  We propose a new algorithm for estimating NARMAX models with $L_1$
regularization for models represented as a linear combination of basis
functions. Due to the $L_1$-norm penalty the Lasso estimation tends to produce
some coefficients that are exactly zero and hence gives interpretable models.
The novelty of the contribution is the inclusion of error regressors in the
Lasso estimation (which yields a nonlinear regression problem). The proposed
algorithm uses cyclical coordinate descent to compute the parameters of the
NARMAX models for the entire regularization path. It deals with the error terms
by updating the regressor matrix along with the parameter vector. In
comparative timings we find that the modification does not reduce the
computational efficiency of the original algorithm and can provide the most
important regressors in very few inexpensive iterations. The method is
illustrated for linear and polynomial models by means of two examples.
