When a decision made on a variable, some of the related variables would be effected through time. For example, when income tax rate is increased, this would reduce expenditures of consumers on goods and services, which reduces profits of suppliers, which reduces the demand for productive inputs, which reduces the profits of the input suppliers, and so on (Judge and Griffiths, 2000). These effects occur over the future time periods; hence, they are distributed across the time.
In a distributed-lag model, the effect of an independent variable \(X\) on a dependent variable \(Y\) occurs over the time. Therefore, DLMs are dynamic models. A linear finite DLM with one independent variable is written as follows:
$$
Y_{t} = \alpha +\sum_{s = 0}^{q}\beta_{s}X_{t-s}+\epsilon_{t},
$$
where \(\epsilon_{t}\) is a stationary error term with \(E(\epsilon_{t})=0, Var(\epsilon_{t})=\sigma^{2},Cov(\epsilon_{t},\epsilon_{s})=0\).
When there is only one predictor series, both of model and formula objects can be used. But when they are supplied, both x and y arguments should be NULL.
The variable names in formula must match with the names of variables in data argument and it must be in the form of a generic formula for R functions.
The argument data contains dependent series and independent series. Required lags of dependent series are generated by the dlm function automatically.
The argument remove is used to specify which lags will be removed from the full model. Each row of remove includes the numbers of lags to be removed in its columns and the rest of the elements should be set to NA. Notice that it is possible to fit a model with different lag lengths for each independent series by removing the appropriate lags of independent series with remove argument.