Title: Two-lasso Bayesian adaption to study financial contagion
Description: Algorithmic advances in the form of Friedman's Coordinate Descent algorithm have allowed the Lasso REML penalty to see fruitful use in selection and regularization problems, such as Tibshirani's 2007 " Glasso " or "Graphical-Lasso" procedure. If we posit a sparse multiplicative inverse of the correlation matrix, or the "concentration matrix", we can interpret this as a Gaussian graphical network. We begin with a treatment of the linear regression problem and our modified penalty, dubbed the "Two-Lasso", a hierarchical mixture of large and small variance LaPlacian priors. This mixture takes advantage of pre-existing Lasso algorithms, as well as the EM missing-data framework, allowing statistician to input familiar selection parameters, such as a prior probability of activation. Though the Two-Lasso threshold function is not necessarily continuous, the second chapter includes bounds for continuity. Six theorems in the first chapter's appendix establish the asymptotic properties of the Two-Lasso estimate and the "Limit-Lasso", which is a Two-Lasso estimate achieved for widely disparate active and de-active set prior variances.

Target data for a Two-Lasso penalized Glasso model, or "Two- Glasso ", is financial markets data collected during a change in correlation structure. This includes the 2008 credit crisis which has recently wrecked havoc with endowments and institutional pension investment funds. We make three select modifications to the Glasso : a Two-Lasso selection penalty, constrained maximization rather than a strong prior on volatility to guarantee positive definite matrix estimates, and a non-zero baseline concentration level added to a sparse network. We fit Two- Glasso estimates for small-sample-size historical financial data and depict output visually with informative figures of our own design. From these graphs, we can posit a hedging strategy to the benefit of locked-in investors during a correlation-heavy crisis.