The analysis of financial data has received considerable attention in the literature during the last 20 years. Several models have been suggested to capture special features of financial data and most of these models have the property that the conditional variance (or the conditional scaling) depends on the past. Some of the well known and most often used examples are the autoregressive conditionally heteroscedastic (ARCH) process introduced by Engle (1982) and the GARCH model introduced by Bollerslev (1986). We investigate parameter estimation for such processes via the so called "quasi-maximum likelihood" method, study the asymptotic behavior of related empirical processes and estimators for the moment index of such processes, another important economical parameter. The ARCH and GARCH processes have short range dependence, but empirical evidence suggests in many typical financial situations a much greater degree of persistence of the process, indicating a long memory behavior. A model describing such long memory behavior was suggested recently by Giraitis, Robinson and Surgailis. We study the asymptotic behavior and parameter estimation for this model, the so called LARCH process.