The linear regression model (LRM)

Summary

Given a random sample \(\left( y_1,x_1 \right),…,(y_n,x_n)\) , a linear regression model can be formulated as follows

\[y_i=β_1+β_2x_i+ε_i , i=1,…,n\]

• \(y\) is called the dependent or explained variable
• \(x\) is called the explanatory or independent variable
• \(ε\) is called the error term
• \(β_1\) and \(β_2\) are two unknown parameters (called simply the beta-parameters)
• It is assumed that \(y\) is explained partly by \(x\) through the linear expression \(β_1+β_2x_i\) and partly by a unexplainable error term.
• The error term \(ε_i\) is an unobserved random variable. The outcome of the error term is not observed as \(β_1\) and \(β_2\) are unknown.

LRM, main point:

• Observe data \(x_1,…,x_n\) and \(y_1,…,y_n\) .
• Make the LRM assumption.
• Find estimates of \(β_1\) and \(β_2\) .