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With the broader availability of panel data, fixed effects (FE) regression models are becoming important and used widely for identifying causal effects. In this post, I discuss a few papers in which the usage of fixed effects is quite illuminating. I have benefited from discussions with Gurpal S. Sran and Jingtao Zheng.

The Basics

In general, one runs into many discussions on what fixed effects (FE) to add and which variation they retain afterwards. Here are some commons from the finance literature:

Some important caveats / tips to keep in mind:

1. FE (without a time subscript) cannot address time-variant unobserved heterogeneity.
   • Often, the fixed effects in Khwaja-Mian (2008) is referred to as adding time-variant fixed effects. These are just firm-year fixed effects.
2. FE restricts analysis to “within” variation. So no-within variation observations do not contribute to the coefficient estimates, which renders the interpretation of the estimate a bit more nuanced.
3. If you have data that has three dimensions you can add three two-way fixed effects. If you have data that has four dimensions, you can add four three-way fixed effects. What this calculation reveals is that it requires a decision on the researcher’s part on which cross-sections to purge and which cross-sections to retain. And naturally, these decisions invite questions regarding external validity.

Example 1. Khwaja and Mian (2009)

I discussed the usage of fixed effects in a separate post earlier. The bottom line is that one can add firm-year-specific effects — if the firm has multiple banking relationships — to control for unobserved credit demand. With enough granularity in data, we can compare the loan growth of a given firm in a given year across two different banks.