First-difference estimator

The FD estimator avoids bias due to some unobserved, time-invariant variable ${\displaystyle c_{i}}$, using the repeated observations over time:

${\displaystyle y_{it}=x_{it}\beta +c_{i}+u_{it},t=1,...T,}$

${\displaystyle y_{it-1}=x_{it-1}\beta +c_{i}+u_{it-1},t=2,...T.}$

Differencing the equations, gives:

${\displaystyle \Delta y_{it}=y_{it}-y_{it-1}=\Delta x_{it}\beta +\Delta u_{it},t=2,...T,}$

which removes the unobserved ${\displaystyle c_{i}}$.

The FD estimator ${\displaystyle {\hat {\beta }}_{FD}}$ is then obtained by using the differenced terms for x and u in OLS:

${\displaystyle {\hat {\beta }}_{FD}=(\Delta X'\Delta X)^{-1}\Delta X'\Delta y=\beta +(\Delta X'\Delta X)^{-1}\Delta X'\Delta u}$

Where ${\displaystyle X,y,}$ and ${\displaystyle u}$, are notation for matrices of relevant variables. Note that the rank condition must be met for ${\displaystyle \Delta X'\Delta X}$ to be invertible (${\displaystyle rank[\Delta X'\Delta X]=k}$) where ${\displaystyle k}$ is the number of regressors.

Let ${\displaystyle \Delta X_{i}=[\Delta X_{i2},\Delta X_{i3},...,\Delta X_{iT}]}$ and define ${\displaystyle \Delta u_{i}}$ analogously. If ${\displaystyle E[u_{it}|x_{i1},x_{i2},..,x_{iT}]=0}$ , by the Central limit theorem, Law of large numbers, and Slutsky's theorem, the estimator is distributed normally with asymptotic variance of ${\displaystyle E[\Delta X_{i}'\Delta X_{i}]^{-1}E[\Delta X_{i}\Delta u_{i}\Delta u_{i}']E[\Delta X_{i}'\Delta X_{i}]^{-1}}$.

Under the assumption of homoskedasticity and no serial correlation, mathematically that, ${\displaystyle Var(\Delta u|X)=\sigma _{\Delta u}^{2}}$, the asymptotic variance can be estimated with

${\displaystyle {\widehat {Avar}}({\hat {\beta }}_{FD})={\hat {\sigma }}_{\Delta u}^{2}(\Delta X'\Delta X)^{-1},}$

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

${\displaystyle {\hat {\sigma }}_{\Delta u}^{2}=[n(T-1)-K]^{-1}\sum _{i=1}^{n}\sum _{t=2}^{T}{\widehat {\Delta u_{it}}}^{2}}$

and ${\displaystyle {\widehat {\Delta u_{it}}}=\Delta y_{it}-{\hat {\beta }}_{FD}\Delta x_{it}}$.

To be unbiased, the fixed estimator (FE) requires strict exogeneity, ${\displaystyle E[u_{it}|x_{i1},x_{i2},..,x_{iT}]=0}$. The first difference estimator is also unbiased under this assumption. Under the weaker assumption that ${\displaystyle E[(u_{it}-u_{it-1})(x_{it}-x_{it-1})]=0}$, the FD estimator is consistent. Note that this assumption is less restrictive than the assumption of strict exogeneity which is required for consistency using the FE estimator when T is fixed. If T goes to infinity, then both FE and FD are consistent with the weaker assumption of contemporaneous exogeneity.

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

Under the assumption of homoscedasticity and no serial correlation in ${\displaystyle u_{it}}$, the FE estimator is more efficient than the FD estimator. This is because the FD estimator induces no serial correlation when differencing the errors. If ${\displaystyle u_{it}}$ follows a random walk, however, the FD estimator is more efficient as ${\displaystyle \Delta u_{it}}$ are serially uncorrelated.

• Factor analysis
• Panel analysis

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