  This paper examines LASSO, a widely-used $L_{1}$-penalized regression method,
in high dimensional linear predictive regressions, particularly when the number
of potential predictors exceeds the sample size and numerous unit root
regressors are present. The consistency of LASSO is contingent upon two key
components: the deviation bound of the cross product of the regressors and the
error term, and the restricted eigenvalue of the Gram matrix. We present new
probabilistic bounds for these components, suggesting that LASSO's rates of
convergence are different from those typically observed in cross-sectional
cases. When applied to a mixture of stationary, nonstationary, and cointegrated
predictors, LASSO maintains its asymptotic guarantee if predictors are
scale-standardized. Leveraging machine learning and macroeconomic domain
expertise, LASSO demonstrates strong performance in forecasting the
unemployment rate, as evidenced by its application to the FRED-MD database.
