Conditions for geometric ergodicity of multivariate autoregressive conditional heteroskedasticity (ARCH) processes, with the so-called BEKK (Baba, Engle, Kraft, and Kroner) parametrization, are considered. We show for a class of BEKK-ARCH processes that the invariant distribution is regularly varying. In order to account for the possibility of different tail indices of the marginals, we consider the notion of vector scaling regular variation (VSRV), closely related to non-standard regular variation. The characterization of the tail behavior of the processes is used for deriving the asymptotic properties of the sample covariance matrices.AMS 2010 subject classifications: 60G70, 60G10, 60H25, 39A50.

We consider inference and testing in extended constant conditional correlation GARCH models in the case where the true parameter vector is a boundary point of the parameter space. This is of particular importance when testing for volatility spillovers in the model. The large-sample properties of the QMLE are derived together with the limiting distributions of the related LR, Wald, and LM statistics. Due to the boundary problem, these large-sample properties become nonstandard. The size and power properties of the tests are investigated in a simulation study. As an empirical illustration we test for (no) volatility spillovers between foreign exchange rates.

This paper considers asymptotic inference in the multivariate BEKK model based on (co-)variance targeting (VT). By de…nition the VT estimator is a two-step estimator and the theory presented is based on expansions of the modi…ed likelihood function, or estimating function, corresponding to these two steps. Strong consistency is established under weak moment conditions, while sixth order moment restrictions are imposed to establish asymptotic normality. Included simulations indicate that the multivariately induced higher-order moment constraints are indeed necessary.

In this paper, we consider asymptotic inference in the multivariate BEKK model based on (co)variance targeting (VT). By definition the VT estimator is a two-step estimator and the theory presented is based on expansions of the modified likelihood function, or estimating function, corresponding to these two steps. Strong consistency is established under weak moment conditions, while sixth-order moment restrictions are imposed to establish asymptotic normality. The simulations included indicate that the multivariately induced higher-order moment constraints are necessary.

The 'fixed regressor' -or 'fixed design' -bootstrap is usually considered in the context of classic regression, or conditional mean (autoregressive) models, see for example, Goncalves and Kilian, 2004). We consider here inference for a general class of (non)linear ARCH models of order q, based on a 'Fixed Volatility' bootstrap. In the Fixed Volatility bootstrap, the lagged variables in the conditional variance equation are kept fixed at their values in the original series, while the bootstrap innovations are, as is standard, resampled with replacement from the estimated residuals based on quasi maximum likelihood estimation. We derive a full asymptotic theory to establish validity for the Fixed Volatility bootstrap applied to Wald statistics for general restrictions on the parameters. A key feature of the Fixed Volatility bootstrap is that the bootstrap sample, conditional on the original data, is an independent sequence. Inspection of the proof of bootstrap validity reveals that such conditional independence simplifies the asymptotic analysis considerably. In contrast to other bootstrap methods, one does not have to take into account the conditional dependence structure of the bootstrap process itself. We also investigate the finite sample performance of the Fixed Volatility bootstrap by means of a small scale Monte Carlo experiment. We find evidence that for small sample sizes, the Fixed Volatility bootstrap test is superior to the asymptotic test, and to the recursive bootstrap-based test. For large samples, both bootstrap schemes and the asymptotic test share properties, as expected from the asymptotic theory. Its appealing theoretical properties, together with its good finite sample performance, suggest that the proposed Fixed Volatility bootstrap may be an important tool for the analysis of the bootstrap in more general volatility models. FIXED VOLATILITY BOOTSTRAP 921With respect to the fixed design bootstrap in regression models, which keep the conditional mean of the bootstrap data fixed among bootstrap repetitions, when dealing with volatility models the fixed design bootstrap maintains the conditional volatility fixed among bootstrap repetitions. This requires generating heteroskedasticity in the bootstrap data by recolouring the (conditionally i.i.d.) bootstrap innovations with some estimates of the volatility process which depend on the original data only. Hence, and in contrast to other bootstrap methods such as the recursive bootstrap, this method -which we label the 'Fixed Volatility' bootstrap in what follows -generates bootstrap samples which, conditional on the original data, features the same conditional volatility process.The recursive bootstrap for ARCH type model has been explored in the existing literature. For instance, Hidalgo and Zaffaroni (2007) propose a recursive bootstrap scheme for model specification tests in ARCH(∞) models. Corradi and Iglesias (2008) compare recursive bootstrap with block bootstrap methods in terms of asymptotic refinements. Pascual et al. (2006) exploit the rec...