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Consider the linear regression model with design matrix $X = [x_1 \mid x_2 \mid \cdots\mid x_p] \in\mathbb R^{n\times p}$. Assume that $x^T_j 1_n = 0$, for $j = 1, \ldots , p$, that is, the columns of $X$ are centered. (Here, and elsewhere $1_n \in\mathbb R^n$ is the vector of all-ones of length $n$). Note that the model does not include an intercept.

Assume that:

$$ \frac{1}{n}X^TX=\begin{bmatrix}I_{p-1} & z\\z & a^2\end{bmatrix}, z\in R^{p-1}, a^2\in R $$ where $a^2$ and $z$ are given quantities. Assume that $a^2=10\|z\|^2$.

Let R$^2_j$ be the $R^2$ obtained when regressing x$_j$ onto the rest of covariates $\{x_1, \ldots , x_{j−1}, x_{j+1}, \ldots , x_p\}.$ Find R$^2_p$, that is, the coefficient of determination resulting from regressing $x_p$ onto $\{x_1, \ldots , x_{p−1}\}.$

I'm not really sure how to convert from one $\mathbb R^2$ to another $\mathbb R^2$. Hopefully someone can point me in the right direction or towards a formula I can use, as I can't seem to find much helpful information.

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  • $\begingroup$ Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. $\endgroup$ – gung - Reinstate Monica Jun 10 '17 at 22:41
  • $\begingroup$ Edited. This is a homework question. I can't find the formula I should be using in the textbook we have and was hoping someone could point me in the right direction. $\endgroup$ – Lebron James Jun 10 '17 at 22:44
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Steps:

1: Reshape $X$ matrix: $X_{-p} = (x_1,\dots,x_{p-1})$ and $Y=(x_p)$.

2: Find total sum of square of $Y$.

3: Find $\hat\beta = (X_{-p}'X_{-p})^{-1}X_{-p}'Y$

4: Find model SS, which is $\hat Y' \hat Y$ given $\bar Y = 0 $

5: The ratio of model SS to total SS is the $R^2$ that you need.

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