Testing Separability of Functional Time Series

We derive and study a significance test for determining whether a panel of functional time series is separable. In the context of this paper, separability means that the covariance structure factors into the product of two functions, one depending only on time and the other depending only on the coordinates of the panel. Separability is a property that can dramatically improve computational efficiency by substantially reducing model complexity. It is especially useful for functional data, as it implies that the functional principal components are the same for each member of the panel. However, such an assumption must be verified before proceeding with further inference. Our approach is based on functional norm differences and provides a test with well-controlled size and high power. We establish our procedure quite generally, allowing one to test separability of autocovariances as well. In addition to an asymptotic justification, our methodology is validated by a simulation study. It is applied to functional panels of particulate pollution and stock market data.