The graphical lasso is a popular method for estimating the structure of undirected Gaussian graphical models from data by penalized maximum likelihood. This paper extends the idea of structure estimation of graphical models by penalized maximum likelihood to Gaussian chain graph models for state space models. First we show how the class of linear Gaussian state space models can be interpreted in the chain graph set-up under both the LWF and AMP Markov properties, and we demonstrate how sparsity of the chain graph structure relates to sparsity of the model parameters. Exploiting this relation we propose two different penalized maximum likelihood estimators for recovering the chain graph structure from data depending on the Markov interpretation at hand. We frame the penalized maximum likelihood problem in a missing data set-up and carry out estimation in each of the two cases using the EM algorithm. The common E-step is solved by smoothing, and we solve the two different M-steps by utilizing existing methods from high dimensional statistics and convex optimization.
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