How can I deal with a multicollinearity problem? (difference-in-differences analysis) 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.
(ITEM1-3 and SIZE1-12 are dummy 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?
 A: 
The occurrence of multicollinearity does not reduce the predictive
  power of the model as a whole, though it affects the reliability of
  the coefficients of individual predictors.

I usually detect Multicollinearity using VIF (Variance inflation factor). So higher the VIF value higher the Multicollinearity. Usually, if we have high VIF value then we can remove it.
But there is a constraint here. If you have very low p-value and high VIF then we can not ignore this variable.  
A: The most popular approach of handling Multicollinearity is Variance Inflation Factor[VIF].
We first take all the variables which have to potential to influence/affect the outcome as explanatory variables, and fit a regression model.

The command vif(model) is then used to determine the factors
  affecting the model,If any VIF>5, we remove the variable with the
  largest VIF, and build the regression model again for remaining
  variables.

Continue the above procedure until all the explanatory variables in the model are less than 5.
After eliminating variables with VIF>5, it suggests that the remaining explanatory variables in the models are independent of each other.
We then reduce the model by excluding each of the explanatory variables (one at a time) that do not have significant contribution to the outcome of interest.]
I hope this explains your conflict!
