LRM with an exogenous explanatory variable

Summary

Exogenous variable

• Given a random sample \(\left( y_1,x_1 \right),…,(y_n,x_n)\) and the LRM assumption

\[y_i=β_1+β_2x_i+ε_i , i=1,…,n\]

• From the LRM assumption,

\[E\left(y_i | x_i \right)=β_1+β_2x_i+E\left(ε_i | x_i \right) , i=1,…,n\]

• We say that the \(x\) -variable is exogenous if

\[E\left(ε_i | x_i \right)=0 , i=1,…,n\]

• If the \(x\) -variable is exogenous then

\[E\left(y_i | x_i \right)=β_1+β_2x_i , i=1,…,n\]

• and

\[y_i=E\left(y_i | x_i \right)+ε_i , i=1,…,n\]

• or

\[ε_i=y_i-E\left(y_i | x_i \right) , i=1,…,n\]

Interpreting \(β\) -parameters

• The exogeneity assumption

\[E\left(y_i | x_i \right)=β_1+β_2x_i , i=1,…,n\]

• relates \(x_i\) to \(E\left(y_i | x_i \right)\) for our sample .
• We also believe that the same relationship exists for arbitrary values of \(x\) and \(y\) and we write

\[E\left(y | x \right)=β_1+β_2x \]

• From this,

\[β_2= \frac{dE\left(y | x \right)}{dx}\]

\[β_1=E\left(y | x=0 \right)\]

Main point:

If the \(x\) -variable is exogenous, we can interpret \(β_2\) in the LRM as the approximate increase in the conditional expectation \(E\left(y | x \right)\) when \(x\) increases by 1 unit. This interpretation is sometimes abbreviated to “ \(β_2\) measure the increase in \(y\) when \(x\) increases by 1 ” .