  In this paper, we develop a complete methodology for detecting
time-varying/non time-varying parameters in ARCH processes. For this purpose,
we estimate and test various semiparametric versions of the time-varying ARCH
model (tv-ARCH) which include two well known non stationary ARCH type models
introduced in the econometric literature. Using kernel estimation, we show that
non time-varying parameters can be estimated at the usual parametric rate of
convergence and for a Gaussian noise, we construct estimates that are
asymptotically efficient in a semiparametric sense. Then we introduce two
statistical tests which can be used for detecting non time-varying parameters
or for testing the second order dynamic. An information criterion for selecting
the number of lags is also provided. We illustrate our methodology with several
real data sets.
