Interpretation of interaction variable when base variable is insignificant

I am investigating the effect of Indian service imports on U.S. service employment, using a balanced panel data set. The employment data are state-level for 5 different business service sectors. In this research the dependent variable is U.S. business service employment. The independent variables are: Indian service imports, Skill intensity within business service sectors(measured in average years of education), Real GDP and an interaction variable IMPORT*Skill. The purpose of this interaction variable is to measure whether the effect of Indian business services is larger in sectors that are more skill intensive. Logs are used to ease the interpretation of results. Furthermore, not all variables were found to be stationary, which is why the first difference is taken of each variable. A dummy for each year has been used to add a time effect.

The results show that there is no significant relationship between Indian service Imports and U.S. business service employment and that there is a significant positive relationship between Skill and business service employment. Furthermore, the interaction variable is found to be significantly negatively related with employment.

The regression I used is:
xtreg DIFvalue DIFimp DIFskill DIFgdp DIFimpXskill dummy1 dummy2 dummy3 dummy4 dummy5 dummy6 dummy7 dummy8 dummy9 dummy10 dummy11 dummy12 dummy13 dummy14 dummy15 dummy16, fe vce (cluster cn_ind)


The output is as following:

R-sq:               Obs per group:
within  =   0.2501  min =         14
between =   0.5817  avg =       14.0
overall =   0.2011  max =         14
F(15,249)         =          .
corr(u_i, Xb)   = 0.0045    Prob > F          =          .

(Std. Err. adjusted for 250 clusters in cn_ind)

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

DIFimp         .045624    .0393354     1.16   0.247    -.0318485    .1230965
DIFskill       8.36e-10   3.32e-10     2.51   0.013     1.81e-10    1.49e-09
DIFgdp        -.0016054   .0273678     -0.06  0.953    -.0555073    .0522965
DIFimpXskill  -.0003645   .0001315     -2.77  0.006    -.0006236   -.0001055


My question is, How do I interpret the interaction variable? Indian imports(DIFimp) are not significantly related with U.S. employment(DIFvalue) and Skill(DIFskill) is positively related with U.S. employment. The interaction variable is negatively related with employment. So does this still mean that as the skill within a sector is higher, that the effect of Indian imports on employment is lower? Or is this impossible as the effect of Indian imports on employment is insignificant?

Could anyone help me with this interpretation?

You should use Stata's factor notation and margins command to interpret the interaction. Assuming your two variables (DIFimp and DIFskill) are continuous you could rewrite your model as:

xtreg DIFvalue c.DIFimp##c.DIFskill c.DIFgdp i.year, fe vce (cluster cn_ind)


Then, you can use the margins command to compute predicted values for various values of the two independent variables. Let us assume both variables have a range from 1 to 10, you can use this code:

margins, at(DIFimp=(1 5 10) DIFskill=(1(1)10))
marginsplot


This will create a plot with three lines showing you the expected value of DIFvalue for each value of DIFskill. The three lines show you how the predicted values differ between the three levels of DIFimp.

You should refrain from interpreting the interaction by looking at each coefficient in isolation. The coefficients of DIFimp and DIFskill only display the effect of these variable when the other variable equals zero. This might not be a plausible value. Even if it is, it is only one part of the interaction.

Hope this helps.