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In statistics and econometrics, the first-difference 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 estimator.
The estimator requires data on a dependent variable, y i t {\displaystyle y_{it}} , and independent variables, x i t {\displaystyle x_{it}} , for a set of individual units i = 1 , … , N {\displaystyle i=1,\dots ,N} and time periods t = 1 , … , T {\displaystyle t=1,\dots ,T}. The estimator is obtained by running a pooled ordinary least squares estimation for a regression of Δ y i t {\displaystyle \Delta y_{it}} on Δ x i t {\displaystyle \Delta x_{it}}.