  Accurately modeling the correlation structure of errors is essential for
reliable uncertainty quantification in probabilistic time series forecasting.
Recent deep learning models for multivariate time series have developed
efficient parameterizations for time-varying contemporaneous covariance, but
they often assume temporal independence of errors for simplicity. However,
real-world data frequently exhibit significant error autocorrelation and
cross-lag correlation due to factors such as missing covariates. In this paper,
we present a plug-and-play method that learns the covariance structure of
errors over multiple steps for autoregressive models with Gaussian-distributed
errors. To achieve scalable inference and computational efficiency, we model
the contemporaneous covariance using a low-rank-plus-diagonal parameterization
and characterize cross-covariance through a group of independent latent
temporal processes. The learned covariance matrix can be used to calibrate
predictions based on observed residuals. We evaluate our method on
probabilistic models built on RNN and Transformer architectures, and the
results confirm the effectiveness of our approach in enhancing predictive
accuracy and uncertainty quantification without significantly increasing the
parameter size.
