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Predicting volatility: a comparative analysis between GARCH Models and
Neural Network Models
MCs Student: Miruna StateSupervisor: Professor Moisa Altar
- Bucharest, June 2002 -
Doctoral School of Finance and Banking 2
Contents
Introduction Models for return series
GARCH modelsMixture Density Networks
Aplication and results Conclusion and further research Selective bibliography
Doctoral School of Finance and Banking 3
1. Introduction
Concepts of risk and volatility Objective:
compare the GARCH volatility models with neural network based models for modeling conditional density
Doctoral School of Finance and Banking 4
2. Models for time series returns
2.1 ARCH(p) models
2 2 20 1 1* ... *t t p t p
22110
2 *...* ptptt 22110
2 *...* ptptt 22110
2 *...* ptptt
0 10, ,..., 0p
Doctoral School of Finance and Banking 5
2.2 GARCH (p,q)
2 2 2 2 21 1 1 1* ... * * ... *t t p t p t q t q
1 10, ,..., , ,..., 0p q
GARCH(1,1)
2 2 21 1* *t t t
0, , , 0
Doctoral School of Finance and Banking 6
The unconditional variance from the GARCH (1,1)
2
1
GARCH (1,1) it can be written as an infinite ARCH model :
2 2 21 1
2 2 21 2 3
2 2 2 21 2 3
* *
* * * * * * ...
* * * ...1
t t t
t t t
t t t
Doctoral School of Finance and Banking 7
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
Doctoral School of Finance and Banking 8
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
Doctoral School of Finance and Banking 10
three common types of non-linearities used in ANNs
Doctoral School of Finance and Banking 11
Multi-Layer Perceptron (MLP) has one hidden layer
The mapping performed by the MLP is given by
1
N
t j j t jj
MLP x g v h w x c b
Doctoral School of Finance and Banking 12
Mixture Density Networkcombines a MLP and a mixture model the conditional distribution of the data -
expressed as a sum of normal distributions
1
( | ) ( ) ( | , )N
jj
p y x g x p y x j
Estimation of MDN - by minimizing the negative logarithm of the likelihood function
- by using backpropagation gradient descendent algorithm
Doctoral School of Finance and Banking 13
RPROP algorithmpartial 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+=1.2η-=0.5
Doctoral School of Finance and Banking 14
3. Application and results
Data used daily closing values of the BET-C from
17.04.1998 to 10.05.2002Returns calculated as follows: rt= ln(Pt/Pt-1)
Two data sets: - a training one
- a testing oneSoftwere used: Eviews, Matlab Netlab
Doctoral School of Finance and Banking 15
GARCH Estimation
-0.10
-0.05
0.00
0.05
0.10
200 400 600 800 1000
The daily BET-C returns
0
50
100
150
200
-0.10 -0.05 0.00 0.05
Series: RETURN_BETCSample 1 1020Observations 1020
Mean -0.000170Median -0.000184Maximum 0.093332Minimum -0.097570Std. Dev. 0.015423Skewness -0.020636Kurtosis 8.409205
Jarque-Bera 1243.601Probability 0.000000
Histogram of the returns series
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Mean equationDependent 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
Doctoral School of Finance and Banking 17
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
Doctoral School of Finance and Banking 18
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
Coefficient Std. Error z-Statistic Prob.
RETURN_BETC(-1) 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 Gaussiansm-dimensional input xt-1,…,xt-m
3n dimensional output : weights, conditional mean, and conditional variance
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Evaluation of the models
Normalized mean absolute error
Normalized mean squared error
2 2
1
2 21
1
ˆN
t ttN
t tt
rNMAE
r r
22 2
1
2 2 21
1
( )
N
t ttN
t tt
rNMSE
r r
Doctoral School of Finance and Banking 21
Hit rate
Weighted hit rate
,1
1 N
tt
HRN
2 2 2 21 1ˆ 0t t t tr r r
2 2 2 2 2 21 1 1
1
2 21
1
ˆsgn ( )( )N
t t t t t tt
N
t tt
r r r r rWHR
r r
Doctoral School of Finance and Banking 22
Results
Model NMAE HR Loss function WHR NMSE
NN Learning sample0.750584 0.592732 2.909279 0.560268 0.888448
Testing sample0.831139 0.578704
2.8200940.587878 0.784555
Garch(1,1) Learning sample0.59435 0.685464 2.932613 0.569474 0.76637
Test sample0.62155 0.712963 2.744326 0.575886 0.983746
GARCH(1,1)0.905486 0.636899 2.896639 0.57564 0.806182
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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
Doctoral School of Finance and Banking 24
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.