My question is regarding the Adaptive LASSO procedure, developed by Zou, the details of which can be found here.
In this paper, I wish to replicate Models 0 and 1. Let us focus on model 0 for the moment. But before that, let us clarify some notation. In this paper, we have a true model, consisting of $p_{0}$ variables, lesser than the total n umber of predictors, $p.$ I could be wrong, but the variance-covariance matrix of regressors, $$ \frac{1}{n}\boldsymbol{X'X\rightarrow}\boldsymbol{C} $$ is partitioned as: $$ \boldsymbol{C=\left[\begin{array}{cc} \boldsymbol{C_{11}} & \boldsymbol{C_{12}}\\ \boldsymbol{C_{21}} & \boldsymbol{C_{22}} \end{array}\right]} $$ Here, the author states on page 2 that $\boldsymbol{C_{11}}$is a $p_{0}\times p_{0}$ matrix. This, I would imagine, corresponds to the variances, of the regressors in the true model. $\boldsymbol{C_{22}}$ would correspond to the $p-p_{0}$ extraneous regressors, and the off-diagonal elements correspond to the covariances. My question pertains to page 6, wherein we replicate Model 0. In model 0, the author says that they simulate data, $y=\boldsymbol{x'\beta+N\left(0,\sigma^{2}\right)}$where the true regression coefficients are $\beta=\left(5.6,5.6,5.6,0\right)$ . The predictors $\boldsymbol{x_{i}}(i=1...n)$ are i.i.d N(0,\textbf{$\boldsymbol{C)}$ }, where \textbf{$\boldsymbol{C}$ }is the corollary matrix with $\rho_{1}$ and $\rho_{2}$ given. The values of $\rho_{1}=-0.39$ and $\rho_{2}=0.23.$
My question is :- how do $\rho_{1}$ and $\rho_{2}$ correspond to the matrix $\boldsymbol{C?}$ Although it is clear that $\boldsymbol{x_{i}}$ are distributed according to mean 0, how does $\rho_{1}$ and $\rho_{2}$ map to variances and covariances? Normally, $\rho$ corresponds to correlation. Without any further information, can we conclude that these are standardized regressors and that the variances are 1? I would greatly appreciate if someone could guide me towards what the matrix $\boldsymbol{C?}$ would look like in this case, so that I can replicate this simulation in Model 0.
