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Let $\hat{\beta}$ be the OLS estimator of $\beta$ in $$y_t = \beta x_t + u_t$$ Let $\tilde{\beta}$ be the OLS estimator of $\beta$ in $$\dfrac{y_t}{\sqrt{k_t}} = \beta \dfrac{x_t}{\sqrt{k_t}} + \dfrac{u_t}{\sqrt{k_t}}$$ $\text{Var}(\hat{\beta}) = \dfrac{\sum x_t^2\sigma_t^2}{\left(\sum x_t^2\right)^{2}}$ $\text{Var}(\tilde{\beta}) = \dfrac{\sigma^2_u}{\sum ... 1 The Ω matrix is the matrix of the variance of the error term for each observation. Since we do not observe the true error term, we cannot find the true Ω, but we can try to estimate it. There are different ways of estimating Ω, and there is some debate in econometrics as to which method is the best but the most common method I think would be to take the ... 1 Let us write$\mathbf{x}\delta = \delta_0 + \delta_1 x_1 + \cdots + \delta_k x_k$for notational brevity. If$u^2 = \sigma^2 \exp(\mathbf{x}\delta) v$, where$E(v|\mathbf{x})=1$, then it is indeed true that $$E(u^2|\mathbf{x}) = \sigma^2 \exp(\mathbf{x}\delta) E(v|\mathbf{x}) = \sigma^2 \exp(\mathbf{x}\delta).$$ The LHS is$Var(u|\mathbf{x})$by definition ... 1 I'm using the 4th edition, but the content should be the same. It sounds like you are asking how he got from $$Var(u|\boldsymbol{x}) = \sigma^2exp(\delta_0 +\delta_1x_1+\delta_2x_2+...+\delta_kx_k)$$ to $$u^2=\sigma^2exp(\delta_0 +\delta_1x_1+\delta_2x_2+...+\delta_kx_k)v$$ As you correctly noted,$Var(u|\boldsymbol{x}) = E(u^2|\boldsymbol{x})$... 1 You can regress residual squares (from RE or FE depending on your estimation) on$X_{it} \hat\beta\$ and its square using the clustered standard errors (the vce(cl id) option), and read the F statistic and the associated p value. This is basically the same as Het test for cross sectional models (White's simplified test). xtreg y x1 x2, re predict uhat, ue ...