# Control for variables: multiple regression vs residuals

I'm trying to understand how to control for variables in linear regression. Say I want to predict $\mathrm{outcome}$ using $x_1$ and control for $x_2$ and $x_3$. The common approach I see is just putting them all in a linear model:

lm(outcome ~ x1 + x2 + x3)


Another approach is to calculate the residuals of $x_1$ and then use them in prediction:

lm(outcome ~ lm(x1 ~ x2 + x3)$residuals)  The coefficient for$x_1$in these methods is the same, but its statistical significance is different. Which approach is better? It seems that from the first we cannot interpret the significance of$x_1$because we have colinearity, is it true? Are there any other things I should notice (assumptions etc) in this analysis? Example of the difference in outcomes: > x2 <- runif(n=100, 0, 10) > x3 <- runif(n=100, 0, 10) > x1 <- 0.3*x2 + 0.6*x3 + 0.1*runif(100, 0, 10) > outcome <- 2 + 0.5*x1 + 0.25*x2 + 0.25*x3 + rnorm(100, 0, 0.5) > summary(lm(outcome ~ x1 + x2 + x3)) Call: lm(formula = outcome ~ x1 + x2 + x3) Residuals: Min 1Q Median 3Q Max -1.38003 -0.27255 0.05079 0.27100 0.88337 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.02248 0.15876 12.739 < 2e-16 *** x1 0.41726 0.15909 2.623 0.0101 * x2 0.27063 0.04612 5.867 6.3e-08 *** x3 0.31244 0.09833 3.177 0.0020 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.4598 on 96 degrees of freedom Multiple R-squared: 0.9377, Adjusted R-squared: 0.9357 F-statistic: 481.3 on 3 and 96 DF, p-value: < 2.2e-16 > summary(lm(outcome ~ lm(x1 ~ x2 + x3)$residuals))

Call:
lm(formula = outcome ~ lm(x1 ~ x2 + x3)$residuals) Residuals: Min 1Q Median 3Q Max -3.7353 -1.0978 0.1281 1.2922 4.4606 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.8806 0.1819 37.835 <2e-16 *** lm(x1 ~ x2 + x3)$residuals   0.4173     0.6292   0.663    0.509
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.819 on 98 degrees of freedom
Multiple R-squared:  0.004467,  Adjusted R-squared:  -0.005691
F-statistic: 0.4398 on 1 and 98 DF,  p-value: 0.5088