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I would like some insights regarding two possible ways to enroll two-way fixed effects. Let's take an example of 30-by-30 panel data (30 groups and 30 periods).

The first way is classic. We control for group fixed effects and period fixed effects separately. My understanding is that this way is equivalent to enrolling 29 group dummies and 29 period dummies (58 dummies in total).

The second way is of my interest. We control for group fixed effects and time fixed effects by enrolling group-by-time fixed effects. Perhaps the second way is equivalent to enrolling 30*30-1 dummies? It should not only control for time-invariant group-specific factors and group-insensitive time trends but also group-specific time-variant factors, in other words, control variables (or covariants).

At least, I found an article possibly holding this opinion: https://www.cambridge.org/core/journals/journal-of-financial-and-quantitative-analysis/article/shadow-banking-in-a-crisis-evidence-from-fintech-during-covid19/3337F225B3CDF66BA7E962B24DA54F76# They controlled for city-by-time fixed effects. That's why they did not control for any city-level features.

I think the second way is fascinating because, if I understand it correctly, I no longer have to struggle with controlling for group-specific features. Reviewers often require you to justify why some dimensions are controlled while others are not, which is a challenge I want to avoid.

Could anyone point out my misunderstanding or suggest resources to refine my understanding of these two ways to control fixed effects? Also, when should we use the first way, and when should we use the second way? I appreciate any help you can provide.

StudentM
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  • Welcome. I delved into a similar problem in my answer here. After reviewing this post, let me know if you're still confused. If so, I would be happy to elaborate on any additional concerns you may have. – Thomas Bilach Nov 27 '23 at 23:38
  • Hi Tom, I have read through it. It's quite intuitive. I think the panel data levels are the key. My tentative conclusion is that to avoid exhausting the degree of freedom, you should not implement joint or concatenated fixed effects with the same granularity level as the Treat * Post interaction term. In that post, if the interaction term is at the state level, using state-by-period fixed effects is likely to fail. However, if the interaction term is at the firm level, using state-by-period fixed effects is possibly feasible. – StudentM Nov 28 '23 at 02:53
  • From my point of view, the benefit of these joint fixed effects, when implemented appropriately, is that it is no longer necessary to enroll any state-level control variables. However, the payoff of protecting the degree of freedom is that firm-level control variables are still needed. – StudentM Nov 28 '23 at 03:02

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