Robust M-Estimation of Multivariate GARCH Models

Transcription

1 Robust M-Estimation of Multivariate GARCH Models Kris Boudt Christophe Croux Faculty of Business and Economics, K.U.Leuven, Belgium Abstract In empirical work on multivariate financial time series, it is common to postulate a Multivariate GARCH model. Due to measurement error or unusual economic events, some observations can be in discordance with the model assumptions. It is desirable that these outliers have little influence on the estimation result. In a Monte Carlo study, we show that the Gaussian quasi-maximum likelihood estimator can be highly affected by outliers. As a more robust alternative, we propose to use M-estimators. By downweighting extreme returns in the loss function of the M-estimator and in the MGARCH equation, we obtain robust parameter estimates and conditional covariance matrix predictions. We prove consistency of a wide class of M-estimators for MGARCH models with elliptical innovations. Simulations and a real data example document the benefits of the robust approach. JEL classification: C13; C32; C51 Keywords: GARCH models, M-estimators, multivariate time series, outliers, robust methods Correspondence to: Kris Boudt, Faculty of Business and Economics, 69 Naamsestraat, B-3000 Leuven, Belgium. Kris.Boudt@econ.kuleuven.be Tel: Fax:

2 1 Introduction It is widely recognized that the volatility and the correlations of daily and weekly financial returns may be time-varying. There is no doubt that incorporating these features into the estimation of the conditional covariance matrix of financial returns can lead to better decisions on portfolio optimization, asset pricing and risk management. One way to do this is to specify the conditional covariance matrix as a deterministic function of the innovations in the mean equation. This approach defines the class of Multivariate GARCH (MGARCH) models (see Bauwens et al., 2006, for a recent survey). We propose a new estimator and a new MGARCH volatility forecasting model that, compared to the classical MGARCH methods, have the advantage of being robust to outliers in the data. It is common to estimate MGARCH models by maximizing a Gaussian likelihood function. Jeantheau (1998) shows that this estimation procedure, called the Gaussian Quasi-Maximum Likelihood (QML) estimation, can produce consistent parameter estimates, even when the true distribution is not Gaussian. This is an important result, since it is common to find that after correcting the returns for the dynamics in the conditional covariance matrix, the marginal distribution of the standardized return series is still heavy tailed. Engle and González-Rivera (1991) show that, for the univariate GARCH model, 2

3 the QML estimator has a degree of inefficiency that increases with the degree of departure from normality. A current practice, therefore, consists in doing ML estimation under the assumption of a multivariate Student t distribution (Fiorentini et al., 2003, and Pesaran and Pesaran, 2007). According to Franses and Ghijsels (1999) the leptokurtosis in the standardized returns can also be interpreted as an indication of outliers in the data set. The effect of outliers on estimating univariate GARCH models has been studied in the literature and robust estimators have been proposed. One approach to estimating GARCH models in the presence of outliers is to prune iteratively the outliers and fit the model to the remaining data until no more outliers are detected (Franses and Ghijsels, 1999). Another approach is to use an estimator that is robust to outliers. Sakata and White (1998), Park (2002) and Peng and Yao (2003) estimate the GARCH model by minimizing a robust measure of scale of the residuals. Mancini et al. (2005) and Muler and Yohai (2002,2007) propose a robust M-estimator that assigns a much lower weight to outliers than the Gaussian ML estimator does. Newey and Steigerwald (1997) prove consistency of the Student t Quasi-Maximum Likelihood (QML) estimator, provided the true innovation density is unimodal and symmetric around zero. All authors find that in the presence of outliers, the Gaussian QML estimator collapses and that robust estimators are more reliable. Van Dijk et 3

4 al. (1999) find that LM-tests for conditional heteroscedasticity perform better when they are based on robust parameter estimates rather than on the Gaussian QML estimates. Muler and Yohai (2002,2007) argue that in the presence of outliers, it may be more prudent to specify the conditional variance as a bounded function of past returns. Doing so, one limits the impact of outliers on subsequent conditional variance predictions which depend in an autoregressive way on past returns. To our best knowledge, we are the first to study the effect of outliers on the Gaussian QML estimator of multivariate GARCH models and to propose robust alternatives. In Section 2 we prove consistency of M-estimators of MGARCH models with elliptical innovations. In Section 3 we argue that the Gaussian QML estimator can be made more robust by replacing its loss function with a loss function whose derivative increases more rapidly, and by bounding the impact of news on the conditional covariance matrix process. Section 4 proposes new diagnostic checks for outlier identification and for correct model specification. Section 5 illustrates the new methodology through a Monte Carlo study and a financial application. We show that our proposal of a robust M-estimator has a reasonably good efficiency at the assumed model and that, in contrast with the Gaussian QML estimator, it remains reliable when the data is outlier contaminated. In the empirical case of the daily returns on the Nasdaq and NYSE indices, we find that our proposal for a robust 4

5 MGARCH volatility prediction model reflects better reality than the classical model, especially regarding the difference in reaction of volatility to the 1987 stock market crash and the burst of the internet bubble in Section 6 summarizes our conclusions and outlines the implications for further research. 2 M-estimation 2.1 Definition We consider a centered, conditionally elliptically distributed N-variate random process {r t } whose conditional covariance matrix process {H t,θ } is an MGARCH process. Under this model, each random variable r t can be expressed as r t = H 1/2 t,θ u t, (1) where u t is a zero mean random variable with an elliptically symmetric distribution. Let I rt 1 be the information incorporated in the past of r t. It follows that under this model P θ, the probability of observing r t conditional on its past is described by the elliptic density function p(r t I rt 1 ; θ) = (det H t,θ ) 1/2 g ( r t H 1 t,θ r t), (2) where the function g : R + R +. The scatter matrix H t,θ and the function g( ) are assumed to be standardized such that H t,θ is the conditional covariance 5

6 matrix of r t. While there exist elliptic distributions without finite second moment, we consider here only distributions for which the covariance matrix exists. A common choice of g( ) is the standard normal density function which we denote by φ( ). Another popular choice for g( ) is the standardized Student t density function with ν > 2 degrees of freedom: t ν (z) = Γ ( ) ν+n 2 Γ ( ) N ν 2 [π(ν 2)] 2 [ 1 + z ] N+ν 2, (3) ν 2 where Γ( ) is the gamma function. A useful result is that under (2), the squared Mahalanobis distance is i.i.d. distributed with density function d 2 t,θ = r t H 1 t,θ r t = u t u t (4) f(z) = 2πN/2 Γ(N/2) z N 1 2 g(z). (5) In the special case where g( ) is the Gaussian density function, f( ) is the chisquared density function with N degrees of freedom. We denote F 1 ( ) the quantile function associated to f( ). Under P θ, the conditional covariance matrix H t,θ is parameterized by a MGARCH model as a measurable function of the past innovation terms u t. More precisely, we assume that H t,θ = H θ (u t 1, u t 2,...), (6) with H θ ( ) a N N positive definite, symmetric matrix function of a sequence 6

7 of N-dimensional vectors. As an illustration, consider the BEKK parameterization for H t,θ (Engle and Kroner, 1995): H t,θ = C C + A H 1/2 t 1,θ u t 1u t 1 H1/2 t 1,θ A + B H t 1,θ B, (7) where A, B and C are N N parameter matrices and C is upper triangular. Denote θ the true, unknown parameter vector belonging to the parameter space Θ and g ( ) the true density function in (2). It is common to estimate θ by the value of θ Θ that maximizes the joint density of the sample S T = {r 1,..., r T } constructed under the nominal assumption that the density function of the data is g( ). This estimation approach yields the QML estimator ˆθ QML = argmax θ Θ ave t [ log det Ht,θ 2 log g ( r t H 1 t,θ r t)], (8) where ave t stands for the arithmetic average over t = 1,..., T. It coincides with the ML estimator when g( ) = g ( ). The ML estimator belongs to the broader class of M-estimators defined as follows. Definition 1 The M-estimate based on a sample S T = {r 1,..., r T } is the value of θ Θ for which the M-function M (S T ; θ, ρ) = ave t [ log det Ht,θ + σ ρ ( r t H 1 t,θ r t)], (9) is minimized, with ρ( ) a positive, non-decreasing scalar function. The scalar σ is a consistency factor. 7

8 The function ρ( ) in the above definition is called the loss function associated to the M-estimator. If ρ( ) = 2 log g( ), with g( ) a specified density function, we obtain a QML estimator. The Gaussian φ and standardized Student t ν density functions are the most popular elliptical density functions used to describe financial return series. The corresponding Gaussian and Student t ν (ν > 2) loss functions are given by [ ρ φ (z) = z ; ρ tν (z) = (N + ν) log 1 + z ]. (10) ν 2 The consistency factor σ depends on the true density function g ( ) and on ψ( ) = ρ ( ), the first derivative of the function ρ( ). Denote E Pθ the expectation under P θ. By the consistency proof in Appendix A and by formula (5), we have that σ equals { 2π σ = N/E Pθ ψ(u t u t)u N/2 t u t = N Γ(N/2) 0 ψ(t 2 )t N+1 g (t 2 )dt} 1. (11) For the M-estimator with Gaussian loss function (being the Gaussian QML estimator) the consistency factor is one for all elliptical density functions g, whatever be the dimension of the series (N). For all loss functions, we have that the consistency factor is one when the loss function is proportional to log g ( ). In this case, the M-estimator coincides with the ML estimator. In Figure 1 we illustrate for two different loss functions ρ tν with ν = 20 and ν = 4, the dependence of the consistency factor on the number of degrees of freedom ν for bivariate Student t ν distributed innovations. Note that 8

9 Figure 1: Sensitivity of the consistency factor σ for two Student t ν loss functions (ν = 20 and ν = 10) to the number of degrees of freedom ν for bivariate Student t ν distributed innovations. σ * ρ t20 ρ t ν * the consistency factor σ is most sensitive to ν when ν is small. When the econometrician believes the innovations in the MGARCH model are Student t distributed but has no idea on the value of ν, then, as a practical guideline, we recommend to calculate the consistency factor for an intermediate value of ν such as ν =. The Monte Carlo study of Subsection 5.3 documents the good estimation properties of this approach. If the return process is conditionally normal, the M-estimator with Gaussian loss function is efficient. The Monte Carlo experiment in Section 5 shows that, in the presence of outliers, the M-estimator with Gaussian loss function 9

10 is no longer reliable and that, the smaller ν is in the loss function ρ tν ( ), the more robust the M-estimator with Student t ν loss function will be. First, we give a general description of the asymptotic properties of M-estimators. 2.2 Asymptotic properties In Appendix A we prove consistency for a wide range of M-estimators, defined above as the value of θ Θ that minimizes the time series average of m (r t ; θ, ρ) = log det H t,θ + σ ρ ( r t H 1 t,θ r t). The consistency result is obtained under the following assumptions on the loss function ρ( ), on the functional form H θ ( ) of the conditional covariance matrix and on the density function g ( ) of the innovations. In parenthesis, we indicate what the assumptions are needed for. A1 (Existence and uniqueness) The loss function ρ( ) has the following properties. 1. Its derivative ψ( ) = ρ ( ) is nonnegative, nonincreasing and continuous. 2. The function ψ(z)z is bounded. Let K = sup z 0 ψ(z)z. 3. The function ψ(z)z is nondecreasing, and is strictly increasing in the interval where ψ(z)z < K. 4. There exists z 0 such that ψ(z0 2)z2 0 > N and that ψ(z)z > 0 for z z There exists a > 0 such that for every hyperplane H, the probability under P θ for an observation to lie in H is at most 1 N/K a. 10

11 A2 (Consistency) p(r t I rt 1 ; θ ) = (det H t,θ ) 1/2 g ( r t H 1 t,θ r t ). A3 (Identifiability) H t,θ = H t,θ θ = θ. A4 (Uniform convergence) The parameter space Θ is compact with E Pθ sup θ Θ m (r t ; θ, ρ) / θ <. A5 (Uniqueness) The function H θ ( ) is such that, with probability one, the gradient of the M-function can only be zero if H t,θ = E Pθ [ σ ψ(r t H 1 t,θ r t)r t r t I rt 1 ]. Proposition 1 Under A1-A5 the M-estimator in Definition 1 is a consistent estimator for θ. The key idea behind this result is that, under these assumptions, the conditional covariance matrix evaluated at the M-estimate for θ will, asymptotically, be equivalent to Maronna (1976) s M-estimator for the scale matrix, conditional on the past realizations. Consistency of the M-estimator for MGARCH models follows from the consistency of Maronna (1976) s M-estimator for the unconditional scale matrix. Assumption A1 and A2 are needed to ensure consistency of the unconditional M-estimator of scale. Note that the Gaussian and Student t loss functions satisfy assumption A1. Consistency is not guaranteed for bounded loss functions, because for bounded ρ( ), it is impossible that ψ(z)z is nondecreasing and strictly increasing in the interval where 11

12 ψ(z)z < K. The assumption A2 of elliptical symmetry guarantees the estimator to be asymptotically unbiased. When the u i s are i.i.d. distributed, but not elliptically symmetric, the M-estimator will still converge (Chapter 8 in Huber, 1981). Finally note that, under regularity conditions, M-estimators are asymptotically normal with asymptotic covariance matrix (Chapter 7 in Hayashi, 2000): { [ ]} 2 1 m(r t ; θ, ρ) E Pθ Σ θ θ { [ ]} 2 1 m(r t ; θ, ρ) E Pθ, (12) θ θ where Σ is the long run covariance matrix of m(r t ; θ ; ρ)/ θ. 3 Robust M-estimation For many applications, a fully specified MGARCH time series model with elliptical innovations is at most a very good approximation of the true data generating process. It may explain well the bulk of the observations, but realworld time series will almost always contain some outlying observations or discrepant substructures. In a conditionally heteroscedastic time series setting, outliers can be defined as observations that are extremely unusual given the past of the series. It is common to do as if there were no outliers in the data and to estimate θ by the value of θ Θ that minimizes the M-function (9). In the Monte Carlo experiment of Section 5, we show that such an approach still yields reliable results in the presence of outliers if (i) a loss function with a derivative 12

13 that decreases sufficiently fast towards zero, is used; (ii) a specification for H t,θ is taken under which innovations have a limited impact on the conditional covariance matrix process. 3.1 The loss function An intuitive view on M-estimation of MGARCH models is that it consists in finding the value of θ Θ for which the estimated conditional covariance matrix equals the conditional expectation of a weighted covariance matrix, with weights proportional to ψ(r t H 1 t,θ r t). Indeed, under the assumptions listed above, the M-estimator is asymptotically equal to the value of θ Θ which solves the following set of equations H t,θ = E Pθ [ σ ψ(r t H 1 t,θ r t)r t r t I r t 1 ]. Note how important the shape of ψ( ), the first derivative of the loss function ρ( ), is. It determines the weight given to each observation as a function of the squared Mahalanobis distance d 2 t,θ in (4). Since outliers are characterized by a large value of d 2 t,θ = r t H 1 t,θ r t, robust M-estimators are characterized by a loss function whose derivative, for large values of its argument, decreases sufficiently fast and tends to zero at infinity. The derivatives of the Gaussian and Student t ν loss functions equal ρ φ (z) = 1 ; ρ t ν (z) = N + ν ν 2 + z. (13) 13

14 We see that the M-estimator with Gaussian loss function is not robust because it gives the same weight to all realizations irrespective of their degree of outlyingness, as measured by d 2 t,θ. However, the derivative of the Student t ν loss functon is decreasing and tends to zero at infinity. The smaller ν is, the more robust the M-estimator with Student t ν loss function will be. 3.2 Impact of news on subsequent volatility predictions Most MGARCH time series models specify the elements of the conditional covariance matrix as an unbounded function of a distributed lag of the squares and cross-products of the components of the innovation vector u t. It follows that the news impact curve, which measures the effect of the innovations on the one-step ahead conditional covariance matrix (Engle and Ng, 1993), is unbounded. Moreover, because of the autoregressive structure of the process, the impact of news propagates to future values of the conditional covariance matrix process. Muler and Yohai (2002,2007) stress that because of this property of unbounded innovation propagation, neglecting outliers renders any inference on autoregressive time series processes extremely sensitive to outliers. For GARCH processes this is aggravated by the fact that the news impact curve is typically quadratic. Hence, neglected outliers may have a very adverse effect on subsequent values of the estimated volatility process. It is possible to bound the impact of outliers on the conditional covariance 14

15 matrix by specifying it as a function H θ ( ) of the weighted innovations ũ t = w(d 2 t,θ )u t instead of the raw innovations u t, where the weight function w( ) downweights extreme observations, that is observations with a large value of d 2 t,θ = r t H 1 t,θ r t = u t u t. When H θ ( ) has the BEKK specification (7), we obtain the following econometric model for conditional covariance matrix: H t,θ = H θ (ũ t 1, ũ t 2,...) = C C + w 2 (d 2 t 1,θ )A H1/2 t 1,θ u t 1u 1/2 t 1 H t 1,θ A + B Ht 1,θ B. (14) For numerical reasons, the weight function w( ) needs to be smooth. In the application we will use Tukey s bisquare weight function: w(z) = min { 1, 1 ( 1 (c/z) 2) 3 }, (15) with c the 95% quantile of u t u t, as defined in (5). Note that only the observations with squared distance above its 95% quantile are downweighted. By construction the weight function tends to zero at infinity such that the impact of outliers on the conditional covariance matrix is limited. The idea of robustifying the conditional covariance matrix against outliers in the data by downweighting past observations of r t in the GARCH specification has already been pursued by Muler and Yohai (2002,2007) in the univariate case. We follow them in calling this the Bounded Innovation Propagation (BIP) GARCH model. 15

16 Muler and Yohai (2002,2007) estimate the parameters of the univariate GARCH model by minimizing the M-function corresponding to the BIP-GARCH model and call the newly obtained estimator the BIP M-estimator. This approach readily extends to the multivariate case in which the BIP M-estimator is defined as the value of θ Θ that minimizes the BIP M-function [ M (S T ; θ, ρ) = ave log det H t,θ t ( + σ ρ r t )] 1 H t,θ r t. (16) 4 Robust diagnostic checks In the absence of outliers and under correct model specification, the squared Mahalanobis distances of the observations in terms of the M-estimate for the conditional covariance matrix, are asymptotically i.i.d. distributed with density function f( ) given in (5). In the same spirit as Rousseeuw and van Zomeren (1990) in the unconditional case, one can use this result to identify MGARCH outliers as these observations for which the estimated squared distance d 2 t,θ exceeds an extreme quantile of f( ). The above result can also be used for verifying correct model specification, using the following test statistic κ = T 1 T T t=1 w(r t H 1 t,θ r t)r t H 1 t,θ r t m, (17) s 16

17 with w( ) a weight function and m = 2πN/2 Γ(N/2) [ 2π N/2 s = Γ(N/2) 0 0 w(t 2 )t N+1 g(t 2 )dt, (18) [ ψ(t 2 )t 2 m ]2 t N 1 g(t )dt] (19) Since under the model assumptions made previously, the distances are conditionally independent and the M-estimator is consistent, one can invoke a central limit theorem to state that this statistic is asymptotically standard normal distributed under correct model specification. If a weight function is used that downweights extreme observations, then this result will be robust to a small proportion of outliers in the data. Its robustness can be increased by using the BIP conditional covariance matrix specification. The null of correct model specification can be tested by comparing κ with the critical values of the standard normal distribution. 5 Application This section illustrates the usefulness of robust estimators and robust MGARCH volatility forecasting models. We do this analysis for the BEKK model, which is a popular and representative member of the family of MGARCH models. We first report the results of a Monte Carlo study on the relative estimation performance of the Gaussian QML estimator, the M-estimator with Student t 4 loss function and the BIP counterparts of these two estimators. We find 17

18 that the BIP M-estimator with Student t 4 loss function calculated under the assumption of normality is robust to additive outliers and misspecification of the consistency factor. We then compare the forecasting performance of the BEKK and BIP-BEKK model for the daily return series of the Nasdaq and NYSE stock market indices. 5.1 A specific MGARCH Model We consider the bivariate symmetric BEKK volatility model of order one, with Student t innovations, proposed by Engle and Kroner (1995). Under this model, the data generating process is given by the following set of equations: i.i.d. r t I rt 1 t ν (0, H t,θ ) ; ν > 2 (20) H t,θ = C C + A r t 1 r t 1A + B H t 1,θ B. The parameter matrices C, A and B all denote 2 2 coefficient matrices. The matrix C is upper triangular and the matrices A and B are symmetric. Let θ be the vector of length 9 that stacks the upper diagonal elements of the matrices C, A and B one on top of the other. Observe that each element of the conditional covariance matrix is specified as a linear function of the lagged squares and cross-products of the elements of r t as well as of the lags of the elements of the conditional covariance matrix. As such, the BEKK model can replicate the stylized fact that volatility clusters over time and co-moves across assets. 18

19 The way in which the conditional covariance matrix process evolves in time depends on the parameter values and the realizations of the stochastic process {u t }. In general, one wants the process H t,θ to be positive definite and covariance stationary whatever be the realizations of {u t }. Engle and Kroner (1995) and Altay-Salih et al. (2003) show that this is guaranteed under the following constraints on the parameter space: C 11 > 0, C 22 > 0, A 11 > 0, B 11 > 0 (21) I 4 A A B B is positive definite. (22) We find the solution of this estimation problem using the DONLP2 sequential quadratic programming algorithm. 5.2 Robustness of M-estimators to additive outliers This section aims to compare the behavior of the (BIP) M-estimators with Gaussian and Student t 4 loss function for simulated data with different levels of outlier contamination. The artificial time series are generated as follows. First outlier-free time series {y t } of length 1000 are generated from model (20), 19

20 with parameter matrices 1 C C = A = B = We then add outliers to the clean series using the following probabilistic setup:. r t = y t + 5ξ t d t. (23) Under this model the occurrence of an outlier in any of the two series is governed by the random process {d t }, modeled as a sequence of i.i.d. draws from a Bernoulli distribution where the occurrence of an outlier (d t = 1) has probability ε. The transmission and the size of the outliers stems from a 2-dimensional i.i.d. vector process {ξ t } constructed such that, when an outlier occurs, it will be either in the first component, second component or both with probability 0.3, 0.3 and 0.4, respectively. The magnitude of the outlier is 5 times the conditional standard deviation of the corresponding element of y t. Since M-estimators are defined by the extremum of the M-function, it is important that the location of the extremum as a function of the parameter vector is robust to outliers in the data. In Figure 2 the M-function (9) with Gaussian and Student t 4 loss function are plotted as a function of the parameter B 22, once in the absence and once in the presence of 5% of outliers. 2 We see 1 The dependence parameter matrices A and B are similar to those in Chapter 18 of Lütkepohl (2005), while the matrix C is chosen such that it has a similar scale as A and B. 2 To save space, we do not report this figure for the BIP M-functions. The main result 20

21 that in the absence of outliers, all M-functions reach an extremum close to the true parameter value, which is 0.8. Because of the quadratic forms in the BEKK specification, the M-function reaches also a local minimum around θ 9 = 0.8. Hence, the importance of having good starting values and of imposing the bound constraints (21) in the optimization process. Note that contaminating 5% of the data with additive outliers increases strongly the variability of the M-function. The magnitude of this outlier induced variability depends on the loss function. For the Gaussian loss function, the variability of the M-function is so large that there is no more visible evidence that the M-function reaches an extremum close to the true parameter value. The outlier induced variability is much smaller for the M-function with Student t 4 loss function and it does not obscure the location of the extremum. is that the M and BIP M-functions that share the same loss function are very similar, even when 5% of the observations are outlier-contaminated. 21

22 Figure 2: Sensitivity of (BIP) M-functions with Gaussian and Student t 4 loss function to the levels of contamination ε. The M-function is taken as a function of B 22 only; the other parameters being fixed at their true values, which is 0.8 for B 22. Gaussian loss: ρ φ Student t 4 loss: ρ t No outliers No outliers 5% outliers 5% outliers M3theta9[, 3] B B 22 A current practice in MGARCH modelling of financial time series is to assume that the innovations are Student t ν distributed and to treat the number of degrees of freedom ν as an additional parameter (Pesaran and Pesaran, 2007). Figure 2 shows that M-estimation with a Student t ν loss function, where the parameter ν in the loss function is an additional estimation parameter, is no longer reliable in the presence of outliers. In Figure 2 the M-functions are 22

23 computed for Gaussian innovations (ν = ). For joint M-estimation of θ and ν to be a reliable estimation procedure, we would need that ρ t (θ) < ρ t4 (θ) for θ close to the true parameter value. We see that this is the case in the absence of outliers, but because outliers blow up the M-function with loss function ρ t ( ) much more than when the loss function ρ t4 ( ) is used, this is no longer true in the presence of outliers. It seems thus that M-estimation of θ and ν jointly is no longer reliable in the presence of outliers. Of course this is only a partial analysis. To study the effect of outlier contamination on the bias and efficiency of the (BIP) M-estimator with (un)bounded Gaussian or Student t 4 loss function, we consider 200 replications of (23) and three levels of outlier contamination (ε = 0, ε = 0.01 and ε = 0.05). For each M-estimator and for each type of simulated series, we aggregate the parameter estimates to compute the mean error (ME) and the root mean squared error (RMSE) as follows, ME = ave ˆθj θ, RMSE = ave ˆθ j θ 2, (24) j j where ave j denotes the average across the 200 replications and is the Euclidian norm operator. These summary statistics are reported in Table 1. We find that in the absence of outliers (ε = 0), the estimation of the BEKK model on the basis of the BIP-BEKK model does not seem to have a large impact on the bias of the M-estimator. 23

24 When the series are outlier contaminated (ε = 0.01 or ε = 0.05), an overall result is that outliers adversely affect the bias as well as the efficiency of all M-estimators. Regarding the relative performance of the M-estimators with Gaussian and Student t 4 loss function, we find that when 1% of the data is outlier contaminated, the M-estimator with Student t 4 loss function is clearly more robust, but when 5% of the observations are affected by outliers, then both estimators collapse. The lack of robustness of the M-estimator with a Student t 4 loss function is because of the unbounded innovation propagation in the conditional covariance matrix process. The BIP version of this estimator should thus be used. A conclusion of this simulation experiment is that, among the considered estimators, no M-estimator yields better robustness results than the BIP M-estimator with Student t 4 loss function. The estimation of the BEKK model is a non-linearly constrained non-linear optimization problem. This is a computationally difficult problem to solve, especially in the real case where data, because of outliers or model misspecification, do not exactly satisfy the model assumptions. Consider e.g. the case where the clean series has BEKK covariance dynamics, but the econometrician assumes a BIP-BEKK model. In this case, the estimation routine does not converge in approximately 10% of the cases. In Table 1 we also see that the convergence frequency of the M-estimator with Gaussian loss function deteriorates when the data contain outliers, while the convergence behavior of the 24

25 Table 1: Mean error (ME), root mean squared error (RMSE) and convergence frequency of the (BIP) M-estimators with Gaussian and Student t 4 loss function, when a proportion ε of the Gaussian BEKK simulated data are outlier contaminated. Loss function Gaussian loss Student t 4 loss BIP no yes no yes ε = 0 ME RMSE conv. freq ε = 0.01 ME RMSE conv. freq ε = 0.05 ME RMSE conv. freq M-estimator with Student t 4 loss function seems to be more robust to outlier contamination. We find thus that weighting the observations in function of their outlyingness, leads to a more stable objective function and renders the optimization procedure feasible in instances in which Gaussian QML estimation is not. 25

26 5.3 Robustness of M-estimators to misspecification of the consistency factor An issue in applying the M-estimator in practice is that it requires knowledge of the true density function through the definition of the consistency factor σ. For the M-estimator to be truly robust, it must be that misspecification of σ should impair the performance only slightly. In Table 2 we report the ME and RMSE for the Gaussian and Student t 4 M-estimator, when the innovation vector in the bivariate BEKK model (20) is Gaussian, Student t 10 or Student t 4 distributed. For each situation, we do M-estimation with a Student t 4 loss function for a consistency factor computed under the nominal assumption that the innovations that are Gaussian, Student t 10 or Student t 4 distributed. Alike Engle and González-Rivera (1991) for the univariate GARCH model, we find that the Gaussian QML estimator has a degree of inefficiency that increases with the degree of departure from normality. For Gaussian innovations, the Gaussian QML estimator has a RMSE of 0.266, while for Student t 4 innovations the RMSE is If the M-estimator with Student t 4 loss function is correctly specified, it has a RMSE of for Student t 4 innovations and a RMSE of for Gaussian innovations. Of course, in real applications the innovation density function is unknown to the econometrician. For financial time series, a Student t density function 26

27 Table 2: Mean error (ME) and root mean squared error (RMSE) of the Gaussian QML estimator and the M-estimators with Student t 4 loss function and consistency factor computed for a Student t density function with ν degrees of freedom. Loss function Gaussian loss Student t 4 loss with σ computed for: ν = ν = 10 ν = 4 Gaussian innovations ME RMSE Student t 10 innovations ME RMSE Student t 4 innovations ME RMSE is often assumed with an unknown number of degrees of freedom ν. When the assumed number of degrees of freedom does not equal the population values, the consistency factor and hence the Student t 4 M-function are misspecified and the consistency result no longer holds. Recall that for the Gaussian QML estimator, there is no problem of correct specification, since for this estimator the consistency factor is always 1. In Table 2 it seems that for the M-estimator with Student t 4 loss function, the estimation performance is more robust to overestimation of the true number of degrees of freedom than to underestima- 27

28 tion of it. Of all M-estimators with Student t ν loss function considered, the estimator with σ calculated for ν = 10 is most robust to misspecification of σ. Its finite sample bias varies between (correct specification) and (Student t 4 innovations), while for the other M-estimators the finite-sample bias more than doubles when σ is misspecified. If the true number of degrees of freedom is unknown, we therefore recommend to calculate the consistency factor σ for an intermediate value of ν such as ν = 10. In Table 2 we do not report the convergence frequencies, because for all M-estimators with Gaussian and Student t 4 loss function, it was close to 1. A failure of convergence is thus not due to the thickness of the tails nor due to misspecification of the consistency factor. The decrease in convergence frequency when the data are contaminated with additive outliers or when the BIP M-estimators are used (Table 1), indicate that a failure to converge is a manifestation of additive outliers and/or MGARCH model misspecification. 5.4 Empirical application We apply the (BIP) M-estimators with Gaussian and Student t 4 loss function to daily returns of the Nasdaq and NYSE composite indices over the period January 2, December 29, The data were downloaded from datastream and consist of 7043 observations. The series are plotted in Figure 3. Note the large negative return corresponding to the stock market crash 28

29 Figure 3: Daily returns on Nasdaq and NYSE composite index. Nasdaq NYSE 12/80 06/86 11/91 05/97 11/02 of October 19, Interestingly, the Nasdaq index had returns of similar magnitude in the year In an unconditional framework, these returns are qualified as outliers, whereas under a conditionally heteroscedastic time series model, some of these extreme returns can be explained by the model. Observe that not only there is volatility clustering in both series, but also that there is a lot of commonality in volatility across the two series. We assume that the very large majority of these observations are generated by the conditionally Gaussian BEKK time series process in (20). Return realizations such as the 1987 stock market crash are clearly outliers under this 29

30 Figure 4: Conditional BEKK and BIP-BEKK standard deviation of the daily Nasdaq and NYSE return series BEKK, Nasdaq BIP BEKK, Nasdaq BEKK, NYSE BIP BEKK, NYSE 12/80 06/86 11/91 05/97 11/02 12/80 06/86 11/91 05/97 11/02 parametric framework. Hence a robust estimator is needed. The M-estimator with Student t 4 loss function yields the following parameter estimate Ĉ C = Â = ˆB = The sequential quadratic programming algorithm failed to converge for the (BIP) M-estimator with Gaussian loss function and for the BIP M-estimator 30.

31 with Student t 4 loss function. Since in this subsection, the focus is on comparing the BEKK and BIP-BEKK volatility forecasting models, given a reliable parameter estimate, we abstract from this problem. It is of particular interest to compare the BEKK and BIP-BEKK volatility predictions for the period of the 1987 stock market crash and the period of the burst of the internet bubble in In each of these periods, large returns of similar magnitude occur but the persistence in volatility is different. The stock market crash in 1987 was not followed by returns of similar size whereas the burst of the internet bubble corresponds to a prolonged period of high volatility. As can be seen in Figure 4, we have that due to the property of a limited effect of innovations on future volatility, the BIP-BEKK model captures well this difference in reaction to shocks. In contrast, the BEKK model overestimates the persistence in the realized return variability in the period subsequent to the 1987 crash. The same interpretation holds for the patch of extreme returns on the Nasdaq index on March 27-28, At all other times, the BEKK and BIP-BEKK conditional standard deviation forecasts are similar. 31

32 6 Concluding remarks This paper introduces a class of M-estimators for multivariate GARCH time series models. We prove that under general conditions M-estimators are consistent. We study the effect of the shape of the loss function and of bounding the news impact curve on the robustness of the M-estimator to outliers. The popular Gaussian quasi-maximum likelihood estimator is shown to be very sensitive to outliers in the data. If the data is suspected to be contaminated by outliers, we recommend the use of a volatility model that has the property of bounded innovation propagation and to use a M-estimator with Student t 4 loss function. The Monte Carlo study and empirical application document the good robustness properties of this approach. This research can be extended by looking at other classes of robust estimators and by considering other multivariate GARCH models. In recent research on these models, the tendency has been to reduce the dimensionality of the problem using principal component analysis or by specifying separate univariate GARCH models for the volatility of each series and a time-varying process for the conditional correlations. Further research could show how well robust methods work for these types of models. 32

33 7 Acknowledgment The authors are grateful to Tim Bollerslev, Geert Dhaene, Sébastien Laurent and Oliver Linton for very helpful comments and suggestions. Financial support from the FWO and Research Fund K.U.Leuven is gratefully acknowledged. The usual disclaimer applies. A Consistency of the M-estimator First we derive the asymptotic limit of the gradient of the M-function. Then we show under which assumptions θ is the only value in Θ for which this probability limit is zero. Denote m (r t ; θ, ρ) = log det H t,θ + σ ρ ( r t H 1 t,θ r t). Its first derivative ṁ i (r t ; θ, ρ) = m (r t ; θ, ρ) / θ i is given by Tr [ I N σ ψ(r t H 1 t,θ r ] t)r t r t H 1 H t,θ t,θ H 1 t,θ θ, (25) i (Comte and Lieberman, 2003). The square N-dimensional matrix H t,θ / θ i holds the i-th partial derivative of the corresponding elements of H t,θ and Tr is the trace operator. Since ṁ i (r t ; θ, ρ) is a measurable function of the strictly stationary and ergodic process {r t }, it is also strictly stationary and ergodic. By the Ergodic Theorem its time series average, which is the score of the M- function, will converge to its ensemble mean, which is the mean of all possible 33

34 realizations of (25) conditional on the past of {u t }. We thus find that the asymptotic limit of the score of the M-function equals { [IN Tr E Pθ σ ψ(r t H 1 t,θ r ] t)r t r t H t,θ H 1 t,θ θ i H 1 t,θ I u t 1 } {[ = Tr E Pθ IN σ ψ(r t H 1 t,θ r ] t)r t r t H 1 H t,θ t,θ Iut 1} H 1 t,θ θ, (26) i since H t,θ = H θ (u t 1,...) is a deterministic N N matrix function of the past realizations of the process {u t }. Under the no dominance assumption A4, convergence in probability will be uniform. Borrowing results from the theory on M-estimation of unconditional scale, it can be shown that under A1-A3, the score function is zero for θ. Indeed, Maronna (1976) proves that under A1, there exists a unique solution to the following implicit equation determining H t,θ : {[ E Pθ IN σ ψ(r t H 1 t,θ r ] } t)r t r t H 1 t,θ Iut 1 = 0. (27) The solution to this equation is called the M-estimator of scatter of r t or also the pseudo-covariance matrix of r t. Since this estimator is affine equivariant and since r t is assumed to be elliptically symmetric with conditional scatter matrix H t,θ, one can show that this solution will be proportional to the true conditional scatter matrix H t,θ, with proportionality factor c equal to c = N σ { 2π N/2 Γ(N/2) 0 ψ(t 2 )t N+1 g (t 2 )dt} 1, (28) (Chapter 13 in Bilodeau and Brenner, 1999, Chapter 6 in Maronna et al., 2006). It follows that M-estimators will be consistent since σ in (11) is such 34

35 that c = 1. By the no observational equivalence assumption A3, it is only for θ that the conditional covariance matrix equals H t,θ. We may thus conclude that for loss functions satisfying A1, the limit of the M-function will have an extremum in θ. This extremum is unique under the assumption A5 that there is no θ Θ for which the score of the M-function is zero while the conditional expectation under P θ of the weighted covariance matrix is not the true conditional covariance matrix H t,θ. A further sufficient condition to ensure that the extremum of the limit is the limit of the extremum of the M-estimator is that the parameter space Θ is compact. 35

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