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Econometrics Procedures
 

CPANEL Procedure

One-Way Fixed-Effects Model, First Differencing (FDONE Option)

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.

You cannot obtain the error variance directly from the OLS regression because that regression is based on the differenced residual e Subscript i t Baseline minus e Subscript i comma t minus 1. Instead, you can estimate sigma Subscript e Superscript 2 by ModifyingAbove sigma With caret Subscript e Superscript 2 Baseline equals normal upper S normal upper S normal upper E slash left-parenthesis upper M minus upper N minus upper K right-parenthesis, where

normal upper S normal upper S normal upper E equals sigma-summation Underscript i equals 1 Overscript upper N Endscripts sigma-summation Underscript t equals 1 Overscript upper T Subscript i Baseline Endscripts left-parenthesis y Subscript i t Baseline minus bold x Subscript i t Baseline ModifyingAbove bold-italic beta With caret Subscript d Baseline minus ModifyingAbove nu With caret Subscript i Baseline right-parenthesis squared

and ModifyingAbove nu With caret Subscript i Baseline equals y overbar Subscript i period Baseline minus bold x overbar Subscript i period Baseline ModifyingAbove bold-italic beta With caret Subscript d.

Estimation and inference for alpha and nu Subscript i 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.

Last updated: July 09, 2026