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Question:

Let $X=(X_1,X_2,\ldots,X_p)'$ be a random vector. What can you say about the change of multiple correlation coefficient of $X_1$ on $X_2,X_3,\ldots,X_p$ if each $X_i$ is multiplied by $\sqrt i$ ($i=1,2,\ldots,p$)? Justify your answer.

I know the formula of multiple correlation coefficient is;

$$\rho_{1\cdot23\ldots p}=\sqrt{1-\frac{|R|}{R_{11}}}\,,$$

where $|R|$ is the determinant of the correlation matrix $R$ and $R_{11}$ is the cofactor of the $(1,1)$ element of $R$.

But I really cannot figure out, how to deal with this variable quantity $\sqrt i$.

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    $\begingroup$ For each $X_i,$ $\sqrt{i}$ is not variable: it's constant. If that doesn't make the answer immediately clear, consider the case $p=2$ in detail. $\endgroup$
    – whuber
    Nov 26 '18 at 14:06
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    $\begingroup$ Thank you..As per your direction, I have calculated and found out that such a change, leaves Correlaion(Xi, Xj) the same as after multiplying sqrt(i)..So the multiple correlation coefficient seems to remain the same as before.. $\endgroup$
    – P db
    Nov 26 '18 at 16:38
  • $\begingroup$ Right. Identify why that is--how you do it depends on your definition of correlation and/or the formulas you use--and see whether you can generalize your observation. $\endgroup$
    – whuber
    Nov 26 '18 at 17:28
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Let $Y_i=\sqrt iX_i$ for all $i$, which directly gives you $$\operatorname{Corr}(Y_i,Y_j)=\frac{\sqrt i\sqrt j\operatorname{Cov}(X_i,X_j)}{\sqrt{i\operatorname{Var}(X_i)}\sqrt{j\operatorname{Var}(X_j)}}=\operatorname{Corr}(X_i,X_j)\quad\forall\,i,j$$

This means the correlation matrix of $(Y_1,\ldots,Y_p)$ is identical to that of $(X_1,\ldots,X_p)$, whence the multiple correlation coefficient of $Y_1$ on $Y_2,\ldots,Yp$ is also the same as that of $X_1$ on $X_2,\ldots,X_p$ using the formula in your post.

We can also possibly justify this by recalling that the multiple correlation coefficient of $X_1$ on $X_2,\ldots,X_p$ is the maximum possible correlation between $X_1$ and any linear function of $X_2,\ldots,X_p$. Intuitively, multiplying each $X_i$ by a constant should not change this correlation.

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