A comparison on GARCH parameter estimation: SVR versus ML

1
A comparison on GARCH parameter estimation SVR
versus ML

• Ramya Ramakrishnan
• Advanced Machine Learning

2
Overview

• GARCH is a well known method in the financial
  community for modeling and predicting the
  conditional volatility of market returns
• Assumes data is normally distributed and
  parameter estimates are based on ML procedures
• Financial data is rarely normally distributed
  leptokurtic
• We use Support Vector Regression (SVR) to
  estimate the parameters of GARCH
• SVR does not assume that there is a probability
  density function over the return series and it
  adjusts the parameters based on empirical risk
  minimization
• SVR defines an insensitivity zone that results in
  its ability to deal with any pdf
• Results on simulation and experimental data show
• GARCH models can be accurately estimated using
  SVR
• SVR estimates have higher predictive ability that
  those obtained using ML methods

3
Methodology
4
Understanding GARCH(1,1)Generalized
Autoregressive Conditional Heteroskedasticity

• GARCH
• heteroskedasticity variances of the error
  terms is not equal the error terms may be
  expected to be larger for some points/ranges of
  the data than for others
• conditional heteroskedasticity
  heteroskedasticity that is not random and has
  autocorrelation
• time varying volatility or volatility
  clustering.

a process yt follows a GARCH(1,1) model if yt
µ stet st2 ? ayt-12 ßst-12 et is an
uncorrelated process with zero mean and unit
variance µ 0 without affecting model
performance Forecast yt,predict2 ?predict
(apredictßpredict)yt-1,actual2
5
Importance of GARCH in Finance

• Financial returns series often clearly exhibit
  conditional heteroskedasticity (volatility
  clustering)
• Being able to accurately forecast volatility is
  especially important in finance for risk
  analysis, portfolio selection, and derivative
  pricing
• Goal of GARCH models is to provide a volatility
  measure

6
Understanding SVRIterated Re-Weighted Least
Squares

• Problem formulation
• min Lp 0.5w2 0.5?(aiei2 ai(ei)2)
• where
• ei e yi ?(xi)w b ei e yi
  ?(xi)w b
• ai 2ai / ei ai 2ai / ei
• Basic Procedure
• 1. Fixing ai and ai minimize Lp
• 2. Recalculate ai and ai from the solution in
  step 1
• 3. Repeat until convergence
• Project Parameters
• RBF kernel exp(-xi-xj2/2s2)
• s and e terms selected by cross validation

7
Simulation Results

• Relative R-squared (R2SVR R2ML)/ R2ML
• As kurtosis increases, SVR estimates provide
  better predictive results
• Performance for normal distribution varies by
  sample size
• 1000 samples ML does marginally better
• 500 samples SVR does better than ML

10 independent trials
8
Empirical Results on SP 100 Returns
ML SVR
? 1.460 E -6 9.20 E -6
a 9.998 E -2 0.0461
ß 0.888 0.0956

Kurtosis (yt/st,predict) 5.37 5.86
LB Statistic (yt/st,predict)2 p-value 3.631 0.6028 3.661 0.599

R2 for forecast in sample out of sample 7.63 5.29 10.394 13.338
9
Conclusions

• SVR based estimates of GARCH parameters produce
  more accurate predictions of financial volatility
  than ML estimates
• ML tries to fit the residuals to a Gaussian
  distribution but if this is not the case it will
  increase the error by forcing the residuals to be
  Gaussian.
• SVR tries to get the best fit with the data, not
  relying on prior knowledge and focuses on
  minimizing the prediction error with a given
  machine complexity