## FANDOM

15 Pages

The First-Difference (FD) estimator is obtained by running a pooled OLS from $\Delta y_{it}$ on $\Delta x_{it}$.

The FD estimator wipes out time invariant omitted variables $c_{i}$ using the repeated observations over time:

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


Differencing both equations, we get:

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


which removes the unobserved $c_{i}$.

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

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


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

Similarly,

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


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

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


## Properties Edit

Under the assumption of $E[u_{it}-u_{it-1}|x_{it}-x_{it-1}]=0$, the FD estimator is unbiased, i.e. $E[\hat{\beta}_{FD}]=\beta$. Note that this assumption is less weaker than the assumption of weak exogeneity required for unbiasedness using the fixed effects (FE) estimator.

## Relation to fixed effects estimator Edit

For $T=2$, the FD and FE estimators are numerically equivalent.

Under the assumption of strong exogeneity, i.e. homoscedasticity and no serial correlation in $u_{it}$, the FE estimator is more efficient than the FD estimator. If $u_{it}$ follows a random walk, however, the FD estimator is more efficient as $\Delta u_{it}$ are serially uncorrelated while strong exogeneity is violated due to the presence of serial correlation in the $u_{it}$.

In practice, the FD estimator is easier to implement without special software, as the only transformation required is to first difference.

Community content is available under CC-BY-SA unless otherwise noted.