Structured distributed lag models come in two types: finite and infinite.
Infinite distributed lags allow the value of the independent variable at a particular time to influence the dependent variable infinitely far into the future, or to put it another way, they allow the current value of the dependent variable to be influenced by values of the independent variable that occurred infinitely long ago; but beyond some lag length the effects taper off toward zero.
Finite distributed lags allow for the independent variable at a particular time to influence the dependent variable for only a finite number of periods.
Finite distributed lags The most important structured finite distributed lag model is the
Almon lag model. This model allows the data to determine the shape of the lag structure, but the researcher must specify the maximum lag length; an incorrectly specified maximum lag length can distort the shape of the estimated lag structure as well as the cumulative effect of the independent variable. The Almon lag assumes that lag weights are related to linearly estimable underlying parameters according to : w_i = \sum_{j=0}^{n} a_j i^j for i=0, \dots , k.
Infinite distributed lags The most common type of structured infinite distributed lag model is the
geometric lag, also known as the
Koyck lag. In this lag structure, the weights (magnitudes of influence) of the lagged independent variable values decline exponentially with the length of the lag; while the shape of the lag structure is thus fully imposed by the choice of this technique, the rate of decline as well as the overall magnitude of effect are determined by the data. Specification of the regression equation is very straightforward: one includes as explanators (right-hand side variables in the regression) the one-period-lagged value of the dependent variable and the current value of the independent variable: : y_t= a + \lambda y_{t-1} + bx_t + \text{error term}, where 0 \le \lambda . 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 :b+ \lambda b + \lambda^2 b + ... = b/(1-\lambda). Other infinite distributed lag models have been proposed to allow the data to determine the shape of the lag structure. The
polynomial inverse lag assumes that the lag weights are related to underlying, linearly estimable parameters
aj according to :w_i = \sum_{j=2}^{n}\frac{a_j}{(i+1)^j}, for i=0, \dots , \infty . The
geometric combination lag assumes that the lags weights are related to underlying, linearly estimable parameters
aj according to either : w_i = \sum_{j=2}^{n} a_j(1/j)^i, for i=0, \dots , \infty or : w_i = \sum_{j=1}^{n} a_j [j/(n+1)]^i, for i=0, \dots , \infty . The
gamma lag and the
rational lag are other infinite distributed lag structures. ==Distributed lag model in health studies==