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Weighted Least Squares is different to Ordinary Least Squares and is used when the assumption of constant variance is violated. You need to read up on the difference but your model assumes that the error variance is directly related to X2. The Weighted Least Squares estimate $\hat{\beta}_{w}$ is obtained by applying OLS to the transformed model:

$$\sqrt{N_tY_t}=\sqrt{N_t}X^{'}_t\beta +v_t$$

The model in both cases fits poorly and you may want to think again about it.

Weighted Least Squares is different to Ordinary Least Squares and is used when the assumption of constant variance is violated. You need to read up on the difference but your model assumes that the error variance is directly related to X2. The Weighted Least Squares estimate $\hat{\beta}_{w}$ is obtained by applying OLS to the transformed model:

$$\sqrt{N_tY_t}=\sqrt{N_t}X^{'}_t\beta $$\sqrt{N_t}Y_t=\sqrt{N_t}X^{'}_t\beta +v_t$$

The model in both cases fits poorly and you may want to think again about it.