Dale–Chall readability formula

Template:Multiple issues 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 $\mathrm{\Delta }{y}_{it}$ on $\mathrm{\Delta }{x}_{it}$.Template:Clarify

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:

$\mathrm{\Delta }{y}_{it}=y_{it}-y_{it-1}=\mathrm{\Delta }{x}_{it}\beta +\mathrm{\Delta }{u}_{it},t=2,...T,$

which removes the unobserved $c_i$.

The FD estimator ${\hat{\beta }}_{FD}$ is then simply obtained by regressing changes on changes using OLS:

${\hat{\beta }}_{FD}=(\mathrm{\Delta }{X}^{\prime }\mathrm{\Delta }X{)}^{-1}\mathrm{\Delta }{X}^{\prime }\mathrm{\Delta }y$

Note that the rank condition must be met for $\mathrm{\Delta }{X}^{\prime }\mathrm{\Delta }X$ to be invertible ($rank[\mathrm{\Delta }{X}^{\prime }\mathrm{\Delta }X]=k$).

Similarly,

$Av\hat{a}r({\hat{\beta }}_{FD})={\hat{\sigma }}_u^2(\mathrm{\Delta }{X}^{\prime }\mathrm{\Delta }X{)}^{-1},$

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

${\hat{\sigma }}_u^2=[n(T-1)-K{]}^{-1}{\hat{u}}^{\prime }\hat{u}.$

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 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 $\mathrm{\Delta }{u}_{it}$ 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

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