I was instructed on an assignment to "calculate variance of the residuals obtained from your fitted equation." It was a simple linear regression, so I thought "ok, it's just the sum of squared residuals divided by $(n - 2)$ since it lost two degrees of freedom from estimating the intercept and slope coefficient." Wrong. He didn't want me to estimate the residual variance. Instead, I was told that I was supposed to divide it by $(n - 1)$. I don't understand why this would be done.
Variance can only be calculated around a parameter, and it is the summed deviations from that (or those) parameters divided by the degrees of freedom resulting from the sample size and the constraints of the parameter. If we're descriptively calculating the variance of one variable in a single population, the parameter would be a mean, so the degrees of freedom would be $(n - 1)$. I understand that, and I understand why it's true. But if the parameter is a "fitted equation" referring to a simple linear model, I don't see any way around using two parameters and therefore having $(n - 2)$ degrees of freedom when discussion variance of the residuals.
Can someone enlighten me as to what I'm misunderstanding, and what the difference between "variance of residuals" and "estimated residual variance" are?