  We study semiparametric time series models with innovations following a
log-concave distribution. We propose a general maximum likelihood framework
which allows us to estimate simultaneously the parameters of the model and the
density of the innovations. This framework can be easily adapted to many
well-known models, including ARMA, GARCH and ARMA-GARCH. Furthermore, we show
that the estimator under our new framework is consistent in both ARMA and
ARMA-GARCH settings. We demonstrate its finite sample performance via a
thorough simulation study and apply it to model the daily log-return of FTSE
100 index and the rabbit population.

Methodology