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.