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Unformatted text preview: ESTIMATING STOCK MARKET
VOLATILITY USING ASYMMETRIC
GARCH MODELS
Dima Alberg, Haim Shalit and
Rami Yosef
Discussion Paper No. 0610
September 2006 Monaster Center for Economic Research
BenGurion University of the Negev
P.O. Box 653
Beer Sheva, Israel
Fax: 97286472941
Tel: 97286472286 September 11, 2006 Estimating Stock Market Volatility Using
Asymmetric GARCH Models. DIMA ALBERG
Department of Economics, BenGurion University of the Negev, Beer Sheva, 84105
Israel
Email: [email protected] HAIM SHALIT
Department of Economics, BenGurion University of the Negev, Beer Sheva, 84105
Israel
Email: [email protected] RAMI YOSEF
Department of Business Administration, BenGurion University of the Negev, Beer
Sheva, 84105 Israel
Email: [email protected] Abstract
A comprehensive empirical analysis of the return and conditional variance of
Tel Aviv Stock Exchange (TASE) indices is performed using GARCH models. The
prediction performance of these conditional changing variance models is compared to
newer asymmetric GJR and APARCH models. We also quantify the dayoftheweek
effect and the leverage effect and test for asymmetric volatility. Our results show that
the EGARCH model using a skewed Studentt distribution is the most successful in
forecasting the TASE indices. Keywords: GARCH, Leverage Effect, Dayof Week Effect, Market Volatility. 2 Estimating Stock Market Volatility Using Asymmetric Changing Variance
Models
1. Introduction Volatility clustering and leptokurtosis are common observations in financial
time series (Mandelbrot (1963)). Another phenomenon often encountered is the socalled “leverage effect” (Black (1976)), which occurs when stock prices changes are
negatively correlated with changes in volatility. Observations of this type in financial
time series have led to the use of various changing variance models.
In his seminal paper, Engle (1982) proposed to model timevarying
conditional variance with AutoRegressive Conditional Heteroskedasticity (ARCH)
processes using lagged disturbances. Early empirical evidence shows that a high
ARCH order is needed to capture the dynamic behavior of conditional variance. The
Generalized ARCH (GARCH) model of Bollerslev (1986) fulfills this requirement as
it is based on an infinite ARCH specification which reduces the number of estimated
parameters from infinity to two. Both models capture volatility clustering and
leptokurtosis, but as their distribution is symmetric, they fail to model the “leverage
effect.” To address this problem, many nonlinear extensions of GARCH have been
proposed, such as the Exponential GARCH (EGARCH) model by Nelson (1991), the
socalled GJR model by Glosten, Jagannathan, and Runkle (1993) and the
Asymmetric Power ARCH (APARCH) model by Ding, Granger, and Engle (1993).
Another problem encountered when using GARCH models is that they do
not always fully embrace the thick tails property of high frequency financial timesseries. To overcome this drawback Bollerslev (1987), Baillie and Bollerslev (1989),
Kaiser (1996) and Beine, Laurent, and Lecourt (2000) have used the Student's tdistribution. Similarly to capture skewness Liu and Brorsen (1995) use an asymmetric stable density. But the variance of such a distribution rarely exists. For
modelling both skewness and kurtosis Fernandez and Steel (1998) used the skewed
Student's tdistribution that was later extended to the GARCH framework by Lambert
and Laurent (2000, 2001).
Forecasting conditional variance with asymmetric GARCH models has been
comprehensively studied by Pagan and Schwert (1990), Brailsford and Faff (1996),
Franses, Neele, and Van Dijk (1998) and Loudon, Watt, and Yadav (2000). A 3 comparison of normal density with nonnormal ones was made by Hsieh (1989),
Baillie and Bollerslev (1989), Peters (2000), Lambert and Laurent (2001).
The purpose of this paper is to characterize a volatility model by its ability to
forecast and capture commonly held stylized facts about conditional volatility, such as
persistence of volatility, mean reverting behavior, and asymmetric impacts of negative
versus positive return innovations. We investigate the forecasting performance of
GARCH, EGARCH, GJR and APARCH models together with the different density
functions: normal distribution, Student's tdistribution, and asymmetric Student's tdistribution. We also compare between symmetric and asymmetric distributions using
the three different density functions.
We forecast two major TelAviv Stock Exchange (TASE) indices: TA100 and
TA25. To compare the results, we use several standard performance measurements.
Our results suggest that one can improve overall estimation by using the asymmetric
GARCH model with fattailed densities for measuring conditional variance.
Moreover, we find that the asymmetric EGARCH model is a better predictor than the
asymmetric GARCH, GJR and APARCH models.
The paper is structured as follows. Section 2 presents the data. In Section 3,
we present the methodology and the GARCH models used in the paper. In Section 4,
we describe the estimation procedures and present the forecasting results. 4 2. Data
The data consist of 3058 daily observations of the TA251 index from
20/10/1992 to 31/5/2005 and 1911 daily observations of the TA1002 index from
02/07/1997 to31/5/2005. To estimate and forecast these indices, we created many
calibrating programs3 and use [email protected] 2.0 by Laurent and Peters (2001), a package
whose purpose is to estimate and forecast GARCH models and many of its
extensions. The code written by Doornik (1999) in the Ox programming language
provides a dialogoriented interface with features that are not available in standard
econometric software.
Parameters were estimated using the QML technique by Bollerslev and
Wooldridge (1992). The optimization algorithm used is the BroydenFletcherGoldfarbShanno (BFGS) quasiNewton method. 1
2
3 The TA25 Index is a valueweighted index of 25 stocks traded on the Tel Aviv Stock Exchange (TASE).
The TA100 Index is a value weighted index of 100 stocks traded on the TASE.
Coded with Visual Basic Applications 5 3. Methodology Early empirical evidence has shown that to capture conditional variance
dynamics one needs to select a high ARCH order. The Bollerslev (1986) Generalized
ARCH (GARCH) model, which is based on infinite ARCH specifications, allows us
to reduce the number of estimated parameters by imposing nonlinear restrictions. The
GARCH (p, q) model expresses the variance as:
q p i =1 j =1 σ t2 = w + ∑ α i ε t2−i + ∑ β jσ t2− j
Using the lag operator L, the variance becomes: σ t2 = w + α (L )ε t2 + β (L )σ t2
with α ( L ) = p q j
∑ α i L and β ( L) = ∑ β j L .
i j =1 i =1 If all the roots of the polynomial 1 − β ( L) = 0 lie outside the unit circle, we have: σ t2 = w[1 − β (L )]−1 + α (L )[1 − β (L )]−1 ε t2 .
This may be envisaged as an ARCH ( ∞ ) process since the conditional variance
depends linearly on all previous squared residuals. As such, the conditional variance
of yt4 can become larger than the unconditional variance. Then, if past realizations of ε t2 are larger than σ 2 it is given by:
σ 2 ≡ E (ε t2 ) = w
q p i =1 j =1 1 − ∑α i − ∑ β j Like ARCH, some restrictions are needed to ensure that σ t2 is positive for all
t. Bollerslev (1986) shows that imposing w > 0, α i ≥ 0 ( for i = 1, K , q ) and β j ≥ 0 ( for j = 1,K , p ) is sufficient for the conditional variance to be positive.
To capture the asymmetry observed in the data, a new class of ARCH models
was introduced: the GJR, the exponential GARCH, and the EGARCH (p,q), models : ( ) = a + ∑ (a ln σ q 2
t 0 i =1 4 Mean equation: p i ( ) z t −i + γ i z t −i ) + ∑ b j ln σ t2− j , where z t −i =
j =1 ε t −i
σ t −i y t = E ( y t Ω t −1 ) + ε t , where Ω t −1 is the information set at time t1 6 The parameters allow us to capture the asymmetric effects. For example, if γ 1 = 0 a
positive surprise ε t > 0 has the same effect on volatility than a negative
surprise ε t < 0 . The presence of a leverage effect can be investigated by testing the
hypothesis that γ 1 < 0 .
Engle's (1982) ARCH model uses the normal distribution of normalized
residuals z t . Bollerslev (1987), on the other hand, proposed a standardized Student's tdistribution with v > 2 degrees of freedom whose density is given by: D(z t ; v ) = Γ((v + 1) / 2 ) z
1 + v−2
Γ(v / 2 ) π (v − 2 ) 2
t − ( v +1)
2 , ∞ where Γ(v ) = ∫ e − x x v −1 dx is the gamma function and v is the parameter measuring the
0 tail thickness. The Student's tdistribution is symmetric around mean zero. For v > 4,
the conditional kurtosis equals 3(v − 2)/(v − 4), which exceeds the normal value of 3.
The common methodology for estimating ARCH is by maximum likelihood
assuming i.i.d. innovations. For D( zt ; v) , the loglikelihood function of {y t (θ )} for the
Student's tdistribution is given by: z 2 1T v +1
v 1 ln σ t2 + (1 + v ) ln1 + t ,
LT ({y t };θ ) = T ln Γ − ln − ln (π (v − 2 )) − ∑ 2 v − 2 2
2 2
t =1 where θ is the vector of parameters to be estimated for the conditional mean, the () conditional variance and, the density function. When v → ∞ we have a normal
distribution, so that the lower v is, the fatter are the tails. Recently, Lambert and
Laurent (2000, 2001) extended the skewed Student's tdistribution proposed by
Fernandez and Steel (1998) to the GARCH framework. Using D( zt ; v) , the loglikelihood function of {y t (θ )} for the skewed Student's tdistribution is given by: ln Γ v + 1 − ln v − 1 ln (π (v − 2 )) + ln 2 + ln(s )
LT ({y t };θ ) = T 1
2
2 2 ξ + ξ − sz + m 1T
∑ ln σ t2 + (1 + v ) ln1 + vt − 2 ξ − It 2 t =1 () , 7 where ξ is the asymmetry parameter, v the degree of freedom of the distribution and v + 1
m Γ v−2
1, if z t ≥ − s 1
1 2 ξ − and s = ξ 2 + 2 − 1 − m 2
It = , m= ξ
ξ
v − 1 if z < − m
π Γ t 2
s (See Lambert and Laurent (2001) for more details.)
Maximum likelihood estimates of parameters are usually obtained using the
BFGS numerical maximization procedure. In our work instead, we use the quasimaximum likelihood estimator (QMLE). According to Bollerslev and Wooldridge
(1992) this estimator is generally consistent, has a normal limiting distribution, and
provides asymptotic standard errors that are valid under nonnormality. 8 4. Estimation Results 4.1 Descriptive Statistics and the Stationarity Constraint
To obtain a stationary series, we use returns rt = 100(log(Pt ) − log(Pt −1 ))
where Pt is the closing value of the index at date t. The samples for TA25 and TA100
have means of 0.0433 and 0.0452; standard deviations of 1.4895 and 1.3198;
skewness of −0.2106 and 0.4546; and kurtosis of 3.4024 and 5.0898. The sample
kurtosis is greater than 3, meaning that return distributions have excess kurtosis for
both indices. Excess skewness is also observed, leading to high JarqueBera statistics
indicating nonnormality. Table 1: Descriptive Statistics for Logarithm Differences 100 ⋅ [ln (Pt ) − ln (Pt −1 )]
Index Average Min. Max. Std. Dev. Kurtosis Skewness Jarque Bera Stat. TA25 0.0433 10.1555 7.1408 1.4895 3.4024 0.2106 43.24 TA100 0.0452 10.3816 7.6922 1.3198 5.0898 0.4546 413.56 As daily stock returns may be correlated with the dayoftheweek effect, we
avoid the potential calendar effect on the volatility analysis by filtering the daily
means and variances using the following two regressions:
(1) rt = α 1 SUN t + α 2 MON t + α 3TUE t + α 4WEDt + α 5THU t + δ t , (2) ˆ
(rt − rt )2 = β1SUNt + β 2 MONt + β3TUEt + β 4WEDt + β5THUt + ε t , where SUNt, MONt, TUEt ,WEDt and THU t are the dummy variables for Sunday,
ˆ
Monday, Tuesday, Wednesday and Thursday; and rt is the ordinary least squares
(OLS) fitted value of rt from regression (1) at date t. Table 2: Regression Coefficients for DayoftheWeek Effect – TA25
Regression
Mean Sunday S.E.
Variance
S.E. (2) Tuesday Wednesday Thursday 0.12** 0.020 0.041 0.0395 0.074 0.06 (1) Monday 0.06 0.06 0.06 0.06 3.41** 1.45** 2.19** 1.91** 2.12** 0.205 0.205 0.207 0.208 0.207 9 Table 3: Regression Coefficients for DayoftheWeek Effect – TA100
Regression Sunday Monday Tuesday Wednesday Thursday Mean 0.178** 0.089 0.040 0.097 0.090 S.E. 0.0663 0.0663 0.0673 0.0673 0.0671 Variance (2) 2.732** 0.878** 1.658** 1.757** 1.626** S.E. 0.231 0.231 0.235 0.235 0.235 (1) The OLS estimates of the two regressions on Tables 2 and 3 show that the
TA25 and TA100 indices have significantly positive daily means on Sunday and
significant daily variations for Sunday through Thursday5.
To eliminate the daily effects, we “standardize” the daily returns using
ˆ
ˆ
ˆ
ˆ
yt = (rt − rt ) / η t , where η t is the fitted value of (rt − rt )2 from regression (2) at
date t. We now substitute the original daily return rt with yt and referred to it as the
daily return at date t. Table 4: Descriptive Statistics for "Standardized" Returns yt
Index Average Min. Max. Std.Dev. Kurtosis Skewness Jarque Bera Stat. TA25 0 6.7131 5.30431 0.9999 3.1528 0.2179 27.17 TA100 0 8.0305 5.8756 1.00026 4.569 0.4561 262.27 Returns yt are thus normalized to zero mean and unit variance. The sample skewness and kurtosis of yt’s are 0.2179 and 3.1528;0.4561 and 4.569 for the two
indices. 4.2 Choosing a Volatility Model
For the TA25 index, convergence could not be reached with the EGARCH
model and a Student's tdistribution. Therefore we turn to the other three models
where all asymmetric coefficients are significant at standard levels. Moreover, the
Akaike information criteria (AIC) and the loglikelihood values indicate that the
EGARCH, APARCH or GJR models better estimate the series than traditional
GARCH.
These models are estimated by the approximate quasimaximum likelihood
estimator assuming normal, Studentt or skewed Studentt errors. Note that it is quite
5 * and **  means significance at 5% and 1% levels, respectively. 10 evident that the recursive evaluation of maximum likelihood is conditional on
unobserved values and therefore the estimation cannot be considered to be perfectly
exact. To solve the problem of unobserved values, we set these quantities to their
unconditional expected values.
When we analyzed the densities we found that the two Student's tdistributions
(symmetric and skewed) clearly outperform the normal distribution. Indeed, the loglikelihood function increases when using the skewed Student's tdistribution, leading
to AIC criteria of 2.701 and 2.730 for the normal density versus 2.665 and 2.697 for
the non normal densities, for the TA 100 and the TA 25 respectively. Table 5: Comparison between the Models for the TA1006
Normal Student's t Skewed t APARCH APARCH EGARCH Q(20) 20.739 20.731 20.313 Q2(20) 24.051 27.762 24.630 P(50) 72.909 48.472 46.326 Prob[1] 0.015 0.494 0.582 Prob[2] 0.002 0.168 0.196 AIC 2.730 2.697 2.697 LogLik 2600.690 2568.060 2566.940 TA100 The skewed Student's tdistribution shows results that are superior to the
symmetric Studentt distribution when modeling the TA 25 and TA100. A possible
explanation for this result is that, if skewness is significant in both series, its
magnitude will be inferior in both indices. It may therefore be necessary to add two
asymmetric parameters (asymmetric GARCH + asymmetric distribution).
6 In Tables 5 and 6 Q(20) and Q2(20) are respectively the BoxPierce statistics at lag 20 of the standardized and squared standardized residuals. P(50) is the Pearson goodnessoffit with 50 cells.
AIC is the Akaike information criterion. LogLik is the loglikelihood value. 11 Table 6: Comparison between the Models for the TA25
Normal Student's t Skewed t TA25
GJR APARCH GJR APARCH EGARCH Q(20) 32.290 32.363 32.080 32.157 32.689 Q2(20) 14.875 17.177 18.705 26.229 24.349 P(50) 58.710 65.806 46.055 50.241 45.564 Prob[1] 0.161 0.055 0.593 0.424 0.613 Prob[2] 0.045 0.008 0.271 0.129 0.218 AIC 2.701 2.699 2.669 2.665 2.665 LogLik 4122.670 4118.770 4073.290 4065.650 4065.430 All the models describing the dynamics of the first two moments of the series
are shown by BoxPierce statistics for residuals and squared residuals. All are nonsignificant at the 5% level. The stationary constraints are observed for every model
and for every density. The values (ranging from 0.831 to 0.982) suggest long
persistence of the volatility for the indices. 12 4.3 Forecasting The forecasting ability of GARCH models has been comprehensively
discussed by Poon and Granger (2001). However Andersen and Bollerslev (1997)
pointed out that the squared daily returns may not be the proper measure to assess the
forecasting performance of the different GARCH models for conditional variance.
Thus, we consider the following five measures to assess forecasting ability:
1. Mean squared error (MSE):
MSE = 1 S +h 2
∑ σˆ t − σ t2
h + 1 t =S ( 2 ) 2. Median squared error (MedSE): ( ˆ
MedSE = Inv( f Med (et )), where et = σ t2 − σ t2 ) 2 and t ∈ [S , S + h ] 3. Mean absolute error (MAE):
MAE = 1 S +h 2
∑ σˆ t − σ t2
h + 1 t =S 4. Adjusted mean absolute percentage error (AMAPE): AMAPE = ˆ
1 S + h σ t2 − σ t2
∑ σˆ 2 + σ 2 ,
h + 1 t =S t
t ˆ
where h is the number of lead steps, S is the sample size, σ t2 is the forecasted
variance and σ t2 is the “actual” variance.
5. Theil's inequality coefficient (TIC)7: TIC = 1 S +h
ˆ
∑ ( y t − y t )2
h + 1 t=S
1 S +h
1 S +h
ˆ
∑ ( y t )2 − h + 1 ∑ ( y t )2
h + 1 t=S
t=S The forecasting ability is reported by ranking the different models with respect
to the five measures. This is done in tables 7 and 8 where we compare the
distributions for the TA 25 and TA 100 indexes. 7 The Theil inequality coefficient is a scale invariant measure that lies always between zero and one, where zero indicates a perfect fit. 13 Table 7: Forecasting Analysis for the TA25 Index: Comparing between Densities
GARCH EGARCH GJR APARCH TA25
Studentt Skewedt Skewedt Studentt Skewedt Skewedt MSE(1) 0.187 0.187 0.188 0.187 0.187 0.188 MSE(2) 0.344 0.343 0.269 0.407 0.415 0.347 MedSE(1) 0.029 0.029 0.033 0.033 0.033 0.033 MedSE(2) 0.223 0.227 0.128 0.309 0.316 0.224 MAE(1) 0.274 0.274 0.272 0.273 0.272 0.272 MAE(2) 0.500 0.499 0.397 0.568 0.575 0.501 RMSE(1) 0.432 0.432 0.433 0.433 0.433 0.434 RMSE(2) 0.587 0.586 0.519 0.638 0.644 0.589 AMAPE(2) 0.782 0.782 0.758 0.795 0.796 0.781 TIC(1) 0.938 0.944 0.983 0.968 0.976 0.981 TIC(2)
0.559
0.559
0.565
(1) Mean Equation, (2)Variance Equation 0.565 0.566 0.560 For the TA 25 index shown on Table 7 the results support the use of the
asymmetric EGARCH model. For most measures in the variance equation, the
EGARCH model outperforms the APARCH model. The GARCH model provides
much less satisfactory results and the GJR model provides the poorest forecasts.
For the TA 100 index shown on Table 8, the EGARCH model gives better
forecasts than the GARCH model while the APARCH and GJR models give the
poorest forecasts. The skewed Studentt distribution is the most successful in forecasting the TA100 conditional variance, contrary to the TA25 where the results
conflict. Therefore we were unable to draw a general conclusion. The skewed
Studentt distribution seems to be the best for forecasting series showing higher
skewness. In fact Lambert and Laurent (2001) found that the skewed Studentt density
is more appropriate for modeling the NASDAQ index than symmetric densities.
14 Table 8: Forecasting Analysis for the TA100 Index: Comparing between Densities
GARCH EGARCH GJR APARCH Skewedt Skewedt Studentt Studentt MSE(1) 0.218 0.220 0.219 0.220 MSE(2) 0.366 0.275 0.444 0.393 MedSE(1) 0.093 0.092 0.093 0.092 MedSE(2) 0.376 0.292 0.466 0.409 MAE(1) 0.379 0.381 0.38 0.381 MAE(2) 0.562 0.480 0.624 0.586 RMSE(1) 0.466 0.469 0.468 0.469 RMSE(2) 0.605 0.525 0.666 0.627 AMAPE(2) 0.648 0.621 0.664 0.654 TIC(1) 0.954 0.969 0.965 0.970 TIC(2)
0.538
0.561
0.509
(1) Mean Equation, (2)Variance Equation 0.547 TA100 5. Conclusion We compared the forecasting performance of several GARCH models using
different distributions for two Tel Aviv stock index returns. We found that the
EGARCH skewed Studentt model is the most promising for characterizing the
dynamic behavior of these returns as it reflects their underlying process in terms of
serial correlation, asymmetric volatility clustering, and leptokurtic innovation. The
results also show that asymmetric GARCH models improve the forecasting
performance. Among the tested models, the EGARCH skewed Studentt model
outperformed GARGH, GJR and APARCH models. This result further implies that
the EGARCH model might be more useful than the other three models when applying
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