As I understand it, a STAR clearance is only a lateral clearance to navigate via a certain route. Unless and until you get clearance to a lower altitude descent is NOT authorised. There may be, however, crossing restrictions to comply with after a descent clearance is received.
This is how it is in LAX to my certain knowledge.
See also Oz AIP ENR 1.5-36 para 12.2.5.
In the USA we used to be cleared via "profile descent" which incorporated both lateral and vertical navigation in the one clearance, but these were separate from STARs.