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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!

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    $\begingroup$ "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. $\endgroup$
    – Giskard
    Mar 9 at 8:58
  • $\begingroup$ @Giskard I think it’s part of the proof to deduct $W(x)$. $\endgroup$
    – Gordon Ji
    Mar 9 at 9:02
  • $\begingroup$ 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 $W(x)$. $\endgroup$
    – Giskard
    Mar 9 at 9:11

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