predicting volatility: a comparative analysis between garch models and neural network models

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


1. Introduction

Concepts of risk and volatility Objective:

compare the GARCH volatility models with neural network based models for modeling conditional density


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


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


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


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


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


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


three common types of non-linearities used in ANNs


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


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


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


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


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


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


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



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







Estimation of GARCH (1,1)

Dependent Variable: RETURN_BETC

Method: ML - ARCH



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







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


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


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


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


4. Conclusion and further research

Recurrent neural networks The structure of the network used Trading or hedging strategies Methodoligies for measuring market risk


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.

predicting volatility: a comparative analysis between garch models and neural network models

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