Identification and Likelihood Inference for Recursive Linear Models with Correlated Errors

In recursive linear models, the multivariate normal joint distribution of all variables exhibits a dependence structure induced by recursive systems of linear structural equations. Such models appear in particular in seemingly unrelated regressions, structural equation modelling, simultaneous equation systems, and in Gaussian graphical modelling. We show that recursive linear models that are ‘bow-free’ are well-behaved statistical models, namely, they are everywhere identifiable and form curved exponential families. Here, ‘bow-free’ refers to models satisfying the condition that if a variable x occurs in the structural equation for y, then the errors for x and y are uncorrelated. For the computation of maximum likelihood estimates in ‘bow-free’ recursive linear models we introduce the Residual Iterative Conditional Fitting (RICF) algorithm. Compared to existing algorithms RICF is easily implemented requiring only least squares computations, has clear convergence properties, and finds parameter estimates in closed form whenever possible.