Vector Multiplicative Error Models: Representation and Inference

Abstract

The Multiplicative Error Model introduced by Engle (2002) for positive valued processes is specified as the product of a (conditionally autoregressive) scale factor and an innovation process with positive support. In this paper we propose a multi-variate extension of such a model, by taking into consideration the possibility that the vector innovation process be contemporaneously correlated. The estimation procedure is hindered by the lack of probability density functions for multivariate positive valued random variables. We suggest the use of copulafunctions and of estimating equations to jointly estimate the parameters of the scale factors and of the correlations of the innovation processes. Empirical applications on volatility indicators are used to illustrate the gains over the equation by equation procedure.

A General Framework for Observation Driven Time-Varying Parameter Models

“…Other frameworks for observation driven models within the exponential family of distributions have been suggested in the literature, including the generalized linear autoregressive (GLAR) models of Shephard (1995), the generalized autoregressive moving average (GARMA) models of Benjamin, Rigby, and Stanispoulos (2003), and the vector multiplicative error models (MEM) of Cipollini, Engle, and Gallo (2006). In contrast to these proposals, GAS models are able to exploit the complete density structure rather than only means and higher moments.…”

Semiparametric Vector MEM

“…In this paper we have presented an extension to the specification of the vector Multiplicative Error Model introduced by Cipollini et al (2006), specifying all matrices of coefficients to be full matrices. In such a specification, the approach proposed in Cipollini et al (2007) to estimate the system with univariate model estimation adopting Gamma marginals, and then using copula functions to retrieve the contemporaneous correlation among innovations would not be applicable.…”

Modelling High-Frequency Volatility and Liquidity Using Multiplicative Error Models

“…Such a specification allows to flexibly capture asymmetries in responses of Y t to small or large lagged innovation shocks, such as, e.g., shocks in liquidity demand, liquidity supply or volatility. A similar specification is considered by Cipollini et al (2006) to capture leverage effects if Y t corresponds to a volatility variable. For more details on extended MEM specifications in the context of ACD models, see Hautsch (2004) or Bauwens and Hautsch (2008).…”

“…As discussed by Cipollini et al (2006), a possible candidate is a multivariate gamma distribution which however imposes severe restrictions on the contemporaneous correlations between the errors ε (i) t . Alternatively, copula approaches can be used as, e.g., proposed by Heinen and Rengifo (2006) or Cipollini et al (2006).…”

“…Engle and Gallo (2006) apply MEM specifications to jointly model different volatility indicators including absolute returns, daily range, and realized volatility. Recently, Cipollini et al (2006) extend the MEM by a copula specification in order to capture contemporaneous relationships between the variables.…”