## 5.3 Weighted Least Squares

1. Estimate the following equation using OLS

$y_i = \mathbf{x}_i \beta + \epsilon_i$ and obtain the residuals $$e_i=y_i -\mathbf{x}_i \hat{\beta}$$

1. Transform the residual and estimate the following by OLS,

$ln(e_i^2)= \mathbf{x}_i\gamma + ln(v_i)$

and obtain the predicted values $$g_i=\mathbf{x}_i \hat{\gamma}$$

1. The weights will be the untransformed predicted outcome,

$\hat{\sigma}_i =\sqrt{exp(g_i)}$

1. The FWLS (Feasible WLS) estimator is obtained as the least squared estimated for the following weighted equation

$(1/\hat{\sigma}_i)y_i = (1/\hat{\sigma}_i) \mathbf{x}_i\beta + (1/\hat{\sigma}_i)\epsilon_i$

Properties of the FWLS

• The infeasible WLS estimator is unbiased under A1-A3 for the unweighted equation.
• The FWLS estimator is NOT an unbiased estimator.
• The FWLS estimator is consistent under A1, A2, (for the unweighted equation), A5, and $$E(\mathbf{x}_i'\epsilon_i/\sigma^2_i)=0$$
• A3a is not sufficient for the above equation
• A3 is sufficient for the above equation.
• The FWLS estimator is asymptotically more efficient than OLS if the errors have multiplicative exponential heteroskedasticity.
• If the errors are truly multiplicative exponential heteroskedasticity, then usual standard errors are valid
• If we believe that there may be some mis-specification with the multiplicative exponential model, then we should report heteroskedastic robust standard errors.