Panel data, simple rearrangement?

Question

Consider the Correlated Random Effects model $y_{it} = \alpha + x_{it}\beta + \bar x \gamma + w_i + \epsilon_{it} $ where $x_{it}$ is a scalar explanatory variable.

The correlated random effects GLS estimator $ \hat \beta_{CRE} $ is the OLS estimator of $\beta$ in the quasi-demeaned regression

$\tilde y_{it} = \delta + \tilde x_{it} \beta + \bar x_i \rho + u_{it} $,

where $\tilde y_{it} = y_{it} - \theta \bar y_i , \tilde x_{it} = x_{it} - \theta \bar x_i $ and $\theta = 1 - (\sigma^2_\epsilon/ (\sigma^2_\epsilon + T\sigma^2_w))^{1/2} $

Question: I need to show that the residuals from the regression of $x_{it} - \bar x_i $ on a constant and $\bar x_i $ is just $x_{it} - \bar x_i $ itself.

Attempt: Regress $x_{it} - \bar x_i = \alpha + \bar x_{i} + \tilde r_{it} $ rearrange to get the residuals, $\tilde r_{it} = (x_{it} - \bar x_i) - (\alpha + \bar x_{i})$ I'm not sure how to proceed.

Answer 1

By stnadard OLS regression results, in the simple regression

$$y_t = \alpha + \beta z_t + v_t, \;\;\;t=1,...,T$$

we have that

$$\hat \beta = \frac {\sum_{i=1}^T (z_t - \bar z)(y_t-\bar y)}{\sum_{i=1}^T (z_t - \bar z)^2}$$

and

$$\hat \alpha = \bar y - \hat \beta \bar z$$

So the residuals are

$$\hat v_t = y_t -(\hat \alpha +\hat \beta z_t)$$

Then, substitute for your particular $y_t$ and $z_t$ in all the above.