Misspecified models

Summary

• If our postulated statistical model fails to hold, then we say that our model is misspecified .
• Example ( \(k=1\) ):
• We postulate (assume) that \(E\left( y_i \right|x_i)=βx_i\)
• In fact, \(E\left( y_i \right|x_i)=βx_i^2\)
• We believe that the error terms are given by \(ε_i^M =y_i-E\left( y_i \right|x_i)=y_i-βx_i\)
• In fact, they are given by \(ε_i^T =y_i-E\left( y_i \right|x_i)=y_i-βx_i^2 \)
• The subscripts on the error terms are \(M\) for “Model” and \(T\) for “True”
• We set up the LRM, \(y_i=βx_i+ε_i^M\)
• However

\[E\left( ε_i^M|x_i \right)=E\left( y_i-βx_i|x_i \right)=E\left( y_i|x_i \right)-βx_i=βx_i^2-βx_i\]

• \(E\left( ε_i^M|x_i \right)\) is not zero and exogeneity fails
• The main conclusion is that the explanatory variables are exogenous if and only if the statistical model is correctly specified .