First-difference estimator

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 $\Delta x_{{it}}$ .

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 ( $rank[\Delta X'\Delta X]=k$ ).

Similarly,

Av{\hat {a}}r({\hat {\beta }}_{{FD}})={\hat {\sigma }}_{{u}}^{{2}}(\Delta X'\Delta X)^{{-1}},

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

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

Properties

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

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

Under the assumption of spherical errors, 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.

References

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

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