  Maximum likelihood estimation of large Markov-switching vector
autoregressions (MS-VARs) can be challenging or infeasible due to parameter
proliferation. To accommodate situations where dimensionality may be of
comparable order to or exceeds the sample size, we adopt a sparse framework and
propose two penalized maximum likelihood estimators with either the Lasso or
the smoothly clipped absolute deviation (SCAD) penalty. We show that both
estimators are estimation consistent, while the SCAD estimator also selects
relevant parameters with probability approaching one. A modified EM-algorithm
is developed for the case of Gaussian errors and simulations show that the
algorithm exhibits desirable finite sample performance. In an application to
short-horizon return predictability in the US, we estimate a 15 variable
2-state MS-VAR(1) and obtain the often reported counter-cyclicality in
predictability. The variable selection property of our estimators helps to
identify predictors that contribute strongly to predictability during economic
contractions but are otherwise irrelevant in expansions. Furthermore,
out-of-sample analyses indicate that large MS-VARs can significantly outperform
"hard-to-beat" predictors like the historical average.
