  The success of large-scale models in recent years has increased the
importance of statistical models with numerous parameters. Several studies have
analyzed over-parameterized linear models with high-dimensional data that may
not be sparse; however, existing results depend on the independent setting of
samples. In this study, we analyze a linear regression model with dependent
time series data under over-parameterization settings. We consider an estimator
via interpolation and developed a theory for the excess risk of the estimator.
Then, we derive bounds of risks by the estimator for the cases where the
temporal correlation of each coordinate of dependent data is homogeneous and
heterogeneous, respectively. The derived bounds reveal that a temporal
covariance of the data plays a key role; its strength affects the bias of the
risk, and its nondegeneracy affects the variance of the risk. Moreover, for the
heterogeneous correlation case, we show that the convergence rate of risks with
short-memory processes is identical to that of cases with independent data, and
the risk can converge to zero even with long-memory processes. Our theory can
be extended to infinite-dimensional data in a unified manner. We also present
several examples of specific dependent processes that can be applied to our
setting.
