<p>I need to prove the following statement, which is <span class="math-container">$\hat{\beta}_{total} = W(x)\hat{\beta}_{within} + (I-W(x))\hat{\beta}_{between}$</span>,</p>
<p>where <span class="math-container">$\hat{\beta}_{within}$</span> is the fixed effect estimator(i.e. the estimator of the linear regression of <span class="math-container">$y_{it} - \bar{y_i}$</span> on <span class="math-container">$x_{it} - \bar{x_i}$</span>), <span class="math-container">$\hat{\beta}_{between} $</span> is the estimator of the linear regression of <span class="math-container">$\bar{y_i}$</span> on <span class="math-container">$\bar{x_i}$</span>, and <span class="math-container">$\hat{\beta}_{total}$</span> is just the regression of y on x, pretending it’s not a panel data. <span class="math-container">$W(x)$</span> is a matrix-valued function of the data <span class="math-container">$\{x_{it}\}_{i=1, t=1}^{N, T}$</span>, and the question need us to find an explicit expression for <span class="math-container">$W(x)$</span> as well.</p>
<p>In the previous part, I have already proved that
<span class="math-container">$$
\sum_{i=1}^N\sum_{t=1}^T (x_{it} - \bar{x_i})’ (x_{it} - \bar{x_i}) + T\sum_{i=1}^N \bar{x_i}’\bar{x_i} = \sum_{i=1}^N\sum_{t=1}^Tx_{it}’x_{it}
$$</span></p>
<p>And</p>
<p><span class="math-container">$$
\sum_{i=1}^N\sum_{t=1}^T (x_{it} - \bar{x_i})’ (y_{it} - \bar{y_i}) + T\sum_{i=1}^N \bar{x_i}’\bar{y_i} = \sum_{i=1}^N\sum_{t=1}^Tx_{it}’y_{it}
$$</span></p>
<p>I don’t know how to connect these two parts to prove the actual statement. The matrix algebra is so confusing, and I don’t know what <span class="math-container">$W(x)$</span> can be. Any help is welcome. Thanks!</p>

CC BY-SA 4.0
2021-03-09T08:58:24.887Z
"*I don’t know what $W(x)$ can be.*" If you **have to** prove a statement where you do not understand the notation, asking for clarification from the lecturer or rereading the textbook is probably a good idea.
Giskard
CC BY-SA 4.0
2021-03-09T09:02:58.913Z
@Giskard I think it’s part of the proof to deduct $W(x)$.
Gordon Ji
CC BY-SA 4.0
2021-03-09T09:11:23.737Z
If you only "*think*" so, asking your lecturer is still a good idea? And if you are right, then the objective is not to "prove" the statement but to find $W(x)$.
Giskard
<econometrics><paneldata>
Econometrics - Panel Data: Relationship between total, within and between
Gordon Ji
# Econometrics - Panel Data: Relationship between total, within and between

(Asked by: username_0 on 2021-03-09)

I need to prove the following statement, which is $\hat{\beta}\_{total} = W(x)\hat{\beta}\_{within} + (I-W(x))\hat{\beta}\_{between}$,

where $\hat{\beta}\_{within}$ is the fixed effect estimator(i.e. the estimator of the linear regression of $y\_{it} - \bar{y\_i}$ on $x\_{it} - \bar{x\_i}$), $\hat{\beta}\_{between} $ is the estimator of the linear regression of $\bar{y\_i}$ on $\bar{x\_i}$, and $\hat{\beta}\_{total}$ is just the regression of y on x, pretending it’s not a panel data. $W(x)$ is a matrix-valued function of the data $\{x\_{it}\}\_{i=1, t=1}^{N, T}$, and the question need us to find an explicit expression for $W(x)$ as well.

In the previous part, I have already proved that
$$
\sum\_{i=1}^N\sum\_{t=1}^T (x\_{it} - \bar{x\_i})’ (x\_{it} - \bar{x\_i}) + T\sum\_{i=1}^N \bar{x\_i}’\bar{x\_i} = \sum\_{i=1}^N\sum\_{t=1}^Tx\_{it}’x\_{it}
$$

And

$$
\sum\_{i=1}^N\sum\_{t=1}^T (x\_{it} - \bar{x\_i})’ (y\_{it} - \bar{y\_i}) + T\sum\_{i=1}^N \bar{x\_i}’\bar{y\_i} = \sum\_{i=1}^N\sum\_{t=1}^Tx\_{it}’y\_{it}
$$

I don’t know how to connect these two parts to prove the actual statement. The matrix algebra is so confusing, and I don’t know what $W(x)$ can be. Any help is welcome. Thanks!

**Tags:** econometrics, paneldata
> **Comments:**
> **username_1**: "*I don’t know what $W(x)$ can be.*" If you **have to** prove a statement where you do not understand the notation, asking for clarification from the lecturer or rereading the textbook is probably a good idea.  
> **username_0**: @Giskard I think it’s part of the proof to deduct $W(x)$.  
> **username_1**: If you only "*think*" so, asking your lecturer is still a good idea? And if you are right, then the objective is not to "prove" the statement but to find $W(x)$.  

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## 0 answers

