Exploring Differentiated Trade Impact of a Continuous Variable on Two Subgroups Using Interaction Terms in PPMLHDFE Regression

[George Mane]

Hello everyone,

I am conducting an analysis using PPMLHDFE regression to explore how the effect of a continuous explanatory variable on trade outcomes differs across distinct subgroups of countries. The subgroup variable used in this analysis is binary/dummy (e.g., 1 for a specific group, 0 for the rest). The objective is to evaluate whether the impact of the continuous variable varies across these groups, not to assess the direct effect of the grouping variable itself on trade.

I have tried two different interaction approaches to address this question and would appreciate insights on their appropriateness and interpretation:

Approach 1: Full Interaction Model (##)
In this approach, I specify the interaction term using two hash marks (##), as in c.ContinuousVariable##GroupingVariable, where:

• ContinuousVariable represents the main explanatory variable.
• GroupingVariable is a binary variable (e.g., 1 for a specific subgroup, 0 for the rest).

This formulation includes:

1. The main effect of the continuous variable for the reference group (baseline subgroup).
2. The main effect of the grouping variable (standalone binary variable).
3. The interaction term, which captures the additional or differential effect of the continuous variable for the subgroup defined by the grouping variable.

While this method is statistically valid, I find the inclusion of the standalone grouping variable challenging to interpret. Since the grouping variable is unrelated to trade in a direct sense, its coefficient can be conceptually ambiguous in the context of trade analysis. For example, what does it mean for the grouping variable itself to "explain" trade?

Approach 2: Simplified Interaction Model (#)
In this approach, I use a single hash mark (#), as in c.ContinuousVariable#GroupingVariable. This simplifies the model by focusing only on the interaction between the continuous variable and the grouping variable, without including the standalone effect of the grouping variable.
In other words:

• The regression estimates the impact of the continuous variable for the baseline subgroup.
• The interaction term captures the differential effect of the continuous variable for the defined subgroup.

This approach avoids the potential issue of interpreting the standalone coefficient of the grouping variable. Instead, it directly examines how the continuous variable’s impact varies between subgroups.

Objective Clarification
The goal of this analysis is to evaluate whether the continuous variable's effect differs across subgroups defined by the binary grouping variable. For example:

• The grouping variable serves as a way to divide the sample into meaningful categories.
• The ## approach provides a more detailed model, including the standalone grouping variable effect.
• The # approach simplifies the interpretation by focusing exclusively on how the continuous variable’s effect changes across subgroups.

Both methods have their merits, and I understand they may yield different insights depending on the context.

Questions for the Community

1. In a PPMLHDGE framework, how do you interpret the standalone coefficient of a grouping variable in the ## approach? Does its inclusion make sense in the context of trade-related analyses, or does it lead to unnecessary complications?
2. Is the # approach a better alternative for understanding differentiated impacts, given its simplicity? Are there any downsides to using this approach compared to the ## method?
3. How would you decide between the two approaches when the objective is to explore how a continuous variable’s effect varies between subgroups? Could the methods complement each other?

Thank you for your insights and feedback! Your suggestions will be invaluable for refining my analysis.

Best regards,

George

[George Mane]

Dear all,

I realise my earlier post might have been too long and confusing. To clarify, I am seeking guidance on how to estimate the differentiated trade impact of a continuous variable (e.g., tariffs) for two subgroups (e.g., low-income exporter countries vs others) using PPMLHDFE regression.

In essence, I want to understand how the effect of a continuous variable on trade outcomes varies between two distinct subgroups. I have tried two approaches:

• Full Interaction (##): Produces three coefficients—the effect of the continuous variable (tariffs), the binary subgroup variable (low-income exporter), and their interaction. However, I’m unsure what question the interaction term truly answers. I don't think it is fit to answer the above question.
• Simplified Interaction (#): Provides two coefficients—one for the tariff variable when the low-income exporters equals 0, and one when it equals 1. This one seems to provide straightforwardly a more direct comparison, although I am not sure if this is correct.

Which approach is the correct one for answering my question? Are there any trade-offs I should consider?

Thank you for being so helpful.

Sincerely,
GM

[George Ford]

Standard procedure is to use ##.

Y = a1*X + a2*Z + a3*X*Z

dy/dX = a1 + a3*Z
dy/dZ = a2 + a3*Z

[George Mane]

Dear george,

Thank you so much for your explanation! It is very helpful to confirm this interaction approach.

However, I still have a concern regarding the inclusion of Z (e.g., a dummy for "low-income country") as an independent explanatory variable.
In my case, Z doesn’t have theoretical or practical relevance in directly explaining trade outcomes (Y). Z serves only to classify observations into subgroups. As such, I find the standalone coefficient for Z (a2) difficult to interpret in this context.

Would a subgroup-specific approach or simplified interaction (Y=b1⋅(X⋅(1−Z))+b2⋅(X⋅Z)) be a better way to estimate X’s effect separately for each subgroup? Or is there a way to use the ## approach while addressing the lack of interpretability for Z?

Thank you again for your insights. I truly appreciate it!

Best regards,

GM

[Clyde Schechter]

Slight correction to #3:
dY/dZ = a2 + a3*X

Concerning the interpretation of the coefficient of Z, it represents the slope of the outcome:Z relationship conditional on X = 0. (For a Poisson relationship, actually, it's the slope of the linear predictor:Z relationship, but this distinction is unimportant for present purposes.) This may or may not be a meaningful statistic. In many situations 0 is not even a possible value for X, or if possible, it is not of any particular interest. Sometimes it is meaningful, for example, if X is a mean-centered transform of another variable U, then it is the slope conditional on U being at its mean value.

So whether you would interpret this coefficient really depends on what the variable X is and what a zero value of X means, if anything. If it is meaningless, you are free to ignore it. But you are not free to omit it from the regression model: it should be there in order for the coefficient of Z#X to properly represent the group-differences in effect of X on Y. In this regard, it is much like the constant term in a simple linear regression like Y = cons + b*X. cons represents the expected value of Y when X = 0. It may or may not be of any interest, and might even be meaningless if X can never be 0 in the real world. But the constant term has to be there; its presence is required (unless you are specifically constraining its value to be 0) so that b will correctly represent the Y:X slope.

[George Mane]

Dear Clyde Schechter,

Thank you so much for your detailed explanation. It has really helped me understand why the ## interaction approach is necessary for correctly estimating the group-differences in the effect of X on Y.

That said, I’m still uncertain whether the ## approach directly answers my objective, which is to estimate X’s impact separately for each subgroup (Z=0 and Z=1). While the interaction term (a3⋅X⋅Z) captures the differential effect, does this approach inherently allow for subgroup-specific estimates of X, or does it remain focused on how Z modifies the relationship between X and Y?

If my aim is purely subgroup-specific effects of X, is the ## approach still appropriate, or should I consider an alternative setup (e.g., subgroup-specific regressions)?

I’d truly appreciate your insights on which approach would be most appropriate for this type of analysis.

Thank you again for your time and expertise!
Sincerely,
GM

[Clyde Schechter]

You can use either one, it's just a matter of correctly interpreting the results and applying the necessary algebra. They are just two different ways of parameterizing the same model.

For the moment, let's put aside the fact that you are using a Poisson regression. Suppose it were a linear regression. The results of the two approaches are equivalent: you just have to know how to translate from one to the other. Or, for the difference in effect between groups, the direct answer is available just by reading out the results for the interaction term in the ## model. Calculating the two group-specific effects requires some additional calculation. By contrast, with the # model, the two group specific effects can be read out directly from the regression output, but calculating their difference requires an additional command. So if it is clear to you that one of these is of interest and the other is not, go for the one that requires the least work to achieve your goal.

If, in fact, you really want both the difference between groups and the two group-specific effects, then your best bet is use the ## approach and follow-up with the margins command:

margins group_var, dydx(continuous_variable)

The group-difference is still read out of the regression output's interaction term coefficient. And -margins- then calculates the group-specific effects for you.

Now, let's take into account the fact that you have a Poisson regression. Many would argue that the coefficients of a Poisson regression are not good indicators of marginal effects, because they ignore the logarithmic link function between the linear part of the regression and the outcome variable. A true marginal effect should be calculated in terms of differences in the actual outcome variable, not the linear part of the regression. That implies that everything should be calculated using margins. Again, the simplest approach for this is to start with the ## model in the Poisson regression. Then, as before

margins group_var, dydx(continuous_variable)

gives you the two group-specific marginal effects, but this time in the outcome variable metric.

To get the group difference, you can no longer rely on the interaction coefficient, nor can you use -lincom-. Instead you do:

margins group_var, dydx(continuous_variable) pwcompare

[George Ford]

Thanks Clyde for the correction.

If it's just subgroups, then # makes sense.

[George Mane]

Dear George Ford and Clyde Schechter ,

Thank you very much for your feedback, it is invaluable to understand the right way to analyze these results.

I think I now understand the differences between the # and ## approaches better but would appreciate some clarification and confirmation.

To explore this further, I think a simpler and practical example would be helpful.

Below is the cpswage dataset available here:

use https://www.stata-press.com/data/r17/cpswage.dta

We can start with a simple regression to understand the impact of education and age on wage:

reg wage educ age

Now, suppose I want to understand the differentiated impact of education for two subgroups in the sample (females and males). This brings me to the # and ## approaches:


the # approach:

reg wage c.educ#i.female  age

This approach gives separate coefficients for educ by subgroup (male and female) without including the standalone effect of educ or female. The output directly provides the subgroup-specific effects of educ:

• For males: Coefficient of educ under Male.
• For females: Coefficient of educ under Female.

To note that this setup does not explicitly include the main effect of female (independent of education) or test for its interaction with educ.

The ## approach:

reg wage c.educ##i.female  age

This approach includes:

• The main effect of educ (overall impact, regardless of gender).
• The main effect of female (independent impact of being female on wage).
• The interaction term for educ and female, showing how gender modifies the effect of education.

Interpretation:

• The main effect of educ represents the impact for the baseline group (males).
• The interaction term tells us how the effect of educ differs for females compared to males.
• The main effect of female represents how being female affects wages when education is 0.

My Understanding and Question:

If I am only interested in the subgroup-specific effects of "educ", it seems that the # approach directly provides this without needing further calculations. However, if I want both subgroup-specific effects and an explicit test for how gender modifies the effect of education, the ## approach seems more appropriate.

Does this understanding sound correct? Also, in contexts where one is purely focused on subgroup-specific effects, is the # approach preferable for simplicity, or would you still recommend using ## for consistency and flexibility?

One final question is whether the coefficient results of the two approaches are comparable. Should they present the same picture?

Thank you again for helping clarify these differences!

Regards,
GM

[George Ford]

This setup would normally use a Oaxaca regression -- different constants and different slopes. Female may have its own effect on the constant, and the effect of age may differ between genders as well (such things are often ignored if not the topic of interest).

I think it is best to let both the constant and slopes vary. If there is no constant effect, then the coefficients will tell you that.

I've converted educ to a college dummy so I don't have to worry about the means. You can see that e4 captures everything in e2 and e3. e5 does not, since the slopes on age are different..
Dropped age for a cleaner look in second set.

summ educ
g college = educ>=16

eststo e1: qui reg wage college age
eststo e2: qui reg wage college age if female
eststo e3: qui reg wage college age if !female
eststo e4: qui reg wage college female c.college#c.female age c.age#c.female
eststo e5: qui reg wage college female c.college#c.female age
eststo e6: qui reg wage c.college#i.female age
esttab e1 e2 e3 e4 e5 e6, mtitle(Pooled Female Male Interactive PartInter Simplest)

* ignore age
eststo g1: qui reg wage college
eststo g2: qui reg wage college  if female
eststo g3: qui reg wage college  if !female
eststo g4: qui reg wage college female c.college#c.female
eststo g5: qui reg wage c.college#i.female
eststo g6: qui reg wage college c.college#c.female
esttab g1 g2 g3 g4 g5 g6, mtitle(Pooled Female Male Interactive Simplest Alt)

[George Mane]

Thank you George Ford , this is very clear and makes sense. So in conclusion, it is best to use G4, interacting the two variables by "##".

[George Ford]

I always do, just in case. But others may have different opinions.

[George Mane]

Dear Clyde Schechter and George Ford ,

Thank you again for the detailed guidance on the # and ## approaches, it has been extremely helpful!

In my analysis, I used the ## approach, but the coefficient for the interaction term is not statistically significant, while the coefficient for the continuous variable remains strongly significant and maintains the same magnitude.

Does this imply that the effect of the continuous variable (𝑋) on the outcome (𝑌) does not differ between the two groups (𝑍=0 and 𝑍=1)?

I’d appreciate your insights on how to best interpret and proceed in this scenario.

Best regards,
GM

[Clyde Schechter]

It is an extremely common error for people to interpret a non-statistically significant difference (in your case, interaction term) as the absence of any difference between the things being compared. It is the result of very widespread, bad teaching of statistics at the introductory level.

When a difference is not statistically significant, the actual absence of a difference is only one of several possible reasons. It is also possible that the analysis lacked statistical power to detect whatever difference there actually is. (This is particularly important with regard to interactions. As a rule of thumb, the sample size needed to adequately power a test of interaction is between 4 and 16 times as large as the sample size needed to test a main effect.) It is also possible that the absence of statistical significance is attributable to noisy outcome data, or a mis-specified statistical model. What you can say is that you did not find convincing evidence of a difference, but that is a (much) weaker conclusion than saying that there is no difference. Another way to say it is that the study is inconclusive with respect to whether those effects are different.

Can you ever say that there is no difference? Yes, but it's rare that the conditions for this are met: you must be able to rule out the alternative explanations for the non-statistical significance of the observed difference.

[George Mane]

Dear Clyde Schechter Thank you for the clarification. Naturally, I leaned toward this explanation. It is good though to have a second perspective on this. Many thanks for your contribution. Best regards, GM