# Clarification for replication in Adaptive Lasso

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.

C_{11} = (1 − \rho_1)I + \rho_1J_1,...

Followed by an explination of C_{12} and C_{22}. From this you should be able to reproduce the result.