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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?

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  • $\begingroup$ Can you please provide more context? What is $\sigma_{it}$? What do the time invariant and time varying variables mean? What is the proposed estimator? Is this a peer effects model? $\endgroup$ Oct 19, 2014 at 22:36
  • $\begingroup$ Unless you forgot the time subscripts also $y$ and $x$ are time invariant and therefore will be absorbed in the individual fixed effects. $\endgroup$
    – Andy
    Oct 19, 2014 at 22:40
  • $\begingroup$ I didn't forget the indices, they are not supposed to be there. I don't know precisely what model it is (it just says time-series regression) nor do I know the complete background. I can check later if I find more info, I will keep you posted. $\endgroup$
    – Tom
    Oct 20, 2014 at 19:00

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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).

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  • $\begingroup$ Ahh now it makes more sense :). My former notations was a little different ($\sum \gamma_i * \alpha_i$) so I thought the Sum only related to $\gamma_i$ and not to both, $\gamma_i \times \alpha_i$. Now it makes more sense. Also thanks Andy for your explanation, helps me quite a bit! $\endgroup$
    – Tom
    Oct 20, 2014 at 19:04

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