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 $).


$ 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.