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I have the following question with regards to Panel Data. Suppose we have the following Data Generating Process: $$ y_{it}=x_{it}'\beta+\alpha_{i}+\epsilon_{it} $$ where the LHS is the dependent variable for individual $i$ at time $t$ , $x$ is a vector of covariates, $\beta$ is a vector of coefficients , $\alpha_{i}$ are the nuisance parameters or the time invariant unobserved individual specific heterogeneity (fixed effects) and $\epsilon_{it}$ represents an time variant unobservables. I know that $\beta$ has units- we input values for $x_{it},$ $\beta$ 'converts' these to $y$ units (I understand $\beta$ as representing $\beta$ units of y per one unit of $x.$

By the same token, what really does $\alpha_{i}$measure? Suppose we put in dummy variables for all individuals such that we get $\hat{\alpha}_{i}$ for each individual. What do those values mean? People call them estimated fixed effects but what units are these coefficients in? How can we interpret them? Do these have units? Thanks a lot!

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These are also called "individual-specific intercepts", because one way to estimate the FE model is to a "least-squares dummy variables regression", in which one regresses $y$ on $x$ and a $n$ dummy variables where each individual on the panel has one dummy that takes the values one if an observation belongs to that person (household, unit, firm,...). The $\hat\alpha_i$ then estimate these intercepts, which may then be interpreted as usual intercepts in regressions, with the only difference that each intercept is specific to a single unit.

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  • $\begingroup$ Thanks. What do those intercepts mean? Are these measured in units of the y-variable? I understand what the fixed effects mean econometrically, but what do they mean intuititively? $\endgroup$
    – ChinG
    Commented Nov 9, 2015 at 15:16
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    $\begingroup$ As I wrote, like other intercepts: so, the predicted values (for individual $i$) if the $x_{it}=0$, thus also the same units. $\endgroup$
    – Christoph Hanck
    Commented Nov 9, 2015 at 15:17
  • $\begingroup$ Thanks. In a sense, they can mean anything that is individual specific that does not vary with time. $\endgroup$
    – ChinG
    Commented Nov 9, 2015 at 15:25
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If you include a constant in the regression, $\hat{\alpha_i}$ gives the difference in $y$, above or below the across-individual-mean of $y$, between person $i$ and the omitted person, holding $x_{it}$ fixed.

Thus, $\hat{\alpha_i}$ is the idiosyncratic effect of person $i$ on $y$, controlling for $x$.

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