The residuals from 2SLS may not be homoskedastic. If the assumption of constant variance is violated, weighted two-stage least square (W2SLS) can give superior estimates. However, it should be stressed that better results will depend on whether the weight can mimic the problem of heteroskedasticity. A poor modeling of covariance matrix can worsen the problem. So, modelling of residual is the key to have better estimates, and that can be inferred from diagnostic tests.

The methods of W2SLS is simply the application of 2SLS on weighted variables. All variables, including the instrument, are multiplied by the weight. Then, the usual 2SLS procedure is applied: (1) regress weighted explanatory variables on weighted instruments to obtain the fitted values, and (2) use the weighted fitted values to replace the explanatory variables in the second-stage regression. All of these steps are taken care by the apps so you do not have to do them manually. All you have to do is follow the menu and instructions, then run the procedure.

## W2SLS estimation

You can run W2SLS in i-Regand as simple as:

Step 1. Tap on the main menu and select W2SLS

Step 2. Define a dependent variable, explanatory variables, and instruments, a weighting variable.

Step 3. Run and see the results.

The output shows regression coefficients (from the second stage) with corresponding t statistic (based on weighted residuals). As in the WLS, both weighted and unweighted statistics are presented. The former corresponds to the weighted residuals and the latter corresponds to the unweighted residuals.

## Diagnostic tests

Various diagnostic tests can be done right after IV/2SLS estimation. You have to be aware that each test is carried out somewhat differently from the same test within OLS or WLS. Notice the header of the test results to get a sense of what is the apps doing for you. The Harvey test is taken as an example:

Step 1. Tap on the auxiliary menu, select the Harvey test.