Modelling Fixed effects in panel data regression models I was given the following equation:
$$\sigma_{it} = \beta_0 + \beta_1 x_i + \beta_2y_i + \beta_3vs_{it} + \beta_4vm_{it} + \sum_{i=1} \gamma_i  \alpha_i + \sum_{t=1} \omega_t \phi_t + \epsilon_{it}$$
with $\phi_t$ as a vector of time fixed-effects and $\alpha_i$ as a vector of individual fixed effects. 
My question is the following: Why are sums used to model the fixed effects here? From my course at university I remember that we modeled fixed effects just with the vector, e.g.  $\phi_t$.
It does not make sense to me to estimate a coefficient or even to multiply it with the sum of all coefficients.
Could anyone clarify this?
 A: For $i = 1,...,I$ individuals you can model individual fixed effects via the inclusion of $I-1$ dummies. This is called the least squares dummy variables estimator (LSDV). This estimator is equivalent to performing the within transformation as was shown by Mundlak (1978).
Instead of vector notation, you would then write
$$\sigma_{it} = \beta_0 + \beta_1 x_{it} + \beta_2y_{it} + \beta_3vs_{it} + \beta_4vm_{it} + \gamma_1  \alpha_1 +  ... + \gamma_{I-1}  \alpha_{I-1} + \sum_{t=1} \omega_t \phi_t + \epsilon_{it}$$
where each $\alpha_i$ is a dummy variable that equals one if a given individual is $i$, and zero if he/she is not $i$. Then it is obviously more convenient to use the summation sign for better notation. The same holds for the time period dummies for which I have left the summation sign.
So in this case, rather than subtracting (within transformation) or differencing out (1st differences) the fixed effects, the LSDV estimator directly estimates them. Hence you have a coefficient for each individual. This can be computationally burdensome but sometimes it is interesting to estimate these fixed effects as it was done for example by Card et al. (2013).
