Proceedings of the World Congress on Engineering 2011 Vol I
WCE 2011, July 6 - 8, 2011, London, U.K.
On the Nonlinear Estimation of GARCH Models
Using an Extended Kalman Filter
Sebastián Ossandón and Natalia Bahamonde
Abstract—A new mathematical representation, based on a
discrete-time nonlinear state space formulation, is presented
to characterize a Generalized Auto Regresive Conditional Heteroskedasticity (GARCH) model. Nonlinear parameter estimation and nonlinear state estimation, for this state space model,
using an Extended Kalman Filter (EKF) are described. Finally
some numerical results, which make evident the effectiveness
and relevance of the proposed nonlinear estimation are given.
Index Terms—GARCH models; Discrete-time nonlinear state
space model; Nonlinear parameter estimation; Nonlinear state
estimation; Extended Kalman Filter.
I. I NTRODUCTION
During the last to decades GARCH type modeling, Engle [5] and Bollerslev [2], has been an extremely active
area of research. These models are often used in financial
econometrics literature because their properties are close to
the observed properties of empirical financial data. Also
these propeties can capture various stylized facts. These
models have gained popularity, specially due to their easy
applicability and flexibility, allowing simple extensions that
better fit the empirical financial data.
For the class of GARCH models, the most commonly used
estimation procedure has been the Quasi Maximum Likelihood (QMLE) aproach. Weiss [14] was the first to study
the asymptotic properties of QMLE in GARCH models. The
asymptotic properties of the QMLE for classical GARCH
models have been extensively studied; see, for recent references, Berkes et al [1], Francq and Zakoian [6], Hall and
Yao [8].
Alternatively, other estimation procedures are available based
on the Autoregressive Moving Average Model (ARMA)
representation of the squared GARCH process. This idea
was taken by Giraitis and Robinson [7] who studied the
Whittle estimator of parametric Auto Regresive Conditional
Heteroskedasticity (∞) (ARCH(∞)) models, which involve
the GARCH(p, q) case. Recently Kristensen and Linton [11]
have proposed the use of the Yule Walker estimator for
the GARCH(1,1) model. In Bose and Mukherjee [3] the
asymptotic properties of two–stage least–squares estimator
of the parameters of ARCH models is investigated, which
has a closed–form expression and is computationally easy to
Manuscript received Mars 23, 2011; revised April 06, 2011. Natalia
Bahamonde was supported in part by FONDECYT Grant 3080009.
S. Ossandón is with the Institute of Mathematics, Pontificia Universidad
Católica de Valparaı́so, Blanco Viel 596, Cerro Barón, Valparaı́so, Chile.
E-mail: sebastian.ossandon@ucv.cl.
N. Bahamonde is with the Institute of Statistics, Pontificia Universidad
Católica de Valparaı́so, Blanco Viel 596, Cerro Barón, Valparaı́so, Chile.
E-mail: natalia.bahamonde@ucv.cl.
ISBN: 978-988-18210-6-5
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
obtain. Simulation results show that for small samples size,
this estimator has a better performance than the QMLE.
On the other hand, space state models are a flexible family of
models which fits for the modelling of many scenarios. These
models, which were introduced in Kalman [9] and Kalman
and Bucy [10], are frequently constructed and applied by
modern stochastic controllers. In Durbin and Koopman [4],
state space models was applied to time series analysis
treatment.
This work proposes a novel estimation procedure for nonlinear time series models based on the EKF. It is shown that
for GARCH processes, it is possible to have a novel state
space formulation and an efficient approach, based on the
EKF, in order to obtain an estimation for the parameters and
predictions for the states. The EKF proposed for GARCH
models is derived from a discrete-time nonlinear state space
formulation of the studied model. This method is adequate to
obtain initial conditions for a maximum-likelihood iteration,
or to provide the final estimation of the states and the parameters when maximum-likelihood is considered inadequate or
costly.
The structure of the paper is as follows. Section II defines
the basic notation and the nonlinear problem. In Section III
the EKF methodology is derived. In Section IV, nonlinear
estimation algorithm is presented and Sections V their performance are evaluated in finite samples. Some concluding
remarks are given in Section VI.
II. T HE DISCRETE - TIME NONLINEAR PROBLEM
Let us consider the following discrete-time nonlinear state
space mathematical model:
(
x(k)=f (x(k−1), u(k−1),θ)+σ(u(k−1),θ)·w(k),
y(k) = h(x(k), u(k), θ) + ν(k),
(1)
where x(k) ∈ Rn is the state unknown vector, u(k) ∈ Rr is
the input known vector, y(k) ∈ Rm is the noisy observation
vector or output vector of the stochastic process, w(k) ∈ Rn
and ν(k) ∈ Rm are, respectively, the process noise (due,
mainly, to disturbances and modelling inaccuracies of the
process) and the measurement noise (due, mainly, to sensor
inaccuracy). Moreover θ ∈ Rp is the parameter vector
that is generally unknown, f (·) ∈ Rn , σ(·) ∈ Rn×n and
h(·) ∈ Rm are nonlinear functions that characterize the
stochastic system.
WCE 2011
Proceedings of the World Congress on Engineering 2011 Vol I
WCE 2011, July 6 - 8, 2011, London, U.K.
With respect to the noises of the process, we assume the
following assumptions:
The vector w(k) is assumed to be Gaussian, zero-mean
E(w(k)) = 0 and white noise with covariance matrix
E(w(k) · w(j)T ) = Q · δ(k − j).
• The vector ν(k) is assumed to be Gaussian, zero-mean
E(ν(k)) = 0 and white noise with covariance matrix
E(ν(k) · ν(j)T ) = R · δ(k − j).
Where δ(k − j) = identity matrix when k = j, otherwise,
δ(k − j) = zero matrix.
•
A. The nonlinear state space formulation of the GARCH
model used
The GARCH model (GARCH(1,1)), that we will use
throughout this work, is characterized by the following
discrete-time equations:
x(k) = σ(k)ε(k),
σ 2 (k) = α0+α1 x2 (k − 1)+β1 σ 2 (k − 1),
(2)
(3)
where x(k) and σ(k) > 0 are, respectively the return
and the volatility, in the discrete-time k ∈ Z, associated
to a financial process, and (ε(k))k∈Z is a i.i.d. Gaussian
sequence, with E(ε(k)) = 0, E(ε(k) · ε(j)) = Qδ(k − j),
and parameters α0 > 0, α1 ≥ 0 and β1 ≥ 0. Moreover x(0)
is independent of sequence (ε(k))k>0 .
The only state space representation of equations (2) and (3)
is,
X1 (k)
f1 (X1 (k−1), X2 (k−1), θ)
0
=
+
ε(k)
X2 (k)
0
1
p
Y (k) = x(k) = X2 (k) X1 (k),
(5)
III. T HE E XTENDED K ALMAN F ILTER
The discrete-time EKF generalizes, for a discrete-time nonlinear stochastic process, the standard Kalman Filter (KF)
used in discrete-time linear stochastic process. This extension
is based on a successive linearization of the nonlinear state
space model proposed for the stochastic process under study
(see Wan and Nelson [12] and Wan et al [13]).
The functions f and h (see equation (1)) are used to compute
the predicted state and the predicted measurement from
the previous estimate state. The following equation shows
the computation of the predicted state from the previous
estimate:
ISBN: 978-988-18210-6-5
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
P(k|k − 1)=A(k − 1)P(k − 1|k − 1)A> (k − 1) + Q. (7)
After making the prediction stage, we need to update the
equations. So we have the residual measure innovation
ỹ(k)=y(k) − h(x̂(k|k − 1), u(k), k, θ)
(6)
(8)
and the conditional covariance innovation
S(k|k − 1)=C(k)P(k|k − 1)C> (k) + R,
(9)
where C is a linearized version of the nonlinear function h
around the current estimate.
The Kalman gain is given by
K(k)=P(k|k − 1)C> (k)S−1 (k|k − 1),
(10)
and the corresponding updates by
x̂(k|k)=x̂(k|k − 1) + K(k)ỹ(k)
(11)
P(k|k)=(I − K(k)C(k))P(k|k − 1).
(12)
and
The state transition and observation matrices (the linearized
versions of f and h) are defined, respectively, by
(4)
where θ = (α0 , α1 , β1 ), f1 (X1 (k), X2 (k), θ) = α0 +
α1 X22 (k)X1 (k) + β1 X1 (k), X1 (k) = σ 2 (k) and X2 (k) =
x(k)/σ(k). Let us notice the obvious nonlinearity of this
state space representation, due to the nonlinearity of the
process and observation equations.
x̂(k|k − 1)=f (x̂(k − 1|k − 1), u(k − 1), k, θ).
To compute the predicted estimate covariance a matrix A
of partial derivatives (the Jacobian matrix) is previously
computed. This matrix is evaluated, with the predicted states,
at each discrete timestep and used in the Kalman filter
equations. In other words, A is a linearized version of the
nonlinear function f around the current estimate.
A(k − 1)=
∂f
∂x
(13)
x̂(k−1|k−1),u(k−1)
and
C(k)=
∂h
∂x
(14)
x̂(k|k−1)
IV. N ONLINEAR ESTIMATION
A. Nonlinear parameter estimation
Given a discrete-time nonlinear stochastic system, as the
presented in equation (1), the maximum likelihood estimation
technique can be used to find the unknown parameters θ, of
the model, from data of the state and output observations.
In other words, given a sequence of measurement or observations YN = [y(0), y(1), y(2), ..., y(k), ..., y(N )], the
likelihood function is given by the following joint probability
density function:
L(θ; YN ) = p(YN |θ),
(15)
or equivalently:
L(θ; YN ) = p(y(0)|θ)
N
Y
p(y(k)|Yk−1 , θ).
(16)
k=1
WCE 2011
Proceedings of the World Congress on Engineering 2011 Vol I
WCE 2011, July 6 - 8, 2011, London, U.K.
Since the dynamics of the stochastic system presented in
equation (1) depends of Gaussian, white noise processes,
it seems reasonable to assume that under certain regularity
conditions, the probability density functions p(y(k)|Yk−1 , θ)
can be approximated by functions of Gaussian probability
densities. Therefore we can rewrite equation (16) as follows:
L(θ; YN )=
N
g(k)
p(y(0)|θ) Y
,
(2π)m/2 k=1 (det(S(k|k − 1))1/2
(17)
where g(k)=exp{−0.5ỹ > (k)·S−1 (k|k−1)·ỹ(k)}, ỹ(k) is the
b (k|k −
residual measure innovation defined in equation (8), y
1) = E(y(k)|Yk−1 , θ) is the conditional mean of y(k) given
y(0), y(1), y(2), ..., y(k − 1) and θ, and finally S(k|k − 1)
is the conditional covariance innovation, defined in equation
(9), given y(0), y(1), y(2), ..., y(k − 1) and θ.
Conditioning on y(0), and considering the function:
l(θ) = − ln(L(θ; Yk |y(0))),
Figure 1.
Nonlinear Parameter Estimation
Figure 2.
Nonlinear State 1 Estimation
(18)
the maximum likelihood estimator of θ can be obtained
solving the following nonlinear optimization problem:
b = arg min(l(θ)).
θ
θ
(19)
Let us remark that for a fixed θ, the values of ỹ(k) and
S(k|k − 1), at each discrete timestep, are obtained from the
Kalman filter equations, described in section III, and subsequently used in the construction of the likelihood function.
Therefore the success of the optimization of likelihood function depends strictly on the behavior of the EKF designed.
B. Nonlinear state estimation
Once estimated the parameters of the nonlinear stochastic
process, such as described in the previous subsection, using
the Extended Kalman Filter, the goal is to calculate from
the observations an estimation of the state of the nonlinear
system. For this calculation, we use again Kalman filter
equations described in section III (see particularly equation
(11)).
V. N UMERICAL RESULTS
In this section a numerical example is presented in order to
make evident the effectiveness and relevance of the proposed
nonlinear estimation method.
A GARCH(1,1) model, with θ = (α0 , α1 , β1 ) = (1, 0.3, 0.5)
and noise process covariance Q = 0.1, is considered. The
sample size used is 1000. Similar results are obtained with
different initial values.
Figure V shows the nonlinear parameter estimation of
GARCH(1,1) model described below.It can be seen the fast
convergence of the parameter evolution (see Table 1 for
details). For the parameter estimation, it is assumed that the
ISBN: 978-988-18210-6-5
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
states of the model is always known. As seen in Figures 2
and 3, the simulated states are very close to the estimated
states (using the EKF technique), being the Mean Squared
Errors 3.2484 and 0.2332 respectively.
Table I
M EAN AND M EAN S QUARE E RROR (MSE)
OF THE NONLINEAR
PARAMETER ESTIMATION
MEAN
MSE
α0
0.9817
0.0062469
α1
0.3049
0.00086165
β1
0.5084
0.0013252
VI. C ONCLUSIONS
This work presents for the first time a state space representation for GARCH family of time series models. Moreover,
WCE 2011
Proceedings of the World Congress on Engineering 2011 Vol I
WCE 2011, July 6 - 8, 2011, London, U.K.
[13] E. A. Wan, R. van der Merwe, and A. T. Nelson. “Dual Estimation
and the Unscented Transformation, ” In S. Solla, T. Leen, and K.-R.
Muller, editors, Advances in Neural Information Processing Systems,
12, pp. 666-672. MIT Press, 2000.
[14] A. Weiss, “Asymptotics theory for Arch models: estimation and
testing,” Econometric Theory, vol. 2, pp. 107-131, 1986.
Figure 3.
Nonlinear State 2 Estimation
an efficient numerical method, for nonlinear estimation of
GARCH processes, is presented. This procedure is implemented using a EKF technique. The numerical results
demonstrate the effectiveness of state representation, and
shows that it’s appropriate when the objective is to estimate
the parameters and the state of a nonlinear system.
ACKNOWLEDGMENT
The authors would like to thank S. Eyheramendy and C.
Reyes for many fruitful discussions related to this work.
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[3] A. Bose and K. Mukherjee, “Estimating the ARCH parameters by
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[4] J. Durbin and S.J. Koopman, “Time Series Analysis by State Space
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[5] R. Engle, “Autoregressive conditional heteroscedasticity with estimates
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[6] C. Francq and J-M. Zakoian, “Maximum Likelihood Estimation of Pure
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[7] L. Giraitis and P. M. Robinson, “Whittle estimation of arch models,”
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[8] P. Hall and Q. Yao, “Inference in ARCH and GARCH models with
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[9] R.E. Kalman, “A new approach to linear filtering and prediction
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[11] D. Kristensen and O. Linton, “A closed-form estimator for the Garch
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[12] E. A. Wan and A. T. Nelson. “Neural dual extended Kalman filtering:
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ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
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