Interactions in a FE-model: scale of the variables to be interpreted (demeaned or orginal?)

[Pablo Christmann]

Hello,

in an interaction in a Fixed Effects model, do I use the original scale of the modifying variable z to interpret the marginal effect of x on y? Or should I use the demeaned scale of the variable z that will actually be used in the fixed effects model? The interaction is between two continuous variables.

The difference became apparent when I calculated the same fixed effects model with xtreg, fe and reg for which I calculated the demeaned variables directly. Using reg with the demeaned variables, I would naturally inspect the demeaned variable and not the original variable.

Many thanks,
Pablo

[Carlo Lazzaro]

Pablo:
why making things more difficult with a pooled OLS (POLS) when you can exploit -xtreg, fe- with all the interactions you want in their original scale?
POLS makes sense when the F-test at the foot of the -xtreg,fe- outcome table is not significant.

[Pablo Christmann]

Dear Carlo,

many thanks for your reply. No, unfortunately a pooled OLS is not an option for the data at hand (the F-test you proposed for xtreg, fe is also highly significant). Likewise a Random Effects model does not seem advisable. I have a TSCS dataset with some 70 countries over a 30 year period. Initially, I started with a multilevel model (xtmixed) where I added the demeaned and group-mean centered terms of the variables to capture the 'between' country and 'within' country effects. To simplify my paper I now only use the 'within' predictors of the FE model.

Regarding my initial question: So do you think I would need to inspect the demeaned variables in order to interpret the interaction effect?

Many thanks,
Pablo

[Sebastian Kripfganz]

Notice that the interaction effect of demeaned variables is not the same as the demeaned interaction effect of the original variables. The fixed-effects model computes the latter.

[Carlo Lazzaro]

Pablo:
you should follow the approach that is consistent with what Others did in your research field when presented with the same research topic.
Hiowever, I have two asides:
- if -xtreg, re- was not the best specification for your model, -xtmixed- (now -mixed-) suffers from the same disease;
- with 30 data waves, I would consider serial correlation as an issue that deserves attention.

PS: Crossed in the cyberspace with Sebastian's very good point.

[Pablo Christmann]

@ Carlo:
many thanks for your helpful responses. My DV is relatively volatile public opinion data and serial correlation is not that much of an issue (also theoretically somewhat difficult to imagine).
Yes, you are right but my question is more generally about how to best interpret interactions in FE models, more in detail about how to plot the effects. For this i need to know the scale of the variables, so I can chose appropriate _at values for margins. I am afraid that people using Stata might not actually pay attention to this as xtreg, fe does all the transformations for you.

@ Sebastian: I fear I do not understand your reply. Could you clarify it a little bit? My intuition is that I need to inspect the scale of the demeaned variables in order to chose appropriate _at values for margins, so I can interpret the substantial effect of the interaction. Would that be correct?

Below is an example. I believe that only the second plot is the correct one. Could you confirm it?

[Pablo Christmann]

Actually I have already posted the example a week ago. Could you have a look on it and tell me what you think?

http://www.statalist.org/forums/foru...after-xtreg-fe

[Carlo Lazzaro]

Pablo:
admittedly I cannot follow your question in full.
That said, I would pay attention to the first one of your -marginsplot- graph (the one after -xtreg, fe-).

[Pablo Christmann]

Many thanks for your reply. My intuition was that the second interaction plot with the demeaned scale of the modifying variable z should be the correct one.

Maybe someone with a third opinion can help us to settle the issue.

[Sebastian Kripfganz]

Your original model looks as follows:
\[y_{it} = \beta_0 + x_{it} \beta_1 + z_{it} \beta_2 + x_{it} z_{it} \beta_3 + u_i + e_{it}\]

To obtain the fixed-effects estimator, the following transformed (de-meaned) model is estimated:
\[(y_{it} - \bar{y}_i) = (x_{it} - \bar{x}_i) \beta_1 + (z_{it} - \bar{z}_i) \beta_2 + (x_{it} z_{it} - \bar{xz}_i) \beta_3 + (e_{it} - \bar{e}_i)\]
where \(\bar{xz}_i = \frac{1}{T} \sum_{t=1}^T x_{it} z_{it}\) is the within-groups average of the product of \(x_{it}\) with \(z_{it}\). Consequently,
\[(x_{it} z_{it} - \bar{xz}_i) \neq (x_{it} - \bar{x}_i) (z_{it} - \bar{z}_i)\]

[Pablo Christmann]

Dear Sebastian,

many thanks for your response. Yes, this is true, I know that I need to use the original variables to estimate the interaction term and then demean it (and not use the demeaned variables to estimate the interaction term).

But my question is about the constitutive terms x and z. I want to plot the marginal effect of variable x conditional on the values of the modifying variable z. But which values to choose? I think i need to interpret the demeaned scale and not the scale of the original variables, am I correct? (Maybe have a look on the example I made: http://www.statalist.org/forums/foru...after-xtreg-fe)

Many thanks!

[Sebastian Kripfganz]

The marginal effects are computed for your specified values of the variable \(z_{it}\), based on the original untransformed model. Your second plot is not meaningful because the variable \(z_{it}\) does not assume such large negative values.

[Sebastian Geiger]

If I understand you correctly, Pablo: Try typing

margins, dydx(*) atmeans

Stata will tell you at the top of the table which values it used to compute the marginal effects. As Sebastian already stated, Stata uses the untransformed values (which is appropriate).

I don't know your research project, but if you just want to choose appropriate values of the covariates to estimate the marginal effects, I would choose average marginal effects instead of fixing a covariate to a specific value. The following command estimates average marginal effects:

margins, dydx(*)

Of course, evaluating the marginal effect at specific values may be interesting in some cases.