I am attempting to duplicate Belmonte et al. 2014. The paper applies a Bayesian Lasso to state space models to get better out of sample prediction. Bitto et al 2019 expands on this model. Their expansion includes a simulation study showing that the Bayesian Lasso shrinkage estimator recovered the true coefficients. Both papers, along with Park and Casella 2008, assume that the covariates are standardized mean zero and variance one. However, in most cases data we have is not standardized. How is it possible for data to first be standardized, then analyzed with a Bayesian Lasso and still recover the true DGP in a simulation study? Why is it when I standardize the variables the Bayesian Lasso returns the wrong coefficients and when I don't standardize it returns the right coefficients?
Bitto et al 2019: https://www.sciencedirect.com/science/article/pii/S0304407618302070
Belmonte et al. 2014: https://onlinelibrary.wiley.com/doi/full/10.1002/for.2276
Park and Casella 2008: https://www.tandfonline.com/doi/abs/10.1198/016214508000000337