Heteroskedasticity due to external variable

Problem

Consider \(y_i=β_1+β_2x_i+ε_i\) where \(x\) is exogenous. \(Var\left( ε_i|x_i \right)=σ^2z_i^2\) where \(z_i\) is a different variable not included in the regression model.

  1. Is the error term homoscedastic or heteroscedastic?
  2. How would you test for heteroscedasticity if you suspected, but was not sure, that the variance of the error term was dependent on \(z\) ?

Solution

  1. Heteroscedastic since \(Var\left( ε_i|x_i \right)\) is different between different individuals
  2. By running a regression of \(e_i^2\) on \(z_i^2\) and a constant and testing if \(z_i\) was significant.