It has long been known that, because of aggregation problems and the Cambridge Capital Theory Controversies, the aggregate production function cannot theoretically exist. Nevertheless, the concept is still widely and uncritically used, presumably because it gives good statistical fits to the data with plausible results. It is shown that this occurs because of the existence of an underlying accounting identity. A suitable mathematical transformation of this identity ensures that it is always possible to specify an "aggregate production function" where the putative output elasticities equal the factor shares, even though the aggregate production does not exist. This is illustrated by reference to a simulation exercise by Felipe and McCombie (2006) and a study by Oulton and O'Mahony (1994). The latter reject the hypothesis that capital is "special", in that their regression estimates demonstrate that the "output elasticity" of capital does not significantly differ from its factor share. However, it is shown in this paper why the data could not have given any other result.