Interactions and time effects in a fixed-effects model

[Beate Schmidt]

Hi everyone,

I’m currently analyzing how job satisfaction has changed since the COVID-19 pandemic. My dataset includes four survey waves: 2019/2020, 2020/2021, 2021/2022, and 2022/2023.

To examine within-person changes, I’m estimating a fixed-effects model, and I also want to compare group-specific differences, particularly between men and women. For that purpose, I’m using the following model with an interaction term between wave and sex:
xtreg jobsatisfaction i.wave##i.sex, fe vce(cluster ID_t) My questions are:

1. Does this specification qualify as a two-way fixed-effects model (i.e., including both individual and time fixed effects)?
2. Should I include all four waves in the model, or would it be more appropriate to focus on a pre- and post-COVID comparison (e.g., 2019/20 vs. 2020/21)?

Any advice on model specification and interpretation in this context would be greatly appreciated! Best Beate

[Carlo Lazzaro]

Beate:
1) yes, yours is a two-way fixed effect model (even though I would cluster the SEs on -panelid- only);
2) I would include all the four waves (one of them will be considered as the reference wave and omitted) and add a categorical variable pre/post COVID-19 (the previous comment about the reference category still holds);
3) your regression probably needs more predictors;
4) causal inference (-xtdidregress-) can be an altervative, assuming that the same individuals are the hypothetical control of the real participants.

[Beate Schmidt]

Hello Carlo,

thank you very much for your helpful response — it really clarified a lot!

I have two follow-up questions:

Do you think it makes sense to include a separate categorical variable for pre-/post-COVID (e.g., post_covid = wave >= 13) if I'm already including i.wave in the model?

Currently, my specification looks like this:

gen post_covid = wave >= 13
label define covidlbl 0 "Pre-COVID" 1 "Post-COVID"
label values post_covid covidlbl

xtreg jobsatisfaction ///
i.wave##i.sex ///
i.post_covid, ///
fe vce(cluster ID_t)

I’d also like to include several other predictors, such as job type, education level, and birth cohort.
Would you recommend estimating them all in one model, possibly with interactions like i.job##i.wave, i.education##i.wave, etc.?

Or would it be better to run separate models to avoid overfitting or interpretational issues?

Thanks again for your guidance and time. It’s very much appreciated!

Kind regards,
Beate

[Carlo Lazzaro]

Beate:
1) including a separate categorical variable for pre-/post-COVID will probably create some collinearity issue with -i.wave-. Perhaps you may want to revise your previous interaction as -i.wave##i.post_covid-, leaving -i.sex- as a standalone predictor;
2) I'd recommend plugging all the predictors in the same regression, keeping the interactions at minimum, as they may not be that easy to explain, if several.

[Beate Schmidt]

Hello Carlo,

thanks a lot for your response!
Just to make sure I understand it correctly:

If I conceptualize COVID-19 as an external shock and want to examine its impact on job satisfaction, it would make sense to focus only on the two relevant waves (pre- and post-COVID), e.g.:
keep if inlist(wave, 12, 15) gen post_covid = wave >= 13
Then I define the panel structure:
xtset ID_t wave
and estimate:
xtreg jobsatisfaction i.wave, fe
If I also want to look at gender differences, I would run:
xtreg jobsatisfaction i.wave##i.sex, fe vce(cluster ID_t)
Is that correct?
Or would it be sufficient for examining gender differences to simply compare pre- and post-COVID periods using a categorical variable (e.g. i.post_covid##i.sex) instead of all waves? Do you need more than two waves to run a two-way fixed effects (TWFE) model? In my case, I’m comparing only two time periods (2019/2020 vs. 2022/2023).



If I only keep these two waves (pre and post COVID) and estimate xtreg jobsatisfaction i.wave, fe, do I in fact create a First-Difference model (since there are only two time points), or is that still something different in Stata?

Thanks again for your help and clarification!

Best

Beate

[Carlo Lazzaro]

Beate:
I would go:
-xtreg jobsatisfaction i.wave##i.sex, fe vce(cluster ID_t)-.
To avoid perfect collinearity issue, I would not create a pre- and post-COVID caterìgorical variable.

[Jeff Wooldridge]

The only problem with i.wave##i.sex generally is that Stata is choosing the reference period for you. In this case there is only one pre-intervention period, and I assume that’s chosen by Stata as the reference period. If so, I would go with your original thought as confirmed by Carlo.