  This paper considers the statistical inference of the class of asymmetric
power-transformed $\operatorname{GARCH}(1,1)$ models in presence of possible
explosiveness. We study the explosive behavior of volatility when the strict
stationarity condition is not met. This allows us to establish the asymptotic
normality of the quasi-maximum likelihood estimator (QMLE) of the parameter,
including the power but without the intercept, when strict stationarity does
not hold. Two important issues can be tested in this framework: asymmetry and
stationarity. The tests exploit the existence of a universal estimator of the
asymptotic covariance matrix of the QMLE. By establishing the local asymptotic
normality (LAN) property in this nonstationary framework, we can also study
optimality issues.
