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 on .[clarification needed]

The FD estimator wipes out time invariant omitted variables using the repeated observations over time:

Differencing both equations, gives:

which removes the unobserved .

The FD estimator is then simply obtained by regressing changes on changes using OLS:

Note that the rank condition must be met for to be invertible ().


where is given by


Under the assumption of , the FD estimator is unbiased and consistent, i.e. and . 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 follows a random walk, the usual OLS standard errors are asymptotically valid.

Relation to fixed effects estimator[edit]

For , the FD and fixed effects estimators are numerically equivalent.

Under the assumption of spherical errors, i.e. homoscedasticity and no serial correlation in , the FE estimator is more efficient than the FD estimator. If follows a random walk, however, the FD estimator is more efficient as are serially uncorrelated.

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