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In section 3.2.3 of Elements of Statistical Learning (Link), there's this statement on multiple regression coefficients on Page 54

we have shown that the $j^{th}$ multiple regression coefficient is the univariate regression coefficient of $\mathbf{y}$ on $\mathbf{x_{j·012...(j−1)(j+1)...,p}}$, residual after regressing $\mathbf{x}_j$ on $\mathbf{x}_0, \ldots, \mathbf{x_{j-1}}, \mathbf{x}_{j+1}, \ldots, \mathbf{x}_p$

If I go by the definition of "regress $\mathbf{b}$ on $\mathbf{a}$" (introduced on Page 53), the statement above would mean that $j^{th}$ multiple regression coefficient is given by

$$ \hat{\beta}_j = \frac{<\mathbf{x}_j, \mathbf{r} >}{<\mathbf{r}, \mathbf{r}>} $$ where $\mathbf{r} = \mathbf{x}_j - \frac{<\mathbf{x}_j, \mathbf{x}_0>}{<\mathbf{x}_0, \mathbf{x}_0>} \mathbf{x}_0 - \ldots - \frac{<\mathbf{x}_j, \mathbf{x}_{j-1}>}{<\mathbf{x}_{j-1}, \mathbf{x}_{j-1}>} \mathbf{x}_{j-1} - \frac{<\mathbf{x}_{j}, \mathbf{x}_{j+1}>}{<\mathbf{x}_{j+1}, \mathbf{x}_{j+1}>} \mathbf{x}_{j+1} - \ldots - \frac{<\mathbf{x}_{j}, \mathbf{x}_{p}>}{<\mathbf{x}_{p}, \mathbf{x}_{p}>} \mathbf{x}_{p} $.

Edit: I understand that the expression for $\hat{\beta}_j$ is wrong. However, I would like to understand what am I interpretig wrong in the notation "residual after regressing $\mathbf{x}_j$ on $\mathbf{x}_0, \ldots, \mathbf{x_{j-1}}, \mathbf{x}_{j+1}, \ldots, \mathbf{x}_p$"?

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  • $\begingroup$ See stats.stackexchange.com/a/46508/919 and stats.stackexchange.com/questions/17336, inter alia. I hope they clarify how your formula isn't quite right. $\endgroup$
    – whuber
    Mar 22 '21 at 17:39
  • $\begingroup$ I had the feeling that it's not right. Trying to understand what the statement I quoted from the textbook means instead. Specifically, what would "residual after regressing $ \mathbf{𝐱}_𝑗$ on $\mathbf{𝐱}_0,…,\mathbf{𝐱}_{𝐣−1}, \mathbf{𝐱}_{𝑗+1},…,\mathbf{𝐱}_{𝑝}$" look like as an expression if not what I wrote. $\endgroup$
    – ethelion
    Mar 22 '21 at 17:46
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    $\begingroup$ The notation means you have to perform a multivariate regression of $x_j$ against all the other explanatory variables, then replace $x_j$ by its residuals: that is, you need to "take out" the components of all the other variables from $x_j.$ Your notation describes $p-1$ separate univariate regressions of $x_j$ against each other variable. $\endgroup$
    – whuber
    Mar 22 '21 at 18:29
  • $\begingroup$ @whuber This makes sense and answers my question. I think I should update the question to reflect that I was confused more by the notation and not so much by what the regression coefficient will look like. Also, I would like to mark your comment above as the answer. How do I do that? $\endgroup$
    – ethelion
    Mar 23 '21 at 0:50

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