In OLS, does the uncorrelatedness between regressors and residuals require a constant? I'm reading this PDF.
It shows how to obtain the OLS estimator and its properties.
It is said that from the normal equations we obtain $X' e = 0$.
Where $X$ is the design matrix and $e$ is the vector of residuals.
Then, at page 4, it claims the following property: "The observed values of X are uncorrelated with the residuals."
Proof: Indicating the regressors by $x_k$, $X' e = 0$ implies $x'_k e = 0$.
So far so good.
Then it goes on to say "In other words, each regressor has zero sample correlation with the residuals."
I didn't understand that.
By definition of sample correlation:
$$r_{x_pe} = \frac{\sum_{i=1}^n x_{pi} e_i - n \bar{x_p} \bar{e}}{n s^{'}_{x_p} s^{'}_e}$$
We have proven that $\sum_{i=1}^n x_{pi} e_i = 0$. But then there is another term at the numerator.
Except if the mean of the residuals equal 0. That is, if $\bar{e}=0$.
But that would require the constant, as shown in point 3 of page 4.
 A: You are right.
Maybe because most regressions do contain a constant, the property $X'e=0$ (often called, more precisely, "orthogonality") and the terminology "uncorrelatedness" are often used interchangeably, when they do amount to the same thing only if the regression contains a constant (or, more precisely, if the residuals have mean zero, which can also be the case if the regressors can be linearly combined into a constant, say with an exhaustive set of dummies).
A little numerical illustration:
n <- 10
y <- rnorm(n)
x <- rnorm(n)

regwcst <- lm(y~x)
regwocst <- lm(y~x-1)
d1 <- c(rep(1,5), rep(0,5)) # two exhaustive dummies
d2 <- 1-d1
regwdumm <- lm(y~x-1+d1+d2)

> crossprod(x, resid(regwcst))  # all numerically zero
              [,1]
[1,] -2.081668e-17

> crossprod(x, resid(regwdumm))
              [,1]
[1,] -1.249001e-16

> crossprod(x, resid(regwocst))
             [,1]
[1,] 1.804112e-16

> cor(x, resid(regwcst))        # numerically zero
[1] -2.721791e-17

> cor(x, resid(regwocst))       # not numerically zero
[1] 0.01718539

> cor(x, resid(regwdumm))       # numerically zero

