PANEL Procedure

One-Way Random-Effects Model (RANONE Option)

You perform one-way random-effects estimation by specifying the RANONE option in the MODEL statement. The specification for the one-way random-effects model is

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

where the nu Subscript i are iid with zero mean and variance sigma Subscript nu Superscript 2, and the e Subscript i t are iid with zero mean and variance sigma Subscript e Superscript 2. Furthermore, a random-effects specification assumes that the error terms are mutually uncorrelated and that each error term is uncorrelated with bold upper X.

Estimation proceeds in two steps. First, you obtain estimates of the variance components sigma Subscript nu Superscript 2 and sigma Subscript e Superscript 2. Second, with the variance components in hand, you form a weight for each cross section,

ModifyingAbove theta With caret Subscript i Baseline equals 1 minus ModifyingAbove sigma With caret Subscript e Baseline slash ModifyingAbove w With caret Subscript i

where ModifyingAbove w With caret Subscript i Superscript 2 Baseline equals upper T Subscript i Baseline ModifyingAbove sigma With caret Subscript nu Superscript 2 Baseline plus ModifyingAbove sigma With caret Subscript e Superscript 2. Taking ModifyingAbove theta With caret Subscript i, you form the partial deviations:

StartLayout 1st Row 1st Column y overTilde Subscript i t 2nd Column equals y Subscript i t Baseline minus ModifyingAbove theta With caret Subscript i Baseline y overbar Subscript i dot Baseline 2nd Row 1st Column bold x overTilde Subscript alpha comma i t 2nd Column equals bold x Subscript alpha comma i t Baseline minus ModifyingAbove theta With caret Subscript i Baseline bold x overbar Subscript alpha comma i dot EndLayout

The random-effects estimation is then the result of OLS regression on the transformed data.

The PANEL procedure provides four methods of estimating variance components, as described in the following subsections.

Wallace-Hussain Method

You can use the Wallace-Hussain (1969) method of estimating variance components by specifying the VCOMP=WH option in the MODEL statement. The Wallace-Hussain method is part of a class of methods known as analysis of variance (ANOVA) estimators.

ANOVA estimators obtain variance components by solving a system of equations that is based on expected sums of squares. The following quadratic forms correspond to the within and between sums of squares, respectively:

StartLayout 1st Row 1st Column q Subscript e 2nd Column equals bold u Superscript prime Baseline bold upper Q 0 bold u 2nd Row 1st Column q Subscript nu 2nd Column equals bold u Superscript prime Baseline bold upper P 0 bold u EndLayout

In these equations, 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, bold upper P 0 equals normal d normal i normal a normal g left-parenthesis bold upper J overbar Subscript upper T Sub Subscript i Subscript Baseline right-parenthesis, and bold u is the vector of true residuals.

The ANOVA methods differ only in how they estimate bold u. The Wallace-Hussain method uses the residuals from pooled (OLS) regression, ModifyingAbove bold u With caret Subscript p, in both quadratic forms.

The expected values of the quadratic forms are

StartLayout 1st Row 1st Column upper E left-parenthesis ModifyingAbove bold u With caret Subscript p Superscript prime Baseline bold upper Q 0 ModifyingAbove bold u With caret Subscript p Baseline right-parenthesis 2nd Column equals left-parenthesis d 1 minus d 3 right-parenthesis sigma Subscript nu Superscript 2 Baseline plus left-parenthesis upper M minus upper N minus upper K minus 1 plus d 2 right-parenthesis sigma Subscript e Superscript 2 Baseline 2nd Row 1st Column upper E left-parenthesis ModifyingAbove bold u With caret Subscript p Superscript prime Baseline bold upper P 0 ModifyingAbove bold u With caret Subscript p Baseline right-parenthesis 2nd Column equals left-parenthesis upper M minus 2 d 1 plus d 3 right-parenthesis sigma Subscript nu Superscript 2 Baseline plus left-parenthesis upper N minus d 2 right-parenthesis sigma Subscript e Superscript 2 EndLayout

where

d 1 equals normal t normal r StartSet left-parenthesis bold upper X Subscript alpha Superscript prime Baseline bold upper X Subscript alpha Baseline right-parenthesis Superscript negative 1 Baseline bold upper X Subscript alpha Superscript prime Baseline bold upper Z 0 bold upper Z 0 Superscript prime Baseline bold upper X Subscript alpha Baseline EndSet
d 2 equals normal t normal r StartSet left-parenthesis bold upper X Subscript alpha Superscript prime Baseline bold upper X Subscript alpha Baseline right-parenthesis Superscript negative 1 Baseline bold upper X Subscript alpha Superscript prime Baseline bold upper P 0 bold upper X Subscript alpha Baseline EndSet
d 3 equals normal t normal r StartSet left-parenthesis bold upper X Subscript alpha Superscript prime Baseline bold upper X Subscript alpha Baseline right-parenthesis Superscript negative 1 Baseline bold upper X Subscript alpha Superscript prime Baseline bold upper P 0 bold upper X Subscript alpha Baseline left-parenthesis bold upper X Subscript alpha Superscript prime Baseline bold upper X Subscript alpha Baseline right-parenthesis Superscript negative 1 Baseline bold upper X Subscript alpha Superscript prime Baseline bold upper Z 0 bold upper Z 0 Superscript prime Baseline bold upper X Subscript alpha Baseline EndSet

Wansbeek-Kapteyn Method

You can use the Wansbeek-Kapteyn method of estimating variance components by specifying the VCOMP=WK option in the MODEL statement. The method is a specialization (Baltagi and Chang 1994) of the approach used by Wansbeek and Kapteyn (1989) for unbalanced two-way models. The method was also suggested by Amemiya (1971) for balanced data.

The Wansbeek-Kapteyn method is an ANOVA method that uses the within residuals from one-way fixed effects, ModifyingAbove bold u With caret Subscript w, in both quadratic forms.

The expected values of the quadratic forms are

StartLayout 1st Row 1st Column upper E left-parenthesis ModifyingAbove bold u With caret Subscript w Superscript prime Baseline bold upper Q 0 ModifyingAbove bold u With caret Subscript w Baseline right-parenthesis 2nd Column equals left-parenthesis upper M minus upper N minus upper K right-parenthesis sigma Subscript e Superscript 2 Baseline 2nd Row 1st Column upper E left-parenthesis ModifyingAbove bold u With caret Subscript w Superscript prime Baseline bold upper P 0 ModifyingAbove bold u With caret Subscript w Baseline right-parenthesis 2nd Column equals left-parenthesis upper N minus 1 plus d right-parenthesis sigma Subscript e Superscript 2 Baseline plus left-parenthesis upper M minus upper M Superscript negative 1 Baseline sigma-summation Underscript i equals 1 Overscript upper N Endscripts upper T Subscript i Superscript 2 Baseline right-parenthesis sigma Subscript nu Superscript 2 EndLayout

where

d equals normal t normal r StartSet left-parenthesis bold upper X Superscript prime Baseline bold upper Q 0 bold upper X right-parenthesis Superscript negative 1 Baseline bold upper X Superscript prime Baseline bold upper P 0 bold upper X EndSet minus normal t normal r StartSet left-parenthesis bold upper X Superscript prime Baseline bold upper Q 0 bold upper X right-parenthesis Superscript negative 1 Baseline bold upper X Superscript prime Baseline bold upper J overbar Subscript upper M Baseline bold upper X EndSet

Fuller-Battese Method

You can use the Fuller-Battese (1974) method of estimating variance components by specifying the VCOMP=FB option in the MODEL statement. Following Baltagi (2013, sec. 9.2), you obtain ModifyingAbove sigma With caret Subscript e Superscript 2 as the mean square error (MSE) from one-way fixed effects. The cross-sectional variance is

ModifyingAbove sigma With caret Subscript nu Superscript 2 Baseline equals StartFraction upper R left-parenthesis bold-italic nu vertical-bar bold-italic beta right-parenthesis minus left-parenthesis upper N minus 1 right-parenthesis ModifyingAbove sigma With caret Subscript e Superscript 2 Baseline Over upper M minus normal t normal r StartSet bold upper Z 0 Superscript prime Baseline bold upper X Subscript alpha Baseline left-parenthesis bold upper X Subscript alpha Superscript prime Baseline bold upper X Subscript alpha Baseline right-parenthesis Superscript negative 1 Baseline bold upper X Subscript alpha Superscript prime Baseline bold upper Z 0 EndSet EndFraction

where

upper R left-parenthesis bold-italic nu vertical-bar bold-italic beta right-parenthesis equals upper R left-parenthesis bold-italic beta vertical-bar bold-italic nu right-parenthesis plus upper R left-parenthesis bold-italic nu right-parenthesis minus upper R left-parenthesis bold-italic beta right-parenthesis

for

StartLayout 1st Row 1st Column upper R left-parenthesis bold-italic nu right-parenthesis 2nd Column equals bold y Superscript prime Baseline bold upper Z 0 left-parenthesis bold upper Z 0 Superscript prime Baseline bold upper Z 0 right-parenthesis Superscript negative 1 Baseline bold upper Z 0 Superscript prime Baseline bold y 2nd Row 1st Column upper R left-parenthesis bold-italic beta vertical-bar bold-italic nu right-parenthesis 2nd Column equals bold y Subscript w Superscript prime Baseline bold upper X Subscript w Superscript prime Baseline 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 Baseline 3rd Row 1st Column upper R left-parenthesis bold-italic beta right-parenthesis 2nd Column equals bold y Superscript prime Baseline bold upper X Subscript alpha Superscript prime Baseline left-parenthesis bold upper X Subscript alpha Superscript prime Baseline bold upper X Subscript alpha Baseline right-parenthesis Superscript negative 1 Baseline bold upper X Subscript alpha Superscript prime Baseline bold y EndLayout

Nerlove Method

You can use the Nerlove (1971) method of estimating variance components by specifying the VCOMP=NL option in the MODEL statement. The Nerlove method provides a simple alternative to the previous three estimation strategies. You estimate sigma Subscript nu Superscript 2 as the sample variance of the cross-sectional effects, estimated from a one-way fixed-effects regression. Specifically, ModifyingAbove sigma With caret Subscript nu Superscript 2 Baseline equals left-parenthesis upper N minus 1 right-parenthesis Superscript negative 1 Baseline sigma-summation Underscript i equals 1 Overscript upper N Endscripts left-parenthesis ModifyingAbove nu With caret Subscript i Baseline minus nu overbar right-parenthesis squared, where nu overbar is the mean of the estimated fixed effects. You estimate sigma Subscript e Superscript 2 by taking the error sum of squares from one-way fixed-effects regression and then dividing by M.

Selecting the Appropriate Variance Component Method

By default, variance components are estimated by the Fuller-Battese method (VCOMP=FB) when the data are balanced, and by the Wansbeek-Kapteyn method (VCOMP=WK) when the data are unbalanced.

Baltagi and Chang (1994) conducted an extensive simulation study of the finite-sample properties of the variance estimators that the PANEL procedure supports. The choice of method has little bearing on estimates of regression coefficients, their standard errors, and estimation of the error variance sigma Subscript e Superscript 2. If your goal is inference on bold-italic beta, then the variance-component method will matter little.

The methods have varying performance in how they estimate sigma Subscript nu Superscript 2, the cross-sectional variance. All four methods tend to perform poorly if either the data are severely unbalanced or the ratio sigma Subscript nu Superscript 2 Baseline slash sigma Subscript e Superscript 2 is much greater than 1.

Of these four methods, the Nerlove method is the only one that guarantees a nonnegative estimate of sigma Subscript nu Superscript 2; the other three methods reset a negative estimate to 0. However, the Nerlove method is particularly unsuitable for unbalanced data because the sample variance that it computes is not weighted by upper T Subscript i.

Last updated: June 19, 2025