Matrix algebra for higher order moments

A large part of statistics is devoted to the estimation of models from the sample covariance matrix. The development of the statistical theory and estimators has been greatly facilitated by the introduction of special matrices, such as the commutation matrix and the duplication matrix, and the corresponding matrix algebra. Some more extensive models require, however, estimation based on higher order moments, typically third- and fourth-order moments. An example is the popular Kenny-Judd model that includes interactions between latent variables. This paper introduces some special matrices that can be used to simplify the model expressions for third-, fourth-, and higher order moments, gives some relationships between these matrices and related matrices, and gives some formulas for Kronecker products of three and four matrices. The theory is applied to derive convenient expressions for third- and fourth-order moments of some structural equation models.