within-between-Interations in hybrid models for panel data

[Guest]

Dear Statalists,

I have a question concerning the possibility of using interaction terms in a hybrid model for panel data. Especially, I'm interested in modeling a between-within-interaction effect.

Let's say, I have the following variables:

x - independent variable 1: time-variant
z - independent variable 2: time-variant
y - dependent variable
with id as the identification number of each individual.

The variables could be prepared for the hybrid-model following Schunck (2013):

mark nonmiss
markout nonmiss y x z 

by id, sort: center x if nonmiss == 1, prefix(d) mean(m)
by id, sort: center z if nonmiss == 1, prefix(d) mean(m)

with the hybrid-model:

xtreg y dx dz mx mz, re

What I want to do, is to include an interaction term of this form:
y = b0 + b1*dx + b2*dz + b3*mx + b4*mz + b5*dx*mz, so an interaction of the within-effect of x with the between-effect of z. E.g. if x is the age and z is the smoking behaviour and y is the weight of a person, I want to know how the effect of age on weight differs between the levels of smoking.

Following Schunck (2013), it is incorrect to calculate the interaction term using ## and it is necessary to calculate them "by hand" in advance in order to obtain the correct results:

gen xXz = x*z
by id, sort: center xXz if nonmiss == 1, prefix(d) mean(m)

Now we can calculate the hybrid-model with the two created interaction terms mxXz and dxXz:

xtreg y dx dz mx mz dxXz mxXz, re

Following Schunck's method we have one between-interaction, mxXz, and one within-interaction ,dxXz, but there is no between-within-interaction.
So my question is now: Since it is incorrect to calculate the dx*mz term using ## and the method of Schunck only calculates one betweeninteraction-effect and one within-interaction-effect, how it is possible to calculate a between-within-interaction-effect in a hybrid model?

Thank you very much for your help.


Literature:

Schunck (2013) Within and between estimates in random-effects models: Advantages and drawbacks of correlated random effetcs and hybrid models. In: The Stata Journal 13(1), 65-76.

[Clyde Schechter]

Well, from a strictly mathematical point of view there is no objection to adding a dx#mz interaction to your model. But it does not seem to me that it would be meaningful in your context. A dx#mz effect would be an estimate of the extent to which the age-related change in BMI as a person ages is modified by how much that life-time average smoking behavior deviates from the lifetime average smoking behavior of others. So if a person is a smoker at age 20 and quits at age 35, somehow the trajectory of his BMI between 20 and 35 is going to be affected that person's future destiny of quitting. How does that work? I can't wrap my mind around it. It seems to me that if there is going to be an interaction here it would either be dx#dz or just dx#z. But maybe there's some aspect to this that I'm missing.

[Guest]

Dear Mr. Schechter,
thank you for your kind reply.

Yes, the given example does not fit very well the problem. Actually I was thinking of the following variables (I really don't know why I posted it with other variables):
The independent variables:
time-variant "x": type of contract (temporary/permanent)
time-variant "z": education (measured by the CASMIN-classification)

And the dependent variable:
y: ln(wage)

So I was thinking of an interaction like dx#mz to test the assumption that the wage-effect of entering temporary employment may differ between the educational levels. And since Schunck (2013) says that calculating the interaction terms in a hybrid model using the factor variable notation (#) will lead to incorrect results, I was wondering how to calculate it. But, if I understand your response correctly, you don't think that there will be a mathematical problem using # to calculate interaction terms in hybrid model?

Kind regards,
Guest

[Clyde Schechter]

Schunck is correct What you will have to do here is create a new variable, -gen dxXmz = dx*mz- and then center that, and include the mean and centered values of dxXmz in your model.

[Guest]

Thank you very much!

[Sapna Goel]

On a related note : kindly help out .
Hello everyone


I am trying to understand the effect of male-out migration on labor force participation of left behind women.

Specifically, I want to know if (say) any male member migrated in a given month, then how does it change the monthly days spent by a women in farm activities.
So my dependent variable is : number of days spent on farm in a month

And my independent variable is household level monthly migration status. It takes value 1 it any member of the household migrated in that month , 0 otherwise.


I have data with monthly frequency for 5 years. I plan to use both month level and year level fixed effects.

Now, the point is that i also want to know how the effect of migration may vary across women from different caste. Caste is a categorical variable and is time-invariant.
So I want to interact caste with household level migration status.

Now my query is : can I still use fixed effects model?

Or if I decide to use hybrid model, then we divide the total effect into (with-in) and (in-between) effect.

But caste doesn't have any with-in effect. So , then do I need to divide total effect into between and within effect?

[Clyde Schechter]

You do not have to do all this "by hand." There is an -xthybrid- command that implements the hybrid model, and it is available from SSC. If you just specify caste as one of the variables, it will be included as a between-only variable. One limitation of -xthybrid- is that it predates factor variable notation and has never been updated to handle it. So if you need to use factor variable notation, then you are back to doing it by hand. In that case, you need not bother with creating a within-household variable for caste.

[Sapna Goel]

Thank you. However, just to clarify, I am posting my code here:

I am following Schunck (2013):

For generating the interaction:
I wrote: generate (caste*household_level_migration_status) = caste*household_level_migration_status.
Caste take value 1 for backward caste and 0 for forward caste.

I also take the deviation as specified in Schunck (2013).
By individualID: center (caste*household_level_migration_status), prefix(d) mean(m).

So hybrid regression code:

1st approach: I take mean values and deviation scores.

xtreg monthly_farm_days dhousehold_level_migration_status d_caste d_caste*household_level_migration_status mcaste mhousehold_level_migration_status mcaste*household_level_migration_status, i(individualID) re



My doubt is : how do I interpret dcaste_householdlevelmigrationstatus. its supposed to give the with-in effects. But there is no with-in effect of caste variable as caste remains same for any individual.


For _caste*household_level_migration_status, i can understand. It tells us how the effect of household level migration varies as per caste.

But how to interpret the deviation scores. Or do I even need the deviation scores here.

Can I just write this code:

2nd approach: i only take the mean values and not the deviation scores.

xtreg monthly_farm_days dhousehold_level_migration_status mhousehold_level_migration_status mcaste mcaste*household_level_migration_status, i(individualID) re

So I basically deleted the deviation scores from my code. As there is no with-in effect of caste. So I only take mcaste and same for the interaction term.

Is this right?.


Or, another confusion: as we know already there is no with-in effect. Then can we simply interact caste and household level migration status. Why we need to take the mean?.


For instance, i can just write:

3rd approach: I don't take mean values or deviation scores. I simply interact my time-invariant caste variable with household level migration status.

xtreg monthly_farm_days dhousehold_level_migration_status m_household_level_migration_status caste caste*household_level_migration_status, i(individualID) re


Kindly explain which approach is the correct. Thank you in advance.

[Sapna Goel]

So, basically what I want to ask is this:
for hybrid models:

When I have an exploratory variable which are time-invariant (say, caste , religion etc.) , there will not be any with-in household effect. So , in this case, I don't need to take mean and deviation score. Isn't it?.

And if I have variables that are time-variant, for example, household income , family type (nuclear, joint family etc.), in this case there can be both with-in and in-between effects. So here I need to bring both mean and deviation score.

For example:
I will have to create the interaction:
(household_level_migration_status)×(family_type)

Then I need to take mean and deviation for each of these variables and for the interaction too.

Whereas for time-invariant variable (caste) , I can just take the interaction and the terms. No need to take the mean and deviation score.

Like, I can just have caste, household_level_migration_status and caste×household_level_migration_status as the explanatory variables.

Kindly clarify.
Thank you.

[Clyde Schechter]

I don't want to comment on the specific conclusions you are drawing about your variables because I don't necessarily understand whether they are time-invariant or not and am not in any position to verify that. Suffice it to say that your conclusions about how to handle time-invariant and time-varying variables in these models are correct.

I will also add that if you are uncertain which way to handle a variable, it is always safe to include the mean and deviation in the model: if they are time invariant, Stata will recognize that fact and will omit those terms. No harm done. By contrast, if you mistakenly omit those terms on a variable that does vary over time, Stata will not recognize or correct your mistake and your model will be misspecified.

[Sapna Goel]

Thank you so much for the help.

[Sapna Goel]

---

Hello I am really confused about interpretation of interaction terms. Let me clarify with an example. Suppose I want to know how the effect of migration differs according to age group. Dependent variable: number of days spent on farm.
Age group: 25-35, 15-25(base).
I have multiple interaction terms in my model, so I won't be able to figure out the effect of migration for (15-25) age group as such.

Now, suppose that the interaction coefficient is negative, (say) , (-2 days).

If the effect of migration for base category is 5 days , then the effect for (25-35) age will be 3 days. So , it means the effect of migration is 2 days less for the (25-35) category.

If the effect of migration for base category is (-5) days , then the effect for (25-35) will be (-7 days). Which means the effect is 2 days more for the (25-35) category.

So, how exactly do I write it?.

Do I just say that effect of migration is 2 days less for (25-35) category. But that doesn't seem right , because if the base effect is negative, then the effect for (25-35) age group is actually 2 extra days of decrease. I am just struggling with the language here. Kindly help.


Another: I am interaction the migration variable with household and individual level characteristics. Would it be conceptually okay if I include one interaction in one regression. Then include another interaction in a separate regression. Like , I first interact age with migration. Then I may interact caste with migration.

Kindly confirm.
Thank you.

[Clyde Schechter]

I have multiple interaction terms in my model, so I won't be able to figure out the effect of migration for (15-25) age group as such.

Yes, you will be able to. -margins age_group, dydx(migration)- will give you the marginal effect of migration, separately, in each age group.

Now, suppose that the interaction coefficient is negative, (say) , (-2 days).

If the effect of migration for base category is 5 days , then the effect for (25-35) age will be 3 days. So , it means the effect of migration is 2 days less for the (25-35) category.

Correct.

If the effect of migration for base category is (-5) days , then the effect for (25-35) will be (-7 days).

So far, so good.

Which means the effect is 2 days more for the (25-35) category.

Wrong! The effect is still 2 days less for the 25-35 category because -7 is less than -5. What is confusing you is that the magnitude of the effect is greater in the 25-35 category, because when speaking of negative numbers, magnitude increases as the number itself decreases.

I am just struggling with the language here.

As will a typical non-specialist audience. So avoid dealing explicitly with negative numbers in a presentation to people who are not quantitatively oriented. So if your interaction is -2, and the base migration effect is 5 days, you can say that you can say that in the older age group, the migration effect is reduced to 3 days. If the base migration effect itself is -5 days, describe that effect as an outcome reduction of 5 days, and say that in the older age group, the migration effect changes [note avoidance of directionality in this word] to a reduction of 7 days. In other words, don't describe effects in terms of signed numbers. Describe them as increases or decreases of some (positive) amount. And then you can speak of the interaction as changing [direction-neutral word] those effects to whatever the other group's result is.

Needless to say, if you are presenting the same material to a quantitatively savvy audience, do use negative numbers where applicable. Even here, though, it is easier to refer to a change from -5 to -7 rather than speaking of "more," or "less," or "increase," or "decrease."

[Sapna Goel]

Hello. Thank you for the clarification and suggesting the margins command. I probably still have some doubts. But I will try the margins command first and then let's see.
thank you.

[Sapna Goel]

So, I am sorry. I didnt clarify all the details in my query.
I am working with a panel data set.

I am using a hybrid model.
My dependent variable is : farm days
Independent variable is: household level migration status.

In the regression equation, I take both mean(hh level migration status) and the deviation score so that I have both within and in-between effect.

Now I have various Household and individual characteristics as the control variables.

Specifically, there are some variables for which i consider the interaction terms.
I have interacted age with migration.
so I have 4 categories: (15-25, 25-35, 35-45, 45-59).
I take (15-25) as the base category.

I create separate dummies for rest of three age categories.

I then interact migration with age (25-35), then i interact migration with age (35-45) and then with (45-59).
in the regression equation, I will have mean and deviation scores for all the three interaction categories i.e. for (25-35, 35-45 and 45-59).

I also interact household level migration status with farm size.
So I divide farm size into landless, small, medium and large.

I take landless as the base category.
Here also, i interact migration with small, medium and large farm size category.

So, now, I believe I can't apply the margins command. i don't have one particular age variable as such. I have created each category separately.

So, what to do here. My main idea is to understand what is the effect of household level migration status on left behind women. Then to see if the effect of migration varies as per age group or farm size. I have some other interactions as well.

As i discussed before, if I could get the effect of migration for the base category (15-25), I could then look at the coefficient of the interaction term and easily figure out the effect for the (25-35) age category.


So like, let me write the equation here:
Farm days= alpha + beta1(migration status of household) +beta3(migration*25-35) +beta4(migration*35-45)+beta5(migration*45-59) + beta6(migration*small farm)+beta7(migration*medium farm)+beta8(migration*large farm).

Now, what is the effect of migration for the (15-25) category?

beta 1? But we get beta1 if all other variables are also at the base value.

But I just want to know the effect for (15-25) age group, I don't care from which category (small farm or medium farm or large farm) those women are.

This is why I feel like I can't get the effect of migration for the base category.

So I wanted to confirm:
1) is it true that i cannot have a number as such for the effect of migration for (15-25) age category?
2) if not, then I just have to rely on the coefficient of interaction terms , which would serve my purpose though. But still I wanted to know if I am making any mistake.

Thank you
I really apologies for not clarifying my query in the previous post.