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From Wikipedia for fixed effect model:

Consider the linear unobserved effects model for N observations and T time periods: $$ y_{it} = X_{it}\mathbf{\beta}+\alpha_{i}+u_{it} \quad t=1,..,T, \quad i=1,...,N $$ where $y_{it}$ is the dependent variable observed for individual $i$ at time $t$, $X_{it}$ is the time-variant $1\times k$ regressor matrix, $\alpha_{i}$ is the unobserved time-invariant individual effect and $u_{it}$ is the error term. Unlike $X_{it}$, $\alpha_{i}$ cannot be observed by the econometrician. Common examples for time-invariant effects $\alpha_{i}$ are innate ability for individuals or historical and institutional factors for countries.

Unlike the Random effects (RE) model where the unobserved $\alpha_{i}$ is independent of $x_{it}$ for all $t=1,...,T$, the FE model allows $\alpha_{i}$ to be correlated with the regressor matrix $x_{it}$. Strict exogeneity, however, is still required.

Since $\alpha_{i}$ is not observable, it cannot be directly controlled for. The FE model eliminates $\alpha_{i}$ by demeaning the variables using the within transformation: $$ y_{it}-\overline{y_{i}}=\left(X_{it}-\overline{X_{i}}\right) \beta+ \left( \alpha_{i} - \overline{\alpha_{i}} \right ) + \left( u_{it}-\overline{u_{i}}\right) = \ddot{y_{it}}=\ddot{X_{it}} \beta+\ddot{u_{it}} $$ where $\overline{X_{i}}=\frac{1}{T}\sum\limits_{t=1}^{T}X_{it}$ and $\overline{u_{i}}=\frac{1}{T}\sum\limits_{t=1}^{T}u_{it}$. Since $\alpha_{i}$ is constant, $\overline{\alpha_{i}}=\alpha_{i} $ and hence the effect is eliminated. The FE estimator $\hat{\beta}_{FE}$ is then obtained by an OLS regression of $\ddot{y}$ on $\ddot{X}$.

On one hand, it says "the FE model allows $\alpha_{i}$ to be correlated with the regressor matrix $x_{it}$", which I understand to be that both $\alpha_{i}$ and $x_{it}$ are random variables. On the other hand, it says "$\alpha_{i}$ is constant", and my previous impression is that $\alpha_{i}$ is nonrandom in fixed effect model, and random in random effect model. So I was wondering how to correctly understand "fixed effect"?

Thanks and regards!

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My answer would be that $\alpha_i$, usually interpreted as the unobserved heterogeneity, is a random variable in the sense that it varies over individuals. In that sense, you can say that it maybe correlated with other variables. However in the fixed effect model, $\alpha_i$ is not treated by the model as a random variable. It's just a set of parameters and in a frequentist framework, parameters are not considered as random variables.

I'm not a specialist in bayesian statistics, but I can imagine that in a bayesian framework, you would not have this "puzzle" since parameters are themselves treated as random variables.

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  • $\begingroup$ Thanks! I wonder why the random effect model require the random parameters to be uncorrelated with the input variables, while the fixed-effect model allow the parameters to be correlated with the input variables? $\endgroup$ – Tim Jul 31 '13 at 14:38
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I think the main difference between fixed and random effects is that the unobserved individual effect $\alpha_i$ is purely random in the random effect paradigm in the sense that its distribution does not depend on the observed $X_{it}$; under the fixed effect settings, the distribution of $\alpha_i$ depends on $X_{it}$. The fixed effect $\alpha_i$ became a constant after sampling. For example, let $\alpha_i$ be one's ability. The ability $\alpha_i$ among the population is surely a random variable regardless of the dependence of its distribution upon $X_{it}$. Given sampled data, the ability, $\alpha_i$, of a particular individual $i$ is constant, and the fixed effect estimator applies.

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    $\begingroup$ @semebruin : your assertion is not true. In the random effect model, you assume that $E(\alpha_i|X_{i,t}) = 0$ but does not reject heterogeneity. For instance, you can imagine that $V(\alpha_i|X_{i,t})$ depends on $X_{i,t}$. $\endgroup$ – PAC Jul 30 '13 at 21:42
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    $\begingroup$ @PAC Right, you made a good point. "Mean independence" is the key issue. Let me edit the answer. $\endgroup$ – semibruin Jul 30 '13 at 21:55
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    $\begingroup$ @PAC I just checked the textbook (p.g. 700 of Cameron and Trivedi: Microeconometrics). The standard random effect model does requires that the distribution of $\alpha_i$ is independent of $X_{it}$. Specifically, $\alpha_i$ has a constant mean and a constant variance. I agree that under the mean independence assumption, $E(\alpha_i \mid X_i)=0$, the random effect estimator is still consistent. But it won't be efficient without the homoskedasticity assumption. $\endgroup$ – semibruin Jul 30 '13 at 23:21
  • $\begingroup$ You're right. In the standard textbook model, we assume that $\alpha_i$ has a constant variance. However, in general, we can imagine to allow $V(\alpha_i|X_{i,t})$ to be a function of $X_{i,t}$ and to choose a weighting matrix such as the estimator is efficient. This would still be a random effect model. $\endgroup$ – PAC Jul 31 '13 at 12:54

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