  Mixed Probit models are widely applied in many fields where prediction of a
binary response is of interest. Typically, the random effects are assumed to be
independent but this is seldom the case for many real applications. In the
credit risk application considered in this paper, random effects are present at
the level of industrial sectors and they are expected to be correlated due to
inter-firm credit links inducing dependencies in the firms' risk to default.
Unfortunately, existing inferential procedures for correlated mixed Probit
models are computationally very intensive already for a moderate number of
effects. Borrowing from the literature on large network inference, we propose
an efficient Expectation-Maximization algorithm for unconstrained and penalised
likelihood estimation and derive the asymptotic standard errors of the
estimates. An extensive simulation study shows that the proposed approach
enjoys substantial computational gains relative to standard Monte Carlo
approaches, while still providing accurate parameter estimates. Using data on
nearly 64,000 accounts for small and medium-sized enterprises in the United
Kingdom in 2013 across 14 industrial sectors, we find that accounting for
network effects via a correlated mixed Probit model increases significantly the
default prediction power of the model compared to conventional default
prediction models, making efficient inferential procedures for these models
particularly useful in this field.

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