# OLS with nested into the regressor

Given the following model:

$$y_t = \alpha x_t + \epsilon_t~~~~,~~~\epsilon_t \sim NID(0,\sigma^2) ~\text{and}~~x_t = y_t + z_t$$

Given that $z_t$ is a non covariate variable with $\epsilon_t$, how is it possible to derive the following unbiased OLS estimator of $\alpha$:

$$1 - \dfrac{\sum_{t=1}^T z_t^2}{\sum_{t=1}^T z_t y_t + \sum_{t=1}^T z_t^2}$$

Substitute $x_t$ in the model to obtain $y_t=\alpha y_t + \alpha z_t + \varepsilon_t$. Manipulating, $y_t = \frac{\alpha}{1-\alpha}z_t + \frac{1}{1-\alpha}\varepsilon_t$. Rename the errors $\delta_t=\frac{1}{1-\alpha}\varepsilon_t$, then $\delta_t\sim NID(0,\tau^2)$, where $\tau^2=\frac{\sigma^2}{(1-\alpha)^2}$, so homoscedasticidy still holds. Rename $\beta=\frac{\alpha}{1-\alpha}$. You now have the model $y_t=\beta z_t+\delta_t$, with the usual assumptions. The OLS estimator of $\beta$ is $$\hat{\beta}=\frac{\sum_{t=1}^T z_t y_t}{\sum_{t=1}^T z_t^2}.$$ Hence, $$\frac{\hat{\alpha}}{1-\hat{\alpha}}=\frac{\sum_{t=1}^T z_t y_t}{\sum_{t=1}^T z_t^2}.$$ Solving for $\hat{\alpha}$, $$\hat{\alpha}=\frac{\sum_{t=1}^T z_t y_t}{\sum_{t=1}^T z_t y_t+\sum_{t=1}^T z_t^2}=\frac{\sum_{t=1}^T z_t y_t + \sum_{t=1}^T z_t^2 - \sum_{t=1}^T z_t^2}{\sum_{t=1}^T z_t y_t+\sum_{t=1}^T z_t^2},$$
so $$\hat{\alpha}=1-\frac{\sum_{t=1}^T z_t^2}{\sum_{t=1}^T z_t y_t+\sum_{t=1}^T z_t^2}.$$
• Multiply both sides by $1-\hat{\alpha}$ and by $\sum_{t=1}^T z_t^2$ and rearrange terms. And sorry for mixing the $\hat{\alpha}$ notation and the $a$ notation, let me correct it. – Anna SdTC Mar 14 '17 at 9:02