Øystein Skregelid Thesis Advisor

110 Pages·2009·3.97 MB·English

NORGES HANDELSHØYSKOLE
Bergen, Spring 2009

Estimation and selection of time-varying
volatility models

Master Thesis within the main profile of Finance

By: Øystein Skregelid

Thesis Advisor: Maytinee Wasumadee

This thesis was written as a part of the master program at NHH. Neither the institution, the
advisor, nor the censors are - through the approval of this thesis - responsible for neither the
theories and methods used, nor results and conclusions drawn in this work.

Abstract
This paper describes methods that can be applied to select the best conditional volatility
model for an individual asset. Three exchange traded funds (ETFs) for the financials, energy
and utilities sectors of the Dow Jones Total Market Index are evaluated to illustrate the
complexity of model selection. For the univariate series, the symmetric GARCH model and
three asymmetric models (EGARCH, GJR-GARCH and TGARCH) with a variety of lag structures
are parameterized under the assumption of both normal and t-distributed errors. The
ranking of these models are based on how well the parameters of each model fit to the
underlying data set (the likelihood), on selection criterions (AIC and BIC) and on their
forecasting ability (through statistic and economic loss functions). The results show that
different volatility models with different lag structures are selected for each of the three
sectors. For the financial sector a t-distributed EGARCH(1,2,1) model gives the most
satisfying results. The energy sector is best described by a t-distributed GJR-GARCH(1,1,2)
model, while a normal distributed GJR-GARCH(1,1,1) model is recommended for the utilities
sector.

In addition to the selection of univariate models, multivariate models are described and
tested. The main focus in this part is on the Dynamic Conditional Correlation model that
builds on univariate parameterizations of the volatility. A DCC model based on three
univariate normal distributed GJR-GARCH(1,1,1) models is compared to the BEKK model and
to a multivariate EWMA model. This comparison shows that while the DCC model performs
best when it comes to minimizing the risk of a portfolio, the BEKK model is superior when
evaluated on the reward-to-variability ratio (Sharpe). This is mainly due to the fact that the
DCC model is unable to catch the time-varying correlation between the three chosen assets.

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Foreword
Writing this master thesis has been a challenging, but highly interesting process. It has given
me valuable insight and knowledge around topics that are not devoted much time in the
obligatory courses within the Finance profile at the Norwegian School of Economics and
Business Administration (NHH).

Another valuable experience is the knowledge and ability I have obtained by using Matlab. I
wish to thank Kevin Sheppard for making his Oxford MFE Toolbox1 for Matlab available at his
homepage. This is a toolbox I would recommend to anyone who wants to study time-varying
volatility.

My interest around the topic of conditional volatility modeling is due to one lecture around
this theme in the course Applied Finance. Still, I was unaware of the complexity of time-
varying volatility, the richness of alternative models to choose from, and the high amount of
research being done around it. Due to this, a large part of this thesis is built on theory and
literature review. Although the initial working title was “Testing volatility forecasting models
in a portfolio optimization framework”, it was the process of selecting the “best” univariate
volatility models that fascinated me most, something that is reflected in the final result of
this thesis.

I am grateful to Professor Richard D.F. Harris for suggesting this fascinating topic to me. I
also want to thank my thesis advisor Maytinee Wasumadee for her help and inputs while
writing this thesis. Finally I wish to express my gratitude to Hella for her invaluable support
and for motivating me throughout the whole writing process.

Øystein Skregelid

Bergen, June 2009

1 Can be found at http://www.kevinsheppard.com/wiki/MFE_Toolbox (Accessed February 12, 2009)
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Contents

Abstract ..................................................................................................................................... iii
Foreword .................................................................................................................................... v
List of figures ..............................................................................................................................ix
List of tables ...............................................................................................................................xi
1. Introduction ........................................................................................................................ 1
2. Background ......................................................................................................................... 3
2.1 Defining volatility ......................................................................................................... 5
2.2 Stationarity .................................................................................................................. 7
2.3 White noise .................................................................................................................. 8
2.4 Stylized facts ................................................................................................................ 9
2.4.1 Thick tails .............................................................................................................. 9
2.4.2 Volatility clustering ............................................................................................. 10
2.4.3 Leverage effects ................................................................................................. 10
2.5 Conditional mean models .......................................................................................... 11
2.6 Historical volatility models ........................................................................................ 12
2.7 Time-varying conditional volatility models ............................................................... 14
3. Methods ........................................................................................................................... 17
3.1 Maximum Likelihood and parameter estimation ...................................................... 17
3.2 Symmetric GARCH models ......................................................................................... 18
3.2.1 ARCH ................................................................................................................... 18
3.2.2 GARCH ................................................................................................................ 20
3.2.3 T-distributed GARCH .......................................................................................... 24
3.2.4 Integrated GARCH (IGARCH) .............................................................................. 25
3.3 Assymetric GARCH models ........................................................................................ 26
3.3.1 Exponential GARCH (EGARCH) ........................................................................... 27
3.3.2 GJR-GARCH ......................................................................................................... 28
3.3.3 Threshold GARCH (TARCH) ................................................................................. 29
3.3.4 Asymmetric Power ARCH (APARCH) .................................................................. 29
3.4 Forecasting performance of the various GARCH models .......................................... 30
3.5 Realized volatility ....................................................................................................... 31
3.6 Testing appropriateness of GARCH class modeling ................................................... 31
3.6.1 Pre estimation analysis ....................................................................................... 32
3.6.2 Post estimation analysis ..................................................................................... 35
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3.7 Evaluation of volatility forecasts ............................................................................... 36
3.8 Multivariate models .................................................................................................. 40
3.9 Portfolio optimization ................................................................................................ 45
4. Data .................................................................................................................................. 49
5. Results .............................................................................................................................. 51
5.1 Descriptive statistics .................................................................................................. 51
5.2 iShares Dow Jones US Financial sector (IYF) .............................................................. 54
5.2.1 Pre-estimation analysis ...................................................................................... 54
5.2.2 Model parameterization and selection .............................................................. 56
5.2.3 Post-estimation analysis ..................................................................................... 59
5.2.4 Student’s T distribution ...................................................................................... 61
5.2.5 Visual comparision of the time-varying volatilities ............................................ 64
5.2.6 Forecast evaluation ............................................................................................ 66
5.3 iShares Dow Jones US Utilities sector (IDU) .............................................................. 71
5.3.1 Pre estimation analysis ....................................................................................... 71
5.3.2 Model parameterization and selection .............................................................. 72
5.3.3 Post estimation analysis ..................................................................................... 75
5.3.4 Student’s T distribution ...................................................................................... 76
5.3.5 Forecast evaluation ............................................................................................ 78
5.4 iShares Dow Jones US Energy sector (IYE) ................................................................. 80
5.4.1 Pre estimation analysis ....................................................................................... 80
5.4.2 Model parameterization and selection .............................................................. 80
5.4.3 Post estimation analysis ..................................................................................... 82
5.4.4 Forecast evaluation ............................................................................................ 83
5.5 Multivariate models .................................................................................................. 84
6. Conclusions ....................................................................................................................... 93
Literature .................................................................................................................................. 95

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List of figures
Figure 3.1 Comparing ARCH(p) models with a GARCH(1,1) model for IYF. ........................................................... 24
Figure 5.1 Price development of the three assets. Initial prices set to 100. .......................................................... 51
Figure 5.2 Plots of daily raw returns scaled by 100 (left) and corresponding absolute returns (right) for IYF, IYE
and IDU respectively.............................................................................................................................................. 53
Figure 5.3 Kernel density plot (left) and QQ-plot (right) for the IYF raw returns against a normal distribution. .. 55
Figure 5.4 Autocorrelations of squared returns with heteroskedastic robust errors (left) and Engle’s ARCH test
(right). ................................................................................................................................................................... 56
Figure 5.5 Plots of AIC (left) and BIC (right) for IYF. .............................................................................................. 58
Figure 5.6 Kernel density plot (left) and QQ-plot (right) of the standardized residuals from an EGARCH(1,2,1)
model against a normal distribution. .................................................................................................................... 60
Figure 5.7 ARCH LM test of the standardized residuals (left) and autocorrelations of squared standardized
residuals (right) for EGARCH(1,2,1). ...................................................................................................................... 60
Figure 5.8 Plots of AIC (left) and BIC (right) for IYF for t-distributed errors. ......................................................... 62
Figure 5.9 Kernel density plot (left) and QQ-plot (right) of the standardized residuals from an t-distributed
EGARCH(1,2,1) model against a t(15.2). ............................................................................................................... 63
Figure 5.10 Estimated time-varying volatility using a TARCH(1,1,1) specification for both normal and t-
distributed errors................................................................................................................................................... 64
Figure 5.11 Estimated time-varying volatility using a symmetric GARCH(1,1) and an asymmetric EGARCH(1,1,1),
both t-distributed. ................................................................................................................................................. 65
Figure 5.12 Estimated time-varying volatility from RiskMetricsTM versus the normal distributed GJR(1,1,1). These
are compared to the unconditional volatility calculated at 6 month intervals. .................................................... 65
Figure 5.13 Time-varying volatility and volatility forecasts for a normal distributed EGARCH(1,1,1) process for
IYF. ......................................................................................................................................................................... 66
Figure 5.14 Time-varying volatility and volatility forecasts for a t-distributed EGARCH(1,2,1) and a normal
distributed GJR(1,1,1) process for IYF. The unconditional volatility is updated every sixth month for the five years
of collected data. ................................................................................................................................................... 67
Figure 5.15 Estimated % VaR using historical simulation, parametric and semi-parametric VaR models for IYF at
the 90% confidence level. The period Jan06 – Jan07 is the out-of-sample period. ............................................... 68
Figure 5.16 P-values from the ARCH LM tests before (left) and after (right) removing outliers. .......................... 72
Figure 5.17 Plots of AIC (left) and BIC (right) for IDU for normal distributed errors. ............................................ 74
Figure 5.18 Kernel density plot (left) and QQ-plot (right) of the standardized residuals from an GJR(1,1,1) model
for IDU against a normal distribution. .................................................................................................................. 75
Figure 5.19 Plots of AIC (left) and BIC (right) for IDU for t-distributed errors. ...................................................... 77
Figure 5.20 Kernel density plot (left) and QQ-plot (right) of the standardized residuals from a t-distributed
GJR(1,1,1) model for IDU against a t(13.4) distribution. ....................................................................................... 77
Figure 5.21 Plots of AIC (left) and BIC (right) for IYE under the assumption of normal distributed errors. ........... 81
Figure 5.22 Estimated % VaR based on a normal distributed GJR(1,1,1) model, using historical simulation,
parametric and semi-parametric VaR models for α=10%. .................................................................................... 84
Figure 5.23 Development in annualized standard deviations for the three assets given by the DCC(1,1),
RiskMetrics and BEKK estimations. ....................................................................................................................... 87
Figure 5.24 Time-varying correlations between each pair of assets given by the DCC(1,1), BEKK and RiskMetrics
models. .................................................................................................................................................................. 87
Figure 5.25 Volatility of the global minimum variance portfolio from DCC(1,1) compared to the three individual
assets given by univariate GJR(1,1,1) processes. .................................................................................................. 88
Figure 5.26 Daily optimal GMVP weights for IYF (top), IYE (middle) and IDU (bottom) given by the three
multivariate models. ............................................................................................................................................. 89

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