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Borusyak, 2021 has a sentence

While our baseline setting is for panel data with two-way fixed effects, we show how our formal results extend naturally in a number of ways. We allow for more general panel-type specifications that can include, for instance, unit-specific trends or time-varying covariates

It is quite contradictory to my understanding. Because to me, when you consider "two-way fixed effects", you also control for unit and period fixed effects. Why the author used the word "more general panel-type specifications that can include, for instance, unit-specific trends or time-varying covariates" here. I mean, the "more general panel-type specifications" as he described is "two-way fixed effects", not "more general"

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A two-way FE model is:

$y_{it} = \beta_0 +\beta_1 x_{it} +\gamma_i + \delta_t +u_{it}$

The $\delta_t$ absorb average time effects, but there may still be unit-specific trends, which could be modelled with:

$y_{it} = \beta_0 +\beta_1 x_{it} +\alpha_i t +\gamma_i + \delta_t +u_{it}$

There is not a violation of multicollinearity and it is more general than before. If instead we had a general time trend, rather than unit specific, it would violate multicollinearity.

$y_{it} = \beta_0 +\beta_1 x_{it} +\alpha t +\gamma_i + \delta_t +u_{it}$

Also, there has been some literature that has cautioned against the use of individual-specific time trends because such can absorb part of a heterogeneous treatment effect. Meer and West 2016.

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  • $\begingroup$ Thank @Micheal , can I ask an example of controlling for "unit-specific trends" is unit's lag variable? $\endgroup$ Sep 16, 2021 at 9:44
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    $\begingroup$ "general time trend" would be perfect colinear with time fixed effect, is it what you mean then? Thanks $\endgroup$ Sep 16, 2021 at 9:46
  • $\begingroup$ Lag variable is different than time trends. A lag variable might be as follows: $y_{it} = \beta_0 +\beta_1 x_{it} + \beta_2 Z_{it-1} +\gamma_i +\delta_t +u_{it} $Z$ is used generically. $Z$ could be a lagged $X$ variable, or instead a lagged outcome variable. There are issues with lagged outcome variables, and I'm guessing that's not relevant to your case. $\endgroup$ Sep 16, 2021 at 9:59
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    $\begingroup$ ""general time trend" would be perfect colinear with time fixed effect, is it what you mean then?" Yes. $\endgroup$ Sep 16, 2021 at 9:59
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    $\begingroup$ The second equation in my answer does this. The coefficient, $\alpha_i$, differs for each individual, measuring an unit-specific time trend. $\endgroup$ Sep 16, 2021 at 10:15

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