First-difference estimator

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,

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

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

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

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

For ${\displaystyle T=2}$, the FD and fixed effects estimators are numerically equivalent.

Under the assumption of homoscedasticity and no serial correlation in ${\displaystyle u_{it}}$, the FE estimator is more efficient than the FD estimator. If ${\displaystyle u_{it}}$ follows a random walk, however, the FD estimator is more efficient as ${\displaystyle \Delta u_{it}}$ are serially uncorrelated.

• Panel analysis

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