ABSTRACT
We start this chapter by considering the first and second order moments of time series – their means, variances, covariances and correlations. If the series are Gaussian, i.e., the joint distribution of any set of values is multivariate normal, these moments completely determine their distribution. We do not consider other forms of distribution, in large part because of the practical difficulties this raises, but also because assuming that the series are Gaussian can deliver solutions which are reasonably robust to departures from this assumption. In particular, the estimation of model parameters by maximum likelihood under the Gaussian assumption can be justified under much wider assumptions about the distribution. See, for example, Section 10.8 of Brockwell and Davis (1987) for relevant results in the context of univariate time series and Chapter 6 of Hannan (1970) in the multivariate case.
