Dynamic conditional eigenvalue GARCH

Abstract In this paper we introduce a multivariate generalized autoregressive conditional heteroskedastic (GARCH) class of models with time-varying conditional eigenvalues. The dynamics of the eigenvalues is derived for the cases with underlying Gaussian and Student’s t -distributed innovations based on the general theory of dynamic conditional score models by Creal, Koopman and Lucas (2013) and Harvey (2013). The resulting time-varying eigenvalue GARCH models – labeled ‘ λ -GARCH’ models – differ for the two cases of innovations, similar to, and generalizing, univariate linear Gaussian GARCH and Student’s t -based Beta- t -GARCH models. Asymptotic theory is provided for the Gaussian-based quasi-maximum likelihood estimator (QMLE). In addition, and in order to test for the number of (linear combinations of) the time-varying eigenvalues, we consider testing and inference under the hypothesis of reduced rank of the GARCH loading matrices. The conditional Gaussian and Student’s t distributed λ -GARCH models are applied to US return data, and it is found that the eigenvalue structure for the sample considered indeed satisfies the hypothesis of reduced rank. Specifically, it is possible to disentangle time-varying linear combinations of the eigenvalues, or factors, from time-invariant factors which drive the dynamics of the conditional covariance.

[1]  N. Shephard,et al.  Multivariate Rotated ARCH Models , 2014 .

[2]  S. T. Jensen,et al.  ASYMPTOTIC NORMALITY OF THE QMLE ESTIMATOR OF ARCH IN THE NONSTATIONARY CASE , 2004 .

[3]  B. M. Brown,et al.  Martingale Central Limit Theorems , 1971 .

[4]  Christian M. Hafner,et al.  On asymptotic theory for multivariate GARCH models , 2009, J. Multivar. Anal..

[5]  Roy van der Weide,et al.  GO-GARCH: a multivariate generalized orthogonal GARCH model , 2002 .

[6]  Anders Rahbek,et al.  Bootstrap Determination of the Co-Integration Rank in Vector Autoregressive Models , 2012 .

[7]  Heino Bohn Nielsen,et al.  Unit Root Vector Autoregression with Volatility Induced Stationarity , 2012 .

[8]  ASYMPTOTIC THEORY FOR A FACTOR GARCH MODEL , 2009, Econometric Theory.

[9]  Offer Lieberman,et al.  Asymptotic theory for multivariate GARCH processes , 2003 .

[10]  Douglas M. Bates,et al.  Unconstrained parametrizations for variance-covariance matrices , 1996, Stat. Comput..

[11]  R. Pedersen.  Inference and Testing on the Boundary in Extended Constant Conditional Correlation GARCH Models , 2015 .

[12]  Jianqing Fan,et al.  Modelling multivariate volatilities via conditionally uncorrelated components , 2005, math/0506027.

[13]  Pentti Saikkonen,et al.  A Multivariate Generalized Orthogonal Factor GARCH Model , 2007 .

[14]  C. Francq,et al.  Testing the Nullity of GARCH Coefficients: Correction of the Standard Tests and Relative Efficiency Comparisons , 2009 .

[15]  P. Zaffaroni,et al.  ON MOMENT CONDITIONS FOR QUASI-MAXIMUM LIKELIHOOD ESTIMATION OF MULTIVARIATE ARCH MODELS , 2013, Econometric Theory.

[16]  S. Johansen,et al.  Asymptotic Inference on Cointegrating Rank in Partial Systems , 1998 .

[17]  Drew D. Creal,et al.  Generalized autoregressive score models with applications ∗ , 2010 .

[18]  Thierry Jeantheau,et al.  STRONG CONSISTENCY OF ESTIMATORS FOR MULTIVARIATE ARCH MODELS , 1998, Econometric Theory.

[19]  R. Engle.  Dynamic Conditional Correlation , 2002 .

[20]  D. Andrews.  Testing When a Parameter Is on the Boundary of the Maintained Hypothesis , 2001 .

[21]  Siem Jan Koopman,et al.  A Dynamic Multivariate Heavy-Tailed Model for Time-Varying Volatilities and Correlations , 2010 .

[22]  TESTING GARCH-X TYPE MODELS , 2018, Econometric Theory.

[23]  Christian Conrad,et al.  NEGATIVE VOLATILITY SPILLOVERS IN THE UNRESTRICTED ECCC-GARCH MODEL , 2008, Econometric Theory.

[24]  Eric Renault,et al.  Testing for Common Conditionally Heteroskedastic Factors , 2013 .

[25]  Rasmus Søndergaard Pedersen,et al.  Multivariate Variance Targeting in the BEKK–GARCH Model , 2014 .

[26]  H. P. Boswijk,et al.  UvA-DARE ( Digital Academic Repository ) Method of moments estimation of GO-GARCH models , 2009 .

[27]  J. Zakoian,et al.  GARCH Models: Structure, Statistical Inference and Financial Applications , 2010 .

[28]  T. Bollerslev,et al.  Modelling the Coherence in Short-run Nominal Exchange Rates: A Multivariate Generalized ARCH Model , 1990 .

[29]  Christian Francq,et al.  QML ESTIMATION OF A CLASS OF MULTIVARIATE ASYMMETRIC GARCH MODELS , 2011, Econometric Theory.

[30]  Christian Francq,et al.  Variance targeting estimation of multivariate GARCH models , 2014 .

[31]  Bootstrap inference on the boundary of the parameter space, with application to conditional volatility models , 2018 .