# Panel data, simple rearrangement?

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.

## migrated from math.stackexchange.comNov 22 '15 at 10:57

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