$Y_i = beta_0 + beta_1 X_i + u_i, i = 1,2,ldots,N$. Model is heteroskedastic.

I am using an iterative version of weighted least squares, in which I iteratively perform weighted regression, then fit the resulting residuals using some nonparametric approach, and then get back to weighted regression and so on.

Assuming that the process converges, yielding estimates $hat{beta}_0,hat{beta}_1$, how do I generate an estimate of their variances?

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#### Best Answer

The regression estimates for a weighted regression model is $$hat{beta} = (X'WX)^{-1}X'WY$$

Now, $Var(hat{beta}) = (X'WX)^{-1}X'W Var(Y)W'X(X'WX)^{-1}$

Assuming $Var(u_i) = sigma_i^2$ and $Cov(u_i,u_j)=0$ for $ine j$ $$Var(Y) = diag(sigma_i^2)$$

You can use the below formulation to calculate the variance of the estimates,$$Var(hat{beta}) = (X'WX)^{-1}X'W diag(sigma_i^2) W'X(X'WX)^{-1}$$

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