The majority of these studies rely on formulations within the Autoregressive Conditional. Heteroskedastic class of models. Most of them however investigate ...
Piotr Fiszeder
FORECASTING VOLATILITY WITH GARCH MODELS
1. INTRODUCTION
Volatility permeates modern financial theories and decision making process. For example, expected future volatility of financial market returns is the main ingredient in assessing asset or portfolio risk and plays a key role in derivatives pricing models. There is a large literature on volatility forecasting, but meanwhile it is difficult to extract a coherent set of prescriptions concerning the most appropriate empirical procedure for tackling this issue. The majority of these studies rely on formulations within the Autoregressive Conditional Heteroskedastic class of models. Most of them however investigate predictive ability of one or two selected specifications of GARCH models and compare it with other selected volatility forecasting methods (see Poon and Granger, 2003). The purpose of this study is to compare performance of seven GARCH models and five other methods for predicting volatility of Polish stock index WIG20. The GARCH models include: GARCH, IGARCH, GARCH-M, EGARCH, GJR, TGARCH, FIGARCH. The other models are: random walk, historical average, moving average, exponential smoothing and stochastic volatility. Different measures are employed to evaluate forecast accuracy, namely, the mean error, the mean absolute error, the root mean squared error, the heteroskedasticity adjusted mean absolute error, the heteroskedasticity adjusted root mean squared error, the logarithmic loss function, the LINEX loss function and the coefficient of determination in a regression of the ex-post realized values of variance on its forecast values. Intraday returns are also used to improve the measuring of volatility. The plan for the rest of the paper is as follows. Section 2 outlines the competing models used in the analysis. In section 3 the measures used to assess the performance of the candidate models are presented. Section 4 describes the empirical results and section 5 concludes.
2. COMPETING MODELS
Various conditional variance specifications within the parametric GARCH class of models were proposed in the literature, but there is no consensus on the relative quality of outof-sample forecasts of those formulations. In this article seven specifications of GARCH models were analysed. Linear GARCH ( p, q) model proposed by Bollerslev (1986) is defined as: ε t = z t ht ,
(1)
q
ht = αo +
∑α ε
2 i t-i
i=1
p
+
∑β h
j t -j
,
(2)
j=1
where z t is a series of independent, identically distributed random disturbances and z t ~ N (0,1) .
Nelson and Cao (1992) give necessary and sufficient conditions to ensure nonnegativity of conditional variance. For the simple GARCH(1,1) model positivity of ht requires that α 0 > 0 , α1 ≥ 0
and
β1 ≥ 0 .
The
process
is
covariance
stationary,
if,
and
only
if,
α1 + α 2 + ... + α q + β1 + β 2 + ... + β p < 1 . The GARCH model with conditionally normal errors results in a leptokurtic unconditional distribution. However, the degree of leptokurtosis induced by the time-varying conditional variance often does not capture all of the leptokurtosis present in high frequency financial data. Bollerslev (1987) suggested using a standardized t-distribution with unknown degrees of freedom (ν > 2 ) that may be estimated from the data. In many applications, especially with daily frequency financial data, the estimate for
α 1 + α 2 + ... + α q + β1 + β 2 + ... + β p in GARCH(p,q) model turns out to be very close to unity. Engle and Bollerslev (1986) were the first to consider GARCH processes with
α1 + α 2 + ... + α q + β1 + β 2 + ... + β p = 1 as a distinct class of models which they termed integrated GARCH (IGARCH). For α 0 > 0 IGARCH process is not covariance stationary, but it is strictly stationary. In the GARCH-M ( p, q ) model proposed by Engle, Lilien and Robins (1987) the conditional mean is an explicit function of the conditional variance: y t = δ ht + ε t .
(3)
The δ parameter measures the impact of the conditional variance on the excess return.
2
There are many extensions of the GARCH model, which account for asymmetric impact of news on the conditional variance. In this study three asymmetric GARCH models are explored: EGARCH, GJR and TGARCH. The EGARCH ( p, q ) model of Nelson (1991) is: q
p
[
∑ {
ln ht = α 0 +
α i θ z t −i + γ z t −i − 2 / π
] }+ ∑ β
i =1
j
ln ht − j .
(4)
j =1
The GJR ( p, q ) model of Glosten, Jagannathan and Runkle (1993) is: q
q
p
i =1
i =1
j =1
ht = α 0 + ∑α iε t2− i + ∑ ωi I t − iε t2− i + ∑ β j ht − j ,
(5) 1 if ε t -i ≤ 0, 0 if ε t -i > 0.
where α 0 > 0, α i ≥ 0, ω i ≥ 0 for i = 1,..., q , β j ≥ 0 for j = 1,..., p and I t −i = The TGARCH ( p, q ) model of Zakoian (1991) is: q
ht = α 0 +
∑ i =1
p
(α i+ ε t+−i − α i− ε t−−i ) +
βj ∑ j =1
ht − j ,
(6)
where ε t+−i = max (ε t −i ,0) , ε t−−i = min (ε t −i ,0) and α 0 > 0 , α i+ ≥ 0 , α i− ≥ 0 for i = 1, ..., q , β j ≥ 0 for j = 1, ..., p . Baillie, Bollerslev and Mikkelsen (1996) introduced FIGARCH model, which has many attractive features that seem consistent with documented long-run dependencies in absolute and squared asset returns. The FIGARCH ( p, d , q ) model is defined by
φ ( L)(1 − L) d ε t2 = α 0 + [1 − β ( L)]ν t ,
(7)
where 0 < d < 1 , φ ( L) (1 − L) d = 1 − α ( L) − β ( L) and all the roots of φ ( L) and [1 − β ( L)] lie outside the unit circle. For the FIGARCH ( p, d , q) model it is difficult to establish general conditions for positivity of ht . For instance for the FIGARCH (1, d ,1) model the inequality constraints, β 1 − d ≤ φ1 ≤ (2 − d ) / 3 and d [φ1 − (1 − d ) / 2] ≤ β 1 (φ1 − β 1 + d ) , are sufficient to ensure the non-negativity of conditional variance. It should be noted, however, that the results of Nelson and Cao (1992) suggested that weaker sufficient conditions may be obtained if desired. In this paper predictive ability of GARCH models is investigated and compared with other volatility forecasting methods. Under a random walk model, the best forecast of volatility is the latest realized volatility: σˆ t2+1 = σ t2 .
(8)
3
If the conditional expectation of volatility is assumed to be constant, the optimal forecast of future volatility would be the historical average of all past volatility realizations: σˆ t2+1 =
1 t
t
∑σ
2 i
.
(9)
i =1
In the moving average model only k latest realizations of volatility are used in estimation: t
σˆ t2+1 =
∑
1 σ i2 . k i =t −k +1
(10)
In the exponential smoothing model the forecast of volatility is a function of the immediate past forecast and the immediate past observed volatility: σˆ t2+1 = α σˆ t2 + (1 − α ) σ t2 ,
(11)
where 0 < α < 1 . The choice of the moving average estimation period (k ) in model (10) and value of smoothing parameter (α ) in model (11) are arbitrary and should be determined empirically. Performance of another sophisticated model, namely stochastic volatility model (SV) is also evaluated. A simple structure for the SV model may be written as: y t = σ exp (0,5 ht ) ε t ,
(12)
ht = φ ht −1 + η t ,
(13)
where φ ≤ 1 , ε t ,η t are series of independent, identically distributed random disturbances and ε t ~ N (0,1) , η t ~ N (0, σ η2 ) . The SV model is supposed to describe financial time series better than the ARCH-type models, since it essentially involves two noise processes.
3. EVALUATION MEASURES
It is generally impossible to specify a forecast evaluation criterion that is universally acceptable (see, e. g., Diebold, Gunther and Tay, 1998). This problem is especially acute in the context of volatility forecasting. Highly nonlinear and heteroskedastic environment may render the usual measures based on root-mean squared errors unreliable. That is why eight measures are used to evaluate the forecast accuracy, namely, the mean error (ME), the mean absolute error (MAE), the root mean squared error (RMSE), the heteroskedasticity adjusted mean absolute error (HAMAE), the heteroskedasticity adjusted root mean squared error (HARMSE), the logarithmic loss function (LL), the LINEX loss function and the R 2 in a
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regression of the ex post realized values of variance on its forecast values. They are defined by ME =
1 m 2 (σ T2 − σ Tp ), ∑ m T =1
MAE =
(14)
1 m ∑ σ T2 − σ Tp2 , m T =1
(15)
1 m 2 2 (σ T2 − σ Tp ) , ∑ m T =1
RMSE =
(16)
2
HMAE =
σ Tp 1 m 1− 2 , ∑ m T =1 σT
HRMSE =
2 1 m σ Tp ∑ 1− m T =1 σ T2
2 1 m σ Tp LL = ∑ ln m T =1 σ T2
LINEX =
(17) 2
,
(18)
,
(19)
1 m ∑ {exp [ a (σ T2 − σ Tp2 )] − a (σ T2 − σ Tp2 ) − 1} , m T =1
(20)
2 where σ T2 , σ Tp are ex-post realized volatility measure and forecast of volatility, respectively.
The coefficient of determination of
the regression of realized volatility on forecasted
volatility is used:
σ T2 = α + βσ Tp2 + ε T .
(21)
The mean error and the percentage of over-predictions and under-predictions can be used as a general guide as to the direction of over/under-prediction. The RMSE and MAE are two of the most popular measures to test the forecasting power of a model. The heteroskedasticity adjusted mean absolute error and the heteroskedasticity adjusted root mean squared error are used to better accommodate the heteroskedasticity in the forecast errors. The logarithmic loss function exaggerates the influence of low volatility scenarios and reduces the impact of outliers. Some investors will not attribute equal importance to both over- and underpredictions of volatility of similar magnitude. For example the under-estimate of volatility is more likely to be of greater concern to a writer than a buyer of a call option. The LINEX loss function is asymmetric and hence penalizes under-predictions and over-predictions differently. If a > 0 , for example, positive errors receive more weight than negative errors. This implies that an under-prediction of volatility needs to be taken into consideration more 5
seriously. Similarly, positive errors receive less weight than negative errors when a < 0 . There are other asymmetric loss functions in the literature but for the LINEX function there is analytical solution for the optimal prediction under conditional normality (see Christoffersen and Diebold, 1997). The R 2 of the regression of realized volatility on forecasted volatility is also used to measure the quality of forecasts. Although this is arguably the most commonly employed criterion in the existing literature, it is not necessarily the best practice to adopt when evaluating nonlinear volatility forecasts.
4. DATA AND FORECAST RESULTS
Several indices are available for the Warsaw Stock Exchange. The index used in this study is WIG20. It is a portfolio index of the 20 largest and most actively traded stocks listed on the main market. The sample consists of 622 daily returns over the period from January 2, 2001 to June 30, 2003. Out-of sample one-day ahead forecasts are constructed for the first half of the year 2003. The continuously compounded rates of return were calculated as rt = 100 ln ( Pt / Pt −1 ) , where Pt is the closing price of the index at date t . Since there was no
significant autocorrelation in returns, the mean equation is given by rt = µ + ε t . The initial data used for the estimation of the models parameters are drawn from the period January 2, 2001 to December 31, 2002. Thus the first day for which an out-of-sample forecast is obtained is 2nd January 2003. As the sample is extended, the models are re-estimated and sequential one-day ahead forecasts are calculated. Hence, in total, 124 daily volatilities are forecasted. Parameters of the GARCH models were estimated by quasi maximum likelihood method (except the GARCH model with conditional Student-t distribution for which maximum likelihood method was used). The choice of the GARCH orders for p and q (the lag lengths) was based on the minimization of the Bayesian information criterion (BIC). The GARCH(1,1) model has been found to be adequate in this study. Parameters of the stochastic volatility model were estimated by quasi maximum likelihood approach, that relies on a transformation of the model to a state-space form to apply the Kalman filter (see Harvey, Ruiz and Shephard, 1994). Despite its inefficiency, the QML method is consistent and very easy to implement numerically. The moving average estimation period in the moving average model and the value of smoothing parameter in the exponential smoothing model are chosen to produce the best fit by minimizing the RMSE in pre-sample.
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Two different measures of ex-post realized volatility are used in this study: squared daily return and sum of squared intraday returns. Forecast results for squared daily return used as a realized variance are presented in Tables 1-3. The forecasting methods in all tables are divided into three groups. The first one includes GARCH models, the second one contains other methods for predicting volatility. In Table 1 the percentage of over-predictions, the values of ME and the values and rankings of all competing models under the logarithmic loss function and the coefficient of determination in a regression of the ex-post realized values of variance on its forecast values are reported. All models except the random walk model overpredict volatility. The historical average model over-predicts volatility most frequently and has the largest negative sample ME. The logarithmic loss function favours the random walk model while the SV model is second best. These models provide the most accurate forecast of low volatility. Under the coefficient of determination in a regression of the ex-post realized values of variance (squared daily returns) on its forecast values the historical average model is the most accurate model, while the GJR model ranks second. The estimates of R 2 for all models are very low and never exceed 0.05. However low R 2 ’s are not an indication of poor forecasts, but rather a direct implication of standard volatility models (see Andersen and Bollerslev, 1998). Table 1 The percentage of over-predictions and the sample forecast evaluation criteria: ME, LL and
R 2 for competing
models. Squared daily return used as a realized variance Models GARCH GARCH-t IGARCH GARCH-M EGARCH GJR TGARCH FIGARCH Random walk Hist. average Mov. average Exp. smoothing SV
% of overpredictions 77.42 77.42 74.19 77.42 77.42 74.19 73.39 74.19 50.00 84.68 71.77 75.00 70.16
ME -0.559 -0.565 -0.343 -0.559 -0.876 -0.257 -0.262 -0.330 0.001 -1.395 -0.231 -0.320 -0.094
R2
LL Value 1.715 1.718 1.594 1.715 1.861 1.533 1.545 1.593 -0.016 2.101 1.524 1.588 1.437
Rank 10 11 8 9 12 4 5 7 1 13 3 6 2
Value 0.015 0.012 0.017 0.017 0.014 0.028 0.022 0.019 0.009 0.041 0.005 0.003 0.017
Rank 8 10 6 5 9 2 3 4 11 1 12 13 7
Table 2 presents the values and rankings of competing models under the MAE, RMSE, HMAE and HRMSE. All those statistics indicate that the GJR model provides the most accurate forecast. The TGARCH model ranks second (except the MAE statistic). The
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historical average model, the random walk model and EGARCH model provide the worst forecasts. According to those four error statistics the rankings of the competing models are very similar except the SV model, which ranks fourth under the MAE and barely tenth under the RMSE. Table 2 The sample forecast evaluation criteria: MAE, RMSE, HMAE and HRMSE for competing models. Squared daily return used as a realized variance Models GARCH GARCH-t IGARCH GARCH-M EGARCH GJR TGARCH FIGARCH Random walk Hist. average Mov. average Exp. Smooth. SV
MAE Value 1.661 1.667 1.547 1.660 1.835 1.497 1.504 1.538 1.972 2.193 1.501 1.553 1.505
Rank 9 10 6 8 11 1 3 5 12 13 2 7 4
RMSE Value 2.207 2.214 2.160 2.204 2.322 2.134 2.141 2.153 3.176 2.548 2.164 2.177 2.219
Rank 8 9 4 7 11 1 2 3 13 12 5 6 10
HMAE Value Rank 133.1 8 133.6 10 117.2 4 133.1 9 149.7 12 106.7 1 108.7 2 118.5 5 144.5 11 218.0 13 116.8 3 127.0 7 123.7 6
HRMSE Value Rank 854.3 7 858.6 8 745.5 3 854.2 6 922.2 11 655.2 1 682.2 2 772.2 5 1050.4 12 1569.1 13 764.5 4 882.1 9 912.9 10
Table 3 Forecasting performance of competing models under LINEX loss function. Squared daily return used as a realized variance Models
GARCH GARCH-t IGARCH GARCH-M EGARCH GJR TGARCH FIGARCH Random walk Hist. average Mov. average Exp. Smooth. SV
LINEX a = −20 Value Rank 5.170e+19 9 8.905e+19 10 1.070e+18 5 4.825e+19 8 6.029e+23 12 1.645e+17 4 2.667e+16 2 1.616e+16 1 6.993e+103 13 5.093e+23 11 2.577e+18 7 6.233e+16 3 1.902e+18 6
LINEX a = −10 Value Rank 1.852e+9 9 2.315e+9 10 2.528e+8 6 1.770e+9 8 1.634e+11 11 8.579e+7 4 3.956e+7 2 3.934e+7 1 7.510e+50 13 2.977e+11 12 2.588e+8 7 6.589e+7 3 1.848e+8 5
LINEX a = 10 Value Rank 1.212e+42 4 1.580e+42 5 9.001e+42 7 9.543e+41 3 1.842e+41 2 7.187e+42 6 3.971e+43 10 1.130e+43 8 1.846e+51 13 6.736e+38 1 2.816e+43 9 1.148e+44 11 2.412e+47 12
LINEX a = 20 Value Rank 9.641e+85 4 1.659e+86 5 5.052e+87 6 5.707e+85 3 4.103e+84 2 5.354e+87 7 1.896e+89 10 9.941e+87 8 4.227e+104 13 4.799e+79 1 6.243e+88 9 1.427e+90 11 7.208e+96 12
In Table 3 the values and rankings under four LINEX loss functions are reported ( a = −20, − 10, 10 and 20). The LINEX with a = −20 and a = −10 , which penalize overpredictions of volatility more heavily, rank the FIGARCH model first, and the TGARCH model second, while the random walk, EGARCH and historical average models provide the worst forecasts. The LINEX with positive number ( a = 10 , a = 20 ), which penalize under-
8
prediction of volatility more heavily, favours the historical average and the EGARCH models while the random walk and SV are the worst performing models. Andersen and Bollerslev (1998) show that although the daily squared return is an unbiased estimator of the realized daily volatility, it is also generally very noisy. The sum of squared intraday returns substantially reduces the noise and should therefore be used for realized volatility. The results of theoretical analyses have shown that the higher the frequency of returns used, the better the out-of-sample forecast performance (the sum of squared intraday returns better approximate true latent volatility). However in empirical studies for the very highest frequency data the effects of market microstructure biases, such as bid-ask bounces and discrete price observations are distorting. Secondly, when the intra-daily returns are correlated, realized variance will either overestimate (with negative autocorrelation) or underestimate (positive autocorrelation) the average daily return variance (see Oomen, 2001). In the actual application below, the latent volatility factor is approximated by the sum of the ex-post squared 5-minute returns. Forecast results for the sum of squared intraday returns used as a realized variance are presented in Tables 4-6. The same evaluation criteria as before are applied. The percentage of over-predictions (Table 4) indicate that all models except the random walk model over-predict volatility more frequently in comparison with the results from Table 1. Table 4 The percentage of over-predictions and the sample forecast evaluation criteria: ME, LL and
R 2 for competing
models. The sum of squared intraday returns used as a realized variance Models GARCH GARCH-t IGARCH GARCH-M EGARCH GJR TGARCH FIGARCH Random walk Hist. average Mov. average Exp. Smooth. SV
% of overpredictions 85.48 85.48 83.06 85.48 91.13 76.62 79.03 81.45 41.13 93.55 78.23 82.26 74.19
ME -0.745 -0.752 -0.530 -0.745 -1.063 -0.444 -0.449 -0.516 -0.186 -1.582 -0.417 -0.506 -0.281
R2
LL Value 0.652 0.655 0.531 0.651 0.798 0.470 0.481 0.530 -1.079 1.038 0.461 0.525 0.373
Rank 10 11 8 9 12 4 5 7 1 13 3 6 2
Value 0.068 0.066 0.072 0.069 0.065 0.060 0.049 0.062 0.032 0.080 0.031 0.040 0.012
Rank 4 5 2 3 6 8 9 7 11 1 12 10 13
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According to the ME statistic which takes into account the magnitude of forecasting errors over-prediction of volatility is higher when the sum of squared intraday returns is used for realized volatility rather than daily squared return. However all other evaluation criteria indicate substantially better out-of-sample forecast performance. Table 5 The sample forecast evaluation criteria: MAE, RMSE, HMAE and HRMSE for competing models. The sum of squared intraday returns used as a realized variance Models GARCH GARCH-t IGARCH GARCH-M EGARCH GJR TGARCH FIGARCH Random walk Hist. average Mov. average Exp. Smooth. SV
MAE Value Rank 0.972 8 0.978 10 0.820 7 0.973 9 1.215 11 0.771 2 0.776 3 0.814 6 1.378 12 1.694 13 0.780 4 0.812 5 0.752 1
RMSE Value 1.114 1.120 0.977 1.113 1.372 0.946 0.946 0.967 2.169 1.781 0.947 0.975 0.978
Rank 9 10 6 8 11 1 2 4 13 12 3 5 7
HMAE Value Rank 1.281 9 1.289 10 1.058 7 1.281 8 1.613 12 0.979 2 0.988 3 1.050 6 1.370 11 2.295 13 0.989 4 1.048 5 0.916 1
HRMSE Value Rank 1.662 9 1.674 10 1.401 7 1.662 8 2.067 11 1.313 3 1.311 2 1.371 5 2.172 12 2.849 13 1.323 4 1.384 6 1.255 1
Table 6 Forecasting performance of competing models under LINEX loss function. The sum of squared intraday returns used as a realized variance Models GARCH GARCH-t IGARCH GARCH-M EGARCH GJR TGARCH FIGARCH Random walk Hist. average Mov. average Exp. Smooth. SV
LINEX a = −20 Value Rank 2.117e+15 8 2.941e+15 10 3.862e+13 7 2.642e+15 9 3.732e+18 11 5.932e+12 5 1.020e+12 3 9.651e+11 2 1.343e+87 13 5.847e+19 12 1 4.422e+11 4 2.362e+12 8.724e+12 6
LINEX a = −10 Value Rank 7.636e+6 8 8.873e+6 10 9.728e+5 7 7.824e+6 9 4.325e+8 11 4.989e+5 5 2.318e+5 3 1.959e+5 2 3.291e+42 13 2.167e+9 12 1 1.742e+5 4 4.291e+5 6.163e+5 6
LINEX a = 10 Value Rank 8.801e+9 4 1.137e+10 5 5.396e+10 6 6.318e+9 3 1.959e+9 2 9.768e+10 7 4.409e+11 10 1.068e+11 8 1.988e+19 12 7.518e+6 1 9 1.686e+11 13 1.172+12 2.594e+15 11
LINEX a = 20 Value Rank 8.190e+21 4 1.442e+22 5 2.578e+23 6 4.022e+21 3 4.744e+20 2 7.002e+23 7 2.197e+25 10 1.052e+24 8 4.888e+40 13 5.561e+15 1 9 1.575e+24 11 1.643e+26 8.336e+32 12
The rankings of the competing models are very similar irrespective of the measure of ex-post realized volatility. The meaningful exception is the SV model which performs considerably better when the sum of squared intraday returns is used as a realized variance. It means that intraday returns should be used to evaluate the out-of-sample forecasts of
10
volatility, however if such data are not available the ranking of competing models can be based on daily data. The ranks of any forecasting model vary depending upon the choice of error statistic. The results suggest that no single model is clearly superior. This sensitivity in rankings highlights the potential hazard of selecting the best model on the basis of an arbitrarily chosen error statistic. Moreover, the choice of error measure should reflect an appropriate underlying loss function which in turn depends on the ultimate purpose of the forecasting procedure. If one should indicate the best model according to all analysed criteria it would be the GJR model. Some other important conclusions result from this analysis. All GARCH models overpredict volatility but the EGARCH model is the one which over-predicts most frequently. It results from the logarithmic construction of this model. The weak performance of the EGARCH model is evident in comparison with the GJR and the TGARCH models which also capture negative correlation between lagged returns and conditional variance. The GARCH model with conditional Student-t distribution better fits the data in sample period (according to the BIC criterion) than the GARCH model with normal distribution, but it does not provide more accurate forecasts. IGARCH model ranks better than the GARCH model in most evaluation criteria (the estimate for α 1 + β 1 in GARCH model was very close to unity). However the FIGARCH model which captures the long-run dependencies in volatility ranks even better than the IGARCH model. The GARCH-M model does not provide more accurate forecasts than the GARCH model, but it must be remembered that the relation between expected returns and conditional volatility was not significant in sample period. The historical average model considerably over-predicts volatility. The SV model provides accurate forecasts of low volatility. It is worth to emphasize a relatively good performance of the moving average model in comparison with other models (better than performance of the exponential smoothing model).
5. CONCLUSIONS
This study evaluates the performance of seven GARCH models and five other methods for predicting volatility of Polish stock index WIG20. The performance of these models is evaluated using several criteria such as symmetric conventional error statistics and asymmetric error statistics that penalize under-predictions and over-predictions differently. The ranks of any forecasting model vary depending upon the choice of error statistic. The 11
results suggest that no single model is clearly superior. Capturing some features of financial time series increases forecasting performance (like long-run dependencies in volatility), other regularities have no significant influence on it (like fat tails of conditional distributions or positive relation between expected stock returns and their conditional volatility). Evaluating these results one must remember that volatility of WIG20 stock index in the analyzed period was relatively low and stable.
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