This exercise 4.14 from the text Regression Analysis by Example 5E by Chatterjee and Hadi. The data set can be obtained using the following code in R:

data = fread(paste0("http://www1.aucegypt.edu/faculty/hadi/RABE5/Data5/", "P128.txt"))

The part that is causing confusion for me is bolded.

Consider fitting the model $Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \epsilon$ to the data set in Table 4.8. Now let $u$ be the residuals obtained from regressing $Y$ on $X_1$. Also, let $X_2$ and $v$ be the residuals obtained from regressing $X_3$ on $X_1$. Show (or verify using the data set in Table 4.8 as an example) that:

(a) $\hat \beta_3 = \sum_{i=1}^{n} u_iv_i / \sum_{i=1}^{n} v_i^2$

(b) the standard error of $\hat \beta_3$ is $\hat \sigma / \sqrt{\sum_{i=1}^n v_i^2}$

How can we let a predictor be residuals?

I am wondering if it should read as follows: "Also, let $v$ be the residuals obtained from regressing $X_3$ on $X_1$ and $X_2$".


1 Answer 1


Yes, this appears to be a typo. We can let residuals (from one model) be predictors in other models, but you are correct that the terminology, as presented in the problem, is incorrect. Furthermore, your suggested correction is appropriate in this context. Using R syntax:

v <- lm(X3~X1+X2,data)$residuals

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