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Econometrics Procedures
 

CPANEL Procedure

Two-Way Fixed-Effects Model (FIXTWO Option)

You perform two-way fixed-effects estimation by specifying the FIXTWO option in the MODEL statement. The error specification for the two-way fixed-effects model is

u Subscript i t Baseline equals nu Subscript i Baseline plus lamda Subscript t Baseline plus e Subscript i t

where the nu Subscript i and lamda Subscript t are nonrandom parameters to be estimated. Assume the identifiability restriction sigma-summation Underscript t Endscripts lamda Subscript t Baseline equals 0.

Estimation is similar to that for one-way fixed effects, for which a within transformation is used to convert the problem to OLS regression. For two-way models under the general case of unbalanced data, the within transformation is more complex.

Following Wansbeek and Kapteyn (1989) and Baltagi (2013, sec. 9.4), let bold upper X Superscript asterisk and bold y Superscript asterisk be versions of bold upper X and bold y whose rows are sorted by time period, and by cross section within each time period. With the data sorted in this manner, define bold upper D Subscript upper N to be the upper M times upper N design matrix for cross sections. Each row of bold upper D Subscript upper N contains a 1 in the column that corresponds to that observation’s cross section, and 0s in the remaining columns. Similarly, define bold upper D Subscript upper T to be the upper M times upper T design matrix for time periods. In balanced data, bold upper D Subscript upper N Baseline equals bold j Subscript upper T Baseline circled-times bold upper I Subscript upper N and bold upper D Subscript upper T Baseline equals bold upper I Subscript upper T Baseline circled-times bold j Subscript upper N.

Define the following:

StartLayout 1st Row 1st Column bold upper Delta Subscript upper N 2nd Column equals bold upper D Subscript upper N Superscript prime Baseline bold upper D Subscript upper N Baseline 3rd Column left-parenthesis upper N times upper N right-parenthesis 2nd Row 1st Column bold upper Delta Subscript upper T 2nd Column equals bold upper D Subscript upper T Superscript prime Baseline bold upper D Subscript upper T Baseline 3rd Column left-parenthesis upper T times upper T right-parenthesis 3rd Row 1st Column bold upper A 2nd Column equals bold upper D Subscript upper T Superscript prime Baseline bold upper D Subscript upper N Baseline 3rd Column left-parenthesis upper T times upper N right-parenthesis 4th Row 1st Column bold upper D overbar 2nd Column equals bold upper D Subscript upper T Baseline minus bold upper D Subscript upper N Baseline normal upper Delta Subscript upper N Superscript negative 1 Baseline bold upper A Superscript prime Baseline 3rd Column left-parenthesis upper M times upper T right-parenthesis 5th Row 1st Column bold upper Q 2nd Column equals normal upper Delta Subscript upper T Baseline minus bold upper A normal upper Delta Subscript upper N Superscript negative 1 Baseline bold upper A Superscript prime Baseline 3rd Column left-parenthesis upper T times upper T right-parenthesis 6th Row 1st Column bold upper P 2nd Column equals bold upper I Subscript upper M Baseline minus bold upper D Subscript upper N Baseline normal upper Delta Subscript upper N Superscript negative 1 Baseline bold upper D Subscript upper N Superscript prime Baseline minus bold upper D overbar bold upper Q Superscript negative 1 Baseline bold upper D overbar Superscript prime Baseline 3rd Column left-parenthesis upper M times upper M right-parenthesis EndLayout

The matrix bold upper P provides the two-way within transformation. If the data are balanced, this amounts to transforming any data value z Subscript i t to z Subscript i t Baseline minus z overbar Subscript i period Baseline minus z overbar Subscript period t plus z overbar Subscript period period.

Applying the two-way within transformation means that you can use OLS regression of bold upper P bold y Superscript asterisk on bold upper P bold upper X Superscript asterisk to obtain ModifyingAbove bold-italic beta With caret Subscript f, Var left-parenthesis ModifyingAbove bold-italic beta With caret Subscript f Baseline right-parenthesis, and fit statistics such as mean-square error (MSE), provided that you adjust the error degrees of freedom to equal upper M minus upper N minus upper T minus upper K plus 1.

Define the residual vector bold r Superscript asterisk Baseline equals bold y Superscript asterisk Baseline minus bold upper X Superscript asterisk Baseline ModifyingAbove bold-italic beta With caret Subscript f. Estimates of the time effects are ModifyingAbove bold-italic lamda With caret equals bold upper Q Superscript negative 1 Baseline bold upper D overbar Superscript prime Baseline bold r Superscript asterisk, and estimates of the cross-sectional effects are ModifyingAbove bold-italic nu With caret equals left-parenthesis normal upper Theta 1 minus normal upper Theta 2 plus normal upper Theta 3 right-parenthesis bold r Superscript asterisk, where

StartLayout 1st Row 1st Column normal upper Theta 1 2nd Column equals normal upper Delta Subscript upper N Superscript negative 1 Baseline bold upper D Subscript upper N Superscript prime Baseline 2nd Row 1st Column normal upper Theta 2 2nd Column equals normal upper Delta Subscript upper N Superscript negative 1 Baseline bold upper A Superscript prime Baseline bold upper Q Superscript negative 1 Baseline bold upper D Subscript upper T Superscript prime Baseline 3rd Row 1st Column normal upper Theta 3 2nd Column equals normal upper Delta Subscript upper N Superscript negative 1 Baseline bold upper A Superscript prime Baseline bold upper Q Superscript negative 1 Baseline bold upper A normal upper Delta Subscript upper N Superscript negative 1 Baseline bold upper D Subscript upper N Superscript prime EndLayout

If the model includes an intercept, then ModifyingAbove alpha With caret equals y overbar Subscript period period Baseline minus bold x overbar Subscript period period Baseline ModifyingAbove bold-italic beta With caret Subscript f, and the ModifyingAbove nu With caret Subscript i are shifted to exclude ModifyingAbove alpha With caret.

Variance and covariance estimates for ModifyingAbove alpha With caret, ModifyingAbove bold-italic nu With caret, and ModifyingAbove bold-italic lamda With caret are obtained by the delta method, because each of these quantities is a linear transformation of bold y Superscript asterisk and ModifyingAbove bold-italic beta With caret Subscript f.

Last updated: July 09, 2026