LRM with an exogenous explanatory variable
Summary
Exogenous variable
- Given a random sample (y1,x1),…,(yn,xn)(y1,x1),…,(yn,xn) and the LRM assumption
yi=β1+β2xi+εi,i=1,…,nyi=β1+β2xi+εi,i=1,…,n
- From the LRM assumption,
E(yi|xi)=β1+β2xi+E(εi|xi),i=1,…,nE(yi|xi)=β1+β2xi+E(εi|xi),i=1,…,n
- We say that the xx -variable is exogenous if
E(εi|xi)=0,i=1,…,nE(εi|xi)=0,i=1,…,n
- If the xx -variable is exogenous then
E(yi|xi)=β1+β2xi,i=1,…,nE(yi|xi)=β1+β2xi,i=1,…,n
- and
yi=E(yi|xi)+εi,i=1,…,nyi=E(yi|xi)+εi,i=1,…,n
- or
εi=yi−E(yi|xi),i=1,…,nεi=yi−E(yi|xi),i=1,…,n
Interpreting ββ -parameters
- The exogeneity assumption
E(yi|xi)=β1+β2xi,i=1,…,nE(yi|xi)=β1+β2xi,i=1,…,n
- relates xixi to E(yi|xi)E(yi|xi) for our sample .
- We also believe that the same relationship exists for arbitrary values of x and y and we write
E(y|x)=β1+β2x
- From this,
β2=dE(y|x)dx
β1=E(y|x=0)
Main point:
If the x -variable is exogenous, we can interpret β2 in the LRM as the approximate increase in the conditional expectation E(y|x) when x increases by 1 unit. This interpretation is sometimes abbreviated to “ β2 measure the increase in y when x increases by 1 ” .