2
$\begingroup$

$Y_i = \beta_0 + \beta_1 X_i + u_i, i = 1,2,\ldots,N$. 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?

$\endgroup$
-1
$\begingroup$

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 $i\ne j$ $$Var(Y) = diag(\sigma_i^2)$$

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

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.