  In this paper the class of ARCH$(\infty)$ models is generalized to the
nonstationary class of ARCH$(\infty)$ models with time-varying coefficients.
For fixed time points, a stationary approximation is given leading to the
notation ``locally stationary ARCH$(\infty)$ process.'' The asymptotic
properties of weighted quasi-likelihood estimators of time-varying ARCH$(p)$
processes ($p<\infty$) are studied, including asymptotic normality. In
particular, the extra bias due to nonstationarity of the process is
investigated. Moreover, a Taylor expansion of the nonstationary ARCH process in
terms of stationary processes is given and it is proved that the time-varying
ARCH process can be written as a time-varying Volterra series.
