First-difference estimator


The first-difference 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.
The FD estimator avoids bias due to some omitted, time-invariant variable 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.
Similarly,
where is given by

Properties

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 strict exogeneity required for unbiasedness using the fixed effects estimator. If the disturbance term follows a random walk, the usual OLS standard errors are asymptotically valid.

Relation to fixed effects estimator

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