PANEL Procedure

One-Way Fixed-Effects Model, First Differencing (FDONE and FDONETIME Options)

You perform one-way fixed-effects estimation via first differencing by specifying the FDONE option in the MODEL statement. The method of first differencing offers an alternative to the within estimator ModifyingAbove bold-italic beta With caret Subscript w. Consider the following one-way fixed-effects model:

y Subscript i t Baseline equals alpha plus bold x Subscript i t Baseline bold-italic beta plus nu Subscript i Baseline plus e Subscript i t

For this model, the fixed effects are removed by subtracting first-order lags from both sides of the equation:

y Subscript i t Baseline minus y Subscript i comma t minus 1 Baseline equals left-parenthesis bold x Subscript i t Baseline minus bold x Subscript i comma t minus 1 Baseline right-parenthesis bold-italic beta plus left-parenthesis e Subscript i t Baseline minus e Subscript i comma t minus 1 Baseline right-parenthesis

Define normal upper Delta y Subscript i t Baseline equals y Subscript i t Baseline minus y Subscript i comma t minus 1 and normal upper Delta bold x Subscript i t Baseline equals bold x Subscript i t Baseline minus bold x Subscript i comma t minus 1, for i equals 1 comma ellipsis comma upper N and t equals 2 comma ellipsis comma upper T Subscript i Baseline. You obtain the first-differenced estimator, ModifyingAbove bold-italic beta With caret Subscript d, and its variance by performing OLS regression of normal upper Delta y Subscript i t on normal upper Delta bold x Subscript i t.

The estimation and parameterization of left-parenthesis alpha comma nu Subscript i Baseline right-parenthesis are identical to that described in the section One-Way Fixed-Effects Model (FIXONE and FIXONETIME Options), with ModifyingAbove bold-italic beta With caret Subscript w replaced by ModifyingAbove bold-italic beta With caret Subscript d.

The FDONETIME option works similarly, switching the roles of cross sections and time periods in the methodology described previously.

Last updated: June 19, 2025