  We extend the theory from Fan and Li (2001) on penalized likelihood-based
estimation and model-selection to statistical and econometric models which
allow for non-negativity constraints on some or all of the parameters, as well
as time-series dependence. It differs from classic non-penalized likelihood
estimation, where limiting distributions of likelihood-based estimators and
test-statistics are non-standard, and depend on the unknown number of
parameters on the boundary of the parameter space. Specifically, we establish
that the joint model selection and estimation, results in standard asymptotic
Gaussian distributed estimators. The results are applied to the rich class of
autoregressive conditional heteroskedastic (ARCH) models for the modelling of
time-varying volatility. We find from simulations that the penalized estimation
and model-selection works surprisingly well even for a large number of
parameters. A simple empirical illustration for stock-market returns data
confirms the ability of the penalized estimation to select ARCH models which
fit nicely the autocorrelation function, as well as confirms the stylized fact
of long-memory in financial time series data.
