I am conducting difference-in-differences analysis. The first difference is from time, treatment/control, and the second is from treated subjects, A and B. There are two control variables, ITEM and SIZE. The A and B have one and two ITEMs, respectively. For each ITEM, SIZE divides into three categories. The attached picture is a frequency table of these variables.
A dependent variable, log price, is obviously affected by ITEM and SIZE. So I want to control these variables, but there is a multicollinearity issue. For instance, SIZE1-3 combined is the same as ITEM1 which is the same as A. To avoid this problem, what I have in mind is to control SIZE2-3, SIZE5-7, SIZE9-12. In other words, I control all SIZE dummy variables but SIZE1, SIZE4, and SIZE8. Is this right specification? Can a D-in-D coefficient capture treatment effect without bias in this specification?
Thank you much in advance.