First-difference estimator

The first-difference (FD) estimator is an approach used to address the problem of omitted variables in econometrics and statistics with panel data. The estimator is obtained by running a pooled OLS estimation for a regression of ${\displaystyle \Delta y_{it}}$ on ${\displaystyle \Delta x_{it}}$.

The FD estimator avoids bias due to some omitted, time-invariant variable ${\displaystyle c_{i}}$ using the repeated observations over time:

${\displaystyle y_{it}=x_{it}\beta +c_{i}+u_{it},t=1,...T,}$

${\displaystyle y_{it-1}=x_{it-1}\beta +c_{i}+u_{it-1},t=2,...T.}$

Differencing both equations, gives:

${\displaystyle \Delta y_{it}=y_{it}-y_{it-1}=\Delta x_{it}\beta +\Delta u_{it},t=2,...T,}$

which removes the unobserved ${\displaystyle c_{i}}$.

The FD estimator ${\displaystyle {\hat {\beta }}_{FD}}$ is then simply obtained by regressing changes on changes using OLS:

${\displaystyle {\hat {\beta }}_{FD}=(\Delta X'\Delta X)^{-1}\Delta X'\Delta y}$

Note that the rank condition must be met for ${\displaystyle \Delta X'\Delta X}$ to be invertible (${\displaystyle rank[\Delta X'\Delta X]=k}$).

Similarly,

${\displaystyle Av{\hat {a}}r({\hat {\beta }}_{FD})={\hat {\sigma }}_{u}^{2}(\Delta X'\Delta X)^{-1},}$

where ${\displaystyle {\hat {\sigma }}_{u}^{2}}$ is given by

${\displaystyle {\hat {\sigma }}_{u}^{2}=[n(T-1)-K]^{-1}{\hat {u}}'{\hat {u}}.}$