  Linear ARCH (LARCH) processes were introduced by Robinson [J. Econometrics 47
(1991) 67--84] to model long-range dependence in volatility and leverage. Basic
theoretical properties of LARCH processes have been investigated in the recent
literature. However, there is a lack of estimation methods and corresponding
asymptotic theory. In this paper, we consider estimation of the dependence
parameters for LARCH processes with non-summable hyperbolically decaying
coefficients. Asymptotic limit theorems are derived. A central limit theorem
with $\sqrt{n}$-rate of convergence holds for an approximate conditional
pseudo-maximum likelihood estimator. To obtain a computable version that
includes observed values only, a further approximation is required. The
computable estimator is again asymptotically normal, however with a rate of
convergence that is slower than $\sqrt{n}.$
