Applying the Expectation-Maximization Algorithm to Multivariate Signal Extraction

Understanding time series signal extraction is vital for federal agencies, especially as data collection accelerates. One natural extension is moving from univariate to multivariate signal extraction, which offers the promise of reducing extraction error by exploiting cross-sectional relationships. However, such an extension ushers in new computational challenges, viz. larger parameter spaces and more complicated objective functions. The Expectation-Maximization (EM) algorithm provides a methodology to implicitly compute (or approximate) maximum likelihood estimates (MLEs). This paper provides methodology for applying the EM algorithm to a class of latent component multivariate time series models that allow for a nuanced specification of the unobserved signal. We derive an explicit formula for the maximization step, which facilitates computation speed while also improving the stability of the algorithm. Numerical studies demonstrate EM’s ability to efficiently compute MLEs in low-dimensional systems, while also providing feasible estimates in moderate-dimensional systems where MLEs are infeasible to compute. Applications to monthly housing starts and daily immigration are provided.