Distributed lag

The most important structured finite distributed lag model is the Almon lag model.[3] 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 k + 1 lag weights are related to n + 1 linearly estimable underlying parameters (n < k) a_j according to

${\displaystyle w_{i}=\sum _{j=0}^{n}a_{j}i^{j}}$

for ${\displaystyle i=0,\dots ,k.}$

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:

${\displaystyle y_{t}=a+\lambda y_{t-1}+bx_{t}+{\text{error term}},}$

where ${\displaystyle 0\leq \lambda <1}$. 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

${\displaystyle b+\lambda b+\lambda ^{2}b+...=b/(1-\lambda ).}$

Other infinite distributed lag models have been proposed to allow the data to determine the shape of the lag structure. The polynomial inverse lag[4][5] assumes that the lag weights are related to underlying, linearly estimable parameters a_j according to

${\displaystyle w_{i}=\sum _{j=2}^{n}{\frac {a_{j}}{(i+1)^{j}}},}$

for ${\displaystyle i=0,\dots ,\infty .}$

The geometric combination lag[6] assumes that the lags weights are related to underlying, linearly estimable parameters a_j according to either

${\displaystyle w_{i}=\sum _{j=2}^{n}a_{j}(1/j)^{i},}$

for ${\displaystyle i=0,\dots ,\infty }$ or

${\displaystyle w_{i}=\sum _{j=1}^{n}a_{j}[j/(n+1)]^{i},}$

for ${\displaystyle i=0,\dots ,\infty .}$

The gamma lag[7] and the rational lag[8] are other infinite distributed lag structures.