Avoiding arbitrary exclusion restrictions using ratios of reduced-form estimates

We show how to obtain coherent structural-form (SF) exclusion restrictions using the reduced-form (RF) parameter ratios. It will be shown that an over-identified SF corresponds to a group of regressors sharing the same RF ratio value; those regressors should be excluded jointly from the SF. If there is no group structure, then the SF is just-identified; in this case, however, it is no longer clear which regressor should be excluded. Hence, just-identified SF’s are more arbitrary than over-identified SF’s in terms of exclusion restrictions. This is in stark contrast to the notion that the former is less arbitrary than the latter, because the former excludes fewer regressors. We formalize these points, and then suggest to find the number of modes in the estimated RF ratios as a way to find groups in the ratios. For this purpose, an informal graphical method using a kernel nonparametric method and a formal modality test are employed. An empirical example with selling price in a residential real estate market and duration on the market as two endogenous variables is provided.