Fixed Effects Model with Time Invariant Independent Variables

[Morgan Pincombe]

I am working with a panel data set in Stata 16.1 that has 14,690 observations and 19 variables. I am running regress on a dependent variable of policy stringency index, which varies by day during the 130 day study period, and a series of independent variables. My main independent variable of interest is World Bank income classification groups, which is a time invariant categorical variable. Several of my other independent variables - used as controls - are also time invariant. I want to know the coefficient for each of these time invariant independent variables as well as use a fixed effects model to capture the other time invariant effects. However, I am struggling to create a fixed effects model that does not omit my time invariant independent variables. My current code is:

xtreg strin_ind i.income_cat i.n_polity over65 covid_cases_prev_day ghsindex i.date, fe

Stata omits all of my independent variables except for covid_cases_prev_day. I have seen studies that treat the other time invariant independent variables as controls - how can I do this?

Thank you!

[Joro Kolev]

It is not possible to estimate the effects of time invariant variables in the presence of fixed effect--they are perfectly collinear with the fixed effects. The within transformation wipes out both the time invariant fixed effects and the time invariant variables.

[Clyde Schechter]

You are seeking something that does not exist. It is a straight calculation in linear algebra that in any fixed-effects model, time-invariant variables do not have identifiable effects. It is mathematically impossible to get around this. Whatever studies you have seen that provide estimates for the effects of time-invariant predictors are not using a fixed-effects model.

You may have seen studies that use a hybrid model. -xthybrid-, available from SSC can do that for you. But that model is not a true fixed-effects model and it does not automatically adjust for unobserved time-invariant effects the way a true fixed-effects model would.

Added: Crossed with #2, which makes the same point.

[Morgan Pincombe]

Clyde and Joro,

Thank you for clarifying the theory behind the fixed-effects model. I believe the -xthybrid- command would be helpful. How can I control for day and country effects with this hybrid model?

I set my panel data with the following code:

xtset id date, daily

[Chris Boudreaux]

I believe the -xthybrid- command would be helpful. How can I control for day and country effects with this hybrid model?

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

[Morgan Pincombe]

Thank you for providing the citation, Chris. I referenced that paper, but I still do not know how to control for day effects.

I also created dummy variables for my categorical variables, as the Schunck paper suggested, using the code below:

generate lowincome = income_cat==1 if income_cat!=.
generate lowmidincome = income_cat==2 if income_cat!=.
generate upmidincome = income_cat==3 if income_cat!=.
generate highincome = income_cat==4 if income_cat!=.


generate autocracy = n_polity_2018==1 if n_polity_2018!=.
generate anocracy = n_polity_2018==2 if n_polity_2018!=.
generate democracy = n_polity_2018==3 if n_polity_2018!=.

However, when I run -xthybrid-, my highincome and democracy variables are omitted.

xthybrid strin_ind lowincome lowmidincome upmidincome highincome autocracy anocracy democracy over65_2019 ghsindex co
> vid_cases_prev_day date, clusterid(id) se

The variable 'lowincome' does not vary sufficiently within clusters
and will not be used to create additional regressors.
[~0% of the total variance in 'lowincome' is within clusters]
The variable 'lowmidincome' does not vary sufficiently within clusters
and will not be used to create additional regressors.
[~0% of the total variance in 'lowmidincome' is within clusters]
The variable 'upmidincome' does not vary sufficiently within clusters
and will not be used to create additional regressors.
[~0% of the total variance in 'upmidincome' is within clusters]
The variable 'highincome' does not vary sufficiently within clusters
and will not be used to create additional regressors.
[~0% of the total variance in 'highincome' is within clusters]
The variable 'autocracy' does not vary sufficiently within clusters
and will not be used to create additional regressors.
[~0% of the total variance in 'autocracy' is within clusters]
The variable 'anocracy' does not vary sufficiently within clusters
and will not be used to create additional regressors.
[~0% of the total variance in 'anocracy' is within clusters]
The variable 'democracy' does not vary sufficiently within clusters
and will not be used to create additional regressors.
[~0% of the total variance in 'democracy' is within clusters]
The variable 'over65_2019' does not vary sufficiently within clusters
and will not be used to create additional regressors.
[~0% of the total variance in 'over65_2019' is within clusters]
The variable 'ghsindex' does not vary sufficiently within clusters
and will not be used to create additional regressors.
[~0% of the total variance in 'ghsindex' is within clusters]

Hybrid model. Family: gaussian. Link: identity.

+-----------------------------------+
|             Variable |   model    |
|----------------------+------------|
| strin_ind            |            |
|         R__lowincome |   -11.5418 |
|                      |     5.7931 |
|      R__lowmidincome |    -3.6237 |
|                      |     3.9716 |
|       R__upmidincome |     1.5030 |
|                      |     3.3585 |
|        R__highincome |  (omitted) |
|                      |            |
|         R__autocracy |    -5.0748 |
|                      |     4.1473 |
|          R__anocracy |    -3.4765 |
|                      |     2.9943 |
|         R__democracy |  (omitted) |
|                      |            |
|       R__over65_2019 |    -0.5860 |
|                      |     0.2605 |
|          R__ghsindex |    -0.0419 |
|                      |     0.1209 |
| W__covid_cases_pre~y |    -0.0000 |
|                      |     0.0000 |
|              W__date |     0.4418 |
|                      |     0.0052 |
| B__covid_cases_pre~y |     0.0000 |
|                      |     0.0000 |
|              B__date |  (omitted) |
|                      |            |
|                _cons |    69.4817 |
|                      |     6.9082 |
|----------------------+------------|
|        var(_cons[id])|            |
|                _cons |   119.1077 |
|                      |    16.3798 |
|----------------------+------------|
|      var(e.strin_ind)|            |
|                _cons |   521.6886 |
|                      |     6.1107 |
|----------------------+------------|
| Statistics           |            |
|                   ll | -6.700e+04 |
|                 chi2 |  7564.0897 |
|                    p |     0.0000 |
|                  aic |  1.340e+05 |
|                  bic |  1.341e+05 |
+-----------------------------------+
                         legend: b/se
Level 1: 14690 units. Level 2: 113 units.

[Chris Boudreaux]

Stata will omit one dummy if you specify all possible categories. Don't worry. This is normal. You then can interpret the remaining dummies relative to the baseline (i.e., omitted category).

[Morgan Pincombe]

Thank you, Chris. Glad this isn't an error.

[Morgan Pincombe]

What sensitivity, or specification, checks should I perform when using the xthybrid command? I did not find a discussion of sensitivity checks in the Schunck 2013 paper Chris referenced above.