# Interpretation of coefficient for interaction term in regression

The following is a regression of ln(earnings), i.e. the logarithm of avg annual earnings, on age, age squared, woman (dummy variable 1=woman), education (years of schooling), married (dummy variable 1=married) and an interaction term = woman*educ. How would you interpret the coefficient of the interaction term = 0.0110802?

As woman is a dummy variable, you can interpret the interaction coefficient as the average effect of one year of education on the log of earnings for woman. That is, for woman, every year of education is correlated, on average, with an increase of 0.11 on log of earnings.

The log scale is often considered to be an approximate of a percentage scale (see this paper, for example). So you could say that, for woman, every additional year of education is correlated with an increase of approximately 10% on earnings.

The average effect of education for man is given by the coefficient of education alone (educ). When you include an interaction term between education and woman, the coefficient of education becomes the effect of education when variable woman equals zero (that is, when man). See this paper for further details . So, for man, the average effect of additional years of education on log income is 0.09.

By the same logic, the coefficient of woman alone gives you the effect of woman (related to man) when education equals to zero, which maybe it is not very informative. When education equals to zero, woman earn around 21% less than man.

When you have a continuous variable in the interaction term, it is always a good idea to check the distribution of the effect through a plot (I will show how to do this below).

Also, as you included an interaction term, you are probably interested in the different effect of education between man and woman. The p-values you have for the interaction term and for education coefficient alone only tell you that the effect of education for both woman and man are different from zero. But they do not tell you if the effect of education is different between woman and man. One way of finding this is by using the margins command in Stata.

You should first re-write your regression syntax, because once you include the interaction this way woman##c.educ you don't need to include the constitutive terms alone. Stata already includes them. That is why you got two omitted variables on your output.

reg lnearn age age2 i.woman##c.educ married manager



Then, you run the margins command to get the confidence intervals of the difference between woman and man. I am assuming that your education variable is coded between 0 and 15. If it is not the case, you should adjust these values in the code below

margins i.woman, at(educ= (0(1)15))


As you did not share your data, I cannot interpret the output you will have. But a good idea and nowadays regarded as a good practice is to plot the interactions. This can be done by running the marginsplot command right after the margins command.

marginsplot

• The t stat on 'woman' is -5.01 so does this mean that this value is not statistically significant? Also, in my previous regression I had all of the same variables as this regression but I didn't include the interaction term. The root MSE was .53611 and in this regression it is lower at .53593, does this mean that this model is less accurate? Nov 18 '20 at 8:26
• @Guntash, 1) the t-stat (-5.01) on woman shows that result is statistically significant. You can see that on the p-value (less than 0.05, in fact less than 0.001), and on the confidence intervals that do not include zero. 2) The MSE is the average of the squared errors. Each error is the difference between the observed and the predicted value. So, simply, the lower the MSE, probably the better the model fit. You may check for the adjusted R2, which is closely related to the MSE, but penalizes the addition of new variables. The higher the adjusted R2, the better. Nov 18 '20 at 10:25

In your model, the relationship between education and earnings is moderated by gender. Moreover, the coefficients in a log-level regression correspond to a semi-elasticity, so you need to multiply by 100 to put things in percent.

This means that for a man, an extra year of schooling is associated with a 9% bump in earnings according to your model. For a woman, there's an extra 1% on top of that, so it becomes ~10%. That difference between men and women is statistically significant.

You can confirm my interpretation by taking the partial derivative of log earnings with respect to education using the chain rule and then re-arranging the left hand-side into a semi-elasticity.