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I cannot say how helpful this is for you but if your model is $$y_i = a(u_i) + b(u_i)x_i, \;\; u_i|x_i \sim U[0,1] \;\;\forall i$$ Then $$\text{Var}(y_i\mid x_i) = \text{Var}[a(u_i)\mid x_i] + x_i^2\text{Var}[b(u_i)\mid x_i] +2x_i\text{Cov}[a(u_i),b(u_i)\mid x_i]$$ which tells you that $y_i$ is conditionally heteroskedastic. This can acquire a ...

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You are not replacing $y$ with $e$. We are replacing $y$ with $\hat{y}+\hat{e}$, which is the fitted value of $y$, given by $X\hat{\beta}$, plus the estimated value of the residual, given by $y-X\hat{\beta}$. So, by construction, $$X\hat{\beta}+\hat{e}=X\hat{\beta}+y-X\hat{\beta}=y$$.

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$$\big[ I-W(W'W)^{-1}W'+WC\big]y = \big[ I-W(W'W)^{-1}W'+WC\big](W\beta + e)$$ $$=\big[ I-W(W'W)^{-1}W'+WC\big]W\beta + \big[ I-W(W'W)^{-1}W'+WC\big] e$$ Analyzing the first term, $$\big[ I-W(W'W)^{-1}W'+WC\big]W\beta = W\beta - W(W'W)^{-1}W'W\beta + WCW\beta$$ Simplyfying the inverse, we get $$...= WCW\beta$$ So if $$CW = 0$$ the whole first term is ...

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mean command with pweight gives you mean and sd estimates, which in turn gives you estimate of the coefficient of variation. pctile also takes pweight. It will generate percentiles. kdensity only gives point estimates, not confidence intervals of the density estimates, so I think using fweight instead of pweight is fine. But I'm not certain about this. ...

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