  Network (or matrix) reconstruction is a general problem which occurs if the
margins of a matrix are given and the matrix entries need to be predicted. In
this paper we show that the predictions obtained from the iterative
proportional fitting procedure (IPFP) or equivalently maximum entropy (ME) can
be obtained by restricted maximum likelihood estimation relying on augmented
Lagrangian optimization. Based on the equivalence we extend the framework of
network reconstruction towards regression by allowing for exogenous covariates
and random heterogeneity effects. The proposed estimation approach is compared
with different competing methods for network reconstruction and matrix
estimation. Exemplary, we apply the approach to interbank lending data,
provided by the Bank for International Settlement (BIS). This dataset provides
full knowledge of the real network and is therefore suitable to evaluate the
predictions of our approach. It is shown that the inclusion of exogenous
information allows for superior predictions in terms of $L_1$ and $L_2$ errors.
Additionally, the approach allows to obtain prediction intervals via bootstrap
that can be used to quantify the uncertainty attached to the predictions.
