The General Linear Model: The Basics

Consider the following regression equation

7.1

$$y = X\beta + u$$

where

$$y = \left[\begin{array}{c}Y_1\\ Y_2\\ \atop^{.}_{.}\\ Y_n\end{array} \right]; X = \left[ \begin{array}{cccc}X_{11} & X_{12} & \ldots & X_{1k}\\ X_{21} & X_{22} & \ldots & X_{2k}\\ \atop^{.}_{.} & \atop^{.}_{.} & \atop^{.}_{.} & \atop^{.}_{.}\\ X_{n1} & X_{n2} & \ldots & X_{nk} \end{array} \right]; \beta = \left[ \begin{array}{c}\beta_1\\ \beta_2\\ \atop^{.}_{.}\\ \beta_k \end{array} \right]; u = \left[\begin{array}{c}u_1\\ u_2\\ \atop^{.}_{.}\\ u_n \end{array}\right]$$

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.