We fit US stock market volatilities on macroeconomic and financial market indicators and some industry level financial ratios. Stock market volatility is non-Gaussian distributed. It can be approximated by an Inverse Gaussian (IG) distribution or it can also be transformed by Box-Cox transformation to a Gaussian distribution. Hence, we use Box-Cox transformed Gaussian LASSO model and IG GLM LASSO model as dimension reduction techniques and we try to identify some common indicators to help us forecast stock market volatility. Via simulation, we validate that we can use four models, i.e. univariate Box-Cox transformation Gaussian LASSO model, 3-phase iterative grid search Box-Cox transformation Gaussian LASSO model, canonical link and optimal link IG GLM LASSO models to fitting an approximately IG distributed response. Using these four models in an empirical study, we identified three macroeconomic indicators that can help us to forecast stock market volatility. They are credit spread between US Aaa corporate bond yield and 10-year treasury yield, total outstanding nonrevolving consumer credit, and total outstanding nonfinancial corporate bonds.