  Scientists and statisticians often want to learn about the complex
relationships that connect two time-varying variables. Recent work on sparse
functional historical linear models confirms that they are promising for this
purpose, but several notable limitations exist. Most importantly, previous
works have imposed sparsity on the historical coefficient function, but have
not allowed the sparsity, hence lag, to vary with time. We simplify the
framework of sparse functional historical linear models by using a rectangular
coefficient structure along with Whittaker smoothing, then reduce the
assumptions of the previous frameworks by estimating the dynamic time lag from
a hierarchical coefficient structure. We motivate our study by aiming to
extract the physical rainfall-runoff processes hidden within hydrological data.
We show the promise and accuracy of our method using eight simulation studies,
further justified by two real sets of hydrological data.
