Outliers and misleading leverage effect in asymmetric GARCH-type models
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Outliers and misleading leverage e¤ect in asymmetric GARCH-type models M. Angeles Carnero , Ana Pérezy July 24, 2019 Abstract This paper illustrates how outliers can a¤ect both the estimation and testing of leverage e¤ect by focusing on the TGARCH model. Three estimation methods are compared through Monte Carlo experiments: Gaussian Quasi-Maximum Likelihood, Quasi-Maximum Likelihood based on the Student-t likelihood and Least Absolute Deviation method. The empirical behavior of the t-ratio and the Likelihood Ratio tests for the signi…cance of the leverage parameter is also analyzed. Our results put forward the unreliability of Gaussian Quasi-Maximum Likelihood methods in the presence of outliers. In particular, we show that one isolated outlier could hide true leverage e¤ect whereas two consecutive outliers bias the estimated leverage coe¢ cient in a direction that crucially depends on the sign of the …rst outlier and could lead to wrongly reject the null of no leverage e¤ect or to estimate asymmetries of the wrong sign. By contrast, we highlight the good performance of the robust estimators in the presence of one isolated outlier. However, when there are patches of outliers, our …ndings suggest that the sizes and powers of the tests as well as the estimated parameters based on robust methods may still be distorted in some cases. We illustrate these results by analyzing two series of daily returns, namely the exchange rate US Dollar/British Pound and futures contracts of Natural gas, as examples of series which include one isolated and two consecutive outliers, respectively. JEL Classi…cation: C22, G10, Q40 Keywords: Conditional heteroscedasticity, QMLE, Robust estimators, TGARCH, AVGARCH. Corresponding Author: Dpto. Fundamentos del Análisis Económico (FAE), Universidad de Alicante. Campus de San Vicente del Raspeig, Alicante. Spain. Tel: +34 965 903400. E-mail: [email protected] yDpto. Economía Aplicada and IMUVA, Universidad de Valladolid. Spain. E-mail: [email protected] 1 Studies in Nonlinear Dynamics & Econometrics, 20180073, ISSN (Online) 1558-3708, DOI: https://doi.org/10.1515/snde-2018-0073
1 Introduction One of the empirical stylized facts about the series of …nancial returns is the leverage e¤ect, which refers to the asymmetric response of volatility to positive and negative past returns. In particular, the volatility tends to be higher following negative return shocks (‘bad’news) than following positive shocks (‘good’news) of the same magnitude. This feature conveys a generally negative cross-correlation between lagged asset returns and volatility; see Black (1976) who originally put it forward using the debt-to-equity ratio and Engle (2011) who provides the economic underpinning of volatility asymmetries following simple asset pricing theory. See also Hibbert et al. (2008) for a behavioral explanation of the negative asymmetric return–volatility relationship. In the econometric literature there have been several methods proposed to represent and estimate conditional heteroscedasticity and leverage e¤ect, like the asymmetric GARCH-type models (see the review in Rodríguez and Ruiz (2012)), the non-parametric high-frequency methods in Andersen et al. (2001) and the observation-driven models proposed by Creal et al. (2013) and Harvey (2013), to name but a few. In practice, when these methods are applied to estimate and test for the leverage e¤ect, mixed results come up. For example, Andersen et al. (2001) found statistically signi…cant leverage e¤ect for most of the stocks returns of the DJIA stock market index by using realized volatilities, although they point out that this e¤ect has a marginal economic importance. In turn, Zivot (2009) and Rodríguez and Ruiz (2012) also report signi…cant leverage e¤ect by applying GARCH-type methods to di¤erent series of returns, including a particular asset stock, a stock market index and exchange rates. However, Ait-Sahalia et al. (2013) discuss what they call the leverage e¤ect puzzle, which relies on the fact that the empirical correlation between returns and changes in volatility estimated from high frequency data becomes nearly zero for most assets tested, despite the many economic reasons for expecting such correlation to be negative. In Energy markets, it is also common to …nd di¤erent results concerning the leverage e¤ect. For example, Kristoufek (2014) …nds the inverse leverage e¤ect (positive correlation between returns and volatility) in future prices of the Natural gas by using correlation-based methods for non-stationary series, whereas Chkili et al. (2014) …nd the standard leverage e¤ect when asymmetric and long-memory models are estimated to spot and future returns of the same commodity. 2
The mixed results on the sign and the signi…cance of the leverage e¤ect could be partly explained, among other reasons, by the harmful e¤ect of the extreme observations usually encountered in the returns. For instance, it is already well known that the presence of outliers can have misleading e¤ects on the identi…cation of both conditional heteroscedasticity and leverage e¤ect; see Carnero et al. (2007) and Carnero et al. (2016), respectively. Moreover, it is also proved that the presence of outliers renders the Gaussian Quasi Maximum Likelihood (QML) estimators unreliable in symmetric GARCH models; see Sakata and White (1998), Mendes (2000), Carnero et al. (2007) and Muler and Yohai (2008), among others. In this context, robust estimators based on maximizing non-Gaussian heavy-tailed likelihoods have been proposed and their main properties have been established; see, for instance, Newey and Steigerwald (1997), Sakata and White (1998), Berkes and Horvath (2004) and Fan et al (2014). Another robust alternative for GARCH-type models is the log-transform-based Least Absolute Deviations (LAD) estimator proposed by Peng and Yao (2003) and further discussed in Huang et al. (2008). Related works also include Pan et al. (2008), Francq and Zakoian (2013) and Hill (2015), among others. Alternatively, some authors deal with this problem by applying methodologies based on detecting and correcting outliers; see, for example, Laurent et al. (2016) and the references therein. In this paper, we face the problem of how outliers can a¤ect the inference on the leverage parameter in asymmetric GARCH-type models: Does the sign of the outliers matter? How much the size of the outliers worsen the results? Is the e¤ect of one isolated outlier comparable to that of consecutive outliers? Which estimation method is more resistant to outliers? To answer these questions, we conduct an extensive Monte Carlo study that includes models with high, low and none leverage and di¤erent types of outliers (isolated and in patches, positive and negative, big and small). For each setting, we compare the performance of three estimation methods: QML, Quasi-Maximum Likelihood estimator based on maximizing the Student-tlikelihood (QML-t) and LAD. We also analyze the size and power of the t-ratio and the Likelihood Ratio tests for the signi…cance of the leverage parameter. Among the several asymmetric GARCHtype models proposed in the literature, we focus on the Threshold GARCH (TGARCH) model of Zakoian (1994) since, according to Rodríguez and Ruiz (2012), this is more ‡exible than its competitors to properly represent the dynamics of …nancial returns. 3
Previous works related to our paper are Pan et al. (2008), who establish the asymptotic properties of both QML and LAD estimators for a general model that nests the TGARCH and compare, for a particular parametrization, the …nite sample performance of both estimators, and Francq and Zakoian (2013), who compare the asymptotic relative e¢ ciencies of QML and LAD estimators for a model that also nests the TGARCH. Our paper provides new insights on the topic in two ways. First, we also evaluate the QML-t estimator, that turns out to be more robust than LAD in most cases, and second, our Monte Carlo study is exhaustive enough to cover the multiple problems often found in real data where di¤erent type of outliers may come up. Our results show that QML-based methods become unreliable in the presence of outliers and could lead to either hide true leverage e¤ects (when there is one isolated outlier) or detect spurious leverage or leverage of the wrong sign (in the presence of two consecutive outliers). As expected, the robust estimators considered (QML-t and LAD) always outperform QML although they are still slightly biased in the presence of big consecutive outliers. In this case, the bias direction crucially depends on the sign of the …rst outlier and could lead to wrongly reject the null of no leverage e¤ect. These results are further enhanced in our empirical application, where two particular series of …nancial returns are analyzed, namely a daily series of the exchange rate US Dollar/British Pound including one isolated negative outlier and a daily series of futures contracts of Natural gas including two consecutive outliers of opposite sign being the …rst one positive. The rest of the paper is organized as follows. Section 2 reviews the TGARCH model and describes the three estimation methods to be analyzed (QML, QML-t and LAD) and the two signi…cance tests considered (t-ratio and Likelihood Ratio tests). Section 3 is devoted to the …nite sample performance of these estimators and tests in the presence of outliers for di¤erent parameter sets and di¤erent types of outliers. Section 4 illustrates our main results with an empirical application based on the two series of daily returns mentioned above. Finally, Section 5 concludes the paper with a summary of the main conclusions. 4
2 Statistical inference in the TGARCH model 2.1 TGARCH model: de…nition and main properties As Rodríguez and Ruiz (2012) point out, the TGARCH model is an appropriate and ‡exible GARCH-type model to represent the features and dynamic properties of …nancial returns, namely excess kurtosis, conditional heteroscedasticity and leverage e¤ect. In this model, the series of demeaned returns, yt, is speci…ed by the following equation: yt=t"t;(1) where tis the volatility and "tis a sequence of independent and identically distributed random variables with zero mean and unit variance. To accommodate the asymmetric relationship between past returns and volatility, tis parametrized as a function of both the magnitude and the sign of past returns. In particular, the equation for the volatility in the TGARCH(1,1) model, as given in Rodríguez and Ruiz (2012), is the following1 t=0+jyt1j+t1+yt1:(2) When yt1is positive, the volatility response is linear in yt1with slope (+)but if yt1is negative, the slope of the response is (). Thus, the volatility can respond asymmetrically to rises and falls in stock prices and the value of is expected to be negative. Under the constraints 0>0,0and jj,tis always positive and represents the conditional standard deviation of yt. Moreover, the model is covariance stationary if 2<12221, where 1=Ej"tj: An advantage of the TGARCH model is that it parametrizes the conditional standard deviation rather than the conditional variance. This makes it easier to work out analytical expressions for its unconditional higher-order moments and cross-moments; see the results in He and Teräsvirta (1999), Hwang and Basawa (2004) and He et al. (2008). Moreover, as the TGARCH model involves absolute values rather than squares, 1There are other (equivalent) parametrizations of the TGARCH(1,1) model proposed in the econometric literature and in the econometric software packages; see, for instance, Zakoian (1994), Hentschel (1995), He and Teräsvirta (1999) and He et al. (2008). Hence, in order to compare estimates and standard errors from di¤erent papers and/or di¤erent software packages, it is important to be very careful about which parametrization is being used in each case. It is also important to be aware that the GJR model, proposed by Glosten et al. (1993), is sometimes erroneously referred to as TGARCH. 5
it is expected to be less sensitive to extreme observations than similar models involving conditional variances and squared returns, like the GJR and the QARCH models in Glosten et al. (1993) and Sentana (1995), respectively. Another advantage of the TGARCH model is that it contains, as a special case, a model without leverage, namely the absolute-value GARCH (AVGARCH) model of Taylor (1986) and Schwert (1989), that comes up by taking = 0 in (2). Thus, model selection can be easily performed by testing the signi…cance of the leverage parameter with the usual t-ratio test and/or the Likelihood Ratio test. On the other hand, the TGARCH model is a particular case of the Asymmetric Power ARCH (A-PARCH) model proposed by Ding et al (1993). Hence, the asymptotic theory in Pan et al. (2008) and Francq and Zakoian (2013) regarding estimation and hypothesis testing in the A-PARCH model can be applied. By contrast, the asymptotic theory for other popular asymmetric GARCH-type models is scarce; see, for instance, the results for the estimation of EGARCH models in Straumann and Mikosch (2006), Wintenberger (2013) and Hafner and Linton (2017). 2.2 Gaussian QML estimation The TGARCH(1,1) model de…ned in equations (1)-(2) can be estimated by maximizing the conditional log-likelihood function, which, given initial values y0and 0, is as follows L() = T X t=1 lt() = T X t=1 1 2log 2 t+ log fyt t;(3) where = (0; ; ; )0denotes the parameter vector to be estimated and f()is the probability density of "t. In particular, if "tis assumed to be N(0;1), the corresponding Gaussian log-likelihood function, denoted as LG(), will come up, namely LG() = T 2log 21 2 T X t=1 log 2 t+y2 t 2 t: The resultant estimator obtained from maximizing LG()is the well-known QML estimator. This estimator, that will be denoted as bQML, is the most commonly used one for GARCH-type models and, in particular, for the TGARCH model introduced above; see, for instance, Zivot (2009) and Francq and Zakoian (2010, chapter 10). Moreover, 6
this estimator is provided in most software packages, such as E-VIEWS, [email protected], MFE MATLAB Toolbox, SAS, Stata, Splus and R. Obviously, if the true distribution of "tis N(0;1), the resultant estimator will be the Maximum Likelihood estimator. Pan et al. (2008) show that QML is consistent and asymptotically normal for an asymmetric power-transformed GARCH model that includes as a particular case the TGARCH model, provided that "tis symmetrically distributed with E"2 t= 1 and E"4 t< 1and some regularity assumptions hold. In such a framework, we can approximate the asymptotic variance of bQML by the so-called “sandwich”estimator V ar(bQML)H(bQML)1B(bQML)H(bQML)1;(4) where H()denotes the Hessian matrix of the log-likelihood and B()is the inner product of the gradient (or score) of the log-likelihood. On the other hand, Muler and Yohai (2008) show that the QML estimator could be regarded as an M-estimator de…ned as bQML = arg min M0;T (); where M0;T is given by M0;T () = 1 T2 T X t=2 0(xtlog 2 t()); where xt= log(y2 t)and 0is the auxiliary function de…ned as 0(x) = log(p2) + 1 2(exx):(5) This function as well as its 1st derivative, 0 0=1 2(ex1), are both unbounded, rendering QML not robust, i.e., a few outliers can have a large in‡uence on this estimator. The lack of robustness of the QML estimator in symmetric GARCH models is already well documented; see Sakata and White (1998), Mendes (2000), Carnero et al. (2007) and Muler and Yohai (2008), among others. In Section 3 we analyze the e¤ect of the outliers on QML in TGARCH models through Monte Carlo experiments. In particular, we investigate if the sign of the outliers, that is irrelevant in symmetric models, makes any di¤erence when estimating the parameters of asymmetric models. 7
2.3 QML-t estimation In order to gain resistance against outliers, some authors propose to use estimators based on maximizing non-Gaussian heavy-tailed log-likelihoods; see, for instance, Sakata and White (1998). Actually, in the seminal paper of Nelson (1991), the EGARCH model is estimated by maximizing the log-likelihood function in (3) assuming that "tfollows a GED distribution normalized to have zero mean and unit variance. Another common practise is to maximize the conditional log-likelihood in (3) computed as if "tfollowed a Student-t distribution with degrees of freedom normalized to have zero mean and unit variance. In such a case, the log-likelihood function, denoted as LStud, becomes: LStud() = Tlog ((+ 1)=2) p(2)(=2)!1 2 T X t=1 log 2 t+ (+ 1) log 1 + 1 2 y2 t 2 t; where the vector parameter is = (0; ; ; ; )0. The resultant estimator obtained from maximizing LStud()is the so-called QML-t estimator denoted as bQMLt. As far as we know, no asymptotic theory exists for QML-t estimation in the context of asymmetric GARCH models. Hence, we will assume the usual practice of researchers using GARCH-type models and we will approximate its asymptotic variance by the “sandwich” estimator in (4) replacing bQML by bQMLt. Empirical applications performing QML-t estimation of the TGARCH model can be found in Zivot (2009), Francq and Zakoian (2010, chapter 10) and Rodríguez and Ruiz (2012). Muler and Yohai (2008) point out that the QML-t estimator also corresponds to an M-estimator de…ned as bQMLt= arg min M1;T (); where M1;T is given by M1;T () = 1 T2 T X t=2 1;(xtlog 2 t()); where xt= log(y2 t)and 1; is the following auxiliary function 1;(x) = x 2++ 1 2log 1 + ex 2log + 1 2+ log p(2)( 2):(6) This function is unbounded but its 1st derivative, 0 1;(x) = ex+2 2(2+ex);is bounded; thus, QML-t is expected to be more robust than QML, although it can still be a¤ected by some type of outliers. 8
Carnero et al. (2007) analyze the robustness of QML-t for symmetric ARCH and GARCH models. In Section 3 we analyze the …nite sample behavior of the QML-t estimator, as compared to QML, in TGARCH models in the presence of outliers. 2.4 LAD estimation As an alternative to QML, Peng and Yao (2003) propose the LAD estimator and they show that, in the context of GARCH models, this estimator is robust to heavy tails of the innovation distribution and it is asymptotically normal and unbiased under mild conditions for the error distribution. Huang et al. (2008) perform a comparison between both QML and LAD, where the latter is viewed as a quasi-maximum likelihood estimator based on the hypothesis that the log-squared innovations follow a Laplace distribution. Pan et al. (2008) establish the asymptotic properties of both QML and LAD estimators for an asymmetric power-transformed GARCH model that nests the TGARCH. They also compare, for a particular parametrization, the …nite sample performance of both estimators and show that the LAD is more accurate than QML for heavy-tailed errors. For the LAD estimator to be applied, the model should be reparametrized in such a way that the median (instead of the mean) of the squared innovations is equal to 1, while the mean of the innovations remains unchanged and equal to 0. In particular, for the TGARCH model we are interested in, the reparametrization is as follows. Let M=median("2 t)>0and let t=M1=2tand " t=M1=2"t, so that median("2 t) = 1. Then, the TGARCH model de…ned in equations (1)-(2) may now be expressed as yt= t" t; t= 0+jyt1j+ t1+yt1; where 0=M1=20,=M1=2,=M1=2and the parameter vector to be estimated is = ( 0; ; ; )0:Notice that, under this new parametrization, the parameters 0, and di¤er from those in the old setting by a common positive constant factor while the parameter remains unchanged. The LAD estimator is based on the regression equation for the log-squared transformation, namely log y2 t= log 2 t+ log "2 t; 9
positive, pushed QML-t estimated upwards leading to a possible erroneous detection of inverse leverage. Hence, there is still place to improve robustness in the estimation of asymmetric GARCH-type models, a topic that is left for further research. INSERT FIGURE 4 HERE 3.3 Simulation results on signi…cance tests In this section we analyze the empirical properties of both the t-test and LR test using the standard rejection regions (8) and (9), respectively, to test H0:= 0 against H1:6= 0:We only perform this analysis with the two likelihood-based estimators, namely QML and QML-t. In all cases, the nominal size considered is 5%:In order to assess the empirical size of both tests, we simulated 1000 independent samples of size T=1000 of an AVGARCH (Parameter set 3) and, for each sample, we …tted a TGARCH model, by both QML and QML-t, and computed the corresponding t-ratio and loglikelihood. To calculate the LR statistic, we also …tted an AVGARCH and compare its log-likelihood with that of the …tted TGARCH. On the other hand, to analyze the empirical power of the tests, we simulated 1000 independent samples of size T=1000 of a TGARCH with Parameter set 2 (=0:11) and, for each sample, we proceed as before. We have also performed the same experiments with a TGARCH model with Parameter set 1 (=0:05) obtaining similar conclusions. Figure 5 compares the empirical size of the t-test (left-hand side panels) and the LR test (right-hand side panels) based on both QML and QML-t estimates of TGARCH models contaminated with 1 isolated outlier (top panels) and with two consecutive outliers (bottom panels). That is, it represents the proportion of rejections under the null (H0:= 0), based on 5% critical value of tand LR tests using the standard rejection regions (8) and (9), respectively. The main conclusions we can draw from this …gure are as follows. As expected, both the LR and t-test based on QML-t are always more robust to outliers than those based on QML. Moreover, as the outlier size increases, the tests based on QML become more oversized, i.e., they erroneously reject the null more often that they should and so they tend to identify spurious leverage. This problem is especially remarkable in the LR test, which could reach an over-rejection as huge as 80% in the presence of outliers of size larger than 15. By contrast, the tests 16
based on QML-t keep the nominal size quite well and even better in the presence of 2 outliers than in the presence on 1 isolated outlier. It is also worth mentioning that, even with no outliers, the t-test does not reach the nominal size 5%. This could be related to the warning of Francq and Zakoian (2010), mentioned in Section 2.5, regarding the inappropriateness of the standard rejection region (8) in GARCH settings. This topic is out of the scope of this paper but deserves further research. INSERT FIGURE 5 HERE Figure 6 compares the empirical powers of the t-test (left-hand side panels) and the LR test (right-hand side panels) based on both QML and QML-t estimates of TGARCH models contaminated with 1 isolated outlier (top panels) and with two consecutive outliers (bottom panels). That is, this …gure displays, for each test, the relative frequency of rejection of the hypothesis H0:= 0 (no leverage) on 1000 independent realizations of length T= 1000 of the TGARCH model with parameter =0:11 (Parameter set 2). This …gure shows that, in terms of power, both the LR and t-test based on QML-t are again more robust to outliers than those based on QML, as expected. The power of the tests based on QML decreases rapidly with the size of the outlier, although the loss of power is not that big in the LR test, which keeps the power around 80% when it is based on QML, even in the presence of huge outliers. However, the loss of power due to outliers of the t-test based on QML is dramatic. By contrast, the power of the tests based on QML-t is around 1 in all cases, regardless of the size, the sign and the amount of the outliers in the sample. INSERT FIGURE 6 HERE 4 Empirical application In this section we illustrate the previous results by …tting both the AVGARCH and the TGARCH models to two series of daily returns from di¤erent markets by using the estimation methods described above. For each series and estimated model, we check whether the leverage parameter is signi…cant or not by applying both the t-test and LR test. The two series analyzed have been chosen to represent the possible e¤ects than one isolated outlier and two consecutive outliers have on the inferential statistics. 17
4.1 Data description and dynamic properties The series analyzed are daily returns of the exchange rate US Dollar/British Pound observed from January 14, 1999 to January 14, 2019, comprising 5218 observations, and daily returns of futures contracts of Natural gas from January 4, 2000 to June 28, 2013, comprising 3368 observations3. In both cases, the returns are computed as yt= 100(log(Pt)log(Pt1));where Ptis the price at day t. The two series are plotted in Figure 7. As we can see, they both display volatility clustering and some occasional extreme values that could be regarded as outliers. For instance, the exchange rate US Dollar/British Pound returns exhibit several extreme observations, the largest one corresponding to June 24, 2016, when the series drops by 8:3% after the referendum held on the previous day in which 51:9% of those voting supported leaving the EU. The Natural gas returns exhibit two extreme consecutive observations, the …rst one positive and the second one negative, of magnitudes about 12 times the standard deviation of the series. These outliers are due to large changes in the open price of the Natural gas on the 13th, 14th and 15th of April 2006, with values of 6.78, 11.26 and 7.15 dollars, respectively. INSERT FIGURE 7 HERE Table 1 contains descriptive statistics of the two series considered, as well as the Jarque-Bera and the Ljung-Box Q(20) test statistics, the heteroscedastic-corrected Qc(20) test statistic, proposed by Diebold (1988), and the test statistic CH(20), proposed by Cumby and Huizinga (1992), which is also robust to conditional heteroscedasticity. We also include the values of the statistics Qand CH applied to the squared returns, denoted by Q2(20) and CH2(20), respectively, to test for conditional heteroscedasticity. As expected, both series exhibit excess kurtosis and the Jarque-Bera test for Normality always rejects the null. The values of Qc(20) and CH(20) never reject the null at 5% signi…cance level, suggesting that both returns are uncorrelated, as expected. Also, as expected, the values of Q2(20) and CH2(20) are signi…cant at 1% for both series, indicating strong evidence of conditional heteroscedasticity. 3The data for the exchange rate US Dollar/British Pound was downloaded from Datastream and the corresponding one for Natural gas was obtained from the additional …les in Kristoufek (2014). 18
INSERT TABLE 1 HERE Figure 8 displays, in its …rst row, the correlograms of the returns with the corrected 95% con…dence bands proposed by Diebold (1988) for conditionally hereroscedastic series. These bands are wider than the usual Barlett bands and show no evidence of autocorrelated returns. The correlograms of squared returns, displayed in the 2nd row of Figure 8, suggest correlation in the squared returns for both series but the pattern of both correlograms is di¤erent. For the Natural gas, we can see the typical pattern of the correlogram of the squared observations in the presence of two consecutive outliers, that is, a very high positive and signi…cant 1st order correlation while the others are pushed downwards towards zero; see Carnero et al. (2007). However, when the robust autocorrelations of squares proposed by Teräsvirta and Zhao (2011) are computed (see the 3rd row of Figure 8), another picture comes up: the correlograms of both series have the same pattern, with all correlations being signi…cantly di¤erent from zero, suggesting that the conditional variances of both returns are not constant over time, as expected. Notice that, for the US Dollar/British Pound, there is not such a di¤erence between both the sample and the robust autocorrelations of squares, although the former are closer to zero. This is the expected pattern due to one isolated outlier; see Carnero et al. (2016). INSERT FIGURE 8 HERE Finally, the last two rows of Figure 8 display the sample cross-correlations between past and squared returns and their robust counterparts, respectively. The latter are computed using the proposal in Carnero et al. (2016) based on applying the Ramsay weighting scheme to the sample variances and cross-covariances. As expected, for the Natural gas, the picture changes depending on whether we look at the sample or the robust cross-correlations. Whereas all the robust cross-correlations are around zero, indicating no leverage e¤ect, the 1st sample cross-correlation is pushed upwards to a signi…cant positive value, suggesting inverse leverage e¤ect (positive relationship between volatility and past returns). Again, this is the predicted pattern in the presence of two consecutive outliers, being the …rst one positive; see Carnero et al. (2016). Therefore, the inverse leverage e¤ect found in the Natural gas by some authors (see Kristoufek 19
(2014) and the references therein) could be an artifact due to the misleading e¤ect of outliers. By contrast, in the US Dollar/British Pound, the sample and robust crosscorrelograms are similar, in agreement with the results in Carnero et al. (2016), who show that a single outlier needs to be larger to bias the sample cross-correlations. 4.2 Estimating and testing for leverage e¤ect We describe below the results from estimating the TGARCH and AVGARCH models to the two return series described above. Since the results in Section 3 suggest a better performance of QML-t over LAD, we focus our main comparison on QML and QML-t, although the estimation results from LAD will be also commented. Table 2 displays the estimation results obtained by QML, QML-t and LAD as well as some diagnostics based on the residuals, b"t=yt=bt, where btis the estimated volatility for each model. The estimation has been carried out with Matlab. The QML and QML-t estimated parameter values and their corresponding standard errors were computed using the Oxford MFE Toolbox, taking into account the reparametrization used in this package as compared to the parametrization in (2) used in this paper.4The LAD estimated parameter values were computed as explained in Section 2.4 and the method proposed by Zhu and Ling (2015) was used to calculate the standard errors. INSERT TABLE 2 HERE As we can see, there are remarkable di¤erences between the results obtained for the two returns series and, for each series, there are also some di¤erences between the estimation methods that are worth mentioning. Since we are interested in the e¤ects of outliers on the leverage e¤ect, our comments will be focused on the parameter . For the exchange rate series, we can see that, regardless of the estimation method, the parameter in the TGARCH model is always estimated negative, as it is expected if there is leverage e¤ect. Moreover, the leverage coe¢ cient is statistically signi…cant at 5% when both the QML and QML-t based t-tests are applied. In particular, the values (and p-values) of the t-statistics are 2:08 (0:0375) and 3:16 (0:002), respectively. 4To check for the robustness of our results, we have repeated the estimation by QML and QMLt in Stata, obtaining similar results. We have also estimated EGARCH models leading to similar conclusions. 20
Meanwhile, the values (and p-values) of the LR test statistic for H0:= 0 (AVGARCH) against H1:6= 0 (TGARCH) are 11:182 (0:001) and 12:396 (0:001) when the models are estimated by QML and QML-t, respectively, con…rming that is signi…cant when both QML and QML-t are used. By contrast, when the LAD estimator is considered, the leverage coe¢ cient is no longer statistically signi…cant due to the large standard error. Notice also that is estimated closer to zero when QML is used compared to QML-t. This result agrees with our discussion in Section 3 where we show that a single isolated outlier (like the one existing in the exchange rate series) biases the QML estimated towards zero and could hide true leverage. Then, it seems that, in this case, QMLt is more reliable than QML suggesting that there is leverage e¤ect in the exchange rate returns, although the negative return associated to the Brexit seems to be biasing towards zero the leverage coe¢ cient, when the model is estimated by QML. For the Natural gas series, the parameter is estimated positive by the three methods (QML, QML-t and LAD), suggesting inverse leverage e¤ect. However, its statistical signi…cance depends on the estimator used as well as the test statistic chosen. Using the LAD estimator, we can see that the standard error is huge rendering the leverage coe¢ cient not statistically signi…cant. On the other hand, when QML is used, the values (and p-values) of both the t-ratio and LR test statistics are 0:636 (0:5247) and 2:377 (0:123), respectively, indicating no evidence of leverage e¤ect at any reasonable significance level. By contrast, when the model is estimated using QML-t, the values (and p-values) of these two test statistics are 2:550 (0:0108) and 8:458 (0:0036), respectively, showing evidence of inverse leverage e¤ect at 5% signi…cance level (or even at 1% if the LR test is selected). This somehow surprising result could be related to our …ndings in Section 3, where we show that QML-t could be still biased in the presence of two big consecutive outliers, with the estimated being pushed upwards if the …rst of these outliers is positive, as it is the case in the Natural gas series. Therefore, we should be very careful in concluding that there is actually inverse leverage e¤ect in this series, as this could be a misleading e¤ect caused by the two consecutive extreme observations present in this series. Finally, it is also worth mentioning that the thickness parameter of the Student error distribution in QML-t, indicates fat tails in both returns. When looking at residuals diagnostics, as expected, the values of the statistics Q2(20) and CH2(20) for remaining autocorrelation in the squared residuals, have been reduced 21
remarkably in all estimated models, as compared to their values for the returns in Table 1. Only the values of Q2(20) remain signi…cant for some models. However, the robust statistic CH2(20) is not longer signi…cant for the TGARCH model, suggesting that this model has properly captured the dynamics in the conditional variance of the returns. To further illustrate how the potential outliers can bias the estimation and testing of the leverage e¤ect, we consider a rolling window scheme of size T= 1000 through the whole sample, starting at point t= 1 and moving forwards up to covering the last 1000 observations of the full sample. For the exchange rate and the Natural gas series, this amounts to analyzing 4217 and 2368 subsamples, respectively, covering periods of di¤erent volatilities and types and sizes of outliers, according to the following steps: 1. Select the 1st subsample of size T= 1000 for the exchange rate (14/01/1999 - 14/09/2002) and for Natural gas (04/01/2000 - 9/01/2004), estimate the parameter by the three methods considered (QML, QML-t and LAD) and compute the corresponding t-ratio and LR test statistics for these subseries.5 2. Delete the 1st observation, add a new observation at the end of the subsample and re-estimate the model and test again for leverage. 3. Repeat the process until we reach the end of the full samples, where we cover the last subsamples of size T= 1000 for the exchange rate (13/03/2015 - 14/01/2019) and Natural gas (23/06/2009 - 28/06/2013). Figure 9 plots, in its …rst row, the estimated values of across the rolling window for the two series considered and for the three estimation methods applied, i.e., it plots the values of bQML,bQMLtand bLAD for each of the 4217 subseries of the exchange rate series (left-hand side panel) and for each of the 2368 subseries of the Natural gas series (right-hand side panel). As a benchmark, the zero line is also displayed to account for no leverage. Similarly, Figure 9 plots the corresponding p-values of both the t-ratio (2nd row) and the LR (3rd row) tests to test H0:= 0 against H0:6= 0. The horizontal line represents the 5% signi…cance level and consequently, those subsamples 5Notice that the t-ratio test can be computed using the three estimators considered but the LR test statistic is only calculated for QML and QML-t. 22
with p-values below this line are those where the leverage e¤ect is signi…cant at 5% signi…cance level. Several conclusions emerge from this …gure. First, we observe remarkable di¤erences among the three estimators considered: bQML,bQMLtand bLAD:Notice that, in both series, the leverage coe¢ cient is never statistically signi…cant when using the LAD estimator (see the p-values displayed in the 2nd row of Figure 9). This fact seems to be driven by the large standard errors of bLAD, as we mentioned previously. Second, we observe how extreme observations can bias the estimated leverage parameter and the corresponding test statistics and could lead to a wrong conclusion about the sign and magnitude of the leverage e¤ect. As expected, the QML estimated values of present several sharp drops and rises in both series. These sharp changes are usually due to the entrance and/or an exit of outlying observations in the corresponding subsample. For instance, in the exchange rate series (leftt-hand side panels), the entrance of the observation in June 24, 2016, where the series sustained its largest drop (y4551 =8:3), conveys a sudden jump in bQML from a negative value to a positive value around 0 (see the graph in the top left-hand side panel). Notice that, for most of the subseries before that date, where there are no outliers, the estimates bQML and bQMLtare very similar, in agreement with our simulation results in Section 3.2. However, in the last subseries around the end of the full sample, i.e., those including the isolated outlier due to the Brexit, there are big di¤erences between bQML and the robust estimate, bQMLt, which is mainly negative and similar to the other robust estimator, bLAD. Actually, when QML-t is used, the p-values of both signi…cance tests show that is statistically signi…cant for these subsamples, indicating the presence of leverage e¤ect. However, for these subsamples, both QML based tests are unable to reject the null, H0:= 0. Recall that, according to our results in Section 3, these di¤erences are expected in the presence of one isolated outlier, which seems to be the cause here of the upwards bias observed in bQML, as compared to the robust estimator bQMLt. When looking at the results for the Natural gas (right-hand side panels), the most remarkable di¤erences between QML and QML-t arise in the subseries located around the middle of the sample, i.e., in those including the two huge consecutive outliers present in this series (y1566 = 50:73 and y1567 =45:41). In these cases, our results in Section 3 show that even the robust estimators could still be slightly biased. In particular, when 23
these two observations, the …rst one being positive and the next one negative, are in a subsample, we expect bQML but also bQMLtto be upwards biased. Another remarkable feature from the Natural gas results in Figure 9, is that the values of bQML and bQMLt for the subseries at the end of the period, where there are no outliers (see the graph in the top right-hand side panel), are very similar to each other, as expected, but quite di¤erent from the values of bLAD: the former take negative values (standard leverage) whereas the latter estimate > 0(inverse leverage). This feature could be related to the fact that LAD is underperforming if no outliers are present in the series, as discussed in Section 3. Accordingly, for most of the subseries at the end of the period, neither the t-ratio nor the LR test statistics based on QML and QML-t reject the null, H0:= 0, in these subseries. Therefore, we wonder whether the inverse leverage e¤ect found in the Natural gas returns by some authors (see, for example, Kristoufek (2014) and the references therein) could be due to the harmful e¤ect of these consecutive outliers. Alternatively, the patterns of the estimated parameters in Figure 9 could be indicating time-varying leverage e¤ect, as suggested by Bandi and Reno (2012), Yu (2012) and Jensen and Maheu (2014), but this is a problem that is out of the scope of this paper. 5 Conclusions This paper analyzes the e¤ect of outliers on the estimation and testing for the leverage e¤ect in TGARCH models. It is shown that QML-t and LAD always outperform QML in the presence of outliers, as expected. Actually, one isolated outlier could lead QML to hide true leverage e¤ect whereas two consecutive outliers bias the QML estimated leverage in a direction that crucially depends on the sign of the …rst outlier. If this is negative (positive), QML underestimates (overestimates) the leverage parameter. Therefore, in these cases, QML could hide true leverage or estimate spurious asymmetries or asymmetries of the wrong sign. However, both robust estimators perform very well when there is one isolated outlier, but they are still slightly biased in the presence of patches of big outliers, leading, in some cases, to inaccurate estimates of the leverage coe¢ cient. In general, QML-t seems to outperform LAD because it is robust to moderate-big outliers without losing much e¢ ciency, as compared to QML, when there are no outliers. That is not the case for LAD, which performs much worse than QML with no outliers. These 24
results are further illustrated with the empirical analysis of two return series, including one isolated negative outlier and two consecutive outliers, respectively. Acknowledgments Financial support from the Spanish Government under projects ECO2017-87069-P and ECO2016-77900-P is gratefully acknowledged by the …rst and second authors, respectively. As usual, we are responsible for any remaining errors. 25
Table 2: Estimation of the TGARCH and AVGARCH models with QML, QML-t and LAD Estimator Parameter US Dollar/British Pound Natural gas TGARCH AVGARCH TGARCH AVGARCH QML !0:0065 (0:003) 0:0068 (0:003) 0:1019 (0:059) 0:1063 (0:075) 0:0479 (0:012) 0:0501 (0:012) 0:0749 (0:028) 0:0794 (0:036) 0:9519 (0:013) 0:9497 (0:013) 0:9173 (0:033) 0:9128 (0:044) 0:0097 (0:005) 0:0077 (0:018) Log-Likelihood 4109:04114:69028:79029:9 Residuals Q(20) CH(20) 19:434 19:921 19:434 19:844 22:239 15:991 22:167 15:875 Q2(20) CH2(20) 26:596 16:342 28:579 15:054 51:180 11:688 57:706 12:459 QML-t !0:0053 (0:002) 0:0056 (0:002) 0:0833 (0:021) 0:0788 (0:018) 0:0390 (0:006) 0:0418 (0:006) 0:0806 (0:010) 0:0813 (0:010) 0:9608 (0:007) 0:9580 (0:0107) 0:9144 (0:011) 0:9152 (0:011) 0:0111 (0:004) 0:0185 (0:007) 7:7403 7:5673 6:7493 6:7402 Log-Likelihood 4005:14011:38779:88784:0 Residuals Q(20) CH(20) 19:627 19:956 19:642 19:886 20:864 16:273 21:393 15:904 Q2(20) CH2(20) 31:772 16:652 35:232 15:509 25:893 12:371 47:318 14:131 LAD !0:0028 (0:136) 0:0024 (0:244) 0:0957 (0:143) 0:0667 (0:517) 0:0445 (0:020) 0:0428 (0:085) 0:0581 (0:073) 0:0523 (0:168) 0:9559 (0:028) 0:9580 (0:112) 0:9201 (0:073) 0:9343 (0:157) 0:0056 (0:065) 0:0217 (0:108) Residuals Q(20) CH(20) 20:072 20:482 20:277 20:589 20:864 16:273 24:863 16:631 Q2(20) CH2(20) 24:212 17:155 26:590 15:578 25:894 12:371 105:362 12:436 ; ;: statistically signi…cant at 10%, 5%, and 1%, respectively 32
Figure 1: Function (left panels) and its …rst derivative (right panels) in M-estimators -3 -2 -1 0 1 2 3 0 2 4 6 8 10 Function in M-estimators QML QML-t3 LAD -3 -2 -1 0 1 2 3 -2 0 2 4 6 8 10 1st derivative of function in M-estimators QML QML-t3 LAD 33
Figure 2: Boxplots of estimated with QML, QML-t and LAD in the presence of one isolated outlier -1 -0.5 0 0.5 =0 QML with 1 isolated outlier -1 -0.5 0 0.5 QML-t with 1 isolated outlier -1 -0.5 0 0.5 LAD with 1 isolated outlier -1 -0.5 0 0.5 =-0.05 -1 -0.5 0 0.5 -1 -0.5 0 0.5 -1 -0.5 0 0.5 =-0.11 Size of outlier -50 -40 -30 -20 -10 010 20 30 40 50 -1 -0.5 0 0.5 Size of outlier -50 -40 -30 -20 -10 010 20 30 40 50 -1 -0.5 0 0.5 Size of outlier -50 -40 -30 -20 -10 010 20 30 40 50 34
Figure 3: Boxplots of estimated with QML, QML-t and LAD in the presence of two consecutive outliers -1 -0.5 0 0.5 =0 QML with 2 consecutive outliers -1 -0.5 0 0.5 QML-t with 2 consecutive outliers -1 -0.5 0 0.5 LAD with 2 consecutive outliers -1 -0.5 0 0.5 =-0.05 -1 -0.5 0 0.5 -1 -0.5 0 0.5 -1 -0.5 0 0.5 =-0.11 Size of outlier -50 -40 -30 -20 -10 010 20 30 40 50 -1 -0.5 0 0.5 Size of outlier -50 -40 -30 -20 -10 010 20 30 40 50 -1 -0.5 0 0.5 Size of outlier -50 -40 -30 -20 -10 010 20 30 40 50 35
Figure 4: Boxplots of estimated with QML-t and LAD in the presence of one isolated and two consecutive outliers -0.1 0 0.1 =0 QML-t with 1 isolated outlier -0.1 0 0.1 LAD with 1 isolated outlier -0.1 0 0.1 QML-t with 2 consecutive outliers -0.1 0 0.1 LAD with 2 consecutive outliers -0.2 -0.1 0 =-0.05 -0.2 -0.1 0 -0.2 -0.1 0 -0.2 -0.1 0 -0.2 -0.1 0 =-0.11 Size of outlier -30 -20 -10 010 20 30 -0.2 -0.1 0 Size of outlier -30 -20 -10 010 20 30 -0.2 -0.1 0 Size of outlier -30 -20 -10 010 20 30 -0.2 -0.1 0 Size of outlier -30 -20 -10 010 20 30 36
Figure 5: Monte Carlo size. Proportion of rejections based on 5% critical value of t test (left panels) and LR test (right panels) under H0:= 0, with one isolated outlier (top panels) and two consecutive outliers (bottom panels) -50 050 0 0.2 0.4 0.6 0.8 1 One single outlier t-ratio test QML QML-t -50 050 0 0.2 0.4 0.6 0.8 1 LR test -50 050 Size of the outlier(s) 0 0.2 0.4 0.6 0.8 1 Two consecutive outliers -50 050 Size of the outlier(s) 0 0.2 0.4 0.6 0.8 1 37
Figure 6: Monte Carlo power. Proportion of rejections based on 5% critical value of t test (left panels) and LR test (right panels) under H1:=0:11, with one isolated outlier (top panels) and two consecutive outliers (bottom panels) -50 050 0 0.2 0.4 0.6 0.8 1 One single outlier t-ratio test QML QML-t -50 050 0 0.2 0.4 0.6 0.8 1 LR test -50 050 Size of the outlier(s) 0 0.2 0.4 0.6 0.8 1 Two consecutive outliers -50 050 Size of the outlier(s) 0 0.2 0.4 0.6 0.8 1 38
Figure 7: Daily returns for the exchange rate US Dollar/British Pound and future contracts of Natural gas 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 -10 -5 0 5Exchange rate returns (US Dollar/British Pound) 2002 2004 2006 2008 2010 2012 -60 -40 -20 0 20 40 60 Natural gas returns 39
Figure 8: Correlograms of returns and squared returns and cross-correlograms between lagged returns and squared returns US/Pound 0 5 10 15 20 -0.1 0 0.1 r(h) Natural gas 0 5 10 15 20 -0.1 0 0.1 0 5 10 15 20 0 0.2 0.4 r2(h) 0 5 10 15 20 0 0.2 0.4 0 5 10 15 20 0 0.2 0.4 Robust r 2(h) 0 5 10 15 20 0 0.2 0.4 0 5 10 15 20 -0.1 0 0.1 r12 (h) 0 5 10 15 20 -0.1 0 0.1 0 5 10 15 20 Lag -0.1 0 0.1 Robust r 12 (h) 0 5 10 15 20 Lag -0.1 0 0.1 40
Figure 9: Leverage parameter in a TGARCH model estimated with subsamples of size T = 1000 using a rolling window of both the Exchange rate US Dollar/British Pound and Natural gas daily returns (top panels) and the corresponging p-values of the t and LR tests for the statistical signi…cance of . 0500 1000 1500 2000 2500 3000 3500 4000 -0.1 0 0.1 Estimated Exchange rate (US Dollar/British Pound) QML QML-t LADE 0500 1000 1500 2000 -0.1 0 0.1 Natural Gas 0500 1000 1500 2000 2500 3000 3500 4000 0 0.5 1 p-values t-ratio 0500 1000 1500 2000 0 0.5 1 0500 1000 1500 2000 2500 3000 3500 4000 Subseries 0 0.5 1 p-values LR 0500 1000 1500 2000 Subseries 0 0.5 1 41