In chapters 4 and 5 all of our atte

In chapters 4 and 5 all of our attention has been directed to developing, estimating, testing, and
forecasting univariate ( = 1) volatility models only. In this chapter, we broaden our interest to multivariate ( ≥ 2) models that–as far as second moments are concerned–will necessarily also concern covariances and correlations besides variances. We therefore examine three approaches to
multivariate estimation of conditional second moments. First, we deal with an approach that moves
the core of the e ffort from the econometrics to the asset pricing, in the sense that covariances will
predicted off factor pricing models (such as, but not exclusively, the CAPM). The advantage of this
way of proceeding is that some of us prefer to do more economics and less econometrics (and this
seems to be a good idea also to the Author of these notes). Unfortunately, most of the asset pricing
theory currently circulating tends to be rejected (sometimes rather obviously, think of the CAPM,
in other occasions only marginally) by most data sets. As a result, the majority of users of financial
econometrics (risk and asset managers, some quantsy types of asset pricers and structurers) prefer
to derive forecasts from econometric models, vs. incorrect, commonly rejected asset pricing models.
Second, we propose models that directly model conditional covariances following a logic similar to
chapter 4: these are in practice multivariate extensions of ARCH and GARCH models. As we
shall see, the idea is similar to when in chapter 3 you did move from univariate time series models
for the conditional mean to multivariate, vector models (such as vector autoregressions). However,
in the case of covariance matrices, we shall see that extending univariate GARCH models to their
multivariate counterparts will present many practical difficulties, unless a smart approach is adopted.
Therefore the corresponding material is presented only in the final, but rather important Section
6. Third, such a smart approach–dynamic conditional correlations (DCC) models–represents the
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other important, key tool that is described in this chapter.