Abstract :
This paper studies the estimation of a semi-strong GARCH(1,1) model when it does
not have a stationary solution, where semi-strong means that we do not require the
errors to be independent over time. We establish necessary and sufficient conditions
for a semi-strong GARCH(1,1) process to have a unique stationary solution. For the
nonstationary semi-strong GARCH(1,1) model, we prove that a local minimizer of
the least absolute deviations (LAD) criterion converges at the rate √n to a normal
distribution under very mild moment conditions for the errors. Furthermore, when
the distributions of the errors are in the domain of attraction of a stable law with
the exponent κ ∈ (1,2), it is shown that the asymptotic distribution of the Gaussian
quasi-maximum likelihood estimator (QMLE) is non-Gaussian but is some stable
law with the exponent κ ∈ (0,2). The asymptotic distribution is difficult to estimate
using standard parametric methods. Therefore, we propose a percentile-t subsampling
bootstrap method to do inference when the errors are independent and identically
distributed, as in Hall and Yao (2003). Our result implies that the least absolute
deviations estimator (LADE) is always asymptotically normal regardless of whether
there exists a stationary solution or not, even when the errors are heavy-tailed. So the
LADE is more appealing when the errors are heavy-tailed. Numerical results lend
further support to our theoretical results.
