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The Jacobian matrix of backward models with lagged exogenous variables is wrong, columns for the lagged exogenous variables are missing. It is unclear to me why we do not simply add auxiliary endogenous variables for the lagged exogenous variables (as in the case of backward/forward models). I suppose that the simplest fix is to add auxiliary variables (rather than augmenting the size of the Jacobian matrix).

An example is available here