For the simple regression model with heteroskedasticity, (y_{i}=beta_{1}+beta_{2} x_{i}+e_{i}) and (operatorname{var}left(e_{i} mid mathbf{x}_{i} ight)=sigma_{i}^{2}) show that the

For the simple regression model with heteroskedasticity, \(y_{i}=\beta_{1}+\beta_{2} x_{i}+e_{i}\) and \(\operatorname{var}\left(e_{i} \mid \mathbf{x}_{i}\right)=\sigma_{i}^{2}\) show that the variance \(\operatorname{var}\left(b_{2} \mid \mathbf{x}_{i}\right)=\left[\sum_{i=1}^{N}\left(x_{i}-\bar{x}\right)^{2}\right]^{-1}\left[\sum_{i=1}^{N}\left(x_{i}-\bar{x}\right)^{2} \sigma_{i}^{2}\right]\left[\sum_{i=1}^{N}\left(x_{i}-\bar{x}\right)^{2}\right]^{-1}\) reduces to \(\operatorname{var}\left(b_{2} \mid \mathbf{x}\right)=\sigma^{2} / \sum_{i=1}^{N}\left(x_{i}-\bar{x}\right)^{2}\) under homoskedasticity.