A_Bayesian_Learning_Method_for_Financial_Time-Series_Analysis

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Received June 5, 2018, accepted July 1, 2018, date of publication July 9, 2018, date of current version July 30, 2018.
Digital Object Identifier 10.1109/ACCESS.2018.2853998
A Bayesian Learning Method for
Financial Time-Series Analysis
FUMIN ZHU1, WEI QUAN1, ZUNXIN ZHENG 1, AND SHAOHUA WAN 2, (Member, IEEE)
1College of Economics, Shenzhen University, Shenzhen 518060, China
2School of Information and Safety Engineering, Zhongnan University of Economics and Law, Wuhan 430073, China
Corresponding author: Shaohua Wan (shaohua.wan@ieee.org)
This work was supported by the National Natural Science Foundation of China under Grants 71601125 and 71471119.
ABSTRACT This article develops a sequential Bayesian learning method to estimate the parameters and
recover the state variables for generalized autoregressive conditional heteroscedasticity (GARCH) models,
which are commonly used in the financial time-series analysis. This simulation-based method combines
particle-filtering technology with a Markov chain Monte Carlo algorithm when the model is non-linear and
the number of observed variables is relatively sparse.We compare the performance of the sequential Bayesian
learning approach with the numerical maximum likelihood estimation (NMLE) in estimating models based
on S&P 500 return rates. Our research concludes that the sequential parameter learning approach performs
more robustly and accurately than the NMLE, by taking into account the uncertainty of the model. We also
carry out simulation studies to confirm that the sequential Bayesian learning method is extremely reliable
for GARCH models.
INDEX TERMS Sequential Bayesian learning, GARCH models, Markov chain Monte Carlo, particle
filtering, sparse recovery.
I. INTRODUCTION
Time-series analysis is a method that reveals the change rules
of phenomena over time through time-series data and pre-
dicts the future of the phenomena. The most common time-
series analysismodels include autoregressivemoving average
(ARMA) model and generalized autoregressive conditional
heteroskedasticity (GARCH) model. ARMA model cannot
describe the heteroscedastic properties. GARCH model are
frequently used in finance for asset pricing, risk management
and volatility forecasting; see, for instance, [1]–[6]. There is a
consensus that the stylized facts that are frequently observed
should be taken into account when considering stock prices
dynamics, interest rates and volatilities. These stylized facts
include non-normality, jump behavior, volatility clustering,
and the leverage effect. Currently, the advantages of GARCH
dynamics for modeling volatilities are well-documented in
the literature, as GARCH models can vividly capture most
of features observed in stock markets. However, some dif-
ficulties and challenges in implementation remain due to
the complexity in estimation. These difficulties result from
the following. First, GARCH models belong to a family
of non-linear dynamics that obviously increase optimization
complexity. Second, the volatilities in the stock market are
unobserved, and stock prices are collected with substantial
noises, which makes the return rates perform as a state vari-
able. For example, when put time-series models into appli-
cation, we often encounter many difficulties by using the
traditional frequency statistical method.
The Bayesian method can synthesize a variety of informa-
tion in statistical model and analysis, so it is widely used for
the study of various problems. In data analysis, frequency
analysis and subjective (experience) are often combined to
establish a reasonable prior information, and sensitivity anal-
ysis is then performed on statistical inferences made using
prior changes to confirm the reasonableness of the statistical
inferences obtained. Bayesian statistics use probabilistic cal-
culus as a principle of data processing and is derived based on
an appropriate conditional distribution of unknown parame-
ters. In the frequency method, the parameter θ is often seen
as an unknown constant, but the Bayesian method often sees
θ as a random variable that obeys the prior distribution f0(θ ).
At the same time, this a priori settings are independent of the
data being studied. The priori knowledge and information of
the parameter θ are introduced through the prior distribution
setting. The correct information can more accurately infer the
parameter θ . At the same time, considering all the variables
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F. Zhu et al.: Bayesian Learning Method for Financial Time-Series Analysis
in the model as random variables can simplify the analysis
method.
The systematic study of sequential Bayesian method is
dynamic, time-related and can be observed periodically,
according to the observed state and records of previous states
at each time, an optimal decision is selected from a set of
feasible solutions, and then observe the possible state in the
next step, collect new information, and make a new optimal
decision, repeated it. This is called sequential, and the next
possible state of the system is random or uncertain, self-
brief, so the sequential parameter learning approach performs
more robustly and accurately than the numerical maximum
likelihood estimation (NMLE).
Bayesian time-series prediction model has been widely
used in practice. Zellner and Min [7] studies Bayesian
theory in econometrics, including regression model, com-
plete recursive model and Bayesian method of distributed
lag model. Monahan [8] studies the problem of complete
Bayesian analysis of time-series Autoregressive models,
including model identification, diagnostic test, parameter
estimation and prediction. Bauwens and Lubrano [9] stud-
ied the Bayesian theory of dynamic econometric models;
Berger [10], Franses et al. [11], and Sims [12] has studied the
Bayesian method of unit root problem. Compared with the
time series model based on the frequency statistic theory,
it also has its unique characteristics (Phillips [13] ). First, its
methods are more common and can be applied to a wider
range of statistical fields. Second, it allows using the prior
information rationally and directly, explicit to analyze spe-
cific issues. Third, the result of this method is not only a
predictive value, but also the probability distribution of a
complete future economic result, which is more real and
useful than the result produced by other prediction methods.
Finally, it can deal with the problem of uncertainty more
clearly and rationally than the traditional statistical method.
Markov chain Monte Carlo (MCMC) method has become
a main Bayesian computing method. On the one hand it
is very efficient to deal with complex problems, on the
other hand, because its programming method is relatively
easy. At present, the most widely used MCMC methods in
Bayesian analysis are Gibbs sampling method andMetroplis-
Hastings method. Studies of Gelfand et al. [14] show that
the MCMC method has great potential in Bayesian comput-
ing. Jacquier et al. [5] connected Bayesian theory with the
stochastic volatility model, used the MCMC method to per-
form a large number of sampling on the posterior distribution
of the parameters, and used nonlinear filtering as a part of the
Bayesian processing of the overall model. MCMC method
not only can be used to estimate the general ARMA model,
but also can be used to estimate various wave models, and the
MCMC method can be extended to the parameter estimation
of option pricing model and term structure model.
The application of computer technology and Bayesian
method in the field of finance can solve the complexnumerical problems in the model estimation by Monte Carlo
simulation, so that the posterior distribution of the parameters
is conveniently obtained. The structure of this paper as fol-
lows. Section II, Constructing the basic model of the article.
Section III, Numerical maximum likelihood estimation of
fast Fourier transform (FFT), sequential Bayesian learning
of particle filter technique are discussed. Section IV is the
simulation study of S&P500 return rates, simulates the stock
price path, estimates each path independently, obtains the
parameter’s distribution, estimates the N-GARCH dynamic
model according to the NMLE and SBLA of each path,
and analyzes the role and influence of parameter learning to
the model based on the empirical results. Section V is the
conclusion of this paper.
II. MODEL
This paper proposes a sequential Bayesian learning approach
to estimate the asymmetric GARCH model that captures the
leverage effect and describes a negative relationship between
return rates and volatilities. This simulation-based method
combines particle-filtering technology and an MCMC algo-
rithm to sequentially update the estimation. The combina-
tional method is referred to in the literature as sequential
learning. For more details, see [15]–[18]. It is well-known
that particle filtering, or a sequential Monte Carlo, is used
to approximate the posterior density of state variables that
cannot be observed directly, while an MCMC algorithm can
deal with the uncertainty of parameters by providing a pos-
terior contribution. Using an MCMC can provide a sample
of the posterior distribution following a number of steps
by running the chain. The more steps one runs, the more
precisely the distribution of the sample matches the true
distribution. Particle filtering increases signal processing and
Bayesian statistical inference; the filtering problem consists
of estimating the internal states in dynamical systems when
partial observations are made and random perturbations are
present.
Following the assumptions of N-GARCH dynamics, stock
prices can be expressed by
St = St−1ert+λσt+εt−ϕz(σt )
εt = σtzt
σ 2
t = α0 + α1σ
2
t−1(zt−1 − γ )
2
+ βσ 2
t−1, (1)
where rt represents the risk-free rate for an asset without risk,
λ is the market price of risk, and σt is the conditional volatil-
ity. ϕz(σt ) is the mean-correction term, which is computed
from the moment-generating function with the parameters
β, α1, α0, γ ,
ϕz(σt ) = logE[eεt |σt ], (2)
which makes the price process an exponential martingale.
This N-GARCH model also considers non-Gaussian dis-
tributed cases.
We assume the initial parameters of N-GARCH dynamics
as follows: β = 0.85, α1 = 0.06, α0 = 2.5e − 6, λ = 0.04,
γ = 1.2, and the persistence of impact on volatility is 0.9964,
the formula is given by α1(1+γ 2)+β. We set the initial state
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F. Zhu et al.: Bayesian Learning Method for Financial Time-Series Analysis
variables as σ0 = 0.012, z0 = 0, S0 = 1000. These values
are closest to the value that we estimated from the stock
market. To compare the estimation results from simulation
and find the differences between the numerical maximum-
likelihood estimation (NMLE) and the sequential Bayesian
learning approach (SBLA), we simulated 10000 paths for
stock prices (L = 10000), with each simulation path includ-
ing 1000 observations (T = 1000).
III. METHODS
A. MAXIMUM LIKELIHOOD ESTIMATION
The first estimation method was the numerical maximum
likelihood function (NMLE). In non-Gaussian cases, the fast
Fourier transformation based on the given characteristic func-
tion may be used. The residual εt satisfies
E[εt |F t−1] = 0
E[ε2t |Ft−1] = σ 2
t . (3)
Based on equation (3), one can obtain the quasi-likelihood
function iteratively for the return rate using the Gaussian
kernel density, for instance, by using the normal distribution.
The log-likelihood function is given by
log f (εt ; θ ) = −
(yt − rt − λσt + ϕz(σt ))2
2σ 2
t
+ log
1
√
2πσt
,
(4)
where yt ≡ log(St/St−1), t = 1, 2, ...,T , and θ represents
the model parameters. The target function for the estimation
will be
θ̂= argmax
∑T
t=1
log f (εt ; θ ). (5)
When zt follows a discrete-time Lévy jump process,
it is non-Gaussian distributed. By assuming φz(u) is the
characteristic function, the density may be estimated via
fast Fourier transformation (see [19]). We developed a
sequential Bayesian learning approach to jointly estimate
the non-normally distributed GARCH models by filtering
their historical noises, giving the conditional density of
innovations f (εt ) as
f (εt ) =
1
2π
∫
∞
−∞
e−iuztσtφz(σtu)du, (6)
and the conditional density as
f (εt |σt ) =
1
2πσt
∫
∞
−∞
e−iztuσtφz(uσt )d(uσt ) =
f (zt )
σt
. (7)
The conditional volatility σt can be iteratively calculated from
the GARCH dynamics.
B. SEQUENTIAL BAYESIAN LEARNING
It is common to estimate a model using a numerical max-
imum likelihood method for cross-sectional data. However,
themethod usually requires a large sample period to realize its
optimization of likelihood function. Further, investors need to
incorporate prior information during their estimation. Hence,
they can apply Bayesian methods step by step. Based on the
Bayesian rule, the posterior density p(θ |y1:T ) for θ can be
computed by
p(θ |y1:T ) ∝ p(y1:T |θ )p(θ ), (8)
where p(θ ) represents its prior information (or self-brief), and
p(y1:T |θ ) is its likelihood, which can be calculated by particle
filtering as given by
log f (y1:T ; θ ) ≈
T∑
t=1
log{
1
N
N∑
i=1
w(i)
t }. (9)
The importance weights satisfy w(i)
t = p(yt |σ
(i)
t ; θ ) =
f (ε(i)t ) and can be calculated from equation (7). For each sam-
pled θ (m), we also approximate its posterior weights w(θ (m))
from equation (8). Specifically, we only use the MCMC
algorithm when the efficient sample size (ESS) of particles
drops below thresholds (for example, half number of sampled
particles). Parameters are updated with acceptance rates:
α(θ∗) = 1 ∧
p(θ∗)p(y1:T |θ∗)N (θ; θ̂∗, 6̂)
p(θ )p(y1:T |θ )N (θ∗; θ̂ , 6̂)
, (10)
and the estimated parameters with its variance are
θ̂ =
∑M
m=1
θ (m)w̃(θ (m))
6̂ =
∑M
m=1
(θ (m) − θ̂ )w̃(θ (m))(θ (m) − θ̂ )′. (11)
Based on the importance sampling method, we need only
500 particles for parameters and 400 particles for state vari-
ables. The Bayesian learning approach, given model uncer-
tainty, can be updated sequentially with the arrival of new
information. We will show that these number of particles is
sufficiently robust to filter the state variables and parameters
efficiently. For more details about the theoretical background,
see [20]–[24].
IV. SIMULATION STUDY
We first examine the normally distributed N-GARCHmodel,
in which innovation follows discrete-time Brownian motion.
Based on the Gaussian density, we used equation (4) and (5)
to filter the model. We simulate the paths of stock prices
10000 times, estimate each path independently, and obtained
the parameters’ distribution. The initial parameters and states
variables are shown in section I. We estimate the N-GARCH
dynamics separately by NMLE and SBLA from each path.
Figure 1 shows one of the simulations. In this figure, the red
line represents the estimated volatility, and the blue, the sim-
ulated value.
A. NUMERICAL MAXIMUM LIKELIHOOD
ESTIMATION PERFORMANCE
After simulating 10000 paths, we estimate the parameters for
each path and then obtain a sample of parameters for distribu-
tion analysis. The descriptive statistics are shown in Table 1.
Figure 2 depicts the histogram fitting of these parameters.
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F. Zhu et al.: Bayesian Learning Method for Financial Time-Series Analysis
FIGURE 1. Simulations and estimations for the N-GARCH model.
TABLE 1. The estimated parameters and their standard deviations.
FIGURE 2. Histogram fitting for estimators.
Based on the simulation studies of normal N-GARCHmodel,
we find that although these estimators are entirely acceptable,
the NMLE estimator slightly underestimates the coefficient
of the ARCH effect(i.e., α1) and the coefficient of the autore-
gressive effect (i.e., β), while it overestimates the leverage
effect (i.e., γ ). Furthermore, we find that based on the limited
length of 1000 observations, the standard deviation of the
NMLE estimator is a bit greater than we expected.
B. SEQUENTIAL BAYESIAN LEARNING
APPROACH PERFORMANCE
The sequential Bayesian learning approach is more time-
consuming than the maximum likelihood estimation method
due to large simulations. It take us several more days to obtain
the entire distribution (about 10 seconds for each path) for
the 10000 simulated paths. In fact, this is unnecessary, as the
sequential Bayesian learning approach performs very robust,
accurate and convergent calculations at each simulation path
(evidence below). Based on that solid performance, we only
show some of the results. We find the differences between
SBLA and NMLE to be very clear and significant.
For each path (where each simulated scenario is one stock
price path), we draw 500 parameter particles and obtain their
distribution. Table 2 reports the estimation results using the
sequential Bayesian learning approach for one path.
TABLE 2. Estimations by NMLE and SBLA.
FIGURE 3. Estimation by SBLA.
FIGURE 4. Estimation for state variables.
Figures 3 and 4 show the estimation of variables
and parameters with the sequential Bayesian learning
approach. We also plot the simulated stock prices in figure 3.
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F. Zhu et al.: Bayesian Learning Method for Financial Time-Series Analysis
FIGURE 5. Histogram fitting of parameters by SBLA.
FIGURE 6. Box plot of parameters by SBLA.
In this test, we randomly choose one path and find no signif-
icant mismatches for state variables between the simulation
and the estimation, where the state variables include volatili-
FIGURE 7. Box plot for NMLE, 50-path test for N-GARCH
dynamics.
ties and stochastic shocks. In comparing Figures 5 and 6 with
Figure 2, we see that the values of the standard deviation for
the SBLA aremuch smaller than that for theNMLE. Figures 7
and 8 show these findings.
TABLE 3. Estimations for 50 paths by NMLE and SBLA.
We simulate 20 and 50 paths again and apply the two
estimation methods for each path, using different methods to
estimate each on the same simulation path. Table 3 shows
the estimation results for 50 simulation paths. We plot the
performance of estimation for each path. Figures 9 and 10
show the estimated parameters and their likelihood values.
In these figures ‘‘o’’ refers to the simulation value, ‘‘∗’’ refers
to NMLE, and ‘‘+’’ refers to SBLA.
FIGURE 8. Box plot for SBLA, 50-path test for N-GARCH
dynamics.
FIGURE 9. Parameter estimation, 50-path test for N-GARCH
dynamics.
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F. Zhu et al.: Bayesian Learning Method for Financial Time-Series Analysis
FIGURE 10. Persistence and likelihood, 50-path test for N-GARCH
dynamics.
FIGURE 11. Parameter estimation, 20-path test for non-Gaussian
GARCH.
TABLE 4. Estimation for N-GARCH model with CTS innovations.
These figures show that the estimation performance by the
SBLA is very robust at each path, unlike the performance
of the NMLE. We also examine the performance of the
N-GARCH model with CTS innovations, a non-Gaussian
case (see Table 4). Here, we set the parameters to C =
0.6170, λ+ = 1.5903, λ− = 1.0002, and α = 0.9935,
which are estimate by historical filtering noise from S&P
500 return rates in N-GARCH setup. The simulation study
results are plotted in Figures 11 and 12. It is clear that,
compare with NMLE, the SBLA estimator performance is
very stable, robust and convergent.
FIGURE 12. Persistence and likelihood, 20-path test for non-Gaussian
GARCH.
V. CONCLUSION
We propose a sequential Bayesian learning approach that
incorporates particle-filtering technology with a Markov
chainMonte Carlo (MCMC) algorithm to sequentially update
the estimation of a commonly-used time-series asymmetric
GARCHmodel to capture non-normality, volatility clustering
and leverage effect in finance. We present a performance
comparison of the sequential Bayesian learning approach
(SBLA) and the numerical maximum likelihood estimation
(NMLE) in estimating the models using goodness of fit
based on S&P 500 return rates. Our work indicates that the
sequential Bayesian learning approach is more robust and
accurate than the numerical maximum likelihood estimation.
Our simulation study confirms that the sequential Bayesian
learning approach (SBLA) estimates parameters much better
than the numerical maximum likelihood estimation (NMLE),
with fewer standard deviations.
We focus on the time-series models with the sequential
Bayesian learning, the effect is better than the traditional
estimation method. And the future research can be continued
from the following points:
1. Sequential Bayesian analysis method will be used to
analyze more complex models, such as stochastic volatil-
ity models, interest rate term structure models and multiple
factor models. In order to characterize the asymmetry and
long memory characteristics of financial time series, variance
terms tend to be more complicated. The parameter estima-
tion of complex model using Bayesian method is the future
research direction.
2. The application of sequential Bayesian method in model
selection. Sequential Bayesian method has obvious advan-
tages in dealing with uncertainty. So it is also worth study-
ing to consider using sequential Bayesian method for model
selection. Some literature mentions the idea of adopting
Bayesian model averaging to improve the effect of parameter
estimation. This method has not yet been popularized and
further research is still necessary.
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F. Zhu et al.: Bayesian Learning Method for Financial Time-Series Analysis
ACKNOWLEDGMENTS
Fumin Zhu thanks Professors Svetlozar Rachev and Aaron
Kim from the School of Business at Stony Brook Uni-
versity. F. Zhu thanks Dr. Michele L. Bianchi, Professor
Frank J. Fabozzi, Professor Peter Christoffersen, and Profes-
sor Hengyu Wu from the Management School of Jinan Uni-
versity, and the Chinese International Conference of Finance,
2015 and 2018.
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VOLUME 6, 2018 38965
F. Zhu et al.: Bayesian Learning Method for Financial Time-Series Analysis
FUMIN ZHU received the joint-Ph.D. degree in
the quantitative finance program from the Depart-
ment of AppliedMathematics and Statistics, Stony
Brook University, in 2014, and the Ph.D. degree
in management from the Southwestern Univer-
sity of Finance and Economics, Sichuan, China,
in 2015. Since 2015, he has been an Assistant Pro-
fessor with the Center for Finance and Accounting
Research, College of Economics, Shenzhen Uni-
versity. He has published over 20 papers in leading
journals, and his principal research is on Lévy processes, GARCH models,
high-frequency data analysis, FFT methods, particle filtering, and Bayesian
learning approaches. His research interests are asset pricing and financial
econometrics.
WEI QUAN received the bachelor’s degree in
finance from the Kunming University of Sci-
ence and Technology, Yunnan, China, in 2017.
She is currently pursuing the degree with the
College of Economics, Shenzhen University. Her
research interests are asset pricing and financial
econometrics.
ZUNXIN ZHENG received the M.Sc. degree in
economics from Hunan University, Hunan, China,
in 2003, and the Ph.D. degree in management
science and engineering from Shanghai Jiaotong
University, Shanghai, China, in 2007. He was
an Assistant Professor with the College of Eco-
nomics, Shenzhen University, from 2007 to 2008,
and an Associate Professor from 2009 to 2013.
Since 2014, he has been a Full Professor with
the College of Economics, Shenzhen University.
His research interests are financial engineering, asset pricing, and risk
management.
SHAOHUA WAN received the Ph.D. degree
from the School of Computer, Wuhan Univer-
sity, and Department of Electrical Engineering
and Computer Science, Northwestern University,USA, in 2010. In 2015, he was a Post-Doctoral
Researcher with the State Key Laboratory of Dig-
ital Manufacturing Equipment and Technology,
Huazhong University of Science and Technology.
From 2016 to 2017, he was a Visiting Scholar
with the Department of Electrical and Computer
Engineering, Technical University of Munich, Germany. He is currently an
Associate Professor and aMaster Advisor with the School of Information and
Safety Engineering, Zhongnan University of Economics and Law. His main
research interests include massive data computing for Internet of Things and
edge computing.
38966 VOLUME 6, 2018
	INTRODUCTION
	MODEL
	METHODS
	MAXIMUM LIKELIHOOD ESTIMATION
	SEQUENTIAL BAYESIAN LEARNING
	SIMULATION STUDY
	NUMERICAL MAXIMUM LIKELIHOOD ESTIMATION PERFORMANCE
	SEQUENTIAL BAYESIAN LEARNING APPROACH PERFORMANCE
	CONCLUSION
	REFERENCES
	Biographies
	FUMIN ZHU
	WEI QUAN
	ZUNXIN ZHENG
	SHAOHUA WAN