Transcript Predicting volatility: a comparative analysis between

Predicting volatility: a comparative
analysis between GARCH Models and
Neural Network Models
MCs Student: Miruna State
Supervisor: Professor Moisa Altar
- Bucharest, June 2002
-
Contents
Introduction
 Models for return series

 GARCH
models
 Mixture Density Networks
Aplication and results
 Conclusion and further research
 Selective bibliography

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1. Introduction
Concepts of risk and volatility
 Objective:
compare the GARCH volatility models
with neural network based models for
modeling conditional density

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 t2   0  1 *  t21  ...   p *  t2 p
2. Models for time series returns

2.1 ARCH(p) models
  0  1 *
2
t
2
t 1
 ...   p *
2
t p
0  0, 1,..., p  0
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
2.2 GARCH (p,q)
 t2    1 *t21  ...   p *t2 p  1 * t21  ...  q * t2q
  0, 1,..., p , 1,..., q  0
GARCH(1,1)
     *
2
t
2
t 1
  *
2
t 1
  0,  ,  ,  0
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GARCH (1,1) it can be written as an infinite ARCH model :
 t2     *  t21   * t21


    *  t21   *    *  t22   *    *  t23   * ...



  *  t21   *  t22   2 *  t23  ...
1 
The unconditional variance from the GARCH (1,1)
2 

1  
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
2.3 Mixture Density Networks
 Venkatamaran
(1997), Zangari (1996) -used
unconditional mixture densities for calculating VaR

Lockarek-Junge and Prinzler (1998) -used one
neural network to model the density conditionally
 Schittenkopf
and Dorffner(1998, 1999) - concentrated
on the performance of the of neural network based
models to estimate volatility
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
Mixture Densities
 the
random variable is drawn from one out of
many possible normal distributions
 allows for heavy tails
 preserves some convenient characteristics of
a normal distribution
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
Neural Networks
 have
been used for medical diagnostics, system
control, pattern recognition, nonlinear regression,
and density estimation
 relates a set of input variables xt t=1,…,k, to a
set of one or more output variables, yt, t=1,…,k
 it is composed of nodes
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 three
common types of non-linearities used in ANNs
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
Multi-Layer Perceptron (MLP)
 has
one hidden layer
The mapping performed by the MLP is
given by
 N

MLP  xt   g   v j h  w j xt  c j   b 
 j 1

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
Mixture Density Network
 combines
a MLP and a mixture model
 the conditional distribution of the data expressed as a sum of normal distributions
p ( y | x) 
N
g
j 1
j
( x ) p ( y | x, j )
Estimation of MDN - by minimizing the negative logarithm of
the likelihood function
- by using backpropagation gradient
descendent algorithm
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
RPROP algorithm
 partial
derivative of a weight changes its sign
- the update value is decreased by a factor η If the derivative doesn’t change its sign slightly increase the update value by the
factor η+
 0< η- <1< η+
 η+=1.2
 η-=0.5
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3. Application and results

Data used

daily closing values of the BET-C from
17.04.1998 to 10.05.2002
 Returns calculated as follows: rt= ln(Pt/Pt-1)
 Two data sets: - a training one
- a testing one
 Softwere used: Eviews, Matlab Netlab
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
GARCH Estimation
200
0.10
Series: RETURN_BETC
Sample 1 1020
Observations 1020
150
0.05
0.00
100
-0.05
50
-0.10
200
400
600
800
The daily BET-C returns
1000
0
-0.10
-0.05
0.00
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.000170
-0.000184
0.093332
-0.097570
0.015423
-0.020636
8.409205
Jarque-Bera
Probability
1243.601
0.000000
0.05
Histogram of the returns series
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Mean equation
Dependent Variable: RETURN_BETC
Method: Least Squares
Sample(adjusted): 2 1020
Included observations: 1019 after adjusting endpoints
Convergence achieved after 2 iterations
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
-0.000193
0.000654
-0.295154
0.7679
AR(1)
0.294192
0.029949
9.823033
0.0000
R-squared
0.086657
Mean dependent var
-0.000189
Adjusted R-squared
0.085759
S.D. dependent var
0.015418
S.E. of regression
0.014742
Akaike info criterion
-5.594207
Sum squared resid
0.221035
Schwarz criterion
-5.584538
Log likelihood
2852.249
F-statistic
96.49198
Durbin-Watson stat
2.000042
Prob(F-statistic)
0.000000
Inverted AR Roots
.29
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ARCH LM test for serial correlation in the residuals from the mean equation
ARCH Test:
F-statistic
33.88049
Probability
0.000000
Obs*R-squared
120.0804
Probability
0.000000
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Sample(adjusted): 6 1020
Included observations: 1015 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.000126
1.95E-05
6.461881
0.0000
RESID^2(-1)
0.292038
0.031465
9.281234
0.0000
RESID^2(-2)
0.092484
0.032770
2.822200
0.0049
RESID^2(-3)
0.027229
0.032769
0.830916
0.4062
RESID^2(-4)
0.008512
0.031505
0.270192
0.7871
R-squared
0.118306
Mean dependent var
0.000217
Adjusted R-squared
0.114814
S.D. dependent var
0.000573
S.E. of regression
0.000539
Akaike info criterion
-12.20960
Sum squared resid
0.000293
Schwarz criterion
-12.18535
Log likelihood
6201.370
F-statistic
33.88049
Durbin-Watson stat
1.999460
Prob(F-statistic)
0.000000
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Estimation of GARCH (1,1)
Dependent Variable: RETURN_BETC
Method: ML - ARCH
Sample(adjusted): 2 1020
Included observations: 1019 after adjusting endpoints
Convergence achieved after 23 iterations
Bollerslev-Wooldrige robust standard errors & covariance
RETURN_BETC(-1)
Coefficient
Std. Error
z-Statistic
Prob.
0.342440
0.035452
9.659369
0.0000
Variance Equation
C
4.42E-05
1.34E-05
3.303162
0.0010
ARCH(1)
0.345598
0.073219
4.720056
0.0000
GARCH(1)
0.483342
0.111485
4.335486
0.0000
R-squared
0.084258
Mean dependent var
-0.000189
Adjusted R-squared
0.081551
S.D. dependent var
0.015418
S.E. of regression
0.014776
Akaike info criterion
-5.785426
Sum squared resid
0.221616
Schwarz criterion
-5.766087
Log likelihood
2951.675
F-statistic
31.13013
Durbin-Watson stat
2.094529
Prob(F-statistic)
0.000000
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
MDN Estimation
 feed
forward single-hidden layer neural
network
 4 hidden units
 3 Gaussians
 m-dimensional input xt-1,…,xt-m
 3n dimensional output : weights, conditional
mean, and conditional variance
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
Evaluation of the models
N
 Normalized
mean absolute
NMAE 
error
 ˆ
error
NMSE 

t 1
N
 rt 2
rt 2   t2
 (r
t 1
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t 1
r
t 1
mean squared
 rt 2
t 1
N
N
 Normalized
2
t
t
2

2
 rt 21 )2
20
 Hit
rate
1
HR 
N
ˆ
2
t
hit rate WHR 

t 1

t,

 rt 21 rt 2  rt 21  0
 sgn  (ˆ
N
 Weighted
N
t 1
2
t

 rt 21 )(rt 2  rt 21 ) rt 2  rt 21
N
r
t 1
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2
 rt 21
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
Results
Model
NN
NMAE
Learning sample
Testing sample
Garch(1,1)
Learning sample
Test sample
GARCH(1,1)
HR
0.750584
0.592732
0.831139
0.578704
0.59435
0.685464
0.62155
0.905486
Loss function
2.909279
WHR
NMSE
0.560268
0.888448
0.587878
0.784555
2.932613
0.569474
0.76637
0.712963
2.744326
0.575886
0.983746
0.636899
2.896639
0.57564
0.806182
2.820094
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4. Conclusion and further research
Recurrent neural networks
 The structure of the network used
 Trading or hedging strategies
 Methodoligies for measuring market risk

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5. Selective bibliography








Bartlmae, K. and R.A. Rauscher (2000) – Measuring DAX Market Risk: A
Neural Network Volatility Mixture Approach,
www.gloriamundi.org/var/pub/bartlmae_rauscher.pdf.
Bishop, W. (1994) - Mixture Density Network, Technical Report
NCRG/94/004,Neural Computing Research Group, Aston University,
Birmingham, February .
Jordan, M. and C. Bishop (1996)– Neural Networks, in CDR Handbook of
Computer Science, Tucker, A. (ed.), CRC Press, Boca Raton.
Locarek-Junge, H. and R. Prinzler (1998) - Estimating Value-at-Risk Using
Neural Networks, Application of Machine Learning and Data Mining in
Finance, ECML’98 Workshop Notes, Chemnitz.
Schittenkopf, C. and G. Dockner (1999) – Forecasting Time-dependent
Conditional Densities: A Neural Network Approach, Vienna University of
Economic Studies and Business Administration, Report Series no.36.
(1998) – Volatility Prediction with Mixture Density Networks, Vienna
University of Economic Studies and Business Administration, Report Series
no.15.
Venkatamaran, S. (1997) – Value at risk for a mixture of normal distributions:
The use of quasi-Bayesian estimation techniques, Economic Perspectives
(Federal Bank of Chicago), pp. 3-13.
Zangari, P. (1996)- An improved methodology for measuring VaR, in
RiskMetrics Monitor 2.
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