  This paper studies goodness of fit tests and specification tests for an
extension of the log-GARCH model which is stable by scaling. A
Lagrange-Multiplier test is derived for testing the null assumption of extended
log-GARCH against more general formulations including the Exponential GARCH
(EGARCH). The null assumption of an EGARCH is also tested. Portmanteau
goodness-of-fit tests are developed for the extended log-GARCH. Simulations
illustrating the theoretical results and an application to real financial data
are proposed.
