PANEL 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 and are nonrandom parameters to be estimated.

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 and be versions of and whose rows are sorted by time period, and by cross section within each time period. With the data sorted in this manner, define to be the design matrix for cross sections. Each row of contains a 1 in the column that corresponds to that observation’s cross section, and 0s in the remaining columns. Similarly, define to be the design matrix for time periods. In balanced data, and .

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 provides the two-way within transformation. If the data are balanced, this amounts to transforming any data value to .

Applying the two-way within transformation means that you can use OLS regression of on to obtain , , and fit statistics such as mean square error (MSE), provided that you adjust the error degrees of freedom to equal .

Define the residual vector . Estimates of the time effects are , and estimates of the cross-sectional effects are , 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

The full model that contains the intercept, N cross-sectional effects, and T time effects is overidentified, and simultaneous estimation of these quantities is not possible without restrictions. If you specify the PRINTFIXED option, the printed fixed effects reflect these restrictions.

If the model has an intercept, then the PRINTFIXED option output is parameterized as follows:

Intercept:
Cross section i:
Time period t:

If the model does not include an intercept, then the PRINTFIXED option output is parameterized as follows:

Cross section i:
Time period t:

Variance and covariance estimates for the intercept and printed fixed effects are obtained by the delta method, because each of these quantities is a linear transformation of and .

Last updated: June 19, 2025