What cause problem of perfect collinearity problem in this example and how to explain it clearly?

Question

The question gives the following model: $ln(h_i$)=α+$β_1$$ln(I_i$)+$β_2$$D_{2i}$+$β_3$$D_{3i}$+$β_4$$U_i$+$η_i$

This is cross-sectional data 2008-2010 for 5028 individuals living in the UK. h = happiness, I = income, $D_{2i}$ = 1 if year is 2009, $D_{3i}$ if year is 2010 and U is the UK unemployment rate.

The first question is asking what are the problems of estimating this model by OLS.

I think since we have only three values for unemployment, such a model will cause perfect collinearity problem. However, I don't know how to explain the intuition clearly. The answer says 1,$D_{2}$, $D_{3}$ are perfectly collinear with $U_i$, but I don't know what is the reason.

Answer 1

Let unemployment rates be $u_1$, $u_2$ and $u_3$ for year 2008., 2009, 2010. (You have these numbers).

Then $U_i = u_1 + D_{2i} (u_2-u_1) +D_{3i}(u_3-u_1)$ for $i=1,...,5028$. (You can verify this equation). This is why you got "The answer says 1,$D_{2i}, D_{3i}$ are perfectly collinear with $U_i$"

Answer 2

I think you are almost there. The intercept and $D_2$ and $D_3$ together account for all the contrasts between the three years. You cannot then add another variable which effectively tries to contrast the three years.