The General Linear Model: The Basics

Consider the following regression equation $$ y = X\beta + u $$ (7.1) where $$ y = \left[ {\begin{array}{*{20}{c}} {{Y_1}}\\ {{Y_2}}\\ :\\ {{Y_n}} \end{array}} \right];X = \left[ {\begin{array}{*{20}{c}} {{X_{11}}\quad {X_{12}}\quad \ldots \quad {X_{1k}}}\\ {{X_{21}}\quad {X_{22}}\quad \ldots \quad {X_{2k}}}\\ {:\quad \quad :\quad \quad :\quad \quad :}\\ {{X_{n1}}\quad {X_{n2}}\quad \ldots \quad {X_{nk}}} \end{array}} \right];\beta = \left[ {\begin{array}{*{20}{c}} {{\beta _1}}\\ {{\beta _2}}\\ :\\ {{\beta _k}} \end{array}} \right];u = \left[ {\begin{array}{*{20}{c}} {{u_1}}\\ {{u_2}}\\ :\\ {{u_n}} \end{array}} \right] $$ with n denoting the number of observations and k the number of variables in the regression, with n>k. In this case, y is a column vector of dimension (n × 1) and X is a matrix of dimension (n × k). Each column of X denotes a variable and each row of X denotes an observation on these variables. If y is log(wage) as in the empirical example in Chapter 4, see Table 4.1 then the columns of X contain a column of ones for the constant (usually the first column), weeks worked, years of full time experience, years of education, sex, race, marital status, etc.