Neha Patil (Editor)

First difference estimator

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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 .

Contents

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 ^ .

Properties

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.

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 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.

References

First-difference estimator Wikipedia
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