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 $\Delta x_{it}$.

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,

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

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

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

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 spherical errors, 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.

References

• Wooldridge, Jeffrey M. (2001). Econometric Analysis of Cross Section and Panel Data. MIT Press. pp. 279–291. ISBN 978-0-262-23219-7.