How can I assess the strength of the collinearity (if any) between two different sets of categorical dummy variables in an OLS regression? I ran an OLS regression of 'prices' on two sets of dummy variables and nothing else. The dummy variables in question are 'city', with levels A, B, C and D (each corresponds to a different city). And 'items', with levels X, Y and Z (each corresponds to a different item). 
How can I assess the level of imperfect collinearity - if any- between, say, A and X?
 A: We need to specify which collinearity is being examined. The title of the question speaks of collinearity between two sets of dummy variables (each set representing multiple levels of an unordered categorical predictor). The question itself, however, asks about collinearity between specific levels of each of the two categorical predictors, say level "A" of the city predictor and level "X" of items.
With respect to sets of dummy variables, a generalized variance inflation factor (GVIF) describes collinearity when there are more than 2 levels of a categorical predictor in a linear model. GVIF treats all levels of a categorical predictor together, and has the advantage over standard VIF calculations that the result is independent of coding choices such as reference levels for the categories. GVIF, implemented for example in thevif() function in the R car package, provides a useful measure of how collinearity between your two sets of dummy variables might influence the performance of the model.
With respect to "collinearity" between individual levels of 2 categorical predictors, what you have, collected over all the cases, is a 2 X 2 contingency table built from all 4 possibilities. For the example in the question, these possibilities are: neither A nor X, A only, X only, both A and X. This paper describes 21 different measures of association for such tables. So the question is: what type of "collinearity" between A and X are you interested in, and why? The answer to that question will point the way to your choice of association measure.
