I think the first thing you need to ensure is that you're not comparing apples to orangutans. Then we will discuss standard errors, statistical significance, and model selection.
Here's how you might compare OLS/LPM and logit coefficients for dummy-dummy interactions. We will model union membership as a function of race and education (both categorical) for US women from the NLS88 survey.
First, we will use OLS with factor variable notation for the interactions:
. sysuse nlsw88, clear
(NLSW, 1988 extract)
. reg union i.race##i.collgrad
Source | SS df MS Number of obs = 1878
-------------+------------------------------ F( 5, 1872) = 7.02
Model | 6.40214176 5 1.28042835 Prob > F = 0.0000
Residual | 341.434386 1872 .182390164 R-squared = 0.0184
-------------+------------------------------ Adj R-squared = 0.0158
Total | 347.836528 1877 .185315146 Root MSE = .42707
-------------------------------------------------------------------------------------
union | Coef. Std. Err. t P>|t| [95% Conf. Interval]
--------------------+----------------------------------------------------------------
race |
black | .0799445 .0250534 3.19 0.001 .0308089 .1290801
other | .1157454 .1076307 1.08 0.282 -.0953433 .3268342
|
collgrad |
college grad | .0975234 .0261143 3.73 0.000 .0463072 .1487395
|
race#collgrad |
black#college grad | .0415079 .0563381 0.74 0.461 -.0689841 .152
other#college grad | -.0350234 .1867622 -0.19 0.851 -.4013073 .3312606
|
_cons | .1967546 .0136007 14.47 0.000 .1700804 .2234288
-------------------------------------------------------------------------------------
For instance, black women who also graduated from college are 4.15 percentage points more likely to be in a union.
Now we fit a logit model:
. logit union i.race##i.collgrad, nolog
Logistic regression Number of obs = 1878
LR chi2(5) = 33.33
Prob > chi2 = 0.0000
Log likelihood = -1029.9582 Pseudo R2 = 0.0159
-------------------------------------------------------------------------------------
union | Coef. Std. Err. z P>|z| [95% Conf. Interval]
--------------------+----------------------------------------------------------------
race |
black | .4458082 .1361797 3.27 0.001 .178901 .7127154
other | .6182459 .5452764 1.13 0.257 -.4504762 1.686968
|
collgrad |
college grad | .5320064 .1397767 3.81 0.000 .2580491 .8059637
|
race#collgrad |
black#college grad | .0885629 .2791468 0.32 0.751 -.4585548 .6356807
other#college grad | -.2543746 .918575 -0.28 0.782 -2.054748 1.545999
|
_cons | -1.406703 .0801078 -17.56 0.000 -1.563712 -1.249695
-------------------------------------------------------------------------------------
The logit index function coefficients are not particularly meaningful since they are not effects on the probability of union membership. The sign and the significance might tell you something, but the magnitude of the effect is not clear. Also note that the standard errors are large, like in your own data. For instance, the SE of the college graduate of other race coefficient is almost 1.
To get something comparable to OLS, we will use margins with the contrast operator:
. margins r.race##r.collgrad
Contrasts of predictive margins
Model VCE : OIM
Expression : Pr(union), predict()
----------------------------------------------------------------------------------------
| df chi2 P>chi2
-----------------------------------------------------+----------------------------------
race |
(black vs white) | 1 14.34 0.0002
(other vs white) | 1 1.20 0.2725
Joint | 2 15.14 0.0005
|
collgrad | 1 19.09 0.0000
|
race#collgrad |
(black vs white) (college grad vs not college grad) | 1 0.44 0.5085
(other vs white) (college grad vs not college grad) | 1 0.03 0.8666
Joint | 2 0.48 0.7869
----------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------
| Delta-method
| Contrast Std. Err. [95% Conf. Interval]
-----------------------------------------------------+------------------------------------------------
race |
(black vs white) | .0901999 .0238201 .0435134 .1368864
(other vs white) | .1070922 .0976013 -.0842029 .2983873
|
collgrad |
(college grad vs not college grad) | .108149 .0247526 .0596347 .1566633
|
race#collgrad |
(black vs white) (college grad vs not college grad) | .041508 .0627785 -.0815355 .1645515
(other vs white) (college grad vs not college grad) | -.0350233 .2084485 -.4435749 .3735282
------------------------------------------------------------------------------------------------------
These are pretty close to the OLS effects. For instance, black women who graduated from college are also 4.15 percentage points more likely to be in a union according to the logit model. The SEs are somewhat smaller.
Sometimes you can't run the margins command because you don't have the data. All you have are the logit coefficients from someone's paper. While I said they were not particularly meaningful in their raw form, you can transform the logit index function coefficients into a multiplicative effect by exponentiating them, which is easy enough with a calculator. For example, the index function coefficient for black college graduates was .0885629. If I exponentiate it, I get $\exp(.0885629)=1.092603$. This tells me that black college graduates are 1.09 times more likely to be union members compared to a baseline of $\exp(-1.406703)=0.24494955$ (the baseline is the exponentiated constant from the logit). So this means that the union rate for black college graduates will be $0.24\cdot 1.09$ or about $26$%. OLS and logit with margins, will give the additive effect, so there we get about $19.67+4.15=23.87$. That's pretty darn close. It won't always work out so nicely.
Stata will give you exponentiated coefficients when you specify odds ratios option or:
. logit union i.race##i.collgrad, or nolog
or just use logistic:
. logistic union i.race##i.collgrad, nolog
I learned about these tricks from Maarten L. Buis. There are lots of examples with interactions of various sorts and nonlinear models at that link.
In my toy example, I did not cluster my errors, but that doesn't change the main thrust of these results. Some people don't like clustered standard errors in logit/probits because if the model's errors are heteroscedastic the parameter estimates are inconsistent.
After that long detour, we finally get to statistical significance. In all the models above (OLS, logit index function, logit margins, and OR logit), all the interactions are statistically insignificant (though the main effects generally are not). The standard errors are large compared to the estimates, so the data is consistent with the effects on all scales being zero (the confidence intervals include zero in the additive case and 1 in the multiplicative). If we surveyed enough women, it is possible that we would be able to detect some statistically significant interactions. The statistical significance depends in part on the sample size. If you don't have too many Bhutanese students in your data, it will be hard to detect even the main effect, much less the foreign friends interaction. On the other hand, if the effect is huge, you might be able to detect it with only a few students. Perhaps you can try grouping students by continent instead of country, though too much data-driven variable transformation is to be avoided.
Generally, OLS and non-linear models will give you similar results. If they don't, as may be the case with your data, I think you should report both and let you audience pick. Some people believe OLS/LPM is more robust to departures from assumptions (like heteroscedasticity), others disagree vehemently. You can and should justify a preferred model in various ways, but that's a whole question in itself. Personally, I would report both clustered OLS and non-clustered logit marginal effects (unless there's little difference between the clustered and non-clustered versions). You can also use an LM test to rule out heteroscedasticity.
Finally, with dummy-dummy interactions, I believe the sign and the significance of the index function interaction corresponds to the sign and the significance of the marginal effects. For continuous-continuous interactions (and perhaps continuous-dummy as well), that is generally not the case in non-linear models like the logit.
Best Answer
You know that in a logit:
$$Pr[y = 1 \vert x,z] = p = \frac{\exp (\alpha + \beta \cdot \ln x + \gamma z)}{1+\exp (\alpha + \beta \cdot \ln x + \gamma z )}. $$
After some tedious calculus and simplification, the partial of that with respect to $x$ becomes:
$$ \frac{\partial Pr[y=1 \vert x,z]}{\partial x} = \frac{\beta}{x} \cdot p \cdot (1-p). $$
This is (sort of) equivalent to
$$\frac{\Delta p}{\Delta x}=\frac{\beta}{x} \cdot p \cdot (1-p),$$
which can be re-written as
$$\frac{\Delta p}{100 \cdot \frac{ \Delta x}{x}}= \frac{\beta \cdot p \cdot (1-p)}{100}.$$
This is the definition of semi-elasticity, and can be interpreted as the change in probability for a 1% change in $x$.
Here's an example in Stata.* Note that I am using
marginsinstead of the out-of-datemfxto get the average marginal effect of $x$, $\frac{1}{N}\Sigma_{i=1}^N\frac{\beta \cdot p_i \cdot (1-p_i)}{100}$:This means that for a 1% increase in price, the probability that a car is foreign increases by 0.005 on a [0,1] scale. Or a 10% increase in price gives you a 0.05 increase. In this date, about 0.3 of the cars are foreign, so these are economically and statistically significant.
Edit:
A good way to do this in Stata 10 is to install the user-written command
margeff:*This is actually not a great empirical example since the relationship in the data has an inverted-U shape.