  Penalized spline smoothing of time series and its asymptotic properties are
studied. A data-driven algorithm for selecting the smoothing parameter is
developed. The proposal is applied to define a semiparametric extension of the
well-known Spline-GARCH, called a P-Spline-GARCH, based on the log-data
transformation of the squared returns. It is shown that now the errors process
is exponentially strong mixing with finite moments of all orders. Asymptotic
normality of the P-spline smoother in this context is proved. Practical
relevance of the proposal is illustrated by data examples and simulation. The
proposal is further applied to value at risk and expected shortfall.
