This paper studies goodness-of-fit tests and specification tests for an exten-
sion of the Log-GARCH model, which is both asymmetric and stable by scaling. A
Lagrange-multiplier test is derived for testing the extended Log-GARCH against more
general formulations taking the form of combinations of Log-GARCH and exponential
GARCH (EGARCH). The null assumption of an EGARCH is also tested. Portmanteau
goodness-of-fit tests are developed for the extended Log-GARCH. An application to
real financial data is proposed.

• Goodness-of-fit tests for Log-GARCH and EGARCH models