I am working on a graphical LASSO (GLASSO) shrinkage of the variance-covariance matrix of financial log-returns data for 10 years. The objective of the graphical LASSO is:
$$\ell(0,\Sigma) = {-\text{tr}(S_n\Theta) + \log\text{det }\Theta} - \lambda\lVert\Theta\rVert_{od,1}$$
Where:
- Data consists of $X_i \in \mathbb R^p$, for $i=1,\dotsc,n$;
- Data are centered ($E[X_i]=0$), so observations are assumed to be $X_i \sim \mathcal{N}(0,\Sigma)$ distributed;
- $\mathbf{X}_n$ is the $n \times p$ design matrix with rows $X_i^\intercal$;
- $S_n$ is the sample (empirical) covariance of the observations, i.e., $S_n = \mathbf{X}_n^\intercal \mathbf{X}_n/n$;
- $\lambda >0$ is a penalty parameter;
- $\Theta = \Sigma^{-1}$ is the Graphical LASSO precision matrice, assumed symmetric and positive definite ($\Theta \succ 0$);
- $\lVert\Theta\rVert_{od,1}$ is a semi-norm of $\Theta$ (off-diagonal $L_1$ norm such that $\sum_{i \neq j}|\Theta_{ij}|$;
My questions are:
- Point 2. means that data (in my case asset returns) should be standardized?
- What if point 2. is relased and so observations are assumed to be $X_i \sim \mathcal{N}(\mu,\Sigma)$ distributed?
- The graphical LASSO MLE estimator for $\Theta$ would be the same?
