  We consider the problem of forecasting multiple values of the future of a
vector time series, using some past values. This problem, and related ones such
as one-step-ahead prediction, have a very long history, and there are a number
of well-known methods for it, including vector auto-regressive models,
state-space methods, multi-task regression, and others. Our focus is on low
rank forecasters, which break forecasting up into two steps: estimating a
vector that can be interpreted as a latent state, given the past, and then
estimating the future values of the time series, given the latent state
estimate. We introduce the concept of forecast consistency, which means that
the estimates of the same value made at different times are consistent. We
formulate the forecasting problem in general form, and focus on linear
forecasters, for which we propose a formulation that can be solved via convex
optimization. We describe a number of extensions and variations, including
nonlinear forecasters, data weighting, the inclusion of auxiliary data, and
additional objective terms. We illustrate our methods with several examples.
