Forecasting volatility with outliers in GARCH models

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<ul><li><p>Copyright 2008 John Wiley &amp; Sons, Ltd. </p><p>Forecasting Volatility with Outliers in GARCH Models</p><p>AMLIE CHARLESAudencia Nantes, School of Management, France</p><p>ABSTRACT</p><p>In this paper, we detect and correct abnormal returns in 17 French stocks returns and the French index CAC40 from additive-outlier detection method in GARCH models developed by Franses and Ghijsels (1999) and extended to innovative outliers by Charles and Darn (2005). We study the effects of out-lying observations on several popular econometric tests. Moreover, we show that the parameters of the equation governing the volatility dynamics are biased when we do not take into account additive and innovative outliers. Finally, we show that the volatility forecast is better when the data are cleaned of outliers for several step-ahead forecasts (short, medium- and long-term) even if we consider a GARCH-t process. Copyright 2008 John Wiley &amp; Sons, Ltd.</p><p>INTRODUCTION</p><p>In fi nance and fi nancial economics, modelling volatility of returns is fundamentally an important key to risk management, derivative pricing and hedging, market making, market timing, portfolio selection, monetary policy making, and many other fi nancial activities. A good forecast of volatility of prices of asset over the investment holding period is a good starting point for assessing investment risk. Financial stocks returns do not match the familiar bell-shaped normal distribution. Indeed, high-frequency time series of returns on fi nancial assets typically exhibit excess kurtosis. Engle (1982) introduced the ARCH model, which has been generalized by Bollerslev (1986) to capture the excess kurtosis. The GARCH models became popular in both theorical and empirical work. More precisely GARCH(1,1) has been shown to represent adequately the daily returns of most fi nancial time series (Andersen and Bollerslev, 1998; among others). However, even if GARCH processes are able to represent the dynamics of returns, they cannot capture all excess kurtosis.</p><p>This has naturally led to the use of non-normal distributions to model this excess kurtosis. Bollerslev (1987), among others, used a Student distribution, while Nelson (1991) suggested the generalized error distribution. Other propositions include mixture distributions such as the normalPoisson (Jorion, 1988), the normallognormal (Hsieh, 1989) or the Bernoullinormal (Vlaar and Palm, 1993).</p><p>Journal of ForecastingJ. Forecast. 27, 551565 (2008)Published online 5 September 2008 in Wiley InterScience( DOI: 10.1002/for.1065</p><p>* Correspondence to: Amlie Charles, Audencia Nantes, School of Management, 8 route de la Jonelire, BP 31222, 44312 Nantes. E-mail:</p></li><li><p>552 A. Charles</p><p>Copyright 2008 John Wiley &amp; Sons, Ltd. J. Forecast. 27, 551565 (2008) DOI: 10.1002/for</p><p>Excess kurtosis could also be explained by the presence of outliers (Baillie and Bollerslev, 1989). As in linear models, outliers affect the identifi cation and estimation of GARCH models (Carnero et al., 2007). It is known that these observations may wrongly suggest conditional heteroscedasticity (van Dijk et al., 2002) and also may hide true heteroscedasticity (van Dijk et al., 2002). With respect to estimation, Sakata and White (1998) showed that quasi-maximum likelihood (QML) estimators can be severally affected by a small number of outliers such as market crashes and rallies. Carnero et al. (2007) analysed fi nite sample behaviour, in the presence of outliers, of a QML estima-tor based on maximizing Student likelihood when the conditional distribution is Gaussian from Monte Carlo experiments. They concluded that this estimator is robust even when the sample is moderate and the outliers are relatively large. Verhoeven and McAleer (2000) studied, empirically, the effects of outliers on the AR(1)-GARCH(1,1) process by analysing 1000 trading days of fi ve fi nancial time series. They found that outliers tend to dominate QML estimates, resulting in larger ARCH and smaller GARCH estimates, and may give rise to spurious AR(1) and ARCH effects. Recently, Carnero et al. (2007) note that, for all contaminated series, the constant a0 is overestimated without depending on whether the series is contaminated with consecutive or isolated outliers.1 In the case of estimates of a1 and b, biases depend on the nature of outliers: if they are consecutive, a1 is overestimated and b is underestimated, while in the case of isolated outliers the biases are not very clear, and both a1 and b parameters can be overestimated or underestimated.</p><p>There are several procedures to detect outliers in GARCH models. Hotta and Tsay (1999) proposed two test statistics to detect outliers in GARCH processes. They applied their tests to simulated and real examples and concluded that the tests work well in both applications. Franses and Ghijsels (1999) and Franses and van Dijk (2002) proposed to modify the Chen and Liu (1993) method to correct for additive outliers in stock market returns, when GARCH models are used for forecasting volatility. Charles and Darn (2005) extended the procedure of Franses and Ghijsels (1999) to take into account innovative outliers. Indeed, Balke and Fomby (1994) found that most detected outliers in time series are innovative outliers, especially in high-frequency data. Doornik and Ooms (2002) distinguished between outliers that only affect level and those that also affect conditional variance. Other authors proposed robust methods to estimate the parameters, which avoid the problem of identifying outliers (Sakata and White, 1998; Park, 2002; among others).</p><p>Although these robust methods are relatively straighforward, there are several disadvantages: (i) robust methods perform well in some cases but poorly in others; (ii) since outlying observations are not adjusted, the impact of outliers on forecasts remains; (iii) in most cases, only limited infor-mation on the outlier may be obtained (e.g. from the weights applied to the residuals).</p><p>To our knowledge, there are few papers analysing the effects of outliers for out-of-sample fore-casting performance at different horizons. Franses and Ghijsels (1999) showed that outliers biased the estimation of ARCH and GARCH parameters and, consequently, the volatility forecasts. They found a substantial improvement in forecasting using data corrected for outliers over GARCH models with a Student distribution for the original returns. Verhoeven and McAleer (2000) found that, when data are corrected for outliers, volatility forecasts are improved substantially for periods of low volatility clustering but not for periods of high volatility clustering.2 Park (2002) </p><p>1 The authors consider the following process: s 2t = a0 + a1e2t1 + bs 2t1.</p><p>2 The authors consider recursive estimation and forecasting of the volatility of daily returns. They compute one-day-ahead </p><p>volatility forecasts. According to them, deterioration in the forecast accuracy for high-volatility periods can be explained by the fact that outlying observations are frequently clustered with other large observations, so that removing an outlier will reduce the forecast accuracy of subsequent large observations.</p></li><li><p>Forecasting Volatility with Outliers in GARCH Models 553</p><p>Copyright 2008 John Wiley &amp; Sons, Ltd. J. Forecast. 27, 551565 (2008) DOI: 10.1002/for</p><p>proposed a new approach to take into account outliers based on a robust GARCH model (RGARCH). He showed that the RGARCH model outperforms the GARCH model and the random walk model. The author notes that the out-of-sample volatility forecasts of the RGARCH model are superior to those of others competitive models.3</p><p>In this paper we propose to compare the method developed by Charles and Darn (2005) with a GARCH-t model which is very popular. The outline of this paper is as follows. The next section describes modelling outliers in a GARCH model as well as the outlier identifi cation procedure pro-posed by Charles and Darn (2005). We apply this procedure to 17 French daily stock returns and the CAC40 index in the third section, and we examine the effects of outliers on the diagnostics of normality from linear and nonlinear modelling. The fourth section evaluates and compares the forecasting performance of GARCH models from outlier-uncorrected and corrected series and a GARCH-t process using DieboldMariano tests for equal predictive ability. We conclude in the fi fth section.</p><p>METHODOLOGY</p><p>Outliers are aberrant observations that are away from the rest of the data. They can be caused by recording errors or unusual events such as changes in economic policies, wars, disasters, fi nancial crises and so on. They are also likely to occur if errors have fat-tailed distributions, as in the case of fi nancial time series. These observations may take several forms in time series. The fi rst and most usually studied is the additive outlier (AO), which only affects a single observation. In contrast, an innovative outlier (IO) affects several observations. Balke and Fomby (1994) found that many of the detected outliers in fi nancial time series are IOs, especially for data at a high frequency.</p><p>Charles and Darn (2005) extended the additive-outlier detection method in GARCH models developed by Franses and Ghijsels (1999) to innovative outliers,4 using the Chen and Liu (1993) approach.</p><p>Consider the returns series et, which is defi ned by et = log pt log pt1, where pt is the observed price at time t, and consider the GARCH(1,1) model</p><p> t t tz h= , (1)</p><p> et N (0, ht), zt </p><p>i.i.d. N (0, 1)</p><p> h ht t t= + + 0 1 12 1 1 (2)where a0 &gt; 0, a1 0, b1 </p><p>0 and a1 + b1 &lt; 1, such that the model is covariance-stationary.The GARCH(1, 1) model can be rewritten, strictly speaking,5 as an ARMA(1, 1) model for e2t </p><p>(Bollerslev, 1986):</p><p>3 Empirical analysis supports the dominance of the RGARCH model, when it is compared with GARCH, GARCH-t, </p><p>EGARCH and random walk models in terms of one-step-ahead volatility forecasts.4 Some Monte Carlo experiments, which are not reported, have been done to study the power of the test. Results indicate </p><p>that the outliers introduced in a simulated GARCH process are well detected. These Monte Carlo experiments will be the object of future research. In this way, comparisons with other tests will be interesting.5 It is important to note that nt is not a white noise process and thus equation (3) does not exactly represent an ARMA </p><p>process.</p></li><li><p>554 A. Charles</p><p>Copyright 2008 John Wiley &amp; Sons, Ltd. J. Forecast. 27, 551565 (2008) DOI: 10.1002/for</p><p> e2t = a0 +(a1 + b1) e2t1 + nt b1nt1 (3)</p><p>where nt = e2t ht. This analogy of the GARCH model with an ARMA model allows one to directly adapt the method of Chen and Liu (1993) to detect and correct AOs and IOs in GARCH models. Specifi cally, suppose that instead of the true series et one observes the series et, which is defi ned as</p><p> e2t = e2t + wixi(B)It(t) with i = 1, 2 (4)</p><p>where It(t) is the indicator function defi ned as It(t) = 1 if t = t and zero otherwise where t is the date of outlier occurring, wi is the magnitude of the outlier effect, and xi(B) represents their dynamic pattern with x1(B) = 1 for an AO and x2(B) = (1 b1B)(1 (a1 + b1)B)1</p><p>for an IO.An AO is related to an exogenous change that directly affects the series and only its level of the </p><p>given observation at time t = t. An IO is possibly generated by an endogenous change in the series, and affects all the observations after time t through the memory of the process.</p><p>The residuals ht of the observed series e2t are given by</p><p> t t t i i tB B e v B B I=</p><p>+ ( ) = + ( ) ( ) ( )01</p><p>2</p><p>1 (5)</p><p>where p(B) = (1 (a1 + b1)B)(1 b1B)1. The expression (5) can be interpreted as a regression model for ht, i.e.</p><p> ht = wixit + nt (6)</p><p>with xit = 0 for i = 1, 2 and t &lt; t, xit = 1 for i = 1, 2 and t = t, x1,t+k = pk (for AO) and x2,t+k = 0 (for IO) for t &gt; t and k &gt; 0.</p><p>Outlier detection is based on the maximum value of the standardized statistics of the outliers effects:</p><p>AO: </p><p>1 1 12</p><p>1 2</p><p>1 1= ( )( ) = = = =</p><p> v tt</p><p>n</p><p>t tt</p><p>n</p><p>tt</p><p>x x x </p><p>n</p><p>v tt</p><p>n</p><p>v</p><p>x </p><p>= ( ) =</p><p>=</p><p>1</p><p>12</p><p>1 2</p><p>2 2</p><p> IO: v</p><p>where s2v denotes the estimated variance of the residual process.6The outlier detection method for GARCH(1, 1) models then consists of the following steps:</p><p>1. Estimate a GARCH(1, 1) model for the observed series et and obtain estimates of the conditional variance ht and ht = e2t ht.</p><p>2. Obtain estimates w i (i = 1, 2) for all possible t = 1, . . . , n, and compute tmax = max1t nt i. If the value of the test statistic exceeds the critical value C, an outlier is detected at the observation for which t is maximized.</p><p>6 The estimated variance of the residual process is obtained from the omit-one method, which computes the error variance </p><p>from the sample where the observation at t = t has been deleted (Franses and Ghijsels, 1999; Chen and Liu, 1993).</p></li><li><p>Forecasting Volatility with Outliers in GARCH Models 555</p><p>Copyright 2008 John Wiley &amp; Sons, Ltd. J. Forecast. 27, 551565 (2008) DOI: 10.1002/for</p><p>3. Replace e2t with</p><p>AO: *IO: * with </p><p>e e</p><p>e e jj j </p><p>2 21</p><p>2 22 0</p><p>= </p><p>= &gt;+</p><p> where y(B) = p(B)1. The outlier-corrected series e*t is defi ned as</p><p>AO:for </p><p>sign for </p><p>IO:for </p><p>sig</p><p> 2e</p><p>e t</p><p>e e t</p><p>ee t</p><p>t</p><p>t</p><p>t t</p><p>t</p><p>t</p><p>**</p><p>*</p><p>=</p><p>( ) =</p><p>=</p><p> * , 0</p><p>4. Return to step 1 to estimate a GARCH(1, 1) model for the series e*t , and repeat all steps until no tmax test-statistic exceeds the critical value C.</p><p>A critical value C = 10 is used, which considered a low-sensitivity value for our sample size (Verhoeven and McAleer, 2000). This choice for C is based on simulation experiments proposed by Franses and Van Dijk (2002).7 The authors simulate some percentiles of the distribution of the tmax statistic under the null hypothesis that no outliers are present for several values of ARCH and GARCH parameters and for two sample sizes (250 and 500). It is seen that the value of C = 10 is reasonably close to the 90th percentile of this distribution for most parameter combinations.</p><p>APPLICATION TO THE FRENCH STOCK MARKET</p><p>We investigate the daily returns of the CAC40 French index and 17 of the French stocks included in it during the period from 6 January 1997 to 4 April 2002, comprising 1435 observations. From the closing prices, returns are computed as follows: Ri,t =(Pi,t Pi,t1)/Pi,t1, where Pi,t indicates the closing prices for an asset i at time t. The data come from Thomson Financial Datastream.</p><p>As expected, some outliers are found in the daily data when we apply the previous method. Furthermore, many of the detected large shocks seem to be associated with the September 11 terrorist attacks. This result confi rms those of Chen and Siems (2004). The results announcements and the Vivendi Universal business, among others, may explained the abnormal returns.8</p><p>We analyse the descriptive statistics of residuals computed from AR models.9 These analyses are done both for the original series and the outlier-corrected data to see how the evidence of non-normality and conditional heteroscedasticity is altered by taking the outliers i...</p></li></ul>


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