This paper features an application of the Hafner and Herwartz (2006) approach to the analysis of multivariate GARCH models using volatility impulse response analysis. The data set used features ten years of daily return series for the New York Stock Exchange Index and the FTSE 100 index from the London stock Exchange, taken from 3rd January 2005 to January 31st 2015. This period captures both the Global Financial Crisis (GFC) and the subsequent European Sovereign Debt Crisis (ESDC). The attraction of the Hafner and Kerwartz (2006) approach is that it involves a novel application of the concept of impulse response functions, tracing the effects of independent shocks on volatility through time, whilst avoiding typical orthogonalization and ordering problems. Volatility impulse response functions (VIRF) provide information about the impact of independent shocks on volatility. Hafner and Herwartz's (2006) VIRF extends a framework, provided by Koop et al. (1996), for the analysis of impulse responses. This approach is novel because it explores the effects of shocks to the conditional variance, as opposed to the conditional mean. Hafner and Herwartz (2006) utilise the fact that GARCH models can be viewed as being linear in squares, and that multivariate GARCH models are known to have a VARMA representation with non-Gaussian errors. They use this particular structure to calculate conditional expectations of volatility analytically in their VIRF analysis. Hafner and Herwartz (2006) use a Jordan decomposition of St in order to obtain independent and identically defined (hence i.i.d.) innovations. One general issue in the approach is the choice of baseline volatilities. Hafner and Herwartz (2006) define VIRF as the expectation of volatility conditional on an initial shock and on history, minus the baseline expectation that only conditions on history. This makes the process endogenous, but the choice of the baseline shock within the data set still obviously makes a difference. We explore the impact of three different shocks, the first marks the onset of the GFC, which we date as 9th August 2007, (GFC1). This began with the seizure in the banking system precipitated by BNP Paribas announcing that it was ceasing activity in three hedge funds that specialised in US mortgage debt. It took a year for the financial crisis to come to a head, but it did so on 15th September 2008, when the US government allowed the investment bank Lehman Brothers to go bankrupt (GFC2). Our third shock point is May 9th 2010, which marked the point at which the focus of concern switched from the private sector to the public sector. A further contribution of this paper is the inclusion of leverage, or asymmetric effects, after Engle and Ng (1993). Our modelling is undertaken in the context of a multivariate GARCH model featuring pre-whitened return series, which are then analysed via a BEKK model using a t-distribution. A key result is that the impact of negative shocks is larger, in terms of the effects on variances and covariances, but shorter in duration, in this case a difference between three and six months, in the context of our particular return series. An effect previously reported by Tauchen et al., (1996), who use a different theoretical set up.

Asymmetry, BEKK, ESDC, GFC, Volatility Impulse Response Functions
21st International Congress on Modelling and Simulation: Partnering with Industry and the Community for Innovation and Impact through Modelling, MODSIM 2015 - Held jointly with the 23rd National Conference of the Australian Society for Operations Research and the DSTO led Defence Operations Research Symposium, DORS 2015

Allen, D.E, McAleer, M.J, Powell, R.J, & Singh, A.K. (2020). A volatility impulse response analysis applying multivariate GARCH models and news events around the GFC. In Proceedings - 21st International Congress on Modelling and Simulation, MODSIM 2015 (pp. 1008–1014). Retrieved from