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!
Best Answer
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.