<p>By stnadard OLS regression results, in the simple regression</p>

<p>$$y_t = \alpha + \beta z_t + v_t, \;\;\;t=1,...,T$$</p>

<p>we have that</p>

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

<p>and</p>

<p>$$\hat \alpha = \bar y - \hat \beta \bar z$$</p>

<p>So the residuals are</p>

<p>$$\hat v_t = y_t -(\hat \alpha +\hat \beta z_t)$$</p>

<p>Then, substitute for your particular $y_t$ and $z_t$ in all the above.</p>

CC BY-SA 3.0
2015-11-24T11:21:59.210Z
Just a quick question: Since my explanatory variable is $ \bar x_i $ (averaged over time), what is the value of the average of $\bar x_i$ Do I average over time again? (in which case the average of $\bar x_i$ would be itself). Or do i average over individuals i? (so that the average of $\bar x_i$ is $\bar x$?) I need this for the numerator of $ \hat \beta $
damson_jam
CC BY-SA 3.0
2015-11-24T16:09:31.120Z
@user44394 Wait a minute, I must be missing something here. If $\bar x_i$ is the time average per cross-section, then you cannot really regress each cross section separately "on a constant and $\bar x_i$", because both are constant,and so perfectly colinear, and we cannot obtain different estimates for each one.
Alecos Papadopoulos
CC BY-SA 3.0
2015-11-24T16:39:17.103Z
I completely agree. I think there could be a typo in my assignment sheet. This question is one part of a longer question which asks me to use the Frisch-Waugh theorem to show that $ \hat \beta_{CRE} = \hat \beta_{FE}$.  I found this paper: http://econ.msu.edu/faculty/wooldridge/docs/cre1_r4.pdf   , and I am basically trying to prove proposition 2.1 (bottom page 12) but for balanced panels. The first part of the proof obtains the residuals from the regression of $x_{it} - \theta \bar x_i$ on $(1-\theta) \bar x_i$. So I guess there could be a typo in my assignment sheet here?
damson_jam
CC BY-SA 3.0
2015-11-22T02:00:38.233Z
2015-11-22T02:00:38.233Z
Alecos Papadopoulos
<p>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. </p>

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

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

<p>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} $</p>

<p>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. </p>

<p>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. </p>

<econometrics><paneldata>
Panel data, simple rearrangement?
damson_jam
# Panel data, simple rearrangement?

(Asked by: username_0 on 2015-11-21)

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.

**Tags:** econometrics, paneldata


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## 1 answers

### Answer by username_1 on 2015-11-22. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_0**: Just a quick question: Since my explanatory variable is $ \bar x_i $ (averaged over time), what is the value of the average of $\bar x_i$ Do I average over time again? (in which case the average of $\bar x_i$ would be itself). Or do i average over individuals i? (so that the average of $\bar x_i$ is $\bar x$?) I need this for the numerator of $ \hat \beta $  
> **username_1**: @user44394 Wait a minute, I must be missing something here. If $\bar x_i$ is the time average per cross-section, then you cannot really regress each cross section separately "on a constant and $\bar x_i$", because both are constant,and so perfectly colinear, and we cannot obtain different estimates for each one.  
> **username_0**: I completely agree. I think there could be a typo in my assignment sheet. This question is one part of a longer question which asks me to use the Frisch-Waugh theorem to show that $ \hat \beta_{CRE} = \hat \beta_{FE}$.  I found this paper: http://econ.msu.edu/faculty/wooldridge/docs/cre1_r4.pdf   , and I am basically trying to prove proposition 2.1 (bottom page 12) but for balanced panels. The first part of the proof obtains the residuals from the regression of $x_{it} - \theta \bar x_i$ on $(1-\theta) \bar x_i$. So I guess there could be a typo in my assignment sheet here?  

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