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I'm told that $V_{1,-1}V^{-1}_{-1,-1}V_{-1,1}=\sigma_1^2R_1^2$ but I'm having a hard time seeing it. If I understand correctly, I can see that $V_{1,-1}V^{-1}_{-1,-1} = b_{-1}^T$ but I don't see the former.

Understanding the notation: $V$ is the covariance matrix. The part of the matrix being specified is denoted by the subscripts. A positive number means that it includes that row (if it's the first number) or column (if that's the second number). A negative number means it includes all rows/columns except for that number. Thus $V_{1,-1}$ is the first row excluding the first element, V_{-1,1} excludes the first element of the first column, and $V^{-1}_{-1,-1}$ is the portion of the inverse covariance matrix that excludes the first row and column.

I'm using a couple resources that use slightly different notation. It's probably easiest to become familiar with the problem by pulling up this article. The paragraph after equation (4) on page 1823 claims the result but I'd love some help seeing it and building some intuition

Some Working Knowledge:

$\beta = \frac{\sigma_{12}}{\sigma_1^2} = \frac{\sigma_{12}}{\sigma_1^2}\frac{\sigma_{2}}{\sigma_2} = \frac{\sigma_{12}}{\sigma_1\sigma_2}\frac{\sigma_{2}}{\sigma_1} = \rho \frac{\sigma_2}{\sigma_1}$

Another fact I'm aware of (although I'm not sure how to show it) is $\rho_{\hat y,y}^2 = R^2$.

financial_physician
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  • Is it $R_1^2$ or $1-R_1^2$? – papgeo May 13 '21 at 11:17
  • Are you trying to describe the situation at https://stats.stackexchange.com/questions/140080? – whuber May 13 '21 at 13:02
  • @papgeo Well if you have $V_1,1 - V_{1,-1} V_{-1,-1}^{-1} V_{-1,1}$ then it becomes $\sigma_1^2(1-R^2_{1})$ so that implies what I have written. No? – financial_physician May 13 '21 at 15:21
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    @whuber if it is the same question then I may need help understanding how regression coefficients, correlations, and $R^2$ are so interconnected. I know a mathematical definition for each but it's not obvious to me how partial correlations relate to $\sigma^2R^2$ and how $b$ relates to correlation... I'll edit my question to highlight a bit more where I'm at – financial_physician May 13 '21 at 15:33
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    The conditional variance of the first variable given all others is $\sigma^2_c = V_1 - V_{1,-1}V^{-1}_{-1,-1}V_{-1,1}$, where $V_1$ is the total variance and $\sigma^2_c$ is the unexplained variance. Hence, rearranging that equation, $V_1 - \sigma^2_c = V_{1,-1}V^{-1}_{-1,-1}V_{-1,1}$, but the right side $V_1 - \sigma^2_c = V_1 (1 - \sigma^2_c/V_1) = V_1 (1-R^2)$. – papgeo May 14 '21 at 13:00
  • @papgeo I think the last part for me to understand is why $\sigma_c^2$ is to understand why $V_{1,-1}V^{-1}_{-1,-1}V_{-1,1}$ is the explained variation is $V_{1,1}$ and then I'll be golden. Could you help me see that? – financial_physician May 16 '21 at 17:58

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