Incorrect linear model

Problem

Consider the data in the Excel file http://media.nek.lu.se/data/Nonlinear.xlsx where we have fitted a linear trend line through the data. Explain why the exogeneity assumption certainly does not hold here.

Solution

We have fitted a linear trendline to data from a model which is clearly not linear. We say that we hav misspecified the model. It’s clear from the picture that all errors when x is small and when x is large are negative while the errors are positive when x is close to the average. Therefore, \(E\left( ε_i \mid x_i \right)\) cannot be zero for all possible values of \(x_i\) . Exogeneity fails and our estimator is neither unbiased nor consistent.