I am taking an econometrics class, and we are now covering Panel Data (with a focus on Micro-Panel data at the moment). What I am having trouble with is understanding what this data matrix would look like. I have looked through google trying to find some examples, but I have not found one unifying form (I imagine because there are many representations of the data matrix depending on your context). However, I will try to be specific to the context in which I am studying:
If we are given the following regression:
$$y_{it}=\kappa_{i}+\mathbf{x}_{it}'\beta+\varepsilon_{it}, \ \ \ i=1,...,N; \ \ t=1,...,T$$ Where $\kappa_{i}$ is a RV, which is correlated with the predictors under fixed effects, and uncorrelated under random effects.
My first question is, is $\mathbf{x}_{it}$ the following: $$\mathbf{x}_{it}'=\begin{bmatrix} x_{1t} \\ x_{2t} \\ \vdots \\ x_{NT} \end{bmatrix}$$
(I imagine the above is wrong). Additionally, we can rewrite the OLS estimator of $\beta$ in the following way:
$$\mathbf{b}=\beta+(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}\varepsilon=\beta+(\sum_{i=1}^{N}\mathbf{X}_{i}'\mathbf{X}_{i})^{-1}\sum_{i=1}^{N}\mathbf{X}_{i}'\varepsilon_{i}$$
Additionally, what does $\mathbf{X}$ and $\mathbf{X}_{i}$ look like?
I would have thought that $\mathbf{X}$ would be the following:
$$\mathbf{X}=\begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1T} \\ x_{21} & x_{2t} & \cdots & x_{2T} \\ \vdots & \vdots & \ddots & \vdots \\ x_{N1} & x_{N2} & \cdots & x_{NT} \end{bmatrix}$$
However, the above makes no sense since there is no $i$ index to sum over when we want to work with $\mathbf{X}_{i}$ (and is not any different in form from the data matrix of a standard regression model). I apologize profusely for this deep confusion. I usually find stats/econometrics quite intuitive. However, I am having trouble consolidating my understanding of this topic at the moment.
