Population least square first order condition

Question

This question is highly related to this one: Mostly Harmless Econometrics, explanation of solution to the population least squares problem

In the book Harmless Econometrics, page 35, the authors say this :

This section is concerned with the vector of population regression coefficients, defined as the solution to a population least squares problem. At this point we are not worried about causality. Rather, we let the Kx1 regression coefficient vector β be defined by solving $β=argmin b E[(Yi−Xi^′b)^2]$. Using the first-order condition, $E[Xi(Yi−Xi′b)]=0$, the solution can be written $β=E(XiXi′)^{1}E(XiYi)$.

I don't understand how they derive the beta vector under the argmin condition to find $β=argminbE[(Yi−Xi^′b)^2]$. In particular I don't understand the difference between $Xi'$ and $Xi$. I'm sure I'm missing something pretty obvious. I saw that the question I linked had many views, maybe there are other people wondering the same thing as I do.

Answer 1

Be careful about indices. The correct expression is

$$ \beta = \text{argmin}_b E\big[(Y_i-X_i'b)^2\big]. $$

Here $Y_i$ is the $i$-th observation, and $X_i$ is a row vector of explanatory variables. $b$ and the minimizer $\beta$ are column vectors of parameters.

And now the little prime in $X_i'$ simply denotes transposition. That is, $X_i'b$ is the scalar product of $X_i$ and $b$, or just the sum of the pairwise products between entries in the two vectors.

Actually, there is no derivation here. This is just the definition of the OLS parameter estimate as the parameter vector that minimizes the expected squared error between the observations $Y_i$ and the fit $X_i'b$.