Immigrants to Australia are selected on observable characteristics. They may also differ from natives on unobservable characteristics such as ambition or motivation. If we account for unobservable differences, we find a wage gap for immigrant men from English speaking backgrounds, in contrast with previous research which has found no wage gap. Controlling for unobserved heterogeneity also seems important for finding cohort effects. Immigrants that arrived before 1985 faced a larger wage gap compared to native-born Australians than subsequent cohorts. Confirming other research, we find wage gaps for immigrant men and women from non-English speaking backgrounds. Wage assimilation occurs slowly for all groups, but is slowest for those from non-English speaking backgrounds.