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 Δ y i t on Δ x i t .
The FD estimator wipes out time invariant omitted variables c i using the repeated observations over time:
y i t = x i t β + c i + u i t , t = 1 , . . . T , y i t − 1 = x i t − 1 β + c i + u i t − 1 , t = 2 , . . . T . Differencing both equations, gives:
Δ y i t = y i t − y i t − 1 = Δ x i t β + Δ u i t , t = 2 , . . . T , which removes the unobserved c i .
The FD estimator β ^ F D is then simply obtained by regressing changes on changes using OLS:
β ^ F D = ( Δ X ′ Δ X ) − 1 Δ X ′ Δ y Note that the rank condition must be met for Δ X ′ Δ X to be invertible ( r a n k [ Δ X ′ Δ X ] = k ).
Similarly,
A v a ^ r ( β ^ F D ) = σ ^ u 2 ( Δ X ′ Δ X ) − 1 , where σ ^ u 2 is given by
σ ^ u 2 = [ n ( T − 1 ) − K ] − 1 u ^ ′ u ^ . Under the assumption of E [ u i t − u i t − 1 | x i t − x i t − 1 ] = 0 , the FD estimator is unbiased and consistent, i.e. E [ β ^ F D ] = β and p l i m β ^ = β . 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 i t follows a random walk, the usual OLS standard errors are asymptotically valid.
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 i t , the FE estimator is more efficient than the FD estimator. If u i t follows a random walk, however, the FD estimator is more efficient as Δ u i t are serially uncorrelated.
In practice, the FD estimator is easier to implement without special software, as the only transformation required is to first difference.
