A question about the change in value of multiple correlation coefficient on multiplying each value with a variable quantity

Question

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 $\sqrt i$.

Answer 1

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.