21 questions linked to/from Why is RSS distributed chi square times n-p?
5k views

### Linear regression: Unbiased estimator of the variance of outputs [duplicate]

I'm having trouble understanding something from the linear regression chapter of Elements of Statistical Learning. We have a fixed $N\times p$ matrix $\mathbf{X}$ ($N$ inputs with $p$ predictors) ...
584 views

### Residual Sum of Squares degrees of freedom intuition [duplicate]

Let RSS = Residual sum of squares $= \sum (y_i - \hat{y}_i)^2$. Without proof, $\frac{RSS}{\sigma^2} \sim \chi^2_{n-2}$. I do not quite understand why the DoF is $n-2.$ Could someone explain?
647 views

15 views

12k views

### Distribution of sum of squares error for linear regression?

I know that distribution of sample variance $$\sum\frac{(X_i-\bar{X})^2}{\sigma^2}\sim \chi^2_{(n-1)}$$ $$\sum\frac{(X_i-\bar{X})^2}{n-1}\sim \frac{\sigma^2}{n-1}\chi^2_{(n-1)}$$ It's from the ...
6k views

### Why divide by $n-2$ for residual standard errors

I was just watching a lecture on statistics and someone was calculating something called the residual standard error. It looked a lot like finding the average of the square of the residuals, the ...
549 views

### Easy proof of $\sum_{i=1}^n \left(Z_i - \bar{Z}\right)^2 \sim \chi^2_{n-1}$?

Let $Z_1,\cdots,Z_n$ be independent standard normal random variables. There are many (lengthy) proofs out there, showing that  \sum_{i=1}^n \left(Z_i - \frac{1}{n}\sum_{j=1}^n Z_j \right)^2 \sim \...
12k views

### What is “estimated unbiased variance of the error term”?

Disclosure: This is a homework question. I have fit a multiple linear regression model in eviews, and I am asked to calculate "estimated unbiased variance of the error term, i.e., $\hat\sigma^2$". ...
7k views

### Proof that regression residual error is an unbiased estimate of error variance

Consider the least squares problem $Y=X\beta +\epsilon$ while $\epsilon$ is zero mean Gaussian with $E(\epsilon) = 0$ and variance $\sigma^2$. I need to prove that $\frac{V(\hat{\beta})}{N-(n+m)}$ ...
2k views

### Why use the F distribution and F test?

I don't understand why in the F test we calculate the ratio between MSE between subject and MSE within subject. As far as I know, this is due because we want to use the F distribution, which is a rate ...
### Deriving SSE of Simple Linear Regression is $\chi^{2}$
As per my notes, the key step in the proof that the sum of squares of residuals in regression is $\chi^{2}_{n-2}$ is the fact that $e_{i} = y_{i} - \hat{y}_{i}$ has a mean 0 and variance $\sigma^{2}$. ...
### Simple Linear Regression: how does $\Sigma\hat{u_i}^2/\sigma^2$ follow chi squared distribution with df (n-2)?
My question is, as far as i am aware, 1. the residuals($\hat{u_i}$) are not independent of one another 2. the variance of ith residual is \$\sigma\{(1-1/n-(X_i-\overline{X})/\Sigma(X_i-\overline{X})^2\}...