PANEL Procedure

Common Notation

This section presents notation that is common to all subsequent sections. Consider the panel regression:

y Subscript i t Baseline equals alpha plus sigma-summation Underscript k equals 1 Overscript upper K Endscripts x Subscript i t k Baseline beta Subscript k Baseline plus u Subscript i t Baseline i equals 1 comma ellipsis comma upper N semicolon t equals 1 comma ellipsis comma upper T Subscript i Baseline

The total number of observations is upper M equals sigma-summation Underscript i equals 1 Overscript upper N Endscripts upper T Subscript i. For balanced data, upper T Subscript i Baseline equals upper T for all i. For unbalanced data, define T to be the number of unique time periods.

The exact representation of u Subscript i t and the underlying assumptions depend on the estimation method.

In matrix notation the model is

y Subscript i t Baseline equals alpha plus bold x Subscript i t Baseline bold-italic beta plus u Subscript i t

where bold x Subscript i t is a 1 times upper K row vector of independent variables and bold-italic beta is the upper K times 1 vector of coefficients. Let bold y and bold upper X be matrices that are formed by arranging the dependent and independent variables by cross section, and by time within each cross section. Let bold upper X Subscript alpha be the bold upper X matrix augmented by a first column of ones, which corresponds to the intercept term alpha.

Define the following utility matrices:

  • bold upper I Subscript p is an identity matrix of dimension p.

  • bold j Subscript p is a p times 1 column vector of ones.

  • bold upper J Subscript p Baseline equals bold j Subscript p Baseline bold j Subscript p Superscript prime is a matrix of ones of dimension p.

  • bold upper J overbar Subscript p Baseline equals p Superscript negative 1 Baseline bold upper J Subscript p.

  • bold upper E Subscript p Baseline equals bold upper I Subscript p Baseline minus bold upper J overbar Subscript p.

In the following sections, the panel data are assumed to be unbalanced unless otherwise indicated. If the data are balanced, the formulas reduce appropriately.

Last updated: June 19, 2025