  Distributed lag models (DLMs) express the cumulative and delayed dependence
between pairs of time-indexed response and explanatory variables. In practical
application, users of DLMs examine the estimated influence of a series of
lagged covariates to assess patterns of dependence. Much recent methodological
work has sought to de- velop flexible parameterisations for smoothing the
associated lag parameters that avoid overfitting. However, this paper finds
that some widely-used DLMs introduce bias in the estimated lag influence, and
are sensitive to the maximum lag which is typically chosen in advance of model
fitting. Simulations show that bias and misspecification are dramatically
reduced by generalising the smoothing model to allow varying penalisation of
the lag influence estimates. The resulting model is shown to have substantially
fewer effective parameters and lower bias, providing the user with confidence
that the estimates are robust to prior model choice.
