A comparison on GARCH
parameter estimation:
SVR versus ML
Ramya Ramakrishnan
Advanced Machine Learning
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
Methodology
Implement SVR using IRWLS
methodology
Compare Estimation Results
for SVR and ML using
Simulated Data
Compare Estimation Results
for SVR and ML using
Empirical Data
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 + σtεt
σt2 = ω + αyt-12 + βσt-12
εt is an uncorrelated process with zero mean and unit variance
μ ≈ 0 without affecting model performance
Forecast : yt,predict2 = ωpredict + (αpredict+βpredict)yt-1,actual2
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
Log Returns of S&P 500: Jan 02 - Dec 04
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
Understanding SVR
Iterated Re-Weighted Least Squares
• Problem formulation
min Lp = 0.5||w||2 – 0.5∑(aiei2 + ai*(ei*)2)
where:
ei = ε – yi + Ф(xi)w + b
ai = 2αi / ei
ei* = ε + yi + Ф(xi)w + b
ai* = 2αi / 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-xj||2/2σ2)
σ and ε terms selected by cross validation
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
Simulation Results
Relative R-squared
•
•
•
100%
80%
60%
1000 s am ples
40%
20%
0%
-20%
500 s am ples
3
4
5
6
Kurtosis
10 independent trials
7
8
9
Empirical Results on S&P 100 Returns
ML
SVR
ω
1.460 E -6
9.20 E -6
α
9.998 E -2
0.0461
β
0.888
0.0956
Kurtosis (yt/σt,predict)
5.37
5.86
3.631
0.6028
3.661
0.599
7.63%
5.29%
10.394%
13.338%
LB Statistic (yt/σt,predict)2
p-value
R2 for forecast
in sample
out of sample
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