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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 .
The FD estimator avoids bias due to some unobserved, time-invariant variable , using the repeated observations over time:
Differencing the equations, gives:
which removes the unobserved .
The FD estimator is then obtained by using the differenced terms for x and u in OLS:
Under the assumption of homoskedasticity and no serial correlation, mathematically that, , the asymptotic variance can be estimated with
where is given by
To be unbiased, the fixed effects estimator (FE) requires strict exogeneity, . The first difference estimator is also unbiased under this assumption. Under the weaker assumption that , 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 , 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.
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The purpose of this page is to provide supplementary materials for the ordinary least squares article, reducing the load of the main article with mathematics and improving its accessibility, while at the same time retaining the completeness of exposition.
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