Derivation of standard error of regression estimate with degrees of freedom I am taking a course of Econometrics:
I need help to understand as to how do we arrive at the formula for standard error of regression $$\hat{\sigma}^2=\frac{\sum{e_i^2}}{n-k}.$$
I understand the bessel's correction required to remove the bias inherent in sample variance. The proof being available at \href{https://en.wikipedia.org/wiki/Bessel's_correction#Proof_of_correctness_.E2.80.93_Alternate_2}{Bessels Correction Proof of Correctness}. 
I also found \href{https://stats.stackexchange.com/questions/68766/standard-deviation-of-error-in-simple-linear-regression}{Standard deviation of error in simple linear regression}
\href{https://stats.stackexchange.com/questions/85943/how-to-derive-the-standard-error-of-linear-regression-coefficient}{How to derive the standard error of linear regression coefficient}
But I could not find the proof for the above expression (standard error of regression estimate).
I tried to open the equation on the lines of Bessels Correction proof.
$$e_i=\text{Total SS}- \text{Explained SS}$$
Then I try to expand the Explained sum of squares term, but I got stuck at 
$$ \sum _{i=1}^n \operatorname {E} \left((\beta\mathbf{ X}-\bar{y} )^2 \right) = \beta^2 E(x^2)-2\beta\bar{xy}+E(\bar{y}^2)$$
I don't know how to proceed. Can anyone please help ?
Then I read this :
The term "standard error" is more often used in the context of a regression model, and you can find it as "the standard error of regression". It is the square root of the sum of squared residuals from the regression - divided sometimes by sample size n (and then it is the maximum likelihood estimator of the standard deviation of the error term), or by $n−k$ ($k$ being the number of regressors), and then it is the ordinary least squares (OLS) estimator of the standard deviation of the error term.
on \href{https://stats.stackexchange.com/questions/73390/standard-error-vs-standard-deviation-of-sample-mean}{Standard Error vs. Standard Deviation of Sample Mean}
Can anyone suggest a textbook where I can read about these derivations in more details ?
 A: Here's one way.  This will work only if you understand matrix algebra and the geometry of $n$-dimensional Euclidean space.
The model says $y_i = \alpha_0 + \sum_{\ell=1}^k \alpha_\ell x_{\ell i} + \varepsilon_i, \quad i=1,\ldots,n $ where


*

*$y_i$ and $x_{\ell i}$ are observed;

*The $\alpha$s are not observed and are to be estimated by least squares;

*The $\alpha$s are not random, i.e. if a new sample with all new $x$s and $y$s is taken, the $\alpha$ will not change;

*The $x$s are in effect treated as not random.  This is justified by saying we're interested in the conditional distribution of the $y$s given the $x$s.  The $y$s are random only because the $\varepsilon$s are;

*The $\varepsilon$s are not observed. The have expected value $0$ and variance $\sigma^2$ and are uncorrelated.  These assumptions are weaker than those that normality and independence.


The $n\times(k+1)$ "design matrix" is
$$
X= \begin{bmatrix} 1 & x_{11} & \cdots & x_{k1} \\ \vdots & \vdots & & \vdots \\
1 & x_{1n} & \cdots & x_{kn} \end{bmatrix}
$$
with independent columns and typically $n\gg k$.
The $(k+1)\times 1$ vector of coefficients to be estimated is
$$
\alpha= \begin{bmatrix} \alpha_0 \\ \alpha_1 \\ \vdots \\ \alpha_k \end{bmatrix}.
$$
The model can then be written as $Y= X\alpha+\varepsilon$, where $Y, \varepsilon \in\mathbb R^{n\times 1}$.  Then $Y$ has expected value $X\alpha\in\mathbb R^{n\times 1}$ and variance $\sigma^2 I_n\in\mathbb R^{n\times n}$.
The "hat matrix" is $H = X(X^T X)^{-1} X^T$, an $n\times n$ matrix of rank $k+1$.  The vector $\widehat Y = HY$ is the orthogonal projection of $Y$ onto the column space of $X$.  It is also $\widehat Y=HY = X\widehat\alpha$, where $\widehat\alpha$ is the vector of least-squares estimates of the components of $\alpha$.
The residuals are $\widehat\varepsilon_i = e_i = Y_i-\widehat Y_i = Y_i-(\widehat\alpha_0 + \sum_{\ell=1}^k \widehat\alpha_\ell x_{\ell i})$.  These are observable estimates of the unobservable errors.  The vector of residuals is
$$
\widehat\varepsilon = e = (I-H)Y.
$$
This has expected value $(I-H)\operatorname{E}(Y) = (I-H)X\alpha = 0$.
We seek
\begin{align}
& \operatorname{E}(\|\widehat\varepsilon\|^2) = \operatorname{E}(\|e\|^2) \\[10pt]
= {} & \operatorname{E} ( \Big((I-H)Y\Big)^T \Big((I-H)Y\Big)) \\[10pt]
= {} & \operatorname{E} (Y^T (I-H) Y) \qquad \text{since } (I-H)^T = I-H = (I-H)^2. \text{ (Check that.)}
\end{align}
We've projected $Y$ onto the $(n-(k+1))$-dimensional column space of $I-H$.  The expected value of the projection is $0$.
I claim the variance of the projection is just $\sigma^2$ times the identity operator on that $(n-(k+1))$-dimensional space.  The reason for that is that $I-H$ is itself the identity operator on that $(n-(k+1))$-dimensional space, which is the orthogonal complement of the column space of $X$.
So it's as if we have a random vector $w$ in $(n-(k+1))$-dimensional space with expected value $0$ and variance $\sigma^2 I_{(n-(k+1))\times(n-(k+1))}$, and we're asking what $\operatorname{E}(\|w\|^2)$ is.  And that is $\sigma^2(n-(k+1))$.
Hence the expected value of the sum of squares of residuals (which is the "unexplained" sum of squares) is $\sigma^2(n-(k+1))$. 
