# 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 $\Delta y_{it}$ on $\Delta x_{it}$.[clarification needed]

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

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

Differencing both equations, gives:

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

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

Under the assumption of $E[u_{it}-u_{it-1}|x_{it}-x_{it-1}]=0$, the FD estimator is unbiased and consistent, i.e. $E[\hat{\beta}_{FD}]=\beta$ and $plim \hat{\beta}=\beta$. Note that this assumption is less restrictive than the assumption of weak exogeneity required for unbiasedness using the fixed effects (FE) estimator. If the disturbance term $u_{it}$ follows a random walk, the usual OLS standard errors are asymptotically valid.

## Relation to fixed effects estimator

For $T=2$, the FD and fixed effects 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.

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