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In this paper, in column 4 Table 1, the author used firms fixed effects and industry*year fixed effect at the same time. I have two questions here:

  1. What does industry * year fixed effect mean?

  2. What are the difference between (firms fixed effect and industry fixed effects) and ( firms fixed effects and industry * year fixed effect)? Are they all called two-way fixed effects in general?

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What does industry * year fixed effect mean?

$Industry \cdot year$ fixed effect is just an interaction term between industry and dummy year variables. For example, you can have dummy particular industry, let us say finance where $D=1$ if firm is a finance firm and $0$ otherwise, then you can have a year dummy which will be set to equal $1$ for particular year and $0$ otherwise. Note typically you will have a dummies to cover industries in your sample and every year, I just used one industry and one year as an example.

$Industry \cdot year$ Is just multiplication of the two dummies, so you end up with third dummy that will be equal to $1$ only if data point is from both that particular industry and year. So it will tell you what "special" effect the industry and year had on the regressand. The effect of this dummy will be amalgamation of effects of unobserved covariates on the regressand.

What are the difference between (firms fixed effect and industry fixed effects) and ( firms fixed effects and industry * year fixed effect)?

firms fixed effect - it is a firm specific dummy that will tell you what unique effect firm specific and time invariant unobservables are having on the regressand.

industry fixed effect - as above but this tell you the effect of industry specific and time invariant unobservables on the regressand.

firms fixed effects and industry * year fixed effect - this was already covered in the first part, but to recap it tells you the specific effect firm or industry in combination with specific year has on the regressand.

Are they all called two-way fixed effects in general?

We talk about 2 way fixed effects when we use both spatial and temporal fixed effects. So when you use firm and year fixed effects you are talking about two way fixed effects. However, note when we talk about two way fixed effects there should actually be two sets of fixed effects not just some interaction like $industry \cdot time$ (see discussion in Imai & Kim 2020). But if you read the paper you linked in most regression tables they have separate firm fixed effects and year fixed effects. This being said the terminology might not always be used consistently in the applied work.

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    $\begingroup$ @BeautifulMindset, in stata the appropriate way how to use year fixed effects and industry fixed effects is to use i.varname. So for example, to add industry effects (assuming your variable is called industry) and year effects you would do xtreg dep_var ind_var i.industry i.year, options. For the interaction term, I don't remember how to do it in stata I only use stata to teach students at our university not for professional work, but my best guess would be just to have i.industry*i.year, try that if it does not work then just try to google it for stata $\endgroup$
    – 1muflon1
    Jul 4 at 21:16
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    $\begingroup$ @BeautifulMindset spatial means relating to space, so those are firm, individual etc fixed effects $\endgroup$
    – 1muflon1
    Jul 4 at 21:18
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    $\begingroup$ @BeautifulMindset I am not that familiar with reghdfe command, because again I dont use stata for professional work, I prefer python or R, but for most panel models year has to be declared with i.year if you are using it as a fixed effect because in that case you do not want to have it as a numeric variable, but maybe reghdfe does some conversion internally, if in the reghdfe manual it says thats how it should be done do it that way $\endgroup$
    – 1muflon1
    Jul 4 at 21:24
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    $\begingroup$ @BeautifulMindset no, I dont think so. At least most of the time when people talk about two way fixed effects they just mean having a separate spatial and temporal fixed effect. However, as mentioned in my answer sometimes the terminology gets bit muddy, but just casually looking at the paper you linked I think when authors talk about 2way FE they just refer to the fact that virtually all of their model have separate firm and time fixed effects $\endgroup$
    – 1muflon1
    Jul 4 at 21:29
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    $\begingroup$ @BeautifulMindset I am not sure if I understand the second question, what do you mean stricter? you can have both individual effect for each firm, and effects that hold across whole industry. If you want to talk about about whether the anti-trust laws affect some specific industries, which is actually interesting thing to discuss you need to examine what the industry level effect is $\endgroup$
    – 1muflon1
    Jul 4 at 21:32

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