# Intuitive Interpretation of Interaction Term

This is my first time posting on this site so please excuse me if I am not following certain rules or guidelines specific to this forum.

I have a data set on a training program and I want to see whether equally skilled people are paid the same or their demographic influences their future earnings.

Variables:

 assignmt:     treatment group or not.

training:     received the training or not.

afdc:         receiving AFDC (Aid to Families with Dependent Children) at the same time

sex:          male = 1, female = 0

bdate:        date of birth

age:          age at the time of assignment

earn:         total 30 month earnings after the assignment takes place

married:      married and living with spouse or not

hsorged:      high school  diploma or GED (General Educational Development)

black:        black, non-Hispanic

hispanic:     Hispanic

wkless13:     worked less than 13 weeks in the past year


Here is the full regression model:

 . reg earn assignmt training afdc sex bdate age married hsorged black hispanic wkless13

Source |       SS           df       MS      Number of obs   =    11,204

-------------+----------------------------------   F(11, 11192)    =    118.81
Model |  3.2932e+11        11  2.9938e+10   Prob > F        =    0.0000

Residual |  2.8202e+12    11,192   251985269   R-squared       =    0.1046

Total |  3.1495e+12    11,203   281133832   Root MSE        =     15874

------------------------------------------------------------------------------

earn |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]

-------------+----------------------------------------------------------------

assignmt |  -872.3136   397.0343    -2.20   0.028    -1650.571   -94.05645
training |   3139.047   378.3347     8.30   0.000     2397.445     3880.65
afdc |  -2372.576   427.2609    -5.55   0.000    -3210.083    -1535.07
sex |   4598.853   322.3371    14.27   0.000     3967.016    5230.691
bdate |  -1.733176   .7830663    -2.21   0.027    -3.268123    -.198228
age |  -723.1733   286.1614    -2.53   0.012      -1284.1   -162.2466
married |   2491.303   327.5803     7.61   0.000     1849.188    3133.418
hsorged |   3590.436   344.8135    10.41   0.000     2914.541    4266.331
black |  -1744.983   360.6557    -4.84   0.000    -2451.932   -1038.035
hispanic |  -901.8467   496.6624    -1.82   0.069    -1875.392     71.6991
wkless13 |  -6421.666    331.158   -19.39   0.000    -7070.794   -5772.538
_cons |   34420.77   8092.328     4.25   0.000     18558.39    50283.16

------------------------------------------------------------------------------


A general analysis of the regression table indicates that there is some level of discrimination present: however I am trying to build a more specific model to show that equally skilled individuals are not paid the same even with the same training/skill level.

I was considering adding some interaction terms, such as black * hsorged (blck_educ) but the results, assuming I am interpreting this correctly, indicate that black individuals with a GED or diploma earn less than black individuals without an education, which I find very counter-intuitive.

. reg earn assignmt training afdc sex age married hsorged black hispanic

wkless13 blck_educ

Source |       SS           df       MS      Number of obs   =    11,204

-------------+----------------------------------   F(11, 11192)    =    118.53

Model |  3.2863e+11        11  2.9876e+10   Prob > F        =    0.0000

Residual |  2.8209e+12    11,192   252046952   R-squared       =    0.1043

Total |  3.1495e+12    11,203   281133832   Root MSE        =     15876

------------------------------------------------------------------------------

earn |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]

-------------+----------------------------------------------------------------

assignmt |   -881.271   397.0472    -2.22   0.026    -1659.553   -102.9887

training |   3177.937   378.1713     8.40   0.000     2436.654    3919.219

afdc |  -2376.105   427.3079    -5.56   0.000    -3213.704   -1538.507

sex |   4605.594   322.3958    14.29   0.000     3973.641    5237.546

age |  -91.48007   16.00721    -5.71   0.000     -122.857   -60.10312

married |   2527.633   327.4857     7.72   0.000     1885.703    3169.562

hsorged |   3835.689   394.8049     9.72   0.000     3061.802    4609.576

black |   -876.932   685.7989    -1.28   0.201    -2221.219    467.3546

hispanic |  -856.3717   496.8757    -1.72   0.085    -1830.335     117.592

wkless13 |  -6438.197   331.1042   -19.44   0.000     -7087.22   -5789.175

blck_educ |  -1174.194   799.2005    -1.47   0.142    -2740.768    392.3792

_cons |   16387.24   726.8202    22.55   0.000     14962.54    17811.93

------------------------------------------------------------------------------


Is this a reasonable or significant interaction term to incorporate into my analysis? If not, what other interactions would provide a more intuitive and clear answer to me question?

You are not interpreting the outputs correctly. In particular:

1. The p-value for blck_educ is relatively large (e.g., greater than 0.05), so you should be thinking of its coefficient as not being known to be different to 0 rather than being negative.
2. The p-value for black has changed a lot, suggesting something odd. Perhaps multicollinearity. I am not sure. You need to figure this out. The way to test whether the interaction makes a difference to the model as a whole is via an F-test to compare the two models. I am sure Stata will have some nifty way of doing this, but I don't use Stata.
3. You're not comparing like with like, as bdate is in model 1, but not model 2. If you want to test the effect of a single new parameter (such as your interaction), you need to be keeping everything else constant.
4. Ignoring my first three points (e.g., let's pretend everything is significant), the interpretation of your second model is not quite right. The key bits are that having the high school diploma adds 3835.689 to the income of a non-black person, but only (3835.689 + -1174.194) to a black person. That is, it is worth less to a black person. While this effect could be interpreted as a race effect (but, as the effects are not significant, this would be dodgy).
5. This is not my field, but my guess is that with an earnings model you should have a natural log of the dependent variable. Such topics are studied extensively in economics, so I would suggest copying what is the norm there.

In addition to the answer above, I would like to recommend this and this as additional reading. In STATA, it is possible to do an interaction term right off the regression command without generating an additional variable. Instead of having gen blck_educ = black * hsorged, you can run the command like:

reg earn assignmt training afdc sex age married i.hsorged##i.black hispanic


where i indicate dummy/factor variable (assuming both have only 0 or 1 in their values, other wise use c to indicate continuous variable). This way the code will be easier to read. The Interaction and its constitutive terms will be listed together and ease some interpretation when you generate tables etc.

To greatly simplify your model, the first one, you have $$earn = \alpha_1 hsorged + \alpha_2 black$$ In the one with interaction term, it becomes: $$earn = \beta_1 hsorged + \beta_2 black + \beta_3 (hsorged \times black)$$

When the interaction is included, one should look at the statistical significance and coefficients with great attention to their substantive meanings because they are no longer the same. For instance, it is not surprising that the coefficient of black ($\beta_2$) has changed from -1744.983 to -876.932 in your analysis because the number now means a different thing:

• Without interaction, it means the average effect of black on earn
• With interaction, it means the average effect of black on earn when hsorged is set to 0.

This is because when we do a derivative on earn with respect to black on the model with interaction, we will find that $$\frac{d}{d(black)} earn = \beta_2 + \beta_3 hsorged$$

which shows that the effect of black on earn depend not just on $\beta_2$, but also the level of hsorged and its coefficient ($\beta_3$). So $\beta_2$ is now only the effect of black when hsorged = 0. The same logic also shows that $\beta_1$ is no longer the effect of hsorged but its effect when black = 0

Under this situation, it might be better to look at the effect of black when you set the value of hsorged to a range of possible values (in this case perhaps just 0 and 1?) This is doable in STATA with the command (if you specify the interaction term the way I did it above):

margins dydx(hsorged black) at(hsorged=(0 1) black=(0 1))


after which marginsplot can also visualize the interactive relationship. Hope this helps in any way!