I'm seeking clarification on the use of firm fixed effects $a_i$ in the following panel regression equation:
$$y_{it} = \beta_{i}x_{it} + a_{i} + u_{it}, \quad t = 1, 2, \ldots, T$$
From my understanding, a_i represents a dummy variable for each company in the dataset. However, I'm curious about the essence of this term and how it differs from manually including dummy variables in the regression.
If I were to calculate the regression by hand, I would approach it using dummy variables for each firm. How does statistical software, such as Stata or R, handle the inclusion of firm fixed effects? Does it use a different method than manually creating dummy variables?
