A Semiparametric Approach to the Detection of Change-points in Volatility Dynamics of Financial Data

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A Semiparametric Approach to the Detection

of Change-points in Volatility Dynamics of

Financial Data

Huaiyu Hu1and Ashis Gangopadhyay1*

1Department of Mathematics and Statistics, Boston University,

111 Commington Mall, Boston, 02215, MA, USA.

*Corresponding author(s). E-mail(s): ag@bu.edu;

Abstract

One of the most important features of ﬁnancial time series data is volatil-

ity. There are often structural changes in volatility over time, and an

accurate estimation of the volatility of ﬁnancial time series requires care-

ful identiﬁcation of change-points. A common approach to modeling

the volatility of time series data is based on the well-known GARCH

model. Although the problem of change-point estimation of volatility

dynamics derived from the GARCH model has been considered in the

literature, these approaches rely on parametric assumptions of the con-

ditional error distribution, which are often violated in ﬁnancial time

series. This may lead to inaccuracies of change-point detection result-

ing in unreliable GARCH volatility estimates. This paper introduces

a novel change-point detection algorithm based on a semiparametric

GARCH model. The proposed method retains the structural advantages

of the GARCH process while incorporating the ﬂexibility of nonpara-

metric conditional error distribution. The approach utilizes a penalized

likelihood derived from a semiparametric GARCH model along with an

eﬃcient binary segmentation algorithm. The results show that in terms

of the change-point estimation and detection accuracy, the semipara-

metric method outperforms the commonly used Quasi-MLE (QMLE)

and other variations of GARCH models in wide-ranging scenarios.

Keywords: Change-point, Volatility, Semiparametric, GARCH, Binary

Segmentation, Financial Time Series

arXiv:2210.11520v1 [stat.ME] 20 Oct 2022

2Semiparametric Detection of Change-points in Volatility Dynamics

1 Introduction

In time series, particularly in ﬁnancial time series, it is very common to observe

structural changes at various time points in the data. These changes, known

as change-points or breakpoints, separate data into distinct segments. The

problem of change-point detection attracts signiﬁcant attention in widespread

industries, including ﬁnance [1], genomics [2], geology [3], climate data analysis

[4], audio analysis [5], oceanography [6], and many other ﬁelds. In this paper,

we focus on the change-point detection problem in ﬁnancial time series. In

particular, we consider the problem of identiﬁcation of changes in volatility, a

question of utmost importance in ﬁnancial data [7].

The change-point detection can be viewed as a model selection problem

involving the trade-oﬀ between model complexity and the resulting perfor-

mance of the model. Therefore, one general approach is to deﬁne a cost

function for segmentation [8–10], such that both the number and positions of

change-points are decided by minimizing the penalized [11] or constrained cost

function [12]. In the change-point literature, a widely utilized cost function is

twice the negative log-likelihood [13–15]. Therefore, an accurate estimation of

the likelihood function is crucial in change-point detection problems. In this

paper, we focus on the problem of volatility estimation of ﬁnancial data, and

in that context, we introduce a new method of change-point detection via a

semiparametric likelihood.

During the last several decades, a signiﬁcant body of knowledge has been

developed in the area of modeling ﬁnancial time series. In particular, after

the introduction of Autoregressive Conditional Heteroskedasticity (ARCH)

model by Engle [16] and the Generalized ARCH (GARCH) model by Boller-

slev [17], considerable research has been focused on modeling the volatility

of observed ﬁnancial returns. These models have been shown to capture the

stylized features of ﬁnancial data, such as volatility clustering and leptokurto-

sis. Therefore, despite certain limitations, these models have become essential

instruments in understanding the volatility of ﬁnancial data.

The GARCH(p, q) model for the volatility of returns ytassumes the form:

yt=σtt, σ2

t=ω+

i=1

αiy2

t−i+

j=1

βjσ2

t−j(1)

where ω≥0, αi≥0, βj≥0 and Pαi+Pβj<1. The white noise process

t

i.i.d

∼fassumes E(t) = 0 and V ar(t) = 1 . In this model, ytis the centralized

values of the return rt, i.e., yt=rt−µt, where µtstands for a smooth trend, and

σtis a stochastic process known as volatility and assumed to be independent

of t.

When the conditional error distribution fis known, parameters θ=

(ω, α1, ..., αp, β1, ..., βq) in the GARCH model are estimated directly by maxi-

mum likelihood estimation (MLE). However, in practice, the error distribution

is unknown, and the usual approach is to assume a parametric form for f, which

Semiparametric Detection of Change-points in Volatility Dynamics 3

leads to an estimation method called quasi-maximum likelihood estimation

(QMLE). In earlier literature, the conditional error distribution is generally

assumed to be Gaussian [18,19]. The GARCH parameter estimates of the

Gaussian QMLE are consistent regardless of the true error distribution [18].

It is a common approach to the estimation of GARCH parameters. For exam-

ple, in most R packages, such as the fGarch package [20], Gaussian QMLE is

applied by default to estimate parameters θ.

However, for ﬁnancial data, the conditional normality assumption of ytis

usually violated, and the speciﬁcation of an appropriate parametric error distri-

bution is often diﬃcult [21]. In addition, Gaussian QMLE suﬀers from eﬃciency

loss in cases where the true error distribution is non-Gaussian, such as Stu-

dent’s t, generalized Gaussian and other heavy-tailed densities [22]. Hall and

Yao [23] showed that the asymptotic normality and convergence rates QMLE

are incorrect when the error distribution is heavy-tailed. A possible solution to

the problem is to consider a semiparametric estimator of the GARCH param-

eters, where assumption of the error distribution is relaxed by taking it to be

any absolutely continuous pdf f, and the likelihood is constructed by estimat-

ing f, possibly by utilizing the residuals based on an initial estimate of the

model parameters. Some approaches in this vein include discrete maximum

penalized likelihood estimator (DMPLE) [22] and MLE derived from a likeli-

hood based on a nonparametric density estimate (SMLE) [24,25]. A review

of the current state of the semiparametric estimation of the GARCH model is

given by Di and Gangopadhyay [26].

In this paper, we argue that the successful identiﬁcation of change-point of

volatility for ﬁnancial data can be achieved via a semiparametric likelihood,

and we show that the performance of the proposed method is superior to para-

metric likelihood under model miss-speciﬁcation. Even when the likelihood is

correctly speciﬁed, which is an oxymoron in real applications, the performance

of the semiparametric method is at par with its parametric counterpart. A key

step of the analysis is to develop a semiparametric likelihood, and we have uti-

lized a one-step likelihood function of the GARCH model introduced in [25].

The term “one-step” refers to fact that the likelihood is evaluated in a single

step via a search on the parameter space, as opposed to a two-step method that

requires an initial estimate of the model parameters to generate the residuals,

which in turn is used to nonparametrically estimate the error density leading

to the likelihood function [24]. The method is discussed in detail in the next

section. The one-step SMLE is consistent and asymptotically Gaussian [25].

The likelihood function in this model is untrimmed, which implies no informa-

tion is abandoned in the estimating procedure. Simulation results show that

the one-step SMLE approach is more robust with smaller bias and variability

of GARCH estimators than the two-step SMLE and QMLE [25].

The paper address the question of identiﬁcation and estimation of volatil-

ity change-points in ﬁnancial data. This is a fundamental question in volatility

estimation of ﬁnancial time series as ﬁnancial data observed over a long period

4Semiparametric Detection of Change-points in Volatility Dynamics

is likely to have structural changes. The traditional method of volatility esti-

mation based on a single GARCH(1,1) process will not result in a reliable

estimate of volatility. Only a few papers in the literature have attempted to

address this question, and as far as we know, all these works are centered

around Gaussian GARCH(1,1) process. Therefore, in this paper, we address

the critical question based on a novel approach to change-point detection of

volatility based on the one-step SMLE-GARCH model developed in [25]. We

propose an eﬃcient algorithm to search for change-points based on Binary

Segmentation. Since a theoretical analysis of the problem is intractable, we

provide the results of extensive simulations that show the superiority and reli-

ability of the semiparametric algorithm compared to the parametric models.

We believe that the work presented here makes a signiﬁcant contribution to

the realm of volatility estimation of ﬁnancial data.

The rest of this paper is organized as follows. In Section 2, we introduce the

one-step semiparametric GARCH estimator. In Section 3, we review change-

point detection algorithm based on the semiparametric likelihood and describe

a binary segmentation approach of model estimation. Section 4presents the

simulation results of the performance of proposed semiparametric change-point

detection method and applications to the ﬁnancial time series. Discussions and

conclusions are in Section 5.

2 Semiparametric estimation in GARCH Model

In the statistics literature, the term semiparametric has been used to refer to

diﬀerent estimation scenarios. One commonly studied problem is where certain

components of a model are characterized parametrically, while other compo-

nents are described nonparametrically [27]. In GARCH models, this idea is

often incorporated by having a nonparametric functional form for the volatil-

ity function [28–31]. However, in the present work, we focus on the scenario

involving a fully parametrically speciﬁed model and an unknown innovation

pdf. In this section, we describe the one-step semiparametric GARCH estima-

tion introduced by Di and Gangopadhyay [25], which is the foundation of the

semiparametric change-point detection method proposed in the next section.

Let’s denote the parameters of GARCH model as θ=

(ω, α1, ..., αp, β1, ..., βq), and let the observed time series is given by

y= (y1, ..., yn). Based on GARCH model in Equation (1), the residuals are

t(θ) = yt/σt(θ), where σt(θ) is the volatility at time tderived by θ. The true

log-likelihood of this GARCH model is

Ln(θ) = 1

t=1

lt(θ) = 1

t=1

log[1

σt(θ)f(yt

σt(θ))],(2)

where fis uniformly bounded and continuous pdf. Since in practice fis

unknown, we replace it with a nonparametric kernel density estimate ˆ

fngiven

Semiparametric Detection of Change-points in Volatility Dynamics 5

fn(z) = 1

nhn

t=1

K(z−t(θn)

),(3)

where K(·) is a regular density kernel and hnrepresents the bandwidth, or the

smoothing parameter. In the context of nonparametric density estimation, an

important question is the choice of the bandwidth hn. There are many data-

dependent choices of the bandwidth, including nrd0 [32], nrd,ucv,bcv [33],

and SJ [34]. In particular, the so-called normal reference bandwidths nrd0

and nrd are computed based on the prior assumption that the true density

fis Gaussian, and these approaches are probably the easiest and fastest to

implement. After comparing the simulation performances utilizing bandwidth

selection methods nrd0, nrd,ucv,bcv and SJ, we observed no obvious diﬀer-

ences in the results. Thus, due to the computational eﬃciency, in this paper we

utilize nrd, also known as the rule-of-thumb bandwidth selection method [35].

Replacing the true density fby its estimate ˆ

fnin Equation (2), the

corresponding semiparametric log-likelihood function at any given θis

Ln(θ) = 1

t=1

lt(θ) = 1

t=1

log[1

σt(θ)ˆ

fn(yt

σt(θ))].(4)

Therefore, the proposed semiparametric estimator of θ(SMLE) is achieved by

θSMLE

n= argmax

θ∈Θ

Ln(θ).(5)

Note that the QMLE is a special case of the SMLE in the sense that the

ﬁxed parametric form for fassumed in the QMLE is replaced by an estimate

fnin the SMLE. Therefore, SMLE is intuitively better than the QMLE since

a suitable choice of ˆ

fnmay converge to fin some sense, whereas an arbitrary

parametric choice of fdoes not. Hence, with a large sample size (which is gen-

erally the case in the ﬁnancial time series), the semiparametric likelihood based

on ˆ

fncan better reﬂect the true likelihood, compared to a quasi-likelihood

based on a parametric assumption on f.

The properties and applications of such semiparametric estimators have

been investigated in the literature quite extensively. Engle and Gonzalez-

Rivera [22] have discussed an application of the semiparametric volatility

estimate of £/$exchange rate returns. Since the return series exhibits a clear

peak and heavy tail, the validity of the assumption of unconditional Gaussian

or unconditional t−distribution is questionable. They showed that the semi-

parametric approach is more appropriate in representing the distribution of

the return series. Drost and Klaassen [36] studied the exchange rates of a total

of ﬁfteen currencies on the U.S. dollar under their adaptive estimation setup.

They compared the semiparametric approach to the QMLE via bootstrap and

reported that the semiparametric estimators have smaller estimation standard

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摘要：

ASemiparametricApproachtotheDetectionofChange-pointsinVolatilityDynamicsofFinancialDataHuaiyuHu1andAshisGangopadhyay1*1DepartmentofMathematicsandStatistics,BostonUniversity,111CommingtonMall,Boston,02215,MA,USA.*Correspondingauthor(s).E-mail(s):ag@bu.edu;AbstractOneofthemostimportantfeaturesofnanci...

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A Semiparametric Approach to the Detection of Change-points in Volatility Dynamics of Financial Data

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