How can I deal with a multicollinearity problem? (difference-in-differences analysis)

Question

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?

Answer 1

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.

Answer 2

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!

Answer 3

You have two issues but both are not a problem of multicollinearity.

Converting a categorical variable to dummy variables

For instance, SIZE1-3 combined is the same as ITEM1 which is the same as A. To avoid this problem

This is not a problem. Your size categories variable is 'nested' within the item category variable. Categorical variables with $n$ categories always translate into $n-1$ dummy variables. (If you have an intercept in the column space of the design-matrix/model)

There are several questions about this dummy-encoding of categorical variables. Here is one with the design-matrix written out completely: Why there is a dependence between the factors on the same column?

Not enough observations

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.

In this example you do not only have a situation where size is nested inside item. Another issue is that for several sizes you only have a single observation. If you would control for item/size then the degrees of freedom in the model will be severely reduced.

This multicollinearity doesn't arrise because the variables are correlated in the population, but because you do not have enough observations (making them correlated in the sample).