Additive outliers, GARCH and forecasting volatility

International Journal of Forecasting 15 (1999) 1–9

Additive outliers, GARCH and forecasting volatility Philip Hans Franses*, Hendrik Ghijsels Econometric Institute, Erasmus University Rotterdam, P.O.Box 1738, NL-3000 DR Rotterdam, The Netherlands

Abstract The Generalized Autoregressive Conditional Heteroskedasticity [GARCH] model is often used for forecasting stock market volatility. It is frequently found, however, that estimated residuals from GARCH models have excess kurtosis, even when one allows for conditional t-distributed errors. In this paper we examine if this feature can be due to neglected additive outliers [AOs], where we focus on the out-of-sample forecasting properties of GARCH models for AO-corrected returns. We find that models for AO-corrected data yield substantial improvement over GARCH and GARCH-t models for the original returns, and that this improvement holds for various samples, two forecast evaluation criteria and four stock markets.  1999 Elsevier Science B.V. All rights reserved. Keywords: GARCH; Additive outlier; Forecasting volatility

1. Introduction A key property of financial time series is that large (small) absolute returns tend to be followed by large (small) absolute returns, that is, there are periods which display high (low) volatility. As volatility is often perceived as a measure of risk, one is of course interested in forecasting the volatility. A model that is frequently used for this purpose is the Generalized Autoregressive Conditional Heteroskedasticity [GARCH] model introduced in Engle (1982) and Bollerslev (1986). An important feature of a GARCH model is that it can be fitted to data which have excess kurtosis. When the standard GARCH model is extended in order to allow for conditionally t-distributed errors [GARCH-t], see Bollerslev (1987), there are even additional opportunities to capture the typical fea*Corresponding author. Tel.: 131-10-408-1273; fax: 131-10452-7746; e-mail: [email protected]

tures of financial returns. Despite these qualities of the GARCH model, it is frequently observed that estimated residuals from such a model still have excess kurtosis, see, for example, Baillie and Boller¨ slev (1989) and Terasvirta (1996). One possible cause for this outcome is that certain observations on returns are so-called Additive Outliers [AOs], which are not captured by a standard GARCH model. Note that neglecting AOs also leads to biased parameter estimates in conditional mean equations, see for example Fox (1972), and also to biased out-ofsample forecasts, see for example Ledolter (1989). Hence, neglecting AOs in financial returns can lead to biased volatility forecasts. It is the purpose of the present paper to advocate a method to detect AOs in GARCH models, and to reduce the impact of the AOs on parameter estimates and forecasts. Essentially, our method is a modification of the Chen and Liu (1993) method to the GARCH equation. We focus on the consequences of our AO-correction method for the quality of out-of-sample forecasts, when these

0169-2070 / 99 / $ – see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S0169-2070( 98 )00053-3

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P.H. Franses, H. Ghijsels / International Journal of Forecasting 15 (1999) 1 – 9

are compared with the forecasts from GARCH and GARCH-t models in which no account is taken of AOs. The outline of our paper is as follows. In Section 2 we discuss the GARCH model and implement the Chen and Liu (1993) method for AOs. In principle, one can extend our approach to so-called innovative outliers and level shifts, but in order to save space this is not pursued here. Additionally, we confine our analysis to the GARCH(1, 1) model, also since it is often used in practice. In Section 3 we first apply our approach to forecasting the volatility of weekly Amsterdam EOE index returns [AEX]. Our results show that the forecasting performance of a GARCH model can be improved substantially when the impact of AOs is reduced, and that the GARCH model for AO-corrected returns also beats the GARCH-t model. Furthermore, we find that this improvement extends to three other European stock markets. In Section 4, we conclude our paper with some remarks.

y t 5 y t* 1 v It (t ),

(2)

where It (t ) 5 1 It (t ) 5 0

t 5t t ± t,

and y t* is given by (1). Hence, one observes y t instead of y t* . To save space, we analyze the case of a single AO, although the empirical method below can deal with multiple outliers in iterative steps. In case an ARMA( p, q) model is fitted to y t in (2), one obtains the estimated residuals

´ˆ t 5 p (L)yt ,

(3)

where p (L)5 f (L) /u (L)5(12 p1 L2 p2 L 2 2 . . . ). For the moment we assume that the values of the parameters, as well as the orders p and q, are known. For the AO in (2), the equation in (3) amounts to

´ˆ t 5 ´t 1 vp (L)It (t ).

(4)

The expression in (4) can be viewed as a regression model for ´ˆ t , i.e.,

´ˆ t 5 v x t 1 ´t , 2. Additive outliers in the GARCH model

with

In this section we briefly discuss the Chen and Liu (1993) method to deal with additive outliers in an ARMA time series model. In the second part of this section we propose to apply this method to the GARCH model.

xt 5 0 xt 5 1 x t 1k 5 2 pk

2.1. Additive outliers in ARMA models

vˆ (t ) 5

O ´ˆ x YO x . n

Consider a univariate time series y *t , t 5 1, 2, . . . ,n, which can be described by the ARMA( p, q) model fp (L)y t* 5 u (L)´t , i.e., (1 2 f1 L 2 ? ? ? 2 fp L p)y t* (1)

where L is the lag operator defined by L j x t ;x t 2j , and where ´t is a standard zero-mean white noise process with variance s 2 . For notational convenience, we assume that a mean m has been subtracted from y t* . Furthermore, we assume that the roots of fp (z) and uq (z) are outside the unit circle. An additive outlier model can be represented by

t ,t t 5 t and t . t and k 5 1, 2, . . .

The impact v of the AO at time t5t can then be estimated as

t 5t

5 (1 1 u1 L 1 ? ? ? 1 uq L q)´t ,

(5)

n

t t

t5t

2 t

(6)

Chang et al. (1988) suggest to standardize vˆ (t ) such that one can test for the significance of an AO. For this standardization we need an estimate of the variance of the residual process. Preferably, this estimate should not be biased too much because of outliers. Chen and Liu (1993) suggest three methods to estimate a robust error variance. In the empirical section below we will use the so-called ‘omit-one’ method for computational convenience. This method calculates the error variance from the sample where the observation at t5t has been deleted. Denoting the estimated error variance, based on the ‘omit-one’ method, as sˆ a , we can construct the standardized statistic

P.H. Franses, H. Ghijsels / International Journal of Forecasting 15 (1999) 1 – 9

Y FOx G n

tˆ 5 [vˆ (t ) / sˆ a ]

t5t

1/2

2 t

.

(7)

When tˆ exceeds some value C, the impact of the AO is said to be significant. Since tˆ is asymptotically standard normal, the value of C can be fixed at the usual level. However, based on extensive simulations, Chen and Liu (1993) advocate setting C equal to 4. In case tˆ is large and in excess of C, one can adjust the observation y t at t5t to obtain the AOcorrected y *t via (2) using the vˆ obtained from (6), that is y *t 5 y t 2 vˆ It (t ). In case of more than one AO, one can repeat this procedure until any tˆ statistic becomes insignificant. In a final step one can reestimate the parameters for all observations, where some of these have been corrected for AOs. We refer to Chen and Liu (1993) for further details of their method.

2.2. Additive outliers in a GARCH model In this paper we propose to use the Chen and Liu (1993) method for the GARCH equation in order to examine opportunities to improve forecasts of volatility. Consider the returns series r t , which is defined by r t 5log pt -log pt 21 , where pt is the stock price or the stock market index, and consider the GARCH(1, 1) model r t 5 ht h 1t / 2 ,

(8)

h t 5 a0 1 a1 r 2t 21 1 b1 h t 21 .

(9)

When ht is assumed to be distributed as standard normal, we will refer to (8)–(9) as the GARCH model, and when ht is conditionally t-distributed, we refer to the model as GARCH-t, see Bollerslev (1986), (1987). In this paper we focus on modifying the, say, standard normal GARCH model by correcting for AOs in ht . The equation in (9) concerns conditional volatility of the returns, and this model is frequently used for out-of-sample forecasting of volatility. The GARCH equation can be reformulated as 2 t

r 5 a0 1 (a1 1 b1 )r

2 t 21

1 z t 2 b1 z t 21 ,

(10)

under the assumption that z t 5r 2t 2h t . The expression in (10) corresponds to an ARMA(1, 1) model for r 2t ,

3

see Bollerslev (1986). Notice that the z t series, which may be viewed as a residual series, is heterogeneous. The application of the Chen and Liu (1993) method to the GARCH model, which exploits the similarities of (10) with an ARMA model as in (1), proceeds as follows.

2.2.1. Step 1 Estimate the parameters of (8)–(9) using maximum likelihood. This gives aˆ 0 , aˆ 1 , bˆ 1 and the estimated time series hˆ t and hˆ t . Next, construct 2 zˆ t 5 r t 2 hˆ t , which will be used in the second step of examining the presence of AOs. 2.2.2. Step 2 In the notation of (3), we have pˆ (L) 5 (1 2 ( aˆ 1 1 bˆ 1 )L) /(1 2 bˆ 1 L). For each t5t, perform regression (5) for the estimated residuals zˆ t , and calculate vˆ z (t ). Although the zˆ t series is heterogeneous, we calculate the unconditional ‘omit-one’ variance of zˆ t for the sample at hand. As noted above, this method calculates the error variance from the sample where the observation at t5t has been deleted. To allow for time-dependent variation of zˆ t , we recommend estimating this variance for every new sample, that is, in our applications below, we will calculate this variance for every moving window sample (and hence not fix for the entire sample). The tˆ statistics in (7) are calculated for zˆ t , and these are compared with C54. Other choices for C may be entertained, but we find that C54 yields quite favorable results. 2.2.3. Step 3 The observation on zˆ t at t5t with the largest value of the tˆ statistic, which should exceed C54, is replaced by zˆ *t similar to (2), and using the estimated weight using (6). 2.2.4. Step 4 With zˆ *t and hˆ t , construct r *t 2 via r *t 2 5zˆ *t 1hˆ t at time t5t. Furthermore, construct the AO-corrected returns as r *t 5r t for t±t and r t* 5 sign(r t ).(r *t 2 )1 / 2

for t 5 t.

(11)

This expression shows that the sign of r t* equals that of r t at t5t, and hence that there is no sign reversal for AO-corrected returns.

P.H. Franses, H. Ghijsels / International Journal of Forecasting 15 (1999) 1 – 9

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2.2.5. Step 5 Go back to Step 1 with the r t* series, and repeat all steps until no tˆ test statistic values exceed C54, i.e., until there appear to be no more AOs in the data. In the end, one has aˆ 0* , aˆ 1* , bˆ 1* and r t* and hˆ *t . With these, one can generate one-step ahead forecasts for hˆ *n11 at time n using (9). In case r t in (8) is correlated with its own past, one should first filter out these dynamics using for example an ARMA-filter. This yields the residuals ´t which are then to be considered in Step 1. The correction factor v in (2) is however directly applied to r t (and not to ´t ), because this allows for the possibility that the ARMA structure for the returns can be different after correction for AOs.

3. Empirical results In this section we consider the GARCH model for r t , the GARCH-t model for r t and the GARCH model for AO-corrected returns r t* . These models are evaluated for weekly stock market returns (Wednesday to Wednesday) for the period 1983 to 1994. The stock markets are those in The Netherlands

(AEX), Germany (DAX), Spain (MADR) and Italy (MIL). Our empirical procedure is as follows. We estimate the parameters in the models for samples which span four years. We generate a one-step ahead forecast for h t (and hence not for r t ) Next, we delete the first week from the sample and add a week at the end of the sample, and we estimate the parameters and generate a one-step ahead forecast again. Our AOcorrection method is applied time and time again, and therefore, observations may be considered an AO in one sample but not in another sample. For illustrative purposes we first consider the AEX, while we consider the other stock markets later. Also, we first discuss the estimation results for the AEX, after which we address out-of-sample forecasting of volatility. Estimation and diagnostic test results for the GARCH model with the assumption that ht is distributed as standard normal are given in Table 1 for selected samples of AEX returns. The parameters in the model are estimated using the maximum likelihood procedure, as described in Engle (1982) and Bollerslev (1986). The diagnostic results in the last two columns of

Table 1 Estimation results for the AR( p)–GARCH(1, 1) model for AEX returns: r t 2 m 2 f2 r t 22 5 ´t 5ht h 1t / 2 , ht |N(0, 1), h t 5 a0 1 a1 ´ 2t 21 1 b1 h t 21 (t-ratios are given in parentheses) Sample

m 1983–1986 1984–1987 1985–1988 1986–1989 1987–1990 1988–1991 1989–1992 1990–1993 a

Diagnostics b

Parameters

f2 a

0.0040 (2.50) 0.0030 (1.85) 0.0039 a (2.53) 0.0034 a (1.98) 0.0039 a (2.19) 0.0024 (1.55) 0.0011 (0.85) 0.0014 (1.06)

a0

a1 a

0.192 a (2.75)

0.190 a (2.43)

0.0001 (2.04) 0.0002 a (2.79) 0.0001 (1.82) 0.0002 (1.82) 0.0003 a (3.21) 0.0003 a (3.44) 0.0001 a (2.08) 0.0001 a (2.17)

0.170 (1.72) 0.432 a (3.21) 0.299 a (2.66) 0.296 a (2.36) 0.483 a (2.53) 0.224 (1.66) 0.112 (1.51) 0.186 (1.83)

b1

Q 1 (10) a

0.681 (5.84) 0.332 a (2.31) 0.606 a (4.61) 0.436 (1.87) 0.291 a (1.98) 0.149 (1.76) 0.725 a (5.69) 0.616 a (4.57)

Q 2 (10)

5.69

4.85

6.41

4.23

3.45

5.46

2.89

8.27

7.07

5.07

1.51

10.39

4.59

3.01

9.14

2.54

Significant at the 5% level. 2 The diagnostics concern tests for residual autocorrelation up to lag 10 in hˆ t denoted by Q 1 (10), and in hˆ t denoted by Q 2 (10).

b

P.H. Franses, H. Ghijsels / International Journal of Forecasting 15 (1999) 1 – 9

Table 1 indicate that the GARCH models for the various samples seem adequate. The next step is to apply our AO-correction method to the GARCH models in Table 1. In Table 2 we review the number of observations that need to be corrected for AOs for selected samples of data. Clearly, for samples which include 1987 we need to correct more negative returns, where the weeks around October 19, 1987, appear to need most correction. Hence, with our method we are inclined

Table 2 Number of adjusted (positive and negative) returns for the AEX Estimation sample

Number of adjusted returns

1983–1986 1984–1987 1985–1988 1986–1989 1987–1990 1988–1991 1989–1992 1990–1993

Positive

Negative

5 2 5 5 5 3 7 0

3 2 11 12 15 10 9 6

5

to assign the AO-status to the world-wide stock market crash in October 1987. In Table 3 we present some estimation results for the GARCH model for AO-corrected returns r t* . Compared with the estimated parameters in Table 1, one can observe that the estimates for a1 and b1 in Table 3 can differ substantially. For example, for 1987–1990 the value of a1 drops from 0.483 to 0.125 when we take care of AOs, while the b1 value increases from 0.291 to 0.744. Notice that usually the sum aˆ 1 1 bˆ 1 does not seem to change much, i.e., the persistence of shocks seems robust to AOs. Finally, comparing Table 3 with Table 1, it can be seen that outlier removal has led to much better intertemporal stability in the parameter estimates. To illustrate the effects on the returns of correcting for AOs, consider Table 4 where we present a few selected observations on the unadjusted series r t and the adjusted series r t* for 1990. The parameters in the GARCH models are estimated for four selected samples. The observations in 1990 are contained in the four selected samples, and hence we can examine the adjustment of r t across samples. For example, the observation on January 7, 1990, which is 24.23, is

Table 3 Estimation results for AR( p)–GARCH(1, 1) model for the AEX: r *t 2 m 2 f2 r *t 22 5 ´t 5ht h 1t / 2 , ht |N(0, 1), h t 5 a0 1 a1 ´ 2t 21 1 b1 h t 21 where r t* is the AO-corrected returns series (t-ratios are given in parentheses) Sample

m 1983–1986 1984–1987 1985–1988 1986–1989 1987–1990 1988–1991 1989–1992 1990–1993

Diagnostics a

Parameters

0.0041* (2.88) 0.0020 (1.27) 0.0034* (2.58) 0.0028 (1.99) 0.0024 (1.59) 0.0037* (2.98) 0.0010 (0.99) 0.0021 (1.76)

f2

0.106* (1.96)

0.270* (2.70)

a0

a1

b1

Q 1 (10)

Q 2 (10)

0.0001 (1.63) 0.0001 (1.74) 0.0001 (1.48) 0.0002 (1.60) 0.0001 (1.90) 0.0002* (2.29) 0.0005 (1.93) 0.0001* (2.06)

0.110 (1.58) 0.178* (1.95) 0.157 (1.67) 0.128 (1.71) 0.125* (2.08) 0.203 (1.69) 0.219* (2.25) 0.213* (2.17)

0.780* (8.10) 0.591* (3.18) 0.672* (3.75) 0.294 (1.76) 0.744* (7.24) 0.143 (1.52) 0.592* (3.83) 0.571* (3.88)

5.43

4.68

7.22

3.87

4.13

9.51

4.28

2.55

9.02

6.03

4.34

5.62

4.75

4.52

9.14

2.54

*Significant at the 5% level. a 2 The diagnostics concern tests for residual autocorrelation up to lag 10 in hˆ t denoted by Q 1 (10), and in hˆ t denoted by Q 2 (10).

P.H. Franses, H. Ghijsels / International Journal of Forecasting 15 (1999) 1 – 9

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Table 4 A few AO-corrected AEX returns r t* for 1990, based on GARCH models for four different samples Date a

January 7 March 14 August 8 August 22 September 19 September 26

Unadjusted return

Adjusted returns

24.23 4.11 29.31 25.87 24.80 25.14

1987–1990

1988–1991

1989–1992

1990–1993

24.23 4.11 22.17 21.57 22.00 22.39

21.74 4.11 21.79 21.54 21.82 22.16

22.04 1.75 22.32 21.92 22.13 22.71

24.23 4.11 22.21 24.61 24.80 22.20

The numbers in this table are percentage points. a The dates correspond to Wednesdays in 1990.

adjusted towards 21.74 and 22.04 when the GARCH model is estimated for the samples 1988– 1991 and 1989–1992, respectively, while it is not adjusted when the samples 1987–1990 and 1990– 1993 are used. On the other hand, the r t value at August 8, 1990 is corrected using all four samples. Notice that this observation corresponds to the week of the Iraq invasion of Kuwait, and hence that our method assigns the AO-status to this extraordinary event. From the results in Table 3 we observed that the AO-adjusted returns seem adequately described by a GARCH model. An interesting issue is now whether the ht process in (8) may be assumed from the outset to be distributed as standard normal. To obtain a tentative impression, we calculate the

usual statistics reflecting the distributional properties of the unadjusted and adjusted returns, and we present these in Table 5. Again, to save space, we only consider specific samples. For skewness, we find that the AO-adjusted returns are not skewed, as opposed to the unadjusted data. For kurtosis, we find that outlier correction results in a thin-tailed distribution, i.e., the kurtosis is below 3, which is in marked contrast with the thick tails for the unadjusted returns r t . Finally, the results for the normality test statistic in the last two columns of Table 5 indicate that normality can generally not be rejected for the AO-adjusted data, while normality is generally convincingly rejected for the unadjusted returns. In sum, adjustment for a few AOs apparently yields normality of the AEX returns, while the

Table 5 Distributional properties of the original data r t and the AO-corrected data r t* for the AEX Sample

1983–1986 1984–1987 1985–1988 1986–1989 1987–1990 1988–1991 1989–1992 1990–1993

Skewness a

Kurtosis b

Normality c

rt

r *t

rt

r *t

rt

r *t

0.05 21.05* 21.26* 21.25* 21.34* 20.64* 20.70* 20.56*

0.06 20.14 0.07 20.01 20.22 0.01 20.10 0.01

3.16 8.07* 9.72* 9.42* 9.17* 4.13* 4.95* 4.42*

2.14* 2.94 2.33* 2.19* 2.33* 2.26* 2.20* 2.96

0.29 263.4* 4479* 414.0* 392.73* 25.53* 49.92* 28.42*

6.53* 0.70 3.98 5.77 5.53 1.70 5.91 0.02

*Significant at the 5% level. a 11 / 2 21 n i The Skewness measure is calculated as m 3 /m 2 , where m i 5 n o t 51 (b t 2 mˆ ) and mˆ is the estimated mean of b t for b t 5r t or r *t . significance is tested using SK5(n / 6)(m 23 /m 32 ), which is x 2 (1) distributed under the null hypothesis of zero skewness (and autocorrelation). b 2 21 n i The Kurtosis measure is calculated as m 4 /m 2 , where m i 5 n o t 51 (b t 2 mˆ ) and mˆ is the estimated mean of b t for b t 5r t or r t* . 2 2 2 significance is tested using K5(n / 24)((m 4 /m 2 )23) , which is x (1) distributed under the null hypothesis of a kurtosis of 3 (and autocorrelation). c The Normality test is given by SK 1K, which is x 2 (2) distributed under the null hypothesis of normality (and no autocorrelation).

Its no Its no

P.H. Franses, H. Ghijsels / International Journal of Forecasting 15 (1999) 1 – 9

property of GARCH seems to be preserved, according to Table 3.

correct for the possibility that huge misses (because of outliers in the holdout sample) lead to spurious inference using the MSE only. Each measure is calculated for 52 weekly one-step ahead forecasts. The results in Table 6 yield overwhelming evidence that forecasts from a GARCH model for the AO-corrected returns improve on GARCH and GARCH-t models for the original unadjusted data. For both error measures and for most years, we find a substantial improvement in forecasting using the AO-corrected data. (Unreported) pairwise sign-tests on the significance of the differences between the squared forecasting errors support our general finding.

3.1. Forecasting To evaluate the forecasting performance of the three GARCH models, we need an empirical measure of volatility. We follow Pagan and Schwert (1990) when we define this by vt 5 ur t 2 r¯ u,

7

(12)

where r¯ is the long-term average of r t , which is set equal to the continuous weekly return and is calculated along standard lines. Notice that we calculate vt for each considered sample with four years of weekly returns. In Table 6 we report the results of out-of-sample forecasting vt for the AEX using the GARCH models for unadjusted and adjusted data. We summarize our results for the years 1987 to 1994. This allows us to examine developments over time. To evaluate the forecasts, we consider the Mean Squared Error (MSE) and the Median Squared Error (MedSE). The MedSE is calculated in order to

3.2. Three other stock markets Finally, we replicate the above exercise for three other stock markets, i.e., those of Germany, Spain and Italy. The out-of-sample forecasting results are summarized in Table 7. It is clear from these results that again the one-step ahead forecasts of volatility based on AO-corrected returns outperform the forecasts from GARCH and GARCH-t models for unadjusted returns.

Table 6 Results of out-of-sample forecasting of volatility of AEX returns based on GARCH and GARCH-t models for the original data r t and a GARCH model for AO-corrected data r t* Forecasting

Criterion

sample 1987 1988 1989 1990 1991 1992 1993 1994

MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE

Models GARCH for r t

GARCH-t for r t

GARCH for r *t

0.439 0.460 0.358 0.213 0.149 0.059 0.360 0.173 0.217 0.137 0.206 0.162 0.174 0.113 0.179 0.095

1.165 0.255 0.377 0.215 0.179 0.088 0.320 0.130 0.215 0.162 0.213 0.161 0.188 0.113 0.201 0.085

0.376 0.339 0.293 0.121 0.115 0.063 0.299 0.109 0.169 0.103 0.154 0.081 0.148 0.048 0.155 0.093

The evaluation criteria are the Mean Squared Error (MSE) and the Median Squared Error (MedSE) (31000).

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P.H. Franses, H. Ghijsels / International Journal of Forecasting 15 (1999) 1 – 9

Table 7 Results of out-of-sample forecasting of volatility for three European stock market returns, based on GARCH and GARCH-t models for the original data r t and on GARCH models for AO-corrected data r *t Forecasting

Stock a

sample

Market

1987

DAX MADR MIL

1988

DAX MADR MIL

1989

DAX MADR MIL

1990

DAX MADR MIL

1991

DAX MADR MIL

1992

DAX MADR MIL

1993

DAX MADR MIL

1994

DAX MADR MIL

Criterion

MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE MSE MedSE

Models GARCH for r t

GARCH-t for r t

GARCH for r *t

1.030 0.234 1.217 0.350 0.567 0.211 na na 0.297 0.153 0.406 0.374 na na 0.190 0.137 0.273 0.232 0.302 0.198 0.474 0.294 0.360 0.126 0.302 0.198 0.146 0.117 0.273 0.221 0.287 0.181 0.372 0.213 0.465 0.192 0.217 0.172 0.267 0.172 0.417 0.270 0.261 0.202 0.219 0.146 0.414 0.237

1.285 0.274 2.086 0.398 2.197 0.437 na na 0.340 0.208 0.762 0.381 na na 0.223 0.193 0.280 0.219 0.331 0.226 0.477 0.211 0.357 0.139 0.331 0.226 0.196 0.121 0.278 0.281 0.327 0.196 0.669 0.249 0.882 0.258 0.207 0.115 0.508 0.356 0.273 0.795 0.498 0.297 0.320 0.257 0.417 0.314

1.240 0.129 1.160 0.153 0.621 0.230 na na 0.227 0.131 0.344 0.277 na na 0.179 0.137 0.208 0.181 0.248 0.160 0.455 0.195 0.360 0.122 0.248 0.160 0.119 0.075 0.236 0.167 0.254 0.125 0.322 0.141 0.484 0.147 0.171 0.125 0.216 0.075 0.358 0.220 0.249 0.230 0.215 0.121 0.392 0.205

The evaluation criteria are the Mean Squared Error (MSE) and the Median Squared Error (MedSE) (31000). a The stock market indices are the DAX index (Frankfurt, Germany), the MADR SE index (Madrid, Spain) and the Milan BCI index (Milan, Italy). The source of the data is Datastream. The indices are converted to guilders. b ‘na‘ means that for at least one sample the estimation routine does not converge. Diagnostic tests reveal that GARCH does not appear to be present in the corresponding samples.

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4. Concluding remarks In this paper we proposed to apply the Chen and Liu (1993) method to correct for additive outliers in stock market returns, when GARCH models for these returns are used for forecasting volatility. Our research was motivated by empirical evidence that GARCH and GARCH-t models do not seem to ¨ capture all data features, see for example Terasvirta (1996). We examined the forecasting performance of GARCH models for AO-corrected returns. Our empirical outcomes suggested that correcting for AOs results in improved forecasts of stock market volatility. The additive outlier model is not the only model that can represent aberrant data. One may also consider innovative outliers, level shifts and variance changes, see Tsay (1988). In further research, our approach can be extended by correcting for other types of aberrant data.

Acknowledgements This paper is based on the M.Sc. thesis (in Dutch) of the second author, which was written during his internship at the Deutsche Bank / DeBary Asset Management Bv. in Amsterdam. We gratefully acknowledge their support in providing access to the data and to computing facilities. The first author thanks the Royal Netherlands Academy of Arts and Sciences for its financial support. We thank an Associate Editor and three referees for their helpful comments.

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Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307–327. Bollerslev, T. (1987). A conditionally heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics, 69, 542–547. Chang, I., Tiao, G. C., & Chen, C. (1988). Estimation of time series parameters in the presence of outliers. Technometrics, 30, 193–204. Chen, C., & Liu, L. (1993). Joint estimation of model parameters and outlier effects in time series. Journal of the American Statistical Association, 88, 284–296. Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica, 50, 987–1007. Fox, A. J. (1972). Outliers in time series. Journal of Royal Statistical Society B, 34, 350–363. Ledolter, J. (1989). The effect of additive outliers on the forecasts from ARIMA models. International Journal of Forecasting, 5, 231–240. Pagan, A. R., & Schwert, G. W. (1990). Alternative models for conditional stock volatility. Journal of Econometrics, 45, 267– 290. ¨ Terasvirta, T. (1996). Two stylized facts and the GARCH(1, 1) model. Working Paper 96, Stockholm School of Economics. Tsay, R. S. (1988). Outliers, level shifts and variance changes in time series. Journal of Forecasting, 7, 1–20. Biographies: Philip Hans FRANSES is Professor of Applied Econometrics and Director of the Rotterdam Institute of Business Economics, both at the Erasmus University Rotterdam. His research interests include modeling time series, with a specific focus on seasonality, nonlinearity, outliers, and forecasting. Hendrik GHIJSELS finished in 1995 his study Econometrics at the Erasmus University Rotterdam. At present, he is a Senior Trader for Global Equity Derivatives at Rabobank International, Amsterdam.