We study the rate of convergence of moment conditions that have been commonly used in the literature for Generalised Method of Moments (GMM) estimation of short memory latent variable volatility models. We show that when the latent variable possesses long memory, these moment conditions have an n^{1/2-d} rate of convergence where 0GMM, long memory, stochastic volatility and durations

Rohit Deo

Willa Chen

English

Research Papers in Economics

GMM Estimation for Long Memory Latent Variable Volatility
and Duration Models
Willa Chen and Rohit S. Deo∗
Texas A&M University and New York University
January 14, 2005
Abstract: We study the rate of convergence of moment conditions that have been commonly
used in the literature for Generalised Method of Moments (GMM) estimation of short memory
latent variable volatility models. We show that when the latent variable possesses long memory,
these moment conditions have an n1/2−d rate of convergence where 0 <d<0.5 is the memory
parameter. The resulting GMM estimators will thus not be
√
n consistent. We then provide an
alternative set of moment conditions that are
√
n consistent and asymptotically normal under
long memory in the latent variable, thus allowing for
√
n consistent GMM estimation.
JEL: C4, C5
∗Corresponding author. rdeo@stern.nyu.edu, Tel: (212) 998-0469, Stern School of Business, New York Uni-
versity, 8-57 KMEC, 44 W. 4th Street, New York NY 10012, USA.
11 Introduction:
The stochastic volatility (SV) model is one of the popular models used in the literature (see, for
example, Taylor (1986), Harvey (1998)) to model the conditional heteroscedasticity in returns
on ﬁnancial assets. The SV model is given by
rt =e x p( Yt/2)vt, (1)
where vt is a sequence of i.i.d.
¡
0,σ2¢
random variables independent of Yt, and Yt is a stationary
Gaussian process. When the vt are assumed to be an i.i.d. series of random variables with
positive support, independent of Yt, the model is referred to as the stochastic conditional duration
model (SCD) and can be used to model ﬁnancial durations. See Bauwens and Veredas (2004).
When the SV model (1), assuming an AR(1) for Yt, is ﬁt to returns on ﬁnancial assets, the
estimated autoregressive parameter is generally found to be close to unity, suggesting very
strong dependence in squared returns. This has prompted researchers (See Breidt, Crato and de
Lima 1998, Harvey 1998) to consider a long memory process for Yt, such as an Autoregressive
Fractionally Integrated Moving Average Process (ARFIMA), which has slow power law decay in
its autocorrelations. We will refer to an SV model where Yt is a long memory Gaussian process
as a Long Memory SV (LMSV) model. Though it has not been proposed in the literature so far,
a long memory SCD (LMSCD) model would also seem to be a suitable candidate for tick-by-tick
durations of trades. This is due to the fact that durations exhibit the same strong dependence
that is seen in squared returns and that SCD models with AR(1) models for Yt when ﬁtt ot h e
data yield estimated coeﬃcients very close to unity. See Bauwens and Veredas (2004). Thus,
eﬃcient estimation of LMSV and LMSCD models is an important issue.
T h ee s t i m a t i o no fS V / S C Dm o d e l si sm a d ed i ﬃcult by the fact that the dependence is mod-
elled non-linearly through an unobserved latent process. Though this use of a latent process
allows one to obtain a wide range of theoretical properties of the model easily, it is impossible to
write the exact likelihood of the SV/SCD model analytically. Hence, various alternative proce-
dures have been proposed in the literature for estimation, including quasi maximum likelihood
(QML) estimation and generalised method of moments (GMM) estimation. As Andersen and
Sorensen (1996) note, the procedures other than QML and GMM are computationally intensive.
The QML procedure for the SV model exploits the fact that the transformed process logr2
t
may be written as a sum of a Gaussian stationary time series and an independent white noise
series. The QML estimates of the model parameters are obtained by maximising the Gaussian
2likelihood of the logr2
t, even though this series is not Gaussian. Deo (1995) has shown that the
QML estimators of the LMSV model (1) based on the logr2
t are
√
n consistent and asymptotically
normal. Deo’s (1995) result can easily be shown to hold for the LMSCD model too.
Unlike QML estimation, there is currently no known
√
n consistent GMM estimation pro-
cedure of LMSV/SCD models. For GMM estimation, one speciﬁes a set of sample moments
denoted by Mn =( M1n,...,M qn), where Min =
Pn
t=j+1 gi (rt,r t−j)/(n − j),jis the maximum
lag being used, gi is some smooth function and q, the number of selected moments, is at least
as large as the dimension of the parameter vector θ to be estimated. The GMM estimator,
ˆ θ, minimises the distance (Mn − M (θ))
0 Λ−1 (Mn − M (θ)), where M (θ)=Eθ (Mn) and Λ is
some suitably chosen weight matrix. Under suitable regularity conditions, ˆ θ is
√
n consistent
and asymptotically normal (Hansen 1982). These suitable conditions include the requirement
that the vector of moments Mn be a
√
n consistent estimator of M (θ). In the literature on
short memory SV models(see, for example, Andersen and Sorensen 1996 and Jacquier, Polson
and Rossi 1994), the moment conditions that have generally been used are obtained by using
functions gi of the form
gi (rt,r t−j)=|rt|
ai |rt−j|
bi (2)
for some integer valued non-negative ai,b i.W h e n Yt is assumed to follow an AR(1) process,
Andersen and Sorensen (1997) report that GMM estimators based on moments of the form (2)
perform more poorly than QML estimation when the AR(1) coeﬃcient is close to the unit root.
i.e. when the persistence in the volatility is high. Bauwens and Veredas (2004) also report in
their simulations that the sample moments based on functions of the form (2) converge very
slowly to the population analogues in an SCD model with a strongly persistent AR(1) process
for Yt. These two observations indicate that the convergence of the sample moments of the form
(2) will also be very slow if Yt were a genuine long memory process instead of a near unit root
AR(1), thus yielding poor GMM estimators. In the next section, we show that this is indeed
the case. More speciﬁcally, we show that the rate of convergence of the moments (2) is slower
than
√
n and is a decreasing function of the memory parameter. Furthermore, this rate can be
arbitrarily close to a constant. We than provide an alternative set of moment conditions and
prove that the new conditions are indeed
√
n consistent and asymptotically normal. The proofs
of all of our results are in the Appendix at the end of the paper.
32 GMM estimation for the LMSV/SCD model:
We will assume that the spectral density f (·) of Yt in (1) is of the form
fY (λ)=λ−2dg (λ) (3)
for some d ∈ (0,0.5),w h e r eg(·) is a diﬀerentiable function on [−π,π]. The parameter d is
called the memory parameter of the process and controls the rate of decay of the correlations
of the process Yt. Processes such as the well known ARFIMA models have spectral densities
that satisfy (3). It is well known that under (3) the correlations of Yt at lag j, γh (j), decay
hyperbolically in j at a rate given by j2d−1. Furthermore, it can be shown that the correlations
of the transformed processes |rt|
c and logr2
t display the same hyperbolic decay for any c>0.
Thus, the process rt displays very strong conditional heteroscedasticity.
The following theorem establishes the asymptotic behaviour of sample moments of the form
(2) under LMSV/SCD models.
Theorem 1 For any integer valued non-negative a,b and under the conditions (1) and (3),
assuming that vt has all the required moments,
n1/2−d
Ã
1
n
n X
t=1
|rt|
a |rt−j|
b − E
³
|rt|
a |rt−j|
b
´!
D → X,
where X is a zero mean Gaussian random variable.
It is clear from theorem 1 that the rate of convergence of the sample moments is slower
than
√
n. Furthermore, this rate gets worse as d approaches 1/2. This is particularly of concern
since numerous studies (see for example Andersen et al. 2001) have found that high frequency
returns tend to yield estimated values of d which are around 0.3 to 0.45. This problem may
be exarcebated further by the fact that Deo and Hurvich (2001, 2002) have shown that semi-
parametric estimation of d for LMSV models can be negatively biased indicating that the real
values of d may be even greater than the values obtained in the studies.
The particular form of the SV/SCD model can however be exploited to get a set of moment
conditions that retain a
√
n rate of convergence. Using (1) , we can write the transformed series
Zt =l o g r2
t as Zt = µ + Yt + ut where ut =l o g v2
t − E(logv2
t) and µ = E
¡
logv2
t
¢
. Since Yt
4and ut are independent, we get a signal plus noise representation for Zt. QML estimation of
the SV/SCD model is based on precisely this transformation. The transformation Zt has also
been used in the literature to suggest GMM estimators for the LMSV model. Wright (1999) has
proposed using the sample covariances of Zt as the moment conditions to estimate the model
parameters. However, Wright (1999) shows that these moment conditions are
√
n consistent
only when the memory parameter d satisﬁes d<0.25. When d>0.25, t h es a m p l ec o v a r i a n c e s
of Zt can be shown ( Hosking, 1996) to be slower than
√
n consistent. Since, as argued above,
the interval (0.25,0.5) constitutes the more empirically relevant range for d, it is crucial to have
moments which will retain the
√
n convergence rate over the entire parameter space of d. The
following theorem provides precisely such a set of moment conditions.
Theorem 2 Assume the model given by (1) and (3) and that E
©
u8
t
ª
< ∞.L e t
ˆ γj =( n − j)
−1
n X
t=j+1
³
Zt −
_
Z
´³
Zt−j −
_
Z
´
,
γj = Cov(Zt,Z t−j) and Wj =ˆ γ0 − ˆ γj − (γ0 − γj). Then for any integer q
√
nW
D → N(0,Σ),
where W =( w1,...,w q)
0 and Σ = AΣ1A0 + Σ2,
the jth row of A is a0
j =( j/2,(j − 1),(j − 2),...,1, 0,...,0 | {z }
q−jt e r m s
),
the (j,k)th term of Σ1 is 4π
R π
−π cosjλcoskλ|1 − exp(iλ)|
4 f2
Y (λ)dλ,
and the (j,k)th term of Σ2 is Eu4
t − σ4
u + σ2
uCov(2Yt − Yt−j − Yt+j,2Yt − Yt−k − Yt+k).
Using standard Taylor series arguments, it ifollows from Theorem 2 that any diﬀerentiable
transformation of the sample moments provided there will also be
√
n consistent and asymptoti-
cally normal. Thus, one can choose any of these transformations of sample moments to construct
the moment conditions to use for GMM estimation of the model. Needless to say, which moment
conditions one chooses will dictate the eﬃciency of the resulting GMM estimator and a partial
answer regarding this choice may be given by a detailed Monte Carlo simulation. We leave this
issue for further research.
Appendix
5P r o o fo fT h e o r e m1 :L e tµ1 = E {exp(0.5aYt +0 .5bYt−j)} and µ2 = E
n
|vt|
a |vt−j|
b
o
.
Let
T1 = n−1
n X
t=1
|rt|
a |rt−j|
b − n−1
n X
t=1
exp(0.5aYt +0 .5bYt−j)µ2
and
T2 = n−1
n X
t=1
exp(0.5aYt +0 .5bYt−j)µ2 − µ1µ2.
Then n−1 Pn
t=1 |rt|
a |rt−j|
b−E
³
|rt|
a |rt−j|
b
´
= T1+T2. Since vt is an iid series, we get Va r(T1)=
O
¡
n−1¢
and hence T1 = op
¡
nd−1/2¢
. The theorem is thus established if we prove that n1/2−dT2
is asymptotically normal. Let Xt =0 .5aYt+0.5bYt−j.T h e nXt is also a stationary long memory
Gaussian series with a spectral density that satisﬁes (3) and Corr(Xt,X t−s)˜As2d−1 as s →∞
for some constant A. We have
T2 = µ2n−1
n X
t=1
(exp(Xt) − µ1).
Furthermore, the function G(x)=e x p( xσ)−exp
¡
σ2/2
¢
, where σ2 = Va r(Xt), has a Hermite
rank of 1 as deﬁned in Taqqu (1975). Hence, by Theorem 5.1 of Taqqu (1975), n1/2−dT2
D → X
where X is a zero mean Gaussian random variable.
P r o o fo fT h e o r e m2 : We will demonstrate the proof only for Wj since the higher di-
mension case is obtained along similar lines by applying the Cramer Wold device. Letting
ˆ γY,j = n−1 Pn
t=1
³
Yt −
_
Y
´³
Yt−j −
_
Y
´
for j ≥ 0 and using the fact that
_
Y
P → 0, we get by
simple algebra,
ˆ γ0 − ˆ γj =ˆ γY,0 − ˆ γY,j + n−1
n X
t=1
¡
ξ2
t − ξtξt−j
¢
+ n−1
n X
t=1
Yt (2ξt − ξt−j − ξt+j)+op
³
n−1/2
´
=ˆ γY,0 − ˆ γY,j + n−1
n X
t=1
¡
ξ2
t − ξtξt−j
¢
+ n−1
n X
t=1
(2Yt − Yt−j − Yt+j)ξt + op
³
n−1/2
´
.
(4)
From the proof of Theorem 5 of Hosking (1996), we see that
ˆ γY,0 − ˆ γY,j = a0
jX + Op
¡
n−1¢
, (5)
where X =( e γ
(1)
0 −γ
(1)
0 ,e γ
(1)
1 −γ
(1)
1 ,...,e γ
(1)
q−1−γ
(1)
q−1)
0, e γ
(1)
s = n−1 Pn
t=1(Yt−Yt−1)(Yt−j−Yt−1−j) and
6γ
(1)
s = E((Yt − Yt−1)(Yt−j − Yt−1−j)). From (4) and (5), we get
ˆ γ0 − ˆ γj = a0
jX + n−1
n X
t=1
¡
ξ2
t − ξtξt−j
¢
+ n−1
n X
t=1
(2Yt − Yt−j − Yt+j)ξt + op
³
n−1/2
´
.
The limiting distribution result now follows from Hannan (1976).
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GMM Estimation for Long Memory Latent Variable Volatility and Duration Models

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GMM Estimation for Long Memory Latent Variable Volatility and Duration Models

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