  Reduced-rank regressions are powerful tools used to identify co-movements
within economic time series. However, this task becomes challenging when we
observe matrix-valued time series, where each dimension may have a different
co-movement structure. We propose reduced-rank regressions with a tensor
structure for the coefficient matrix to provide new insights into co-movements
within and between the dimensions of matrix-valued time series. Moreover, we
relate the co-movement structures to two commonly used reduced-rank models,
namely the serial correlation common feature and the index model. Two empirical
applications involving U.S.\ states and economic indicators for the Eurozone
and North American countries illustrate how our new tools identify
co-movements.
