Deriving OLS estimator In my course on linear models we derived the OLS estimator by minimizing the residuals $F(\phi) = (Y-X\phi)'(Y-X\phi)$. However there is one step that I do not understand: to find the minimum over all $\phi \in \mathbb{R}^k$ our professor wrote down the following
$$\frac{\partial F(\phi)}{\partial\phi} = \sum_{t = 2}^n 2(y_t-(x_{t1} \phi_1+...+x_{tk}\phi_k))(-x_{ti}) = 0$$  for $i = 1,...,k.$
For me, there are a two questions here:
(1) Why does the sum go from $t = 2$ to $n$ as opposed to $t = 1$?
(2) How does he differentiate the function $F(\phi)$ with respect to a whole vector and get a real number insted of another vector? (In other words: How does this differential work?)
 A: It's just a couple of typos:

*

*Yes, it should start from $t=1$


*Apparently, the derivative is with respect to $\phi_i$ (I guessed it from the multiplicand $x_{ti}$)
A: 
How does he differentiate the function $F(\phi)$ with respect to a whole vector and get a real number insted of another vector? (In other words: How does this differential work?)

Often this means a gradient, i.e.,
$$\frac{\partial F(\phi)}{\partial \phi}\Leftrightarrow \nabla F(\phi)=\left(
\frac{\partial F(\phi)}{\partial \phi_1},...,
\frac{\partial F(\phi)}{\partial \phi_k}\right)$$
In the current context it is however likely a typo, instead of $\frac{\partial F(\phi)}{\partial \phi_i}$.
Regarding the first question - this could be simply a typo, but it could be also due to the conventions stated prior to this equation (possibly orally) - e.g. the data could be shifted to the origin
$$x_t \longrightarrow x_t-x_0, y_t \longrightarrow y_t-y_0,$$
which would avoid the need of dragging everywhere the constant term in regression (i.e., assuming that $\phi_1=1$ and considering only $i=2,...,k$.)
