Signal Processing - May 2017 - 31

three-factor model, including market beta, size beta, and value
beta. In practice, the ordinary least squares (OLS) time-series
regression is used to estimate beta by assuming constant beta
over the observation period. Prior research has shown evidence
of time-varying beta [55]. The OLS time-series regression does
not account for any significant economic event that could affect
systematic risks of an asset.
The piecewise mean-reverting (PMR) model is based on
the observations of the FF three-factor systematic risk behaviors in empirical tests. The betas tend to jump with relevant
significant events and revert to their means with different
rates depending on the type of events. The reverting rate indicates how quickly the systematic risk of a stock recovers from
a sudden change. For the CAPM model, the PMR model consists of a system equation that represents the PMR dynamic
of the hidden time-varying variable beta and an observation
equation of the stock return given by the CAPM. The system
equation is

problem of modeling and predicting time-varying variance of
financial time-series data. In GPRSV models, a GP prior is
placed over the state transition function, and the state transition function is a random function sample from the GP. It is
therefore not limited to a fixed linear AR form, as in GARCHtype models and pure stochastic volatility (SV) models. For
a zero-mean normally distributed asset return time series, Yt,
with time-varying variance v 2t , a GPRSV model is represented
by the following set of equations:

b t = (1 - z t) br + z t b t -1 + z t u t + p t ,

where n is the mean of Yt, a t is the innovation of the return
series, v t is the logarithm of variance v 2t at time t, and f t
and h t are independent and identically distributed zero-mean
standard Gaussian distributed white noises. A well-known
asymmetric effect is called financial leverage, representing a
negative correlation between today's return and tomorrow's
volatility [63], [64]. This asymmetric leverage effect is captured by the correlation t between e t and h t . The unknown
parameters x and t are to be estimated. The hidden state
transition function f is assumed to follow a GP, defined by
the mean function m ^ x h and covariance function k (x, xl ).
The parameters in m ^ x h and k (x, xl ) are called hyperparameters. For example, if the mean function is defined as
m ^ x h = cx, then c is a hyperparameter. The mean function
m ^ x h can encode prior knowledge of system dynamics.
Figure 11 shows the graphical model representation of a
GPRSV model.
Particle-filter-based Markov chain Monte Carlo learning
methods can be used to efficiently estimate the GPRSV model
and make volatility inferences. The GPRSV model demonstrates

where br represents the average b over time and z t is the
mean-reverting rate in the range of [0,1]. The larger is z t,
the slower is the mean reverting. The jump process is based
on the Bernoulli random variable z t, and a zero-mean normally distributed variable, u t ~ N ^0, v 2u h, represents the
amount of jump, i.e., z t takes the value of 1 with a given
probability p and the value of 0 with probability 1 - p;
2
p t ~N ^ 0, v p h represents a random perturbation of b. The
observation equation is simply the CAPM, i.e., the asset
return time series
R t - r f = a t + b t (R M, t - r f ) + f t .
This model assumes that significant economic events can
lead to the abnormal changes in beta. A modified Kalman filter can be used to estimate and track the PMR beta [56], [57].
In addition, the methodology can be extended to multifactor
models, such as the FF three-factor model [57].
As previously discussed, model validation is always an
open issue. To achieve validation, researchers can compare the
model with alternative models and try to obtain the real-world
data to examine whether betas change when major events
occur. However, such case studies are confirmatory rather
than conclusive.

a t = Yt - n = v t f t,

v t = log ^v t2 h = f (v t -1) + xh t ,
f ~ GP (m (x), k (x, xl )),

8 ht B ~ N (0, R),
f

t

1
R =;

Gaussian process regression stochastic volatility model
Volatility modeling is one of the most active research areas of
financial time series. With the recent development of Bayesian
nonparametric modeling in SP and machine-learning communities, flexible tools and modeling methods, such as the
Gaussian process (GP) [58]-[60] and copula process [61], can
be applied to model financial data volatility.
By combining the GP state-space modeling framework
with the stochastic modeling concept [62], a GP regression
stochastic volatility (GPRSV) model can be built to solve the

2 E,

tx

tx x

f (v0)

f (v1)

v0

v1

v2

a0

a1

a2

f (vt-1)

... ...

vt

at

FIGURE 11. A graphical model representation of a GPRSV model of timevarying volatility/ a t : the observation variable at time t, v t : the hidden variable (logarithm of volatility), and ft : the transition function sampled from
a GP. The thick horizontal line represents fully connected nodes.

IEEE Signal Processing Magazine

|

May 2017

|

31



Table of Contents for the Digital Edition of Signal Processing - May 2017

Signal Processing - May 2017 - Cover1
Signal Processing - May 2017 - Cover2
Signal Processing - May 2017 - 1
Signal Processing - May 2017 - 2
Signal Processing - May 2017 - 3
Signal Processing - May 2017 - 4
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Signal Processing - May 2017 - 112
Signal Processing - May 2017 - Cover3
Signal Processing - May 2017 - Cover4
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