The Classical Linear Regression Model

Abstract

Let us consider the following linear model

$$_{n}{{y}_{1}}{{=}_{n}}{{X}_{{kk}}}{{\beta }_{1}}{{+}_{n}}{{\varepsilon }_{1}}$$

((1.1))

where $_{n}{{y}_{1}}=\left[ {\begin{array}{*{20}{c}} {{{y}_{1}}} \\ \cdots \\ \cdots \\ {{{y}_{n}}} \\ \end{array}} \right]$ is a vector of n observations of the dependent variable y, $_{n}{{X}_{k}}=\left[ {\begin{array}{*{20}{c}} 1 \hfill & {{{X}_{{11}}}} \hfill \\ \cdots \hfill & {} \hfill \\ 1 \hfill & {} \hfill \\ 1 \hfill & {{{X}_{{n1}}}} \hfill \\ \end{array}\quad \;\begin{array}{*{20}{c}} {{{X}_{{1k-1}}}} \\ \cdots \\ {} \\ {{{X}_{{nk-1}}}} \\ \end{array}} \right]$ a matrix of n observations on k − 1 non-stochastic exogenous regressors including a constant term, $_{k}{{\beta }_{1}}=\left[ {\begin{array}{*{20}{c}} {{{\beta }_{1}}} \\ {} \\ {} \\ {{{\beta }_{k}}} \\ \end{array}} \right]$ a vector of k unknown parameters to be estimated and $_{n}{{\varepsilon }_{1}}=\left[ {\begin{array}{*{20}{c}} {{{\varepsilon }_{1}}} \\ {} \\ {} \\ {{{\varepsilon }_{n}}} \\ \end{array}} \right]$ a vector of stochastic disturbances. We will assume throughout the book that the n observations refer to territorial units such as regions or countries.