Breusch-Pagan Heteroskedasticity Test

Consider the OLS regression $y_t = x_t' \beta + u_t$, where the error variance may depend on observable variables: $\text{Var}(u_t) = \sigma^2 h(z_t' \gamma)$ with $h(z_t' \gamma) = 1 + z_t' \gamma$. 1. Derive the Breusch-Pagan Lagrange multiplier (LM) test statistic using the auxiliary regression of squared OLS residuals $\hat{u}_t^2$ on $z_t$. Show that the statistic can be written in terms of the $R^2$ of the auxiliary regression. 2. State the asymptotic distribution of the test statistic under $H_0: \gamma = 0$ (homoskedasticity). 3. Discuss finite-sample adjustments and when the standard BP test may perform poorly.