Estimation in Networks with Spatiotemporally Correlated Noise

In this article, we address the problem of estimating a particular transfer function in a dynamic network where the unknown noise processes are potentially correlated across the nodes. It is assumed that the noise correlations are affine in essence. We model the spatial correlation between the noise processes of two nodes of the network as a new hidden node that influences the two nodes. To be able to apply the notion of $d$ -separation from graph theory, we further manipulate the network by adding another fictitious node and slightly altering its structure in a systematic way. The time series generated by a subset of the nodes in this new larger network are equivalent to the time series generated by the original network. In this new larger network, based on the notion of $d$ -separation, we formulate sufficient graphical conditions to select a set of predictor inputs. We prove that the selected set of predictor inputs guarantees a consistent estimation of the transfer function of interest using a prediction error method.