Marginal effect of interaction variable in probit regression using Stata

[David Freyr]

Running the following probit model and computing the marginal effects:

probit y a b a#b

margins, dydx(*)

does not yield marginal effect coefficients for the interaction variable
(a#b could denote dummy-dummy, dummy-continuous or continuous-continuous variable interaction)

So I was wondering if a valid workaround that problem
would be to generate a new variable:

gen c = a*b

and run the following code:

probit y a b c

margins, dydx(*)

Best regards,
Davíð

[Clyde Schechter]

What is the probability that two people would ask a question based on the same misconception within minutes of each other?

See http://www.statalist.org/forums/foru...al-link-probit.

You can run that code if you want, and because you will have defrauded -margins- into thinking that c is a real variable, it will give you a number. That number is meaningless nonsense and a waste of pixels/ink/whatever. There is no such thing as the marginal effect of an interaction term.

[David Freyr]

Thank you for that clarification.

I am trying to replicate the following table from a paper. I gather the variables such as
Maria x Rent/10000 are computed by some other means?

[Richard Williams]

My guess is that the marginal effects are computed incorrectly. But without seeing the whole paper it is hard to say what was done.

[David Freyr]

Thank you for the response Mr. Williams.

The only discussion of this table in the paper is with regards to the interpretation of the coefficients.

Thus, the coefficient of the interaction term in specification (2) that is significant at the 10 percent level
is taken to mean that if Metropolitan == 1, that would increase the probability of Mohammed
getting invited for further contacts or to a showing. This is reported as saying that ethnic discrimination
is strongest outside the metropolitan area since the coefficient for Mohammed is negative.

The following code would seem to accomplish this:

margins, dydx(mohammed) over(r.metropolitan)

This comes from a presentation on this topic (slide 26):

http://www.stata.com/meeting/germany.../de13_jann.pdf

But could this be done for the whole of specification (2) in one line of code?

[David Freyr]

In other words, is there a valid way to obtain the coefficients presented in the table?

[Clyde Schechter]

The table you show in #3, according to its caption, contains coefficients of a probit regression, with corresponding marginal effects in parentheses underneath them. You should be able to replicate the coefficients (and their significance levels) just be running -probit- on the same data.

As for the marginal effects, as already indicated, there is no such thing as the marginal effect of an interaction term. Moreover, notice that all of the parenthesized "marginal effects" are positive, even when the corresponding coefficients are negative. While you can have a positive marginal effect associated with a negative coefficient in a multinomial outcome model, that does not happen in probit models. So I don't think any of the numbers in that entire table are marginal effects. My best guess is that what you are looking at are coefficients and parenthesized standard errors. Further supporting this belief is my observation, spot-checking a few of the table entries, is that the significance stars come with entries where the ratio of the unparenthesized to the parenthesized number is high, and the unstarred entries have low parenthesized to unparenthesized ratios.

Still, I cannot be sure what this is about. If the paper does not explain what they actually have there, I think it is anybody's guess what the numbers in the lower half of the table are. Probably your best bet is probably to contact the authors of the paper and ask them.

[David Freyr]

Thank you very much for your response Mr. Schechter.

It is stated explicitly in the paper (in the text) that the coefficients in the table are indeed marginal effects (and my guess is also that the numbers in parentheses are standard errors).

From the paper, regarding the interpretation of one of the the interaction terms (0.16):

Results from specification (2) reveal that the interaction between the variables Mohammed and Metropolitan area is positive and statistically significant at the 10 per cent level. Thus, our results indicate that ethnic discrimination in the housing market in Sweden is strongest outside of metropolitan areas.

This indicates to me that the interaction term has some meaning.

I found a clearer example of this in an article published in 2011 (https://www.ncbi.nlm.nih.gov/pmc/art...3447245/#SD2):

Here the interaction effect seems also to be of some value, or am I misinterpreting something?

Best regards,
Davíð

[Clyde Schechter]

Yes, you're misunderstanding what was done in the manual. What is calculated there are the marginal effects of age at female = 1 and at female = 2. These are not marginal effects of the age#female interaction--they couldn't be, because there is no such thing. And as the manual's language in the post explains, "...the interaction effect is the difference in the manual effect of age on outcome between men and women..." [emphasis mine]. Note that they do not call this the "marginal effect of the interaction."

Regarding their assertion "Thus, our results indicate that ethnic discrimination in the housing market in Sweden is strongest outside of metropolitan areas", that would be correct if the numbers in the table are coefficients. So everything except the table caption is all consistent with interpreting the table as showing coefficients, notwithstanding their claim that they have shown marginal effects.

I really think you should contact the authors of the paper to ask them what they actually did. Again, my best guess is that the table contains coefficients and their standard errors. But there is no need for any of us to guess here, unless the authors are no longer around.

[David Freyr]

I see. Thank you for the clarification.

Yes, it is quite confusing if they have coefficients in the table, then as to why this is stated in the paper:

The results of the estimates (the marginal effects) are presented in Table 5.

and with regards to specification (4):

It emerges that the interaction between the variables Mohammed and Rent is negative and statistically significant at the 5 per cent level. This implies that the more expensive the apartment is, the larger is the ethnic gap in the probability of being invited to a showing. Thus, ethnic discrimination increases with the apartment’s rent.

But I'm wondering if this is down to wording: would I be right in stating that the incremental effect of gender (being female instead of male) is positive, that is, increases the marginal effect of age?

So although the coefficients in the table are not exactly identical they can be interpreted in relation to one another?

Best regards,
Davíð

[Richard Williams]

I am with Clyde. If these were coefficients and standard errors, I could believe the numbers in the table. But, the references to marginal effects and probability seem weird to me. Maybe they just screwed up the title to the table.

Do we have the citation for the paper in post #3?

[Sebastian Geiger]

I would not go as far as Clyde and say "there is no marginal effect of interaction terms (in non-linear models like probit)". However, I agree that there is a dissent about how to compute those marginal effects (and about what they actually mean). See this article http://www.sciencedirect.com/science...65176503000326 and compare it with http://www.stata.com/statalist/archi.../msg00293.html

[Richard Williams]

I've never quite understood the Norton/Wang approach, but there is a Stata Journal article too:

http://www.stata-journal.com/sjpdf.h...iclenum=st0063

One key thing it says is "The new command inteff calculates the interaction effect, standard error, and z statistic for each observation for either logit or probit when two variables have been interacted." Further, "Some interaction effects are positive, and some are negative, no matter what the sign of the coefficient on the interaction term." So as I understand their method, there isn't a single interaction effect, there is an interaction effect for each case.

[Clyde Schechter]

I don't have access to the article in sciencedirect.com, but the older post in Statalist and the stata-journal article both actually support my position. Within those to articles the closest you will find to a contradiction of my position is a reference by Norton, Wang & Ai to "the marginal effect of the interaction term", for which they give a formula involving a partial first derivative. But they then go on explain that that statistic is the wrong thing to calculate and is not useful. What their program -inteff- calculates is not what they refer to as "the marginal effect of the interaction term" but the marginal effects of changing both of the variables that participate in the interaction at various levels of those variables.

[David Freyr]

Thank you all for your responses. I will read through the referenced reading material.

Mr. Williams, the paper I referenced in post #3 is here: http://www.sciencedirect.com/science...94119008000181
The probit estimates are presented and interpreted on pages 8-10.