# Oaxaca decomposition - Interpretation Interaction

I am running a Oaxaca decomposition on trends in paid work (similar to this paper) The estimates are expressed in minutes.

The total change shows an increase in about 30 minutes between Period 0 and Period 1. I have some difficulties to understand what the Interaction means here. So the Endowments represent the observed characteristics of my model and the Coefficients the unobserved.

What about the Interaction? How should interpret the fact that it is negative ?

                     Decomposition
Period 0                   174.66
Period 1                   204.15
Total Change               29.5
.
Endowments                  22.78
Coefficients                50.38
Interaction                -43.66


## 1 Answer

In principle the idea is that there are three sources of the 29.5 change:

Lets assume we are taking period 1 as the 'baseline'. The first source of the 29.5 change is that the independent variables changed. To see the importance of this effect, you estimate education's effect on time for period 1. Then you apply the estimated model to the period 2 education and you get a predicted increase in time because the period 2 were more educated on average and the model predicts that higher education leads to more time. Suppose that estimate amounts to 22.78 minutes in this case.

The second source of difference is found by estimating the model with period 2 data. Then you look at the different coefficients on education from the period 1 and period 2 models and apply the difference to the period 1 data. In other words, you see how much the function changed, as measured for the period 1 education levels. Suppose that in the second period, the effect of education seems to be larger. This difference in the coefficients explains 50.38 minutes of the difference between 1 and 2.

Finally to interpret the interaction term, you have to imagine that you apply the difference in the models to the difference in the data. While the period 1 education using the period 2 model implies a high time and the period 1 model applied to period 2 education also implies a high time, it turns out that the period 2 model applied to the period 2 education yields a small time. This idea that when you interact the period 2 data with the period 2 model you get a different result than if you had just summed up the education change by the period 1 model and the model change by the period 1 data is what is at stake. In this case it turns out that the combination in yields a lower value than what is predicted by the sum of the differences. The interaction term is then difference between what you expected from the two individual differences and the result. In this case its a negative number, -43.66.

Another way to see this is to imagine we are decomposing the difference between f(x1,x2) and f(x2,y2). As you go from x1,y1 to x2,y2, you will advance over the x axis. You expect the function to change Dx=(x2-x1)*df/dx(x1,y1).AS you go from x1,1 to x1,y2 you will advance over the y axis. You expect the function to change Dy=(y2-y1) *df/dy(x1,x2). Lastly you compare your results and realize that there's an error, xi, so that f(x2,y2)=f(x1,y1)+ Dx+Dy +xi. Dx here corresponds to the endowments effect, Dy to the coefficients effect, and xi to the interaction effect.

• Thank you for your answer. I tried to wrap my head around your answer but I am not sure I still fully understand. What is the baseline here ? Do you have some literature explaining this interaction term ? I cannot find any. Thanks again
– giac
Apr 23 '16 at 16:37
• Thank you for asking again, I'll clarify the answer. The stata command description is here stata-journal.com/sjpdf.html?articlenum=st0151. Apr 25 '16 at 19:29