Abstract.In this paper we establish a statistical methodology for the spectral analysis of stationary multivariate time series via the Walsh‐Fourier transform. Theoretical results pertaining to the definition and estimation of the Walsh‐Fourier spectral matrix and functions of that matrix including cross‐spectra, coherency and phase are given. An example of the statistical techniques developed in this paper is given; in particular, the methodologies are applied to neonatal sleep data collected from a study of the effect of maternal substance use during preg