Dynamic deconvolution and identification of independent autoregressive sources

We consider a multi-variate system (Formula presented.), where the unobserved components (Formula presented.) are independent AR(1) processes and the number of sources is greater than the number of observed outputs. We show that the mixing matrix (Formula presented.), the AR(1) coefficients and distributions of (Formula presented.) can be identified (up to scale factors of (Formula presented.)), which solves the dynamic deconvolution problem. The proof is constructive and allows us to introduce simple consistent estimators of all unknown scalar and functional parameters of the model. The approach is illustrated by an estimation and identification of the dynamics of unobserved short- and long-run components in a time series. Applications to causal models with structural innovations are also discussed, such as the identification in error-in-variables models and causal mediation models.