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First-difference estimator

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In statistics and econometrics, the first-difference (FD) estimator is an estimator used to address the problem of omitted variables with panel data. It is consistent under the assumptions of the fixed effects model. In certain situations it can be more efficient than the standard fixed effects (or "within") estimator.

The estimator requires data on a dependent variable, , and independent variables, , for a set of individual units and time periods The estimator is obtained by running a pooled ordinary least squares (OLS) estimation for a regression of on .

Derivation

The FD estimator avoids bias due to some omitted, time-invariant variable using the repeated observations over time:

Differencing both equations, gives:

which removes the unobserved .

The FD estimator is then simply obtained by regressing changes on changes using OLS:

Where X, y, and u, are notation for matrices of relevant variables. Note that the rank condition must be met for to be invertible (). Let and define analogously. If , by the Central limit theorem, Law of large numbers, and Slutsky's theorem, estimator is distributed normally with asymptotic variance of .

Under the assumption of homoskedasticity and no serial correlation, that, , the asymptotic variance is estimated with

where is given by

and .

Properties

To be unbiased, the fixed estimator requires strict exogeneity, . Under the assumption of , the FD estimator is consistent. Note that this assumption is less restrictive than the assumption of strict exogeneity required for consistency using the fixed effects (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.

Relation to fixed effects estimator

For , the FD and fixed effects estimators are numerically equivalent.

Under the assumption of homoscedasticity and no serial correlation in , 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 follows a random walk, however, the FD estimator is more efficient as are serially uncorrelated.

See also

References

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