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.

I hope I understood your question correct, and that the answer below helps you.
If you read collary 1 in the article it says that:
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.
