Interpreting of continuos interaction in my logit model

[Theo Pollen]

Hi all,

I've been running into a problem and was hoping you could help me.

I'm running a logit model where I'm studying entry modes as binary choices. The main independant variable is economic distance, additionally I include the home nations legal environment in the model. I believe economic distance could have a different impact at different levels of economic distance which is why I also include a interaction term. All three have significant coefficients but I'm really strugeling with the interpretation.

. logit Entry Ec_dist Rule_law Ec_dist#Rule_law

Iteration 0: Log likelihood = -53571.268
Iteration 1: Log likelihood = -49508.953
Iteration 2: Log likelihood = -49487.729
Iteration 3: Log likelihood = -49487.722
Iteration 4: Log likelihood = -49487.722

Logistic regression Number of obs = 79,694
LR chi2(4) = 8167.09
Prob > chi2 = 0.0000
Log likelihood = -49487.722 Pseudo R2 = 0.0762

---------------------------------------------------------------------------------------
Entry | Coefficient Std. err. z P>|z| [95% conf. interval]
----------------------+----------------------------------------------------------------
Ec_dist | -.0018081 .0003688 -4.90 0.000 -.0025309 -.0010852
Rule_law | -.0403795 .000576 -70.10 0.000 -.0415084 -.0392506
Ec_dist#Rule_law | .0004922 .0000214 23.02 0.000 .0004503 .0005341
_cons | -.317896 .0087881 -36.17 0.000 -.3351204 -.3006716
---------------------------------------------------------------------------------------

As you can see Economic distance has a positive effect and legal environment has a negative, this is as expected. However, I also expected the coefficient of the interaction to be negative, with the interpretation that as legal environment increases economic distance loses its force. Could this negative interaction also mean the negative effect of the legal environment is mitigated?
To give further background to my question: All variables are centered; The variable for economic distance seems quite volatile and changes sign often when new variables are included, although it holds the correct (expected) sign in the full model.
Furthermore, these are the marginal effects at different points:

. margins, at(Rule_law=(-30 0 30)) dydx(Ec_dist)

Average marginal effects Number of obs = 79,640
Model VCE: OIM

Expression: Pr(JointVentureFlagDummy), predict()
dy/dx wrt: distc
1._at: Rule_law = -30
2._at: Rule_law = 0
3._at: Rule_law = 30

------------------------------------------------------------------------------
| Delta-method
| dy/dx std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
Ec_dist |
_at |
1 | -.0020415 .0001336 -15.28 0.000 -.0023032 -.0017797
2 | .0007006 .0000958 7.32 0.000 .0005129 .0008883
3 | .0023218 .0001279 18.15 0.000 .0020711 .0025725
------------------------------------------------------------------------------


. margins, at(Ec_dist=(-50 0 50)) dydx(Rule_law)

Average marginal effects Number of obs = 79,640
Model VCE: OIM

Expression: Pr(JointVentureFlagDummy), predict()
dy/dx wrt: ROLc
1._at: Ec_dist = -50
2._at: Ec_dist = 0
3._at: Ec_dist = 50

------------------------------------------------------------------------------
| Delta-method
| dy/dx std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
Rule_law |
_at |
1 | -.0117104 .0001711 -68.42 0.000 -.0120458 -.0113749
2 | -.0087487 .000107 -81.79 0.000 -.0089584 -.0085391
3 | -.0044332 .0002515 -17.63 0.000 -.0049261 -.0039404
------------------------------------------------------------------------------

To repeat my question, how would I interpret the positve interaction coefficient?
1. As economic distance increases the effect of rule of law gets diminished.
2. As Rule of law increases the effect of economic distance gets further reinforced.

Your help is greatly appreciated!
Thanks in advance.

[Clyde Schechter]

As you can see Economic distance has a positive effect and legal environment has a negative..

No, this is not correct. In fact, because you are using an interaction model with continuous variables, it is mathematically impossible to characterize either of those variables as having, overall, a positive or a negative effect. Each of those variables' effects will be sometimes positive and sometimes negative depending on the value of the other variable.

You state that your variables are centered, though you do not state what the centering point is. For purposes of discussion here, I'll assume that you centered them at their means. (Nothing substantive I say here hangs on the specific centering point.) So the direct readout of the regression results is that the effect of economic distance is positive in those situations where legal environment is at its mean value, and, similarly, the effect of rule of law has a negative effect in those situations where legal environment is at its mean value. The effect of each of those variables will differ at other values of the other variables, and for any real number you choose, you can always find a value of economic distance that makes the effect of rule of law equal that number, and vice versa. (Of course, whether the value that you find may not be attained in real life, but that doesn't invalidate the mathematical point.)

As for the interaction term, it conveys the effect modification. Because it is positive, this means that as economic distance increases, the corresponding effect of rule of law also increases. Bear in mind that if the starting point of the rule of law is negative, increasing means, at first, less negative, and then once zero is passed and the effect of rule of law is in positive territory, it means more positive. Similarly, as rule of law increases, the effect of economic distance also increases, again meaning becomes less negative (if we start from a place where it is negative) and then, once into positive, it continues growing by becoming more positive.

Now, ordinary English gets slippery when dealing with negative numbers. If you think of legal environment as having a negative effect (which, as I have already pointed out, you should not do), then increasing economic distance results in the effect of legal environment growing less negative, at least initially. In plain English, one might describe this by saying that the effect of legal environment becoming diminished. By that we mean that the effect, which we are (incorrectly) thinking of as negative becomes less negative. Of course, if we start from a situation where the effect of legal environment is already positive (which will be true if economic distance is sufficiently large) then as the effect becomes even more positive one might, in plain English, refer to this as a reinforced effect! So the same interaction could be described as either diminished or reinforced depending on the starting value of the effect you have in mind!

Because of the inadequacy of English to clearly convey what is going on, I recommend avoiding trying to describe these things in words. Tables such as your -margins- output clearly show what is going on. You might also want to graph those -margins- results, which might give an even clearer picture of what is happening.

[Theo Pollen]

Thank you very much for your response,

You response has lighted to new question which I would be very grateful for if you could take a look.

1.Since the margins output convey such a complicated relationship between the two variables, I have also looked at specifications of the margins command in other models I have been thinking about. I've tried different specifications including a model with a quadratic term of economic distance, This seems to create a more linear marginal effect without sign changes or big drops in effect, would this generally be preffered (better model fit, significant coefficients etc. are ensured but overcomplication of the model is generally advised against by my supervisor)?

2. If the individual effects of the variables keep the same sign with and without interaction term, would it still be wrong to talk about diminishing/reinforced effects?

Your help is greatly appreaciated