Conditional heteroskedasticity properties are derived for some common count data regression and time series models. New extensions are suggested and discussed.Conditional variance; time series; finance; traded stocks; Poisson.

Brännäs, Kurt

English

Research Papers in Economics

Conditional Heteroskedasticity in some Common Count Data
Models for Financial Time Series Data∗
Kurt Br¤ ann¤ as
Department of Economics & USBE, Umeß aU n i v e r s i t y
SE-901 87 Umeß a, Sweden
email: kurt.brannas@econ.umu.se
Abstract
Conditional heteroskedasticity properties are de-
rived for some common count data regression and
time series models. New extensions are suggested
and discussed.
Key Words: Conditional variance, time series, ￿-
nance, traded stocks, Poisson.
JEL Code: C25, G12, G14
Umeß aE c o n o m i cS t u d i e s592, 2002
1. Introduction
This note studies the conditional variance or het-
eroskedasticity properties of some common count
data models for time series data and discusses some
new extensions. Recent ￿nancial research applies
Poisson or other count data models to the number
of traded stocks (e.g., Gourieroux and Jasiak, 2001,
ch. 14). As conditional heteroskedasticity is an im-
portant ingredient in other time series models for
￿nancial markets, the presence of this property in
count data therefore appears of potential interest.
Count data models typically have a het-
eroskedasticity property (e.g., Cameron and
Trivedi, 1998), which automatically implies con-
ditional heteroskedasticity. This is in contrast to
most continuous variable models for, e.g., the stock
price, in which no heteroskedasticity is assumed but
conditional heteroskedasticity is a feature of great
interest (e.g., Engle, 1982).
The starting point in this note is the Poisson re-
gression model and we mainly consider oﬀ-springs
∗Research ￿nancially supported by the Wallander-
Hedelius Foundation and USBE. J¤ orgen Hellstr¤ om is
thanked for comments on a previous version of the paper.
of this model. Hence, directly speci￿ed semipara-
metric models (e.g., Fahrmeier and Tutz, 1994, ch.
6) to be estimated by, e.g., GMM are not consid-
ered.
We start by studying existing count data mod-
els in Section 2. In Section 3 we study ways to
expand these speci￿cations to accommodate condi-
tional heteroskedasticity in alternative ways. The
￿nal section contains a more general discussion.
2. Models
The basic model for most count data regression
modelling is the Poisson model. The Poisson distri-
bution has the property of independent increments
which implies that for a count variable yt at time t
E(yt)=E ( yt|Ft−1)=V ( yt)= V ( yt|Ft−1)=λt,
where Ft−1 =( Yt−1,X t) is the information set with
Yt =( y1,...,y t)a n dXt =( x1,...,xt). Typically,
λt =e x p ( xtβ),
where xt is the vector of exogenous variables and
β is a vector of parameters.
Hence, in this basic model the unconditional and
conditional heteroskedasticities are identical. In
addition, the means and variances are equal. This
is then a very restrictive speci￿cation with respect
to ￿nancial applications as well as for other time
series data.
2.1 Overdispersed Poisson
A common feature of empirical count data is that
the variance exceeds the mean. This is usually
modelled in terms of an overdispersed Poisson
model. Here, yt is Poisson distributed condition-
ally on a latent random variable εt so that
E(yt|εt)=V ( yt|εt)=εtλt.
1Conventionally one assumes {εt} to be an iid se-
quence with E(εt)=1a n dV ( εt)=σ2.
The conditional and unconditional moments are
internally equal, i.e.
E(yt)=E ( yt|Ft−1)=λt
V(yt)=V ( yt|Ft−1)=λt + σ2λ
2
t,
but the means and variances are no longer equal.
When εt is assumed gamma distributed the un-
conditional yt has a negative binomial, NB2, distri-
bution. To estimate, either such a fully parametric
model may be estimated by ML or, e.g., a GMM
estimator based on only the given moments may be
applied.
Even if the variances increase quadratically with
respect to the mean level this is a quite restrictive
speci￿cation.
2.2 Zeger￿s Model
Zeger (1988) suggested an extension of the overdis-
persed Poisson model for time series data. One
sets E(yt|εt)=V ( yt|εt)=εtλt and assumes the
stationary {εt} sequence to again have E(εt)=1
and V(εt)=σ2. Besides implying overdispersion
this model gives serially correlated counts. How-
ever, conditionally on εt and εs, respectively, yt
and ys are independent. The unconditional mean
and variance are those of the previous model. One
may also obtain the time-varying autocovariance
function of the {yt} sequence.
To derive the conditional moments we need the
following result:
E(yk
t |Ft−1)=E εt
£
E(yk
t |εt)|Ft−1
⁄
,
which holds for conventional time series models for
εt. The result holds since
E(yk
t |Ft−1)=
∞ X
y=0
yk
t
Pr(yt,F t−1)
Pr(Ft−1)
=
∞ X
y=0
yk
t
R ∞
0 Pr(yt,εt,Ft−1) dεt
Pr(Ft−1)
=
Z ∞
0
∞ X
y=0
yk
t Pr(yt|εt)f(εt|Ft−1) dεt
=
Z ∞
0
E(yk
t |εt)f(εt|Ft−1) dεt
=E εt
£
E(yk
t |εt)|Ft−1
⁄
.
It follows then that
E(yt|Ft−1)=λtE(εt|Ft−1)
V(yt|Ft−1)=E εt
£
E(y2
t|εt)|Ft−1
⁄
−E2
εt [E(yt|εt)|Ft−1]
= λtE(εt|Ft−1)+λ
2
tV(εt|Ft−1).
Consider as an example the AR(1) model εt =
θεt−1+(1−θ)+ut,w h e r e{ut} is a zero mean ran-
dom sequence with variance σ2
u. The parametriza-
tion is such that E(εt)=1 . T h e nE ( εt|Ft−1)=
θεt−1+(1−θ)a n dV ( εt|Ft−1)=V ( εt)=σ2
u. Hence
E(yt|Ft−1)=[ θεt−1 +( 1− θ)]λt
V(yt|Ft−1)=[ θεt−1 +( 1− θ)]λt + σ2
uλ
2
t.
In this case the conditional mean is aﬀected in the
same way as the conditional variance is.
Consider as another example the MA(1) model
εt = ut + θut−1,w h e r e{ut} is a random se-
quence with mean 1/(1+θ)a n dv a r i a n c eσ2
u.T h e n
E(εt|Ft−1)=1 /(1 + θ)+θut−1 and V(εt|Ft−1)=
σ2
u. Hence
E(yt|Ft−1)=[ 1 /(1 + θ)+θut−1]λt
V(yt|Ft−1)=[ 1 /(1 + θ)+θut−1]λt + σ2
uλ
2
t.
It can also be shown that these results hold when
yt is not only conditional on εt but on Ft−1 as well.
The Zeger speci￿cation is still restrictive in the
sense of not allowing for a less tight relationship
between the conditional mean and variance.
2.3 The Zeger-Qaqish Model
The Zeger and Qaqish (1988) model contains
lagged yt−i,i > 0, in the λt function and speci-
￿es a conditional model for yt given past obser-
vations. This approach can be extended by in-
troducing an εt as in either of the two previous
subsections. It is quite straightforward to demon-
strate that no changes to the conditional moments
of these subsections will arise. The only exception
is the presence of lagged endogenous variables in
λt.
3. Modi￿ed Models
We consider two types of modi￿cations of the ba-
sic models in Section 2. First, we rede￿ne σ2 to
become time dependent and possibly dependent on
previous observations. Second, we alter the basic
conditional expression.
Consider the overdispersed Poisson model (Sec-
tion 2.1) and let all assumptions used above re-
main true, but let the variance of εt be a func-
tion of past observations, i.e. V(εt)=σ2
t(Ft−1).
2This time dependence will not imply dependence
between successive counts nor will it aﬀect the con-
ditional and unconditional means. However, the
conditional variance changes into
V(yt|Ft−1)=λt + σ2
t(Ft−1)λ
2
t.
This then adds ￿exibility for the model speci￿ca-
tion, but suitable speci￿cations of σ2
t(Ft−1) need
to be considered. To guarantee that σ2
t remains
positive an exponential form appears reasonable.
Corresponding to an EGARCH(1,1) speci￿cation
we could specify, say,
σ2
t =e x p
¡
α0 + α1 lnσ2
t−1 + α2u2
t−1
¢
,
where ut = yt − λt is an error term. Given this
speci￿cation GMM estimation or some type of two-
stage estimator of the αi parameters are feasible.
Alternatively with εt gamma distributed yt follows
a NB2 distribution and then ML estimation is feasi-
ble. Within these estimation frameworks LM-type
tests against added conditional heteroskedasticity
(i.e. α1 6=0a n dα2 6= 0) can be constructed.
Corresponding results hold for the more general
Zeger or Zeger and Qaqish models.
If we wish to have identical conditional and un-
conditional means but with a more variable condi-
tional heteroskedasticity we could also start with
E(yt|εt,h t)= V ( yt|εt,h t)=λt +( εt − 1)htλt,
where {εt} is an iid sequence with unit mean and
variance. Then ht is the conditional standard de-
viation of εt and could, e.g., depend on past obser-
vations.
For this model
E(yt)= E ( yt|Ft−1)=λt
V(yt)=λt +E ( h2
t)λ
2
t
V(yt|Ft−1)=λt + h2
tλ
2
t.
An obvious drawback with this type of model arises
from the requirement that λt +( εt − 1)htλt ≥ 0.
This is of importance when εt < 1. If, for example,
εt =0t h e nht < 1m u s th o l d .
Approximately, the same moment properties can
be obtained from the conditional representation
λt exp(εtht). If E(εt) = 0, V(εt)=1a n d
εtht is small, a ￿rst order Taylor expansion gives
exp(εtht) ≈ 1+htεt.T h e n E ( e x p ( εtht)) ≈ 1
and V(exp(εtht)) ≈ 1+h2
t. For this speci￿cation
only size restrictions are involved on εtht. Note
that a conditional speci￿cation λtεtht,w h i c ha p -
pears closer to the continuous variable speci￿ca-
tion, would with E(εt) = 1 result in a model where
it would be diﬃcult to separate mean and variance
eﬀects.
We could obviously also express the model on
a form closer to the mainstream conditional het-
eroskedasticity literature. By using yt =E ( yt)+ut,
where E(ut)=0a n dV ( ut)=λt,w eg e tr e s u l t s
corresponding to the Poisson model. If we set
ut = εtλt with E(εt)=0a n dV ( εt)=σ2
t(Ft−1)w e
get V(yt|Ft−1)=λt+σ2
t(Ft−1)λ
2
t. Distributionally
this route is far from easy.
While in both this and the extended, overdis-
persed Poisson model the resulting conditional vari-
ances are related, the actual data generating pro-
cess for the latter makes it a more appealing ap-
proach.
4. Discussion
In the mainstream literature on conditional het-
eroskedasticity the mean function is not aﬀected.
In restricted versions there is no conditional het-
eroskedasticity. The exception, M-ARCH, con-
tains conditional heteroskedasticity as an explana-
tory variable in the mean function. By contrast all
count data models studied above (and other ones as
well) always contain conditional heteroskedasticity.
In widely used count data models (Sections 2.1-2.3)
there are obviously close relationships between con-
ditional mean and variance functions. Attempts to
relax these ties imply technical diﬃculties in terms
of size restrictions on conditional variance func-
tions. The extension of the overdispersed Poisson
model appears the most reasonable modelling ap-
proach.
Another class of models to consider is the integer-
valued ARMA or INARMA (e.g., McKenzie, 1986).
Br¤ ann¤ as and Hall (2000) gave conditional variance
results for a few alternative INMA models. Their
INMA(1)-Model 1 has the conditional variance
V(yt|Ft−1)=σ2 + θ(1 − θ)εt−1,
where V(εt)=σ2 and θ ∈ (0,1]. Br¤ ann¤ as and
Hellstr¤ om (2001) gave results for generalizations of
the basic INAR(1) model. The standard INAR(1)
model has conditional variance
V(yt|Ft−1)=α(1 − α)yt−1 + σ2,
where V(εt)=σ2 and α ∈ [0,1]. Obviously, this
class can be viewed as an alternative to the ex-
tended, overdispersed Poisson model.
When it comes to estimation it is a general re-
sult that models (the λt part) can be estimated
3consistently by the Poisson ML estimator (Gourier-
oux, Monfort, Trognon, 1984) even if there is added
conditional heteroskedasticity. One would also ex-
pect this pseudo-ML estimator to remain eﬃcient
(cf. Br¤ ann¤ as and Johansson, 1996). When param-
eters characterizing the conditional heteroskedas-
ticity are of interest GMM estimation appears a
reasonable approach. In fact, there will be no loss
in eﬃciency even if these parameters are estimated
separately in a second stage (Ahn and Schmidt,
1995, Br¤ ann¤ as and Johansson, 1996).
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Conditional Heteroskedasticity in some Common Count Data Models for Financial Time Series Data

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Conditional Heteroskedasticity in some Common Count Data Models for Financial Time Series Data

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