[1][2] The starting point for a distributed lag model is an assumed structure of the form or the form where yt is the value at time period t of the dependent variable y, a is the intercept term to be estimated, and wi is called the lag weight (also to be estimated) placed on the value i periods previously of the explanatory variable x.
In the first equation, the dependent variable is assumed to be affected by values of the independent variable arbitrarily far in the past, so the number of lag weights is infinite and the model is called an infinite distributed lag model.
The concept of distributed lag models easily generalizes to the context of more than one right-side explanatory variable.
, assuming independently and identically distributed errors, and imposing no structure on the relationship of the coefficients of the lagged explanators with each other.
However, multicollinearity among the lagged explanators often arises, leading to high variance of the coefficient estimates.
In this model, the short-run (same-period) effect of a unit change in the independent variable is the value of b, while the long-run (cumulative) effect of a sustained unit change in the independent variable can be shown to be Other infinite distributed lag models have been proposed to allow the data to determine the shape of the lag structure.
Distributed lag models were introduced into health-related studies in 2002 by Zanobetti and Schwartz.
[12] The distributed lag model concept was first to applied to longitudinal cohort research by Hsu in 2015,[13] studying the relationship between PM2.5 and child asthma, and more complicated distributed lag method aimed to accommodate longitudinal cohort research analysis such as Bayesian Distributed Lag Interaction Model[14] by Wilson have been subsequently developed to answer similar research questions.