given Zt and by h(zt|yt-1) the conditional p.d.f. of
Zt given Yt-1. Then the conditional distribution of Y1, . . . , YT, Z1, . . . , ZT given Y0 is:
t
l( y1 , . . . , yT , z1 , . . . , zT yo ) = Â [ g( yt zt )h( zt yt -1 )]
t =1
We find that:
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149
l(z1 , . . . , zT y0 , y1 , . . . , yT )
l(y1 , . . . , yT , z1 , . . . , zT yo )
=
l(y1 , . . . , yT yo )
t
’ [g(y z )h(z y
=
’ Â [g(y z)h(z y
t
t =1
t
t =1
t
t
t -1
)]
•
t
t -1
)]
z =0
È
˘
Í g(y z )h(z y ) ˙
t
t t
t t -1
˙ = C l(zt yt , yt -1 ) (say)
= ’t =1 Í •
t =1
Í
˙
ÍÎ Â g(yt z)h(z yt -1 ) ˙˚
z =0
t
Therefore we note that:
(i) Z1, . . . , ZT are independent, conditional on Yo, Y1, . . . , YT.
(ii) The conditional distribution:
l( zt yo , y1 , . . . , yT ) = l( zt yt , yt -1 )
depends on yt-1, yt only.
g(y z )h(zt yt -1 )
(iii) l(zt yt , yt -1 ) = • t t
 {g(yt z)h(z yt -1 )}
z =0
z
d + zt -1
(byt -1 ) t
Ê - yt ˆ y
(
)
b
exp
exp
y
t
1
Ë c ¯ G (d + zt )
G (zt + 1)
cd + zt
=
z
•
d + z -1
Ï 1
(byt -1 ) ¸
Ê - yt ˆ yt
(
)
y
exp
b
exp
Ì d +z
˝
t -1
Â
Ë c ¯ G (d + z)
G (z + 1) ˛
z =0 Ó c
1
zt
1
È byt yt -1 ˘
G (d + zt )G (zt + 1) ÍÎ c ˙˚
=
z
•
Ï
1
Ê byt yt -1 ˆ ¸
Ì
Â
Ë c ¯ ˝˛
z =0 Ó G (d + z )G (z + 1)
APPENDIX E: TRANSITION DENSITY OF THE STATE VARIABLE
We get:
l( zt zt -1 )
= Ú l( zt , yt -1 zt -1 )dyt -1
= Ú h( zt yt -1 )g( yt -1 zt -1 )dyt -1
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C. Gourieroux and J. Jasiak
= Ú exp( - byt -1 )
b zt ytz-t 1
ytd-+1zt -1 -1
yt -1 ˆ
expÊ dyt -1
d + zt -1
Ë
c ¯
G ( zt + 1) c
G (d + zt -1 )
1
1
b zt
expÈÍ - yt -1 Ê b + ˆ ˘˙ ytd-+1zt -1 -1dyt -1
Ë
c¯˚
G ( zt + 1)G (d + zt -1 ) cd + zt -1 Ú
Î
zt
G (d + zt + zt -1 )
b
1
= d + zt -1
d + zt + zt -1
G ( zt + 1)G (d + zt -1 )
c
Ê b + 1ˆ
Ë
¯
c
=
z
=
(bc) t
d + zt + zt -1
(bc + 1)
G (d + zt + zt -1 )
G ( zt + 1)G (d + zt -1 )
APPENDIX F: STATIONARITY CONDITION
We use Proposition 9 to find the conditions, which ensure that the solution of the p-dimensional
recursive system:
X1,t =
cX1,t -1
b1 + X2 ,t -1
1 + cX1,t -1
M
X p-1,t =
X p ,t =
cX1,t -1
b p-1 + X p ,t -1
1 + cX1,t -1
cX1,t -1
bp
1 + cX1,t -1
tends to (0, . . . 0)¢, when t tends to infinity, for any nonnegative initial value (X1,0, . . . , Xp,0)¢. The
system is equivalent to:
X1,t = X p ,t
b1
+ X2 ,t -1
bp
M
X p-1,t = X p ,t
X p ,t = b p -
b p-1
+ X p ,t -1
bp
bp
1 + cX1,t -1
It follows that Xj,t, j = 1, . . . , p take nonnegative values, and that Xp,t is always less than bp.
Moreover, the sequence (Xp,t) satisfies the nonlinear recursive equation:
X p ,t = b p -
bp
b p-1
b1
+ X p ,t - p ˘˙
1 + c ÈÍ X p ,t -1
+ . . . + X p ,t - p+1
bp
bp
Î
˚
Copyright © 2006 John Wiley & Sons, Ltd.
J. Forecast. 25, 129–152 (2006)
DOI: 10.1002/for
Autoregressive Gamma Processes
151
A possible limiting value l of this sequence satisfies:
l = bp -
bp
b p-1 ˘
È b1
+ 1˙
1 + cl Í + . . . +
bp
Îbp
˚
1
È
˘
Therefore, the admissible values are l = 0 and l = b p Í1 .
Î c(b1 + . . . + b p ) ˙˚
If c(b1 + . . . + bp) < 1, the sequence (Xp,t) takes values in the compact set [0, bp], with a unique
admissible limiting value l = 0. We infer its convergence to zero.
1
È
˘
by selectIf c(b1 + . . . + bp) > 1, we can get the convergence of (Xp,t) to b p Í1 Î c(b1 + . . . + b p ) ˚˙
ing the initial value appropriately.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge financial support of the Natural Sciences and Engineering
Research Council of Canada.
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Authors’ biographies:
Joann Jasiak is Associate Professor of Economics at York University, Toronto, Associate Fellow at Centre
Interuniversitaire de Recherche en Economie Quantitative (CIREQ) and Member of Centre de Recherche en
E-Finance (CREF). Her research interest include time series econometrics, financial econometrics, nonlinear
models and high-frequency data.
Christian Gourieroux is Professor of Economics at University of Toronto, Canada, and ENSAE (Ecole Nationale
de Statistique et Economie) in Paris, France. He is director of the Finance and Insurance section of CREST, member
of CEPREMAP, France, and CREF, Canada, and Associate Fellow at CIREQ. His research interest include time
series econometrics, financial econometrics, nonlinear models and high-frequency data.
Authors’ addresses:
Christian Gourieroux, University of Toronto, Toronto, Canada.
Joan Jasiak, Department of Economics, York University, 4700, Keele Street, Toronto, ON, M3J 1P3, Canada.
Copyright © 2006 John Wiley & Sons, Ltd.
J. Forecast. 25, 129–152 (2006)
DOI: 10.1002/for