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This paper applies the Hafner and Herwartz (2006) (hereafter HH) approach to the analysis of multivariate GARCH models using volatility impulse response analysis. The data set features ten years of daily returns series for the New York Stock Exchange Index and the FTSE 100 index from the London Stock Exchange, from 3 January 2005 to 31 January 2015. This period captures both the Global Financial Crisis (GFC) and the subsequent European Sovereign Debt Crisis (ESDC). The attraction of the HH 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, while avoiding typical orthogonalization and ordering problems. Volatility impulse response functions (VIRF) provide information about the impact of independent shocks on volatility. HH’s 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. HH use the fact that GARCH models can be viewed as being linear in the 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. A Jordan decomposition of Σt is used to obtain independent and identically distributed innovations. A general issue in the approach is the choice of baseline volatilities. VIRF is defined as the expectation of volatility conditional on an initial shock and on history, minus the baseline expectation that conditions on history. This makes the process endogenous, but the choice of the baseline shock within the data set makes a difference. We explore the impact of three different shocks, the first marking the onset of the GFC, which we date as 9 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 15 September 2008, when the US government allowed the investment bank Lehman Brothers to go bankrupt (GFC2). The third shock is 9 May 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. Our modelling is undertaken in the context of a multivariate GARCH model featuring pre-whitened return series, which are then analysed using both BEKK and diagonal BEKK models with the 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 the return series.

__Abstract__ Energy and agricultural commodities and markets have been examined extensively, albeit separately, for a number of years. In the energy literature, the returns, volatility and volatility spillovers (namely, the delayed effect of a returns shock in one asset on the subsequent volatility or covolatility in another asset), among alternative energy commodities, such as oil, gasoline and ethanol across different markets, have been analysed using a variety of univariate and multivariate models, estimation techniques, data sets, and time frequencies. A similar comment applies to the separate theoretical and empirical analysis of a wide range of agricultural commodities and markets. Given the recent interest and emphasis in bio-fuels and green energy, especially bio-ethanol, which is derived from a range of agricultural products, it is not surprising that there is a topical and developing literature on the spillovers between energy and agricultural markets. Modelling and testing spillovers between the energy and agricultural markets has typically been based on estimating multivariate conditional volatility models, specifically the BEKK and DCC models. A serious technical deficiency is that the Quasi-Maximum Likelihood Estimates (QMLE) of a full BEKK matrix, which is typically estimated in examining volatility spillover effects, has no asymptotic properties, except by assumption, so that no statistical test of volatility spillovers is possible. Some papers in the literature have used the DCC model to test for volatility spillovers. However, it is well known in the financial econometrics literature that the DCC model has no regularity conditions, and that the QMLE of the parameters of DCC has no asymptotic properties, so that there is no valid statistical testing of volatility spillovers. The purpose of the paper is to evaluate the theory and practice in testing for volatility spillovers between energy and agricultural markets using the multivariate BEKK and DCC models, and to make recommendations as to how such spillovers might be tested using valid statistical techniques. Three new definitions of volatility and covolatility spillovers are given, and the different models used in empirical applications are evaluated in terms of the new definitions and statistical criteria.

__Abstract__ Two of the fastest growing frontiers in econometrics and quantitative finance are time series and financial econometrics. Significant theoretical contributions to financial econometrics have been made by experts in statistics, econometrics, mathematics, and time series analysis. The purpose of this special issue of the journal on “Frontiers in Time Series and Financial Econometrics” is to highlight several areas of research by leading academics in which novel methods have contributed significantly to time series and financial econometrics, including forecasting co-volatilities via factor models with asymmetry and long memory in realized covariance, prediction of Lévy-driven CARMA processes, functional index coefficient models with variable selection, LASSO estimation of threshold autoregressive models, high dimensional stochastic regression with latent factors, endogeneity and nonlinearity, sign-based portmanteau test for ARCH-type models with heavy-tailed innovations, toward optimal model averaging in regression models with time series errors, high dimensional dynamic stochastic copula models, a misspecification test for multiplicative error models of non-negative time series processes, sample quantile analysis for long-memory stochastic volatility models, testing for independence between functional time series, statistical inference for panel dynamic simultaneous equations models, specification tests of calibrated option pricing models, asymptotic inference in multiple-threshold double autoregressive models, a new hyperbolic GARCH model, intraday value-at-risk: an asymmetric autoregressive conditional duration approach, refinements in maximum likelihood inference on spatial autocorrelation in panel data, statistical inference of conditional quantiles in nonlinear time series models, quasi-likelihood estimation of a threshold diffusion process, threshold models in time series analysis - some reflections, and generalized ARMA models with martingale difference errors.

__Abstract__ Of the two most widely estimated univariate asymmetric conditional volatility models, the exponential GARCH (or EGARCH) specification can capture asymmetry, which refers to the different effects on conditional volatility of positive and negative effects of equal magnitude, and leverage, which refers to the negative correlation between the returns shocks and subsequent shocks to volatility. However, the statistical properties of the (quasi-) maximum likelihood estimator (QMLE) of the EGARCH parameters are not available under general conditions, but only for special cases under highly restrictive and unverifiable conditions, such as EGARCH(1,0) or EGARCH(1,1), and possibly only under simulation. A limitation in the development of asymptotic properties of the QMLE for the EGARCH(p,q) model is the lack of an invertibility condition for the returns shocks underlying the model. It is shown in this paper that the EGARCH(p,q) model can be derived from a stochastic process, for which the invertibility conditions can be stated simply and explicitly. This will be useful in re-interpreting the existing properties of the QMLE of the EGARCH(p,q) parameters.

__Abstract__ The paper investigates the impact of jumps in forecasting co-volatility, accommodating leverage effects. We modify the jump-robust two time scale covariance estimator of Boudt and Zhang (2013) such that the estimated matrix is positive definite. Using this approach we can disentangle the estimates of the integrated co-volatility matrix and jump variations from the quadratic covariation matrix. Empirical results for three stocks traded on the New York Stock Exchange indicate that the co-jumps of two assets have a significant impact on future co-volatility, but that the impact is negligible for forecasting weekly and monthly horizons.

__Abstract__ One of the fastest growing areas in empirical finance, and also one of the least rigorously analyzed, especially from a financial econometrics perspective, is the econometric analysis of financial derivatives, which are typically complicated and difficult to analyze. The purpose of this special issue of the journal on “Econometric Analysis of Financial Derivatives” is to highlight several areas of research by leading academics in which novel econometric, financial econometric, mathematical finance and empirical finance methods have contributed significantly to the econometric analysis of financial derivatives, including market-based estimation of stochastic volatility models, the fine structure of equity-index option dynamics, leverage and feedback effects in multifactor Wishart stochastic volatility for option pricing, option pricing with non-Gaussian scaling and infinite-state switching volatility, stock return and cash flow predictability: the role of volatility risk, the long and the short of the risk-return trade-off, What’s beneath the surface? option pricing with multifrequency latent states, bootstrap score tests for fractional integration in heteroskedastic ARFIMA models, with an application to price dynamics in commodity spot and futures markets, a stochastic dominance approach to financial risk management strategies, empirical evidence on the importance of aggregation, asymmetry, and jumps for volatility prediction, non-linear dynamic model of the variance risk premium, pricing with finite dimensional dependence, quanto option pricing in the presence of fat tails and asymmetric dependence, smile from the past: a general option pricing framework with multiple volatility and leverage components, COMFORT: A common market factor non-Gaussian returns model, divided governments and futures prices, and model-based pricing for financial derivatives