I am wondering how the following statement holds.

There is a two-way fixed effects model: $$ y_{it}=\alpha_i + \gamma_t + \beta D_{it}+e_{it} $$ where $\alpha_i$: individual-fixed effects, $\gamma_t$: time-fixed effects $D_{it}$: a dummy variable, and $e_{it}$: the error term.

In this situation, the least square estimator is as follows by the Frisch-Waugh-Lovell theorem:

$\hat{\beta}_{lse}=\frac{\frac{1}{NT}\sum_{i=1}^N\sum_{t=1}^T y_{it}\tilde{D}_{it}}{\frac{1}{NT}\sum_{i=1}^N\sum_{t=1}^T \tilde{D}_{it}}$

where $\tilde{D}_{it}=(D_{it}-\frac{1}{T}\sum_{t=1}^TD_{it})-(\frac{1}{N}\sum_{i=1}^ND_{it}-\frac{1}{NT}\sum_{i=1}^N\sum_{t=1}^TD_{it})$.

I think, this is just the partial regression coefficient considering the fixed effects as individual and time dummy variables.

Is my idea correct?


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