PANEL Procedure

One-Way Fixed-Effects Model (FIXONE and FIXONETIME Options)

You perform one-way fixed-effects estimation by specifying the FIXONE option in the MODEL statement. The error structure for the one-way fixed-effects model is

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

where the nu Subscript i are nonrandom parameters that are restricted to sum to 0, and the e Subscript i t are iid with zero mean and variance sigma Subscript e Superscript 2.

The fixed-effects model can be estimated by ordinary least squares (OLS), treating the nu Subscript i as coefficients on dummy variables that identify the cross sections. However, when N is large, you might want to estimate only bold-italic beta and not nu Subscript i.

Let bold upper Q 0 equals normal d normal i normal a normal g left-parenthesis bold upper E Subscript upper T Sub Subscript i Subscript Baseline right-parenthesis. The matrix bold upper Q 0 represents the within transformation, the conversion of the raw data to deviations from a cross section’s mean. Let bold upper X Subscript w Baseline equals bold upper Q 0 bold upper X and bold y Subscript w Baseline equals bold upper Q 0 bold y. The within estimator of bold-italic beta is

ModifyingAbove bold-italic beta With caret Subscript w Baseline equals left-parenthesis bold upper X Subscript w Superscript prime Baseline bold upper X Subscript w Baseline right-parenthesis Superscript negative 1 Baseline bold upper X Subscript w Superscript prime Baseline bold y Subscript w

The previous estimation does not involve the intercept term because ModifyingAbove bold-italic beta With caret Subscript w is the same whether or not the intercept alpha is included in the model.

Standard errors, t statistics, and fit statistics such as mean square error (MSE) are all equivalent to those obtained from OLS regression of bold y Subscript w on bold upper X Subscript w. The only exception is the error degrees of freedom, which equals upper M minus upper N minus upper K to account for the tacit estimation of the N fixed effects.

Each fixed effect is estimated as

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 w

where y overbar Subscript i period and bold x overbar Subscript i period are cross-sectional means.

The fixed-effects model is parameterized so that the intercept is the fixed effect for the last cross section. That is,

ModifyingAbove alpha With caret equals ModifyingAbove nu With caret Subscript upper N Baseline equals y overbar Subscript upper N period Baseline minus bold x overbar Subscript upper N period Baseline ModifyingAbove bold-italic beta With caret Subscript w

Fixed effects are by default not displayed as part of the regression, but you can obtain them by specifying the PRINTFIXED option in the MODEL statement. In models that have an intercept, the printed fixed effects are the deviations ModifyingAbove nu With caret Subscript i Baseline minus ModifyingAbove nu With caret Subscript upper N. To display the untransformed fixed effects, specify both the NOINT and PRINTFIXED options.

Variance estimates of ModifyingAbove alpha With caret, ModifyingAbove nu With caret Subscript i, and ModifyingAbove nu With caret Subscript i Baseline minus ModifyingAbove nu With caret Subscript upper N are obtained by the delta method.

The FIXONETIME option works similarly, except that the data are grouped by time period instead of by cross section.

Last updated: June 19, 2025