The work summarized in this paper represents the first part of a two-paper analysis of statistical monitoring of complex dynamic multivariate processes. Motivated by recent research highlighting the difficulties of monitoring autocorrelated variables, this first paper revisits the impact of autocorrelation and cross-correlation upon the significance level for hypothesis testing in monitoring statistics. The presented analysis shows that both correlations lead to profound alterations of the significance level, which can manifest themselves in the production of false alarms or an insensitive monitoring scheme. In the research literature on statistical process monitoring, however, only the issue of autocorrelation has received attention thus far. To improve process monitoring of autocorrelated and cross-correlated variables, this article proposes the use of Kalman innovation models to remove these correlations. The utility of this improvement is demonstrated using an application to the Tennessee Eastman simulator and the analysis of recorded data from an industrial distillation unit.