First-difference estimator

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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).


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} .


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[edit]

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.