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.
- Is the error term homoscedastic or heteroscedastic?
- 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
- Heteroscedastic since \(Var\left( ε_i|x_i \right)\) is different between different individuals
- By running a regression of \(e_i^2\) on \(z_i^2\) and a constant and testing if \(z_i\) was significant.